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Distributed adaptive control with application to heating, ventilation and air-conditioning systems
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Distributed adaptive control with application to heating, ventilation and air-conditioning systems
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DISTRIBUTED ADAPTIVE CONTROL WITH APPLICATION TO HEATING, VENTILATION AND AIR-CONDITIONING SYSTEMS by Georgios Lymperopoulos A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2021 Copyright 2021 Georgios Lymperopoulos Dedication To my beloved family. ii Acknowledgements I would like to thank my advisor Professor Petros A. Ioannou for mentoring and supporting me throughout my PhD journey. I believe that he helped me improve as a person and as an academic, as his immense knowledge, plentiful experience and continuous encouragement have been an inspiration to me. It has been an honor and a wonderful experience to work with him. I would also like to thank my dissertation committee members, Professor Viktor K. Prasanna and Professor Henryk Flashner. iii TableofContents Dedication ii Acknowledgements iii ListofTables vi ListofFigures vii Abstract ix Chapter1: Introduction 1 1.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Objectives and Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Chapter2: BuildingTemperatureRegulationusingDistributedAdaptiveControl 16 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Zone Modeling of HVAC Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Design of the Zone Air Temperature Controller . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.1 Controller structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.2 Adaptive estimation of controller gains . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Simulation Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5.1 Case study I: Building with six interconnected thermal zones . . . . . . . . . . . . 29 2.5.1.1 Building description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5.1.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5.2 Case study II: Primary school building . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5.2.1 Building description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5.2.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Chapter3: EectofStrongInterconnectionsandCommunicationDelays 41 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Distributed Control of Interconnected Systems with Delays . . . . . . . . . . . . . . . . . . 44 3.3.1 Controller structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.2 Unknown system parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 iv Chapter4: ActuatorDynamicsofAirHandlingUnits 57 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Description of a Multi-Zone HVAC System with Air Handling Units . . . . . . . . . . . . . 58 4.2.1 Zone model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.2 Air Handling Unit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4 Design of Valve Flow Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4.1 Controller architecture based on backstepping . . . . . . . . . . . . . . . . . . . . . 63 4.4.2 Estimation of controller gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.5 Simulation Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.5.1 Building description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.5.2 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.5.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Chapter5: DistributedFaultDetectioninHVACSystems 83 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2 HVAC System Dynamics for Fault Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.2.1 Thermal Zone model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2.2 Air Handling Unit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3 Fault Diagnosis Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4 Distributed Fault Diagnosis Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.4.1 Temperature estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.4.2 Adaptive threshold design for fault detection . . . . . . . . . . . . . . . . . . . . . 100 5.4.3 Distributed fault detection and isolation logic . . . . . . . . . . . . . . . . . . . . . 104 5.5 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.5.1 Building description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.5.2 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.5.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Chapter6: Conclusion 116 6.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Bibliography 122 Appendix 141 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 v ListofTables 1.1 Advantages and disadvantages of HVAC adaptive control . . . . . . . . . . . . . . . . . . . 10 2.1 Nomenclature of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Performance improvement with adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.1 Nomenclature of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Zones of the school building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3 Neighboring zones of the school building . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.4 Performance improvement in Air handling Unit control . . . . . . . . . . . . . . . . . . . . 80 5.1 Nomenclature of Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.2 Dependency Matrix of Fault Diagnosis Agent . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.3 Fault Signature Matrix of Fault Diagnosis Agent . . . . . . . . . . . . . . . . . . . . . . . . 105 5.4 Design constants for fault diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 vi ListofFigures 1.1 HVAC control categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Adaptive control structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Zone Control Diagram with Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Model of six-zone building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Temperature tracking in the six zone-building . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4 Supply air temperature and gain adaptation for one zone of the six zone-building . . . . . 32 2.5 One day of operation in the six zone-building . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.6 Model of primary school building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.7 Temperature tracking in the school building . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.8 Supply air temperature in the school building . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.9 Temperature tracking and energy consumption versus learning rate . . . . . . . . . . . . . 37 2.10 Adaptation of controller gains in the school building . . . . . . . . . . . . . . . . . . . . . 39 3.1 Performance of fully decentralized control scheme . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Performance of distributed control scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3 Performance of distributed adaptive control scheme with unknown interconnections . . . 54 3.4 Performance of distributed adaptive control scheme with unknown parameters . . . . . . 55 4.1 Schematic diagram of a multi-zone HVAC system . . . . . . . . . . . . . . . . . . . . . . . 58 4.2 Architecture of proposed distributed control scheme for Air Handling Units . . . . . . . . 62 vii 4.3 3D plan of primary school building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4 Performance of proposed scheme for one zone . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.5 Gain adaptation for one zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.6 Temperature tracking in school building zones . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.7 Water mass ow rate of coils in school building zones . . . . . . . . . . . . . . . . . . . . . 82 5.1 Design of Fault Diagnosis Agent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.2 Impact of a zone temperature sensor fault to energy consumption and thermal comfort . . 109 5.3 ARRs of the AHU 10 and 19 for Scenario 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.4 ARRs of the AHU 4 and 6 for Scenario 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.5 ARRs of the AHU 6 and 7 for Scenario 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.6 ARRs of the AHU 10 and 12 for Scenario 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 viii Abstract Buildings are one of the major energy consumers reaching up to 40% of nal energy consumption in de- veloped countries. Heating, Ventilation and Air-Conditioning systems (HVAC) are the largest contributors in building energy consumption, since they are responsible for 50% of nal building energy needs. Such systems are important for improving the quality of life in buildings, since they are necessary for providing comfortable temperature, humidity and air quality indoor conditions. Modern large-scale buildings utilize HVAC systems that are able to provide multiple zones with fresh hot or cold air by generating the ap- propriate heating or cooling loads and distribute them accordingly to the building zones. Enhancing their eciency can lead to reduced energy needs as well as improved climate conditions for occupants. One of the main engineering goals is to guarantee the optimized use of such amounts of energy without sacricing performance. Developing appropriate mechanisms for operating and controlling HVAC systems is a challenging task for engineers and there have been several approaches on the subject, with an accur- ate model of the building being crucial for the design of sophisticated HVAC systems. Existing centralized approaches suer from the disadvantages of centralized schemes, such as higher computational demands and higher susceptibility to faults, aecting temperature control. On the other hand, decentralized control schemes are more robust to faults. However, treating each zone as an isolated subsystem by ignoring the eects of neighboring zones may aect performance negatively. In addition, actuator dynamics are one of the main obstacles in regulating room temperature, with many optimized control algorithms typically ix taking into account the structure of the physical system (i.e., thermal zones and electromechanical equip- ment). Many control approaches require knowledge of system parameters and dynamics, such as surfaces and thermal capacities of materials. However, system dynamics change due to several factors, such as human activity, creation of internal and external openings, leakage, wear and tear and disturbances such as outside weather conditions, solar gains, or heat gains from light or indoor equipment. In this research, we propose a distributed adaptive control approach that assigns to each thermal zone a local controller which takes into account the climate conditions of neighboring zones to combine toler- ance to faults and decreased computational burden, which are characteristics of a decentralized approach, with the high performance of a centralized scheme. By applying on-line learning and assuming exchange of information between neighboring zones, the controller of each zone achieves the local objective of con- trolling zone temperature by compensating for the eects of neighboring zones as well as for possible changes in the parameters of the system. Despite the exchange of information, each local controller does not know how the control actions and temperature of a neighboring zone aect the temperature of its own zone. For this reason, each local controller estimates the parameters of the interconnections in real time and uses them together with the exchanged information to provide more accurate local zone temperature control. The proposed method is illustrated using an example of temperature control in a six-zone building as well as a large school building, which are implemented in a Building Controls Virtual Test Bed (BCVTB) environment using EnergyPlus and MATLAB/Simulink. In real scenarios, HVAC operation is susceptible to disturbances due to occupants, equipment, or solar gains and high parametric uncertainty of heat transfer coecients in equipment, as well as faults in op- eration. Simulations of realistic scenarios have shown that the proposed algorithms regulate temperature satisfactorily and in a energy-ecient manner that could lead to energy savings. Communication between systems or components of a control system is often not ideal; communication channels may encounter delays, noise or data loss and such problems may arise in real world system implementation. Therefore, we x analyze and show how distributed adaptive control algorithms can provide robustness to communication delays, even when system parameters are completely unknown. In addition, we show how the proposed design can be expanded to include actuator dynamics using a backstepping methodology. Air Handling Units (AHUs) are one of the popular types of electromechanical systems for large-scale buildings. The proposed design is based on the modeling of each element of an AHU as well as multi-zone modeling of the building. On-line estimation of controller parameters guarantees that the system can adapt to changes as well as unknown heat gains, and distribute heat across zones in an ecient manner. We present an illustrative simulation example, where the proposed algorithm is applied to a multizone school building and we show how the proposed scheme may have better performance in such a dynamic environment when compared to a control scheme that does not include learning. Finally, we design and analyze a model-based algorithm that detects and identies the location of faults that may occur in the AHU components such as valves or temperature sensors in large-scale buildings. The detection process is based on the calculation of adaptive thresholds, i.e., signals designed to bound the dierence between measured and estimated states under fault-free conditions considering modeling uncertainty and measurement noise, for each equipment component. The presence of a fault in a certain component may cause the violation of the adaptive threshold. Dierent types of faults can trigger the thresholds in a distinct manner, and thus the faults can be located using a combinatorial logic based on the violations. The performance of the proposed algorithm is illustrated using a simulation analysis on the multi-zone school building. xi Chapter1 Introduction 1.1 ProblemDescription Most people spend approximately 90% of their life indoors and it has been shown that indoor climate conditions play a critical role in life quality [1]–[3]. Thermal comfort has a direct eect in several aspects of people’s life: indoor air quality is directly linked to productivity [4]–[9]; proper temperature and humidity conditions can aect health and minimize the growth and spread of bacteria indoors, and especially in hospitals [10]–[13]; human sleep quality depends signicantly on the thermal environment [14]–[16]; air quality and temperature aect school performance [17]–[21], etc. Due to the high importance of proper indoor climate regulation and its direct eect in most aspects of life quality, several dierent types of equipment have been developed, ranging from simple heating and cooling units, such as replaces, gas and electric heaters, fans, air-conditioning units, etc. to complex large-scale systems that regulate temperature, humidity and concentration of air pollutants. Modern large-scale buildings utilize Heating, Ventilation and Air- Conditioning (HVAC) systems that are able to provide multiple zones with fresh hot or cold air by generating the appropriate heating or cooling loads and distribute them accordingly to the building zones. In developed countries HVAC systems have become an integral part of modern buildings. Such wide- spread use has resulted in buildings being one of the most energy consuming sectors, being responsible for 40% of total energy consumption [22]–[24] and 30% of total worldwide CO 2 emissions [25], with HVAC 1 systems requiring up to 50% of the buildings’ energy needs. Thus, it is important to optimize their e- ciency to reduce energy needs and improve climate conditions for occupants. Improved control algorithms can be a more cost-eective solution than replacing equipment with more modern technology [26]. A great amount of scientic interest has focused on optimizing climate regulation and energy eciency and has been facilitated by the advances in sensors, electronics, communications and computational abilities [27]– [30]. HVAC operation is meant to eciently control the indoor climate and air quality. Subsequently, the equipment can be categorized based on the dierent aspect of the indoor climate that it regulates. The most fundamental goal is temperature regulation, i.e., heating and cooling [31]–[33]. In addition, indoor humidity is controlled, i.e., humidication and dehumidication [34]. One important aspect of improv- ing air quality is reducing air pollution, which may be caused by particles, such as CO, lead, smoke, or other chemical compounds, and pathogenic microorganisms, such as bacteria, fungi or viruses [35], [36]. Ventilation and air cleaning provide a safe environment for humans and reduce the concentration of air pollution. In order to achieve such goals, several types HVAC equipment have been invented, such as furnaces, Air Handling Units (AHU), Variable Air Volume (VAV) units, heaters, radiators, with a variety of components, such as heating and cooling coils, thermal storages, pumps, fans, ducts, lters, mixing boxes, air dampers, humidiers and dehumidiers [37], [38]. Such systems are complex systems that are com- posed of a large number of strongly interconnected components, including a plethora of electromechanical equipment. One of the most commonly installed types of HVAC equipment for large-scale buildings is the AHU, since this type of system can serve multiple zones simultaneously and provide either heating or cool- ing loads. An AHU may be composed of either a Fan Coil Unit (FCU) or a VAV unit. FCUs provide supply air with xed ow rate, the temperature of which is regulated by a heat exchanger using a control valve, while VAVs provide supply air with a xed temperature and regulate zone air temperature by adjusting the volume ow rate of supply air using an air damper. 2 Developing appropriate mechanisms for operating and controlling HVAC systems is a challenging task for engineers and there have been several approaches on the subject. An accurate model of the building is crucial for the design of sophisticated controllers. The zone model represents the heat dynamics of a space, the climate of which is controlled by an AHU or an air-terminal device [39], [40]. Such a space is referred to as a thermal zone [41]. Air units are used to add or remove heat in order to modify zone temperature and overcome internal and external heat gains and losses. Although a building is usually divided into several zones, there exist centralized approaches that consider the whole building as an individual system. However, such schemes suer from the disadvantages of centralized schemes, such as higher susceptibility to faults, which directly aect climate control, and higher computational burden. On the other hand, dividing a large building into multiple thermal zones and designing controllers for each zone leads to decentralized control schemes. Such control schemes are more robust to faults, since, if one zone fails locally, the fault does not aect the performance of the other zones. However, treating each zone as an isolated subsystem by ignoring the eects of neighboring zones may aect performance negatively. Strong interconnections may aect system performance if their impact cannot be attenuated, as analyzed in [42]. Air ow due to convection between thermal zones through internal openings is signicant and can have a direct impact on zone temperature, since it results in heat transfer between zones, as explained in [43]–[45]. Neighboring zones also aect each other indirectly due to heat transfer by conduction through separating surfaces, as shown in [45] and in [46], where a feedback linearization scheme that reduces the heat transfer impact due to conduction is proposed. Therefore, a distributed approach that assigns to each thermal zone a local controller which takes into account the climate conditions of neighboring zones can combine tolerance to faults, which is a characteristic of a decentralized approach, with the high performance of a centralized scheme. Choosing the appropriate structure and strategy for HVAC control is a challenge that engineers are faced with. The variety and complexity of these systems lead to several challenges for ecient control. 3 The stochasticity of occupant behavior is one of the main sources of uncertainty in operating buildings, directly aecting performance [47]–[50]. Many control approaches require knowledge of system paramet- ers and dynamics, such as surfaces and thermal capacity of materials. However, dynamics change due to several factors, such as human activity or wear and tear. For example, the opening and closing of doors and windows change drastically the surfaces that separate thermal zones and open doors create internal openings which lead to exchange of heat between zones due to convection. In addition, leakage in in- sulation may result in erroneous measurement of thermal capacity. Thus, a control scheme that needs exact and correct measurement of such parameters may prove to be problematic. Besides, there is often the case that surface temperature is uncontrolled, although it aects air temperature, which, in addition to disturbances such as outside weather conditions, heat gains from light or indoor human activity, can deteriorate system eciency. The impact of the aforementioned uncertainties has been examined in [47]. Actuator dynamics in HVAC system are strong and aect performance signicantly. In regulating room temperature, actuator dynamics are one of the main obstacles. Several authors have proposed methods of optimizing valve control for such systems [51], [52]. Moreover, taking into account the structure of the physical system (i.e., zones are physically interconnected with their neighboring zones and each zone is controlled by its own local AHU), many researchers have tried to optimize control algorithms. A typical AHU consists of a heating and a cooling coil for controlling air temperature, a fan for air circulation, a mixing box for fresh air supply, and possibly lters and a humidier [39]. Controlling an AHU has a lot of challenges, with their operation including non-linearities that have to be overcome. As the system op- erates, its performance degrades with time; a typical example is the heating and cooling coil operation. Their ability to heat or cool the air decreases with time and is reected in their heat transfer coecient [53]. Insulation properties of materials also degrade with time [54], which, along with possible leakage or modeling mismatch, can lead to inaccurate measurement of thermal capacities. 4 Several methods have been proposed, including Model Predictive Control (MPC) techniques, optimal control, machine learning and reinforcement learning [55]–[58], in order to reduce computational com- plexity and fault susceptibility. However, such algorithms ought to consider the eect of neighboring zones, as indirect heat transfer between zones due to conduction through walls or direct heat transfer due to convection through internal openings [46], [59] may deteriorate performance. In addition, HVAC sys- tem performance may be compromised with a control scheme that requires exact and constantly updated measurements of system parameters that are dicult to accurately measure. 1.2 LiteratureReview Several researchers have proposed a number of control designs to improve climate tracking performance and energy eciency. The dierent approaches can be divided into traditional control, model based con- trol, non-model based control, hybrid control, or other techniques. Traditional control is the most widely used in HVAC control industry. The majority of existing industry controllers are based on a PI and PID structure or an On/O methodology. PI strategies for temperature control in large-scale buildings were explored in [60]. PI control has been applied to heating systems [61], industrial supply air systems, evaporators and variable air volume units [62]. One of the main challenges in PID control for HVAC systems is gain selection, since such systems go under signicant parameters changes during operation and may include nonlinearities [63]. Gain scheduling controllers have also been implemented. Model based control techniques consist of advanced methods that are based on control theory. The main categories are robust, adaptive, optimal, model predictive and nonlinear control [64]–[72]. Nonlin- ear control techniques usually focus on plant linearization around the operating points. A PI controller using feedback linearization for evaporators in a multi-unit HVAC system was developed in [73], while the 5 Reinforcement Learning Control Neural Network Control Fuzzy Logic Control Model Predictive Control Adaptive Control PID On / Off Non-Model Based Control Traditional Control Model Based Control Optimal Control Nonlinear Control Robust Control Genetic Algorithms Gain Scheduling Figure 1.1: Main HVAC control categories and hybrids authors in [74] developed a controller for AHUs based on feedback linearization and gain scheduling. Sev- eral researchers have proposed optimal control methodologies [75], with a great emphasis on distributed techniques for multi-zone systems. The authors in [56] formulated the mutli-zone HVAC problem as an event-based optimization problem, and used approximation solutions to nd the optimal local policies for when events occur. In a similar approach [76], the multi-zone control problem was formulated as an optim- ization problem with the target of minimizing energy consumption and maintaining desired temperature using a Computer Fluid Dynamics (CFD) model. A real-time optimal control strategy using an agent-based distributed structure was introduced in [77]. The computational tasks were decomposed into a number of lower-complexity tasks and distributed to dierent agents for online execution. Due to the high level of uncertainty and disturbances in such systems, robust control approaches have also been introduced. In 6 [78] it shown that robust controller improve system performance when compared to traditional PI control, as they provide tolerance to disturbances. Similarly, the author of [79] designed a robust controller for an HVAC system based on the weighted form of the plant model with introduced uncertainty. Among the model based control techniques, the most popular ones follow a Model Predictive Control approach. Model predictive control consists of optimizing a cost function that expresses system performance based on some metric, while satisfying constraints on the feasible control actions. A comprehensive report of the char- acteristics and advantages of MPC can be found in [64], where distributed approaches with zone climate controllers that consider the neighboring zones eect are preferred, such as in [80]. In [81] the authors used sequential quadratic programming and dual decomposition to develop a distributed MPC scheme, while in [82] the authors designed a distributed MPC control scheme based on Benders’ decomposition. The authors in [83] present a two-level hierarchical MPC scheme, with more MPC approaches presented in [55], [84]–[90]. Due to the high complexity of MPC centralized schemes, research interest has turned towards distributed approaches. In [91], a distributed MPC method for forced-air systems in designed and analyzed. Non-model based control methods are based on evolutionary techniques and articial intelligence. The main categories are neural network control, fuzzy logic, genetic algorithms and reinforcement learning. Fuzzy logic control was proposed in [92] for optimizing HVAC system performance as well as lighting and shading operations. The authors in [93] proposed a PID control design for a variable air ow system. The controller self-tunes its parameters based on a set of fuzzy control rules. In [94] the authors proposed a fuzzy logic control algorithm as well as an implementation of the genetic algorithm for controlling the ow of chilled water in FCUs. The advances in articial intelligence have also encouraged approaches that involve the application of neural networks. In [95] a neural network was used for thermal comfort con- trol. The authors in [96] introduced a neural network based reinforcement learning approach for HVAC control. A neural network was utilized to learn system dynamics, which are later used to perform model 7 predictive control. A reinforcement learning on-policy data adaptation was implemented in order to cap- ture potential changes in system parameters and dynamics, indicating the high impact of uncertainty and disturbances in such systems. Fusion or hybrid techniques that attempt to combine the benets of model based and non-model based control have been developed by several researchers. They typically a com- bine PID or model based control structure with a neural network or a set of fuzzy rules that adapt the controller parameters based on simulation or real-time data, in order to account for uncertainty. In [97] the authors tune a PID controller for an AHU using fuzzy logic providing robustness against model-based parametric uncertainties. Similarly, fuzzy logic rules are used to tune a PID variable ow rate controller in [98]. In a dierent approach [99] a PID controller is tuned using a neural network combined with the epsilon-constraint method. The authors in [100] control a discharge air temperature system using a PID structure which is tuned by a neural network. The implementation of hybrid techniques clearly implies the need for the HVAC controllers to be robust and able to adapt to uncertainty and disturbances. According to Ioannou [101], [102], “adaptive control is the combination of a parameter estimator, which generates parameters online, with a control law in order to control classes of plants whose parameters are completely unknown and/or could change with time in an unpredictable manner". This is typically im- plemented by a feedback system that modies the controller parameters in real time in order to satisfy the response requirements based on some performance measure, such as the tracking error and estimation error. Such a feedback system consists of the plant to be controlled, a controller and an appropriate para- meter estimator (adaptive law), as shown in Fig. 1.2. The choice of parameters to be estimated leads to two dierent design approaches. Following the rst approach, the adaptive law estimates the plant parameters online and the controller parameters are calculated using these online estimates. This approach is referred to as indirect adaptive control. On the other hand, if the parametrization is carried out with respect to the closed loop controller-plant system, the adaptive law can estimate the controller parameters directly, without intermediate calculations. This approach is referred to as direct adaptive control. 8 y u ? Input Adaptive Mechanism Controller Plant Figure 1.2: General adaptive control structure Due to the online parameter estimation, adaptive control can accommodate systems with uncertainty [103], [104], time varying parameters, time delay and nonlinearities [105]–[107]. HVAC systems are such systems, having high complexity and a high level of uncertainty, and they can be described as processes in which dynamics change even during normal operating conditions due to the existence stochastic dis- turbances [65], [68]. Buildings can have an unpredictable and unknown behavior as a result of occupants presence and actions, weather conditions, leakage, equipment, operating conditions, etc [45], [108]. This may cause parameters to vary fast in response. Although still not widespread, adaptive control techniques have been shown to be able to react fast to uncertainty in buildings and HVAC systems, improving their performance. Table 1.1 summarizes the advantages and disadvantages of the application of adaptive tech- niques to such systems. It should be noted that such control approaches have shown promising results in energy and cost savings, as well as quicker response to changes in dynamics, faster regulation of input parameters and good stability properties. Notably, adaptive techniques have the unique characteristic of learning the system and disturbances online, and thus being able to modify the control input accordingly. Research eorts have shown the benets of adaptive control when applied to HVAC systems. Earlier eorts focused on self-tuning mechanisms for PI controllers, such as in [114]. The authors designed a PI controller for heating and cooling coils and augmented it with an on-line gain adaptation mechanism 9 Table 1.1: Main advantages and disadvantages of adaptive control in HVAC systems Advantages Disadvantages Adaptive Control Energy savings [109] Appropriate system model is needed [68] Good stability [109] Increased complexity [71] Fast input regulation [109] Fast response to changes in dynamics [65] Improved overshoot, settling time and precision [110] Self-calibration [111] Capability of handling static and dynamic disturbances [112] Capability of coping with time delays [110], [113] based on the Integral Square Error (ISE). The system model consists of a rst order system with delay to represent the coil, but does not include the thermal zone, as the control objective is to track the supply air temperature setpoint. Simulation results showed that the methodology was able to properly self-tune the PI controller. In [115], the authors developed a PI controller, the gains of which are tuned adaptively for an enthalpy dierence method testing platform. The overall system was described by a rst order system plus dead time, with the control element being an electric heater. Adaptation is based on recursive least squares with exponential forgetting. When compared to anH 1 adaptive PI controller, the proposed design produced faster response, smaller overshoot and higher accuracy, while there was no results regarding energy consumption. Another self-tuning PI controller was presented in [116], where an adaptive Smith- based predictor is introduced, in order to compensate for the eect of the inherent system delay. An adaptive law is used to estimate the system parameters online and then the estimates are used in the construction of the Smith predictor. It is shown that the proposed method reduces overshoot settling time and reduces the eect of load disturbances, but may produce oscillations in the transient state. The authors of [117] focused on creating an adaptive algorithm that can calculate the desirable temper- ature reference for a thermal zone by achieving an optimum trade-o between thermal comfort and energy consumption. Application of the adaptive temperature reference prole showed increased thermal com- fort and reduced energy consumption. There is a lot of research on temperature regulation of the thermal 10 zone, as this is the fundamental goal of HVAC systems. The authors of [118] and [119] have modeled the AHUs as switching systems to capture the change in the supply air ow and have provided adaptive solutions that can automatically switch between possible air ow rates to better accommodate the thermal zone’s needs while guaranteeing stability. In [120], an adaptive feedforward control based on the system passivity is designed, reducing energy consumption by 16%. Adaptive control mechanisms that capture the time-varying characteristics of the building can provide better temperature and power tracking with faster temperature regulation when compared to model-free control (MFC) strategies, as shown in [121]. The operation of the HVAC electromechanical equipment is aected by system changes, such as de- gradation and occupant behavior, resulting in increased operational and maintenance costs. Active learn- ing through adaptive control has shown to produce energy savings when applied to heating systems, such as the zone-boiler-radiator system [122]. Similar results have been produced for cooling systems. In [123], an adaptive controller for an air-cooled water chiller is introduced. The performance of the controller was tested with an experimental test facility and the results showed 7-9% improvement with respect to supply water temperature control in the energy eciency rating, which corresponds to the ratio of the cooling capacity over the total power absorption. The real-time performance of an adaptive controller applied to a water chiller was also analyzed in [124]. After estimating online the system parameters and performing a minimum variance performance evaluation, the algorithm self-tunes the parameters of a PID-structured controller. The online adaptation has shown to provide reduced overshoot and improved overall performance when compared to traditional PID control. Adaptive control has also been applied to air-conditioning units, proving that it can make the system meet the requirements more eciently than traditional PID control [125]. Large-scale buildings are one of the biggest energy consumers and their systems require control al- gorithms that are scalable in order to avoid single points of failure and increasing computational burden. Hence, distributed adaptive control algorithms have been developed, such as in [109], where the authors 11 have created a distributed adaptive control algorithm for thermal zone air temperature regulation. The authors developed an algorithm that considers the interactions between zones and adjusts to unknown loads and system changes caused by occupants or other sources. Results showed that energy consump- tion is reduced by approximately 10% under several realistic scenarios, while temperature tracking is also improved by 30-35%. In [126] it is shown how distributed adaptive control can be applied successfully to large-scale buildings that use AHUs in order to control the water ow rate through the coils in an ecient way, achieving simultaneous gains in energy savings as well as temperature tracking. The benets of ad- aptive control in multi-zone systems were also explored in [108] and [127], in which the authors conducted hygrothermal analysis in buildings that contain multiple and small enclosed zones, such as neonatal intens- ive units and museum glass cases. The authors of [112] considered an adaptive controller based on Linear Quadratic Regulator (LQR) theory for a two-zone system in a centralized approach, resulting in increased robustness when compared to a PID design. In [110], a decentralized approach based on frequency-domain adaptation combined with fuzzy feedback control is proposed. An adaptive critic-based control design is proposed in [128] where an event-triggered controller is trained to produce near-optimal control actions while minimizing computations. Adaptive control has been successfully combined with MPC in order to improve setpoint optimiza- tion [129], where an inner adaptive control loop was used to provide information in real time regarding the system model and possible changes. Based on the exibility of the adaptive controller, the MPC op- timizer was shown to produce improved setpoint calculations, resulting in reduced energy costs without compromising occupants comfort. The combination of adaptive control with model predictive control is an emerging eld that has shown promising results but still has a lot of research gaps [68]. As the au- thors of [130] have noted, MPC tuning and optimization may be computationally heavy, and thus they have proposed an online adaptive parameter estimator that can provide a “plug and play” function for such control algorithms, reducing computational burden. Similar results have also been reported in [131]– 12 [133], while eld tests have shown increased energy savings of 28% [134]. In [135] an adaptive predictive control scheme was implemented for a oor heating system, showing to improve thermal performance. In addition to exploiting the cascade topology of AHUs, the control design should overcome the chal- lenges that emerge due to the large scale nature of buildings, and alleviate the computational complexity of traditional centralized control schemes, as well as avoid single points of failure. With the recent ad- vances in the area of Internet-of-Things (IoT), a distributed control design may not suer from the dis- advantages of centralized schemes, but instead it can reduce communication requirements and improve scalability. Therefore, in the last decade the majority of publications on HVAC control propose a distrib- uted design [59], [80], [83], [87], [88], [90], [91], [136]–[140]. Most of the aforementioned distributed control algorithms are based on an MPC design [80], [83], [87], [88], [91], [137], [138] that oers an op- timal solution, but without considering the eects of modeling uncertainty (i.e., occupancy, equipment, openings of doors), unknown disturbances and equipment degradation, while distributed fault diagnosis and accommodation algorithms that can alleviate the eects of faults in large-scale HVAC systems have recently been developed [141], [142]. Taking into account the series conguration in which AHU com- ponents (i.e., mixing box, fan, heating coil and cooling coil) are connected, cascade control may be used. Cascade control is a specialized control architecture formed by inner and outer feedback loops. Several researchers have developed cascade control schemes, the majority of which aim to control supply air tem- perature by regulating the water valve of coils [143]–[145], while only one of them proposed a cascade design for controlling zone air temperature [146], where a genetic algorithm is used. 1.3 ObjectivesandContribution Taking into consideration their importance in the real world, this work focuses on HVAC systems in build- ings, which are one of the three main energy consuming sectors. Understanding the physical system and its energy needs is crucial in designing energy ecient control schemes that exploit the system structure 13 and handle the strong actuator dynamics of such systems. On-line learning of the system using adaptive control techniques has proven to be a powerful tool when dealing with real-world uncertainty and drifting from ideal conditions. This section presents the main research objectives of this work: • Formulate the complex HVAC system, composed of a number of thermal zones and electromechan- ical equipment, as a set of physically interconnected subsystems, that allows the design of agents able to eectively monitor and control the underlying subsystems in a distributed manner, in order to improve scalability and decrease computational demands. • Design a control scheme for multi-zone HVAC systems based on a distributed adaptive control ap- proach to eectively regulate and maintain air temperature in all thermal zones of the building at a desired temperature that is dened by the users of each zone, by online estimating the control gains to respond to parameter changes, modeling uncertainties and unknown disturbances by appropri- ately adjusting supply air temperature. • Evaluate the distributed adaptive control strategy using EnergyPlus simulator, which is a whole building energy simulation program released by the US Department of Energy for modeling en- ergy consumption for heating, cooling and ventilation, as well as other building operations such as lighting and water usage [147]. Simulation results show that the proposed controller is able to provide signicant and consistent improvement in energy savings as well as temperature tracking performance. • Examine the stability and robustness properties of distributed adaptive control systems when there exist network induced communication delays and unknown parameters, and provide stability con- ditions based on communication delay and interconnections between the distributed subsystems. 14 • Exploit the cascade structure of AHUs as well as the network structure of a multi-zone building and design a distributed control scheme for regulating the cooling and heating coils’ valves of each AHU, taking into account the actuator dynamics and providing stability guarantees. • Develop a distributed adaptive algorithm that can learn the electromechanical equipment parameters by exploiting the unique characteristics of the system structure. • Design a distributed diagnosis architecture for sensor and actuator faults that enhances scalability, diagnosability (i.e., detectability, isolability) and robustness in the presence of measurement noise and modelling uncertainties using adaptive thresholds. Accurate fault diagnosis reduces the main- tenance time and, consequently, decreases energy waste and uncomfortable conditions in the build- ings, which can occur until the HVAC system recovers. 1.4 Outline The rest of the report is organized as follows: The primary distributed control scheme for such systems is presented in Chapter 2. The algorithm in this chapter is load-based and can be used to calculate the necessary load that the HVAC system needs to provide, focusing on the interconnections between thermal zones. Chapter 3 presents stability and robustness results of the distributed adaptive control structure in strongly interconnected systems under communication delays, introducing stability conditions in the form of LMIs with respect to the size of communication delays and interconnections. Then, Chapter 4 presents the proposed control scheme adjusted for systems that operate using Air Handling Units, dealing with the strong actuator dynamics by exploiting the cascade structure of the equipment. A detail distributed diagnosis scheme with adaptive thresholds is presented in Chapter 5. Finally, Chapter 6 presents the conclusion and possible future directions. The Appendix includes the proofs of stability and robustness of the proposed control schemes. 15 Chapter2 BuildingTemperatureRegulationusingDistributedAdaptiveControl 2.1 Introduction During recent years there have been considerable research eorts on improving energy eciency of build- ings. Since Heating, Ventilation and Air-Conditioning systems are responsible for a big part of energy consumption, developing ecient HVAC control systems is crucial. In most of the developed approaches, precise knowledge of system parameters and/or adequate historical data is required. However, these ap- proaches may not perform as well in the presence of dynamic parameter changes, due to human activity, material degradation or wear and tear, or disturbances and other operational uncertainties due to occu- pancy, solar gains, electrical equipment, or weather conditions. In this chapter, we consider buildings with several climate zones and propose a distributed adaptive control scheme for a multi-zone HVAC system which can eectively regulate zone temperature by applying on-line learning and assuming exchange of information between neighboring zones. The controller of each zone achieves the local objective of con- trolling zone temperature by compensating for the eects of neighboring zones as well as for possible changes in the parameters of the system. Despite the exchange of information, each local controller does not know how the control actions and temperature of a neighboring zone aect the temperature of its own zone. For this reason, each local controller is estimating the parameters of the interconnections in real time and uses them together with the exchanged information to provide a more accurate local zone 16 temperature control. The proposed method is illustrated using an example of temperature control in a six-zone building as well as a large school building, which are implemented in a Building Controls Virtual Test Bed (BCVTB) environment using EnergyPlus and MATLAB/Simulink. 2.2 ZoneModelingofHVACSystems A thermal zone is dened as a building area, the climate of which is controlled by an AHU [41]. A typical building may consist of multiple interconnected thermal zones. In this section, we consider the dynamical model of heat transfer associated with the HVAC system in a typical building consisting of N thermal zones, as follows [39], [148], [149]: dT z i dt = f sa i a C pa C z i (T sa i T z i ) + X j2M i U w i;j A w i;j C z i T w i;j T z i + X p2N i f i;p a C pa C z i T zp T z i + K o i C z i (T o T z i ) + q i C z i (2.1a) dT w i;j dt = U w i;j A w i;j C w i;j T z i T w i;j + U w i;j A w i;j C w i;j T o T w i;j (2.1b) dT w i;j dt = U w i;j A w i;j C w i;j T z i T w i;j + U w i;j A w i;j C w i;j T zp T w i;j (2.1c) 8i = 1;:::;N Equation (2.1a) states that the rate of change of the zone temperatureT z i is aected by the supply air tem- perature, the temperature of surrounding walls, the temperature of neighboring zones, weather conditions and internal and external heat gains. The various terms in equations (2.1a)-(2.1c) are listed in Table 2.1 and explained as follows: The temperatureT z i in zonei is expressed in o C.C z i is the overall thermal capacity of the zone in J = o C.f sa i denotes the volume ow rate of the supply air in m 3 =sec, whileT sa i represents the supply air temperature in o C.C pa represents the specic heat of air in J =kg o C and a denotes air density in kg =m 3 . The eect of surrounding surfaces temperatureT w i;j on zone temperature is shown withU w i;j being 17 Table 2.1: Nomenclature of Chapter 2 Symbol Denition a Air density in kg =m 3 A w i;j Area ofj th surface that surrounds zonei inm 2 C pa Specic heat of air in J =kg o C C w i;j Overall thermal capacity ofj th surface that surrounds zonei in J = o C C z i Overall thermal capacity ofi th zone in J = o C f sa i Volume ow rate of supply air ofi th zone in m 3 =s f i;p Volume ow rate between zonesi andp due to convection in m 3 =s K o i Thermal transfer coecient between zonei and the outside in W = o C M i Set of surfaces that surround zonei N Total number of zones N i Set of direct neighboring zones ofi th zone q i Internal heat gain of zonei in W T m i Target temperature for zonei in o C T o Outside temperature in o C T s Sampling time ins T sa i Supply air temperature in o C T w i;j Temperature ofj th surface that surrounds zonei in o C T z i Temperature ofi th zone in o C U w i;j Overall heat transfer coecient ofj th surface that surrounds zonei in kW =m 2 o C the overall heat transfer coecient of thej th surface that surrounds zonei in W =m 2 o C andA w i;j denoting the respective surface area inm 2 . M i is the set of surfaces that surround the zone. T zp denotes the zone temperature for a neighboring zonep andf i;p represents volume ow rate in m 3 =sec between zonesi and p2 N i due to convection. N i is the set of neighboring zones of zonei. In addition, outside temperature is represented byT o , withK o i denoting the respective thermal transfer coecient in W = o C. Finally,q i is used to model heat gains and loses, which may emerge due to human activity in the room, lights, electrical equipment, as well as plant and model mismatch, such as wall leakage.q i is considered to be an unknown bounded disturbance that is either constant or slowly varying and is measured in Watt (W). It is assumed that the air in each zonei is fully mixed, zone distribution is uniform, air density is constant, there are no pressure losses across the zone and in the mixing section and AHUs are ideal. The temperatureT w i;j of thej th surface that surrounds zonei is aected by the temperature of the two spaces that it separates. One of the two spaces is the zone, while the other one is either another zone 18 p or the outside. In the case of a surface separating a zone and the outside space, surface temperature satises the dierential equation (2.1b), whereT o represents the outside temperature. For a surface that separates two zones, namelyi andp, surface temperature satises the dierential equation (2.1c). In (2.1b) and (2.1c),C w i;j denotes the overall thermal capacity of the corresponding surface,U w i;j is the overall heat transfer coecient of the respective surface and isA w i;j is the surface area inm 2 . In (2.1c)T zp represents the temperature of the neighboring zonep that is separated by the surface. The supply air from the air unit has a direct impact on zone temperature, while the temperature of the surfaces that surround a thermal zone is also a crucial factor due to heat transfer between air and the walls due to conduction. In addition, open surfaces between zones, such as open doors or windows, let heat transfer between zones due to convection and aect zone climate conditions signicantly. Zone temper- ature is also aected by heat gains caused by human activity, lights, equipment, and weather conditions. In the heat transfer model (2.1a)-(2.1c) it is assumed that all temperatures are measured. For a given building, identication techniques may be used to calculate the various constants that appear in (2.1a)- (2.1c). These parameters may change with time due to various eects described below which could make the initially estimated parameters invalid. For example, the opening of a door due to human activity leads to air ow, and thus, heat transfer between zones due to convection, as analyzed in [43], and may signicantly change the parametersf i;p andA w i j in equations (2.1a),(2.1c). In addition to that, leakages, material degradation and normal wear and tear may alter slowly but steadily thermal capacities, while such changes are not easily measured. The performance of insulation materials may degrade with time sometimes during the rst 2-3 years after installation [54], which may lead to some parameter changes of up to 20%. Air density as well as zone temperature is considered to be uniformly distributed. However, depending on the air temperature, both the air density and the specic heat of air may slightly vary. A control system that is based on some initial estimate of the model parameters may considerably degrade in performance due to large parameter changes over time. In this chapter we design a controller that 19 does not rely on the exact values of the parameters and has the capability to adapt to parameter changes. The structure of the model (2.1a) to (2.1c) motivates the use of a distributed controller structure where a controller is designed for each zone by taking into account possible interactions with neighboring zones in order to meet the objectives of each zone. This distributed structure has better robustness properties than a centralized one where a single central controller controls all zones. In the distributed case, a failure in a particular zone may be easily isolated, whereas in the centralized case it may aect all zones. 2.3 ControlObjective Supply air is used as the control input to regulate zone temperature. HVAC systems can be divided into two major categories according to the way the adjust supply air. A large part of HVAC system design focuses on appropriately adjusting supply air volume ow rate, while keeping supply air temperature constant to some predened value. Such systems are known as Variable Air Volume (VAV) systems. On the other hand, there exist systems that keep supply air volume ow rate constant and adjust supply air temperature to meet the zone temperature objective. Such systems are known as Constant Air Volume (CAV) systems. FCUs are a widely applied type of HVAC equipment that often use constant air volume, or a small set of predened values for supply air volume ow rate, and they operate by properly adjusting supply air temperature [119]. FCUs may often serve as the nal air distribution units that adjust supply air temperature in multi-zone HVAC systems [150]. ControlObjective: Given the measurementsT z i ,T w i;j 8j2M i ,T zp 8p2N i andT o design the supply air temperatureT sa i for each zonei, so that zone temperatureT z i reaches a desired constant temperature target T m i , specied by the users of the system, using the complete zone and wall temperature model, as described in (2.1). In our formulation, we assume constant supply air volume ow rate [148] and the control input is supply air temperature. The controller should be able to accommodate for all uncertainties 20 due to human activity, leakage, thermal capacity computations, material degradation, wear and tear, and disturbances due to occupancy, equipment, solar gains, or other heat sources. 2.4 DesignoftheZoneAirTemperatureController 2.4.1 Controllerstructure Let’s start by dening the zone temperature tracking errore i =T z i T m i 8i = 1;:::;N which denotes how far the zone temperature deviates from the desired one. The control objective is to drive this error to zero or as close to zero as possible for all zones. Based on (2.1a), the sample-data version of the temperature tracking error using sampling timeT s [151] is the following8i = 1;:::;N: e i (k + 1) = d i e i (k) +b d i T sa i (k) + X j2M i U w i;j A w i;j f sa i a C pa T w i;j (k) i C z i f sa i a C pa T m i + X p2N i f i;p f sa i T zp (k) + K o i f sa i a C pa T o (k) + 1 f sa i a C pa q i (k) (2.2) where e(k) is the value of e(t) at time t = kT s and d i = e i Ts ; b d i = 1 i 1e i Ts fsa i aCpa Cz i ; 8i = 1;:::;N and the parameter: i = f sa i a C pa + P p2N i f i;p a C pa + P j2M i U w i;j A w i;j +K o i C z i ;8i = 1;:::;N (2.3) indicates how heat ows in the zone as a result of all contributing factors, such as zone dynamics, neigh- boring zones and surrounding surfaces. 21 The control inputT sa i 8i = 1;:::;N should be chosen, so that errore i 8i = 1;:::;N converges to zero or close to zero despite the eects of neighboring zones, surrounding surfaces, outside weather conditions, etc. This reasoning suggests the following structure for the controller of each zone8i = 1;:::;N: T sa i (k) =K e i e i (k) X j2M i K w i;j T w i;j (k)K m i T m i X p2N i K i;p T zp (k)K out i T o (k)h i G i (k) (2.4a) G i (k) =G i (k 1) +e(k) (2.4b) whereh i is the gain of the accumulator in (2.4b) and the gainsK e i ,K w i;j ,K m i ,K i;p andK out i are calcu- lated as follows: K e i = m i d i b d i ; K w i;j = U w i;j A w i;j f sa i a C pa ; K m i = i C z i f sa i a C pa ; K i;p = f i;p f sa i ; K out i = K o i f sa i a C pa (2.5) where m i is a design parameter which, together with gainh i , will be selected to achieve certain stability properties for the zone temperature. By applying the proposed control input (2.4a) on zonei , the temperature tracking error (2.2) of each zone satises the following error equation: e i (k + 1) = m i e i (k) +b d i h i G i (k) + 1 f sa i a C pa q i (k) ! (2.6) 8i = 1;:::;N. From (2.6), we derive that the temperature tracking error is given by the following transfer function: e i = z 1 z 2 + (h i b d i 1 m i )z + m i b d i f sa i a C pa [q i ] (2.7) 22 wherez is thez-transform. The desired characteristic equation for (2.7) that guarantees stability and a certain desired rise time, overshoot and settling time [152] is of the formz 2 +c 1 z +c 0 = 0, which can be used to choose the parametersh i anda m i . Given the stability of the characteristic equation of (2.7), the zone temperature tracking error converges to zero exponentially fast. The controller described by (2.4a), (2.4b) and (2.5) assumes that the parameters of the model are known exactly which, as we argued before, is not a realistic assumption. Therefore, the signicance of the con- troller (2.4a)-(2.5) which drives the errors to zero is to serve as a reference when we consider the case of unknown model parameters in the following section. 2.4.2 Adaptiveestimationofcontrollergains Since the parametersK e i ,K w i;j ,K m i ,K i;p andK out i are unknown, the controller (2.4a)-(2.5) cannot be implemented. In this case, we use the following control input: T sa i (k) =K e i (k)e i (k) X j2M i K w i;j (k)T w i;j (k)K m i (k)T m i X p2N i K i;p (k)T zp (k)K out i (k)T o (k)K i (k) (2.8) where the gainsK e i (k);K w i;j (k)8j2 M i ;K m i (k);K i;p (k)8p2 N i ;K out i (k) are replacing the gains K e i ,K w i;j ,K m i ,K i;p ,K out i , respectively. The accumulatorh i G i (k) has been replaced withK i (k), with K i being the unknown desired value ofK i (k). The updating rules of the gainsK e i (k), K w i;j (k), K m i (k), K i;p (k), K out i (k) andK i (k) in (2.8) are developed according to the following process. First, the zone temperature tracking error equation (2.2) is re-written as: e i (k + 1) = m i e i (k) +b d i T sa i (k) +K e i e i (k) + X j2M i K w i;j T w i;j (k) 23 +K m i T m i + X p2N i K i;p T zp (k) +K out i T o (k) +K i ! (2.9) which is expressed in the form of the following linear parametric model [102]: z i (k) = > i i (k) (2.10) where z i (k) =e i (k) m i e i (k 1) (2.11a) i = b d i ; b d i K e i ; b d i [K w i;j ] j2M i ; b d i K m i ; b d i [K i;p ] p2N i ; b d i K out i ; b d i K i > = 0 i ; 1 i ; [ 2 i;j ] j2M i ; 3 i ; [ 4 i;p ] p2N i ; 5 i ; 6 i > (2.11b) i (k) =[T sa i (k 1); e i (k 1); [T w i;j (k 1)] j2M i ; T m i ; [T zp (k 1)] p2N i ; T o (k 1); 1] > (2.11c) The linear in the parameters model (2.10) is used to estimate i . Let i (k) = 0 i (k); 1 i (k); [ 2 i;j (k)] j2M i ; 3 i (k); [ 4 i;p (k)] p2N i ; 5 i (k); 6 i (k) > (2.12) be the estimate of i at instantk. Then, the estimates of the controller gains are generated as follows 8i = 1;:::;N,8j2M i and8p2N i : K e i (k) = 1 i (k) 0 i (k) ; K w i;j (k) = 2 i;j (k) 0 i (k) ; K m i (k) = 3 i (k) 0 i (k) ; K i;p (k) = 4 i;p (k) 0 i (k) ; K out i (k) = 5 i (k) 0 i (k) ; K i (k) = 6 i (k) 0 i (k) (2.13) The estimates i (k) can be found using an appropriate adaptive law. One choice would be based on gradient descent with projection [102] and is the following: 24 ~ i (k) = i (k 1) +T s i i (k) i (k) (2.14a) [ i (k)] s = 8 > > > > > > > > > < > > > > > > > > > : [ ~ i (k)] s if [L i ] s [ ~ i (k)] s [U i ] s [L i ] s if [ ~ i (k)] s < [L i ] s [U i ] s if [ ~ i (k)] s > [U i ] s 8s = 0;:::; 4 +jM i j +jN i j (2.14b) i (k) = z i (k) m 2 i (k) = e i (k) m i e i (k 1) m 2 i (k) (2.14c) m 2 i (k) = 1 +T s > i (k) i i (k) (2.14d) where [ i (k)] s denotes thes th element of i (k) and the positive denite diagonal matrix i = diag( b d i , e i , [ w i;j ] j2M i , m i , [ zp ] p2N i ; o i , i ) consists of the positive design constants b d i ; e i , [ w i;j ] j2M i ; m i , [ zp ] p2N i ; o i ; i , referred to as adaptive gains, that aect the speed of adaptation. U i ;L i are upper and lower bounds for the values of the elements of i (k) and [L i ] 0 > 0. The lower bound [L i ] 0 of 0 i (k) has to be strictly positive to guarantee boundedness of the estimated gains. However, this is not constricting, since the real value of 0 i (k) is strictly positive due to system structure. The rest of the bounds [U i ] 0 and [U i ] s ; [L i ] s 8s = 1;:::; 4 +jM i j +jN i j serve as conservative bounds for the gain estimates to guarantee the boundedness of the gains in eq. (2.13). The equations to calculate the gain estimates may also be based on the discrete modied Least Squares algorithm with projection [102]: ^ i (k) =[ ^ 0;i (k); ^ 1;i (k); [ ^ 2;ij (k)] j2M i ; ^ 3;i (k); [ ^ 4;ip (k)] p2N i ; ^ 5;i (k); ^ 6;i (k)] > = i (k 1) + p a i (k)P i (k) i (k) i (k) (2.15a) 25 [ i (k)] s = 8 > > > > > > > > > < > > > > > > > > > : [ ^ i (k)] s ifL s;i [ ^ i (k)] s U s;i L s;i if [ ^ i (k)] s <L s;i U s;i if [ ^ i (k)] s >U s;i 8s = 0;:::; 4 +jM i j +jN i j i (k) = e i (k) m;i e i (k 1) m 2 i (k) (2.15b) ^ P i (k) = 1 i (k) P i (k 1) a i (k)P i (k 1) i (k) > i (k)P i (k 1) m 2 i (k) +a i (k) > i (k)P i (k 1) i (k) (2.15c) P i (k) = 8 > > > > > > > > > < > > > > > > > > > : ^ P i (k) ifL s;i [ ^ i (k)] s U s;i P i (k 1) if [ ^ i (k)] s <L s;i P i (k 1) if [ ^ i (k)] s >U s;i 8s = 0;:::; 4 +jN i j +jM i j (2.15d) m i (k) =1 +T s > i P i (k 1) i (k) (2.15e) a i (k)2(0; 1); i (k)2 (0; 1) (2.15f) withP i (0) = P 0 = P > 0 0,U s;i ;L s;i ;8s = s = 0;:::; 4 +jM i j +jN i j being upper and lower bounds for the values of the elements of i and L 0;i > 0. The lower bound L 0;i of 0;i (k) has to be strictly positive, a restriction which is derived from the structure of the system. The rest of the boundsU s;i ;L s;i are conservative bounds for the gain estimates that guarantee the boundedness of the gains in eq. (2.13). The equations to calculate the gain estimates are based on the discrete modied Least Squares algorithm with projection, which is presented and analyzed in [102].a i (k) is a positive weight factor and i (k) is a positive number that serves as a forgetting factor. Botha i (k) and i (k) can be chosen to be constant. The controller (2.8) with the adaptation laws (2.13),(2.14),(2.15) form the adaptive control system which does not require the knowledge of the model parameters. The following Theorem describes the stability properties of the closed loop system. 26 Theorem2.1. Consider the overall closed-loop system composed of the N subsystems (2.2) using the control scheme (2.8), (2.13). Then,8ie i 2l 1 ande i converges to zero ask!1. Proof. The proof of Theorem 2.1 is presented in Appendix A. Control Mechanism Physical System Air Handling Unit Zone Dynamics Adaptive Law Controller C(? i ) To Tz p, for all p ? Ni Tm i Tw i,j j ? Mi Tsa i Zone i User Input Temperature Sensor Tz i Tz i : Zone Temperature Tw i,j : Surrounding surface Temperature Tz p : Temperature of neighboring zone Tm i : Target Temperature ? i Control Physical System Zone j Control Physical System Zone i Control Physical System Zone k Interaction of Multiple Zones To: Outside Temperature Tsa i : Supply air Temperature ? i : Estimated controller gains Figure 2.1: Zone Control Diagram with Adaptation Theorem 2.1 guarantees that all signals in the closed loop system of each zone are bounded and the zone temperature tracking errors converge to zero with time, asx2l 1 means thatkxk 1 = sup k0 jx(k)j exists 27 for signalx. Thus, the scheme is able to completely cancel the impact of neighboring zones, surrounding surfaces, outside temperature and heat gainq i on zone temperature, making zones reach their temperature objectives satisfactorily. The result holds for unknown constant system parameters and disturbances, but can also be extended to slowly varying parameters or parameters with infrequent nite changes [105]. The proposed control scheme is scalable, since it distributes computational burden and its stability and convergence properties do not depend on the building and HVAC size. It should be noted that the closed loop poles m i 2 (0; 1) should be chosen according to the individual thermal zone’s requirements to achieve fast temperature regulation, without exhibiting unwanted oscillations or overshoot. Fig. 2.1 shows the closed loop conguration of one zone and interaction with other zones. In the next section, we use simulations to demonstrate the results of Theorem 2.1 and evaluate the properties and performance of the proposed control scheme. 2.5 SimulationResultsandDiscussion In this section, we present two case studies to show the eectiveness of the proposed control scheme. In the rst case study, we apply the proposed scheme to regulate temperature in a two-story building with six interconnected thermal zones and we demonstrate the eect on temperature tracking, focusing on the impact of abrupt parameter changes. In the next case study, we apply the proposed scheme to a large primary school building model and we examine the performance of the proposed control scheme under dierent scenarios. Since one of the basic characteristics of the proposed scheme is the utilization of learning (using the gradient descent method in this case), in both cases we compared its performance to the performance of a distributed control scheme that does not include adaptive learning. Both systems are simulated using EnergyPlus for the building and HVAC model and Matlab/Simulink for the control scheme, co-simulated using the Buildings Control Virtual Test Bed (BCVTB). 28 2.5.1 CasestudyI:Buildingwithsixinterconnectedthermalzones 2.5.1.1 Buildingdescription Figure 2.2: Model of six-zone building For our rst study, we consider a building with two stories, each of which is divided into three thermal zones, as shown in Fig. 2.2. The building includes zones that may be separated by walls but also zones that have no wall separation, to show how the proposed algorithm can handle both types of interconnections between zones. Each story has a large zone on the north side with dimensions 12:092 6:096 3:048 m 3 and two smaller zones, one on the west side and one on the east side that have the same size and dimensions 6:096 6:096 3:048m 3 . Any given zone is connected with a neighboring zone via a door of dimension 1 2m 2 . The north zone has a window of dimensions 3 1:333m 2 , while the west zone has a window of dimension 4 1:5m 2 . In addition, there is a permanent horizontal opening of dimensions 10:1924:096m 2 which connects the north zones of the two stories. The HVAC equipment is considered to be ideal, thus able to provide supply air with the desired temperature and volume ow rate. Supply air temperature is going to be calculated by the control scheme, while supply air volume ow rates are predened. The north zones have supply air volume ow rate 0:260 m 3 =s and the west and east ones have 29 supply air volume ow rate 0:220 m 3 =s. Heat gains in each zone consist of heat gains from occupants, lights and electronic equipment and they are scheduled to vary during the day. The control system is implemented using sampling timeT s = 60s. 2.5.1.2 Simulationresults For the purpose of this demonstration, we consider a scenario where the HVAC system is operating, when at 8:00 am doors open and heat gain increases as people start using the building. Open doors result in direct interactions between zones. Several other weather, internal gains and parameter changes occur during the day. The proposed distributed adaptive control scheme regulates the temperature in all zones and its per- formance is compared to a distributed constant-gain control scheme which assumes exact knowledge of the parameters initially without being aware of any parameter changes. In order to quantify the impact of the proposed scheme, we propose the following metrics, which correspond to the change in the overall tracking error of the system and the overall change in energy consumption when compared to the baseline controller: z = N X i=1 P k je z i (k)j P k j~ e z i (k)j P k je z i (k)j (2.16) E = jE N HVAC jj ~ E N HVAC j jE N HVAC j (2.17) for total operating time wheree z i andE N HVAC denote the zone temperature tracking error of zonei and the HVAC energy consumption in allN zones, respectively, for the baseline controller, while ~ e z i and ~ E N HVAC denote the performance and energy consumption, respectively, of the proposed distributed adaptive con- trol scheme. For the current selection of design constants, the proposed control scheme achieves 28:35% 30 08:00 08:15 08:30 08:45 09:00 Time 22.5 23 23.5 24 24.5 25 Zone Temperature Temperature of West zone, 1st floor Constant Gains Adaptation 08:00 08:15 08:30 08:45 09:00 Time 22.5 23 23.5 24 24.5 25 Zone Temperature Temperature of West zone, 2nd floor Constant Gains Adaptation 08:00 08:15 08:30 08:45 09:00 Time 24 24.2 24.4 24.6 24.8 25 25.2 25.4 Zone Temperature Temperature of East zone, 1st floor Constant Gains Adaptation 08:00 08:15 08:30 08:45 09:00 Time 24 24.2 24.4 24.6 24.8 25 25.2 25.4 Zone Temperature Temperature of East zone, 2nd floor Constant Gains Adaptation 08:00 08:15 08:30 08:45 09:00 Time 20.5 21 21.5 22 22.5 23 Zone Temperature Temperature of North zone, 1st floor Constant Gains Adaptation 08:00 08:15 08:30 08:45 09:00 Time 21 21.5 22 22.5 23 23.5 Zone Temperature Temperature of North zone, 2nd floor Constant Gains Adaptation Figure 2.3: Temperature tracking for all zones improvement in overall temperature tracking performance over a one-month period. Based on the En- ergyPlus energy consumption reporting, such improvement may be achieved with 2:05% reduction in total energy consumption. Fig. 2.3 shows the behavior of each zone at 8:00am when each control scheme is applied. The proposed adaptive scheme is able to react faster to the changes in the system and compensate faster for the heat transfer in most of the zones, even though both schemes eventually reach the temperature goal. As shown in Fig. 2.3 and reected in the reduction of the overall tracking error, in most thermal zones the temperature 31 may reach the desired one, within a 2% error margin, several minutes faster when the adaptive scheme is applied, compared to the constant-gain scheme. 08:00 08:15 08:30 08:45 09:00 Time 18 18.5 19 19.5 20 20.5 21 21.5 Supply air Temperature Supply air temperature of East zone, 2ndt floor Constant Gains Adaptation 08:00 08:15 08:30 08:45 09:00 Time -2.08 -2.07 -2.06 -2.05 -2.04 -2.03 K m K m gain change Figure 2.4: Supply air temperature and gain adaptation for East Zone, 2nd oor 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 Time 23 23.5 24 24.5 25 25.5 Zone air Temperature Zone temperature of East zone, 2nd floor Constant Gains Adaptation Figure 2.5: East Zone Temperature, 2nd oor The fast reaction of the proposed control scheme is due to the adaptation of the controller gains to the new dynamics. For instance, Fig. 2.4 shows how the gainK m i changes to accommodate for the dierent dynamics due to a door opening and introduction of heat gains due to human activity. As shown in eq. (2.5), the ideal value ofK m i depends on the parameter i , which is dened in eq. (2.3) and depends on inter- zone air ow due to convection and separating surface area. Thus, when a change occurs, the gain estimate ofK m i adapts to capture the new dynamics and the gain adaptation results in more accurate supply air temperature to be applied to the zone. In the same gure it is also shown how capturing parameter changes by the adaptive control scheme is reected on the applied control inputT sa i . The proposed learning-based 32 scheme adjusts supply air temperature faster to the appropriate value. In several cases fast and accurate temperature regulation is crucial, such as in intensive care units in hospitals or in museums with sensitive exhibits [127]. Satisfactory temperature regulation may also have an impact on learning ability in school [20]. Fig. 2.5 shows the performance of the the proposed scheme compared to a non-learning one for one full day of operation, and how it adapts to several heat gain and parameter changes. 2.5.2 CasestudyII:Primaryschoolbuilding 2.5.2.1 Buildingdescription Figure 2.6: Model of primary school building The second case study refers to the application of the proposed scheme to a large building with mul- tiple thermal zones and varying heat loads, in order to further demonstrate its performance. The al- gorithm is implemented and regulates temperature in a prototype primary school building model, which is the ANSI/ASHRAE/IES Standard 90.1-2016 Primary School model, located in Denver, from the ASHRAE Standard 90.1 prototype buildings suite, as developed by Pacic Northwest National Laboratory [153]. The school building has 25 thermal zones, which vary in size and use. According to their use, the zones have 33 dierent heat load patterns, which is a result of dierent occupancy levels and human activity, lighting or equipment, as well as weather impact. Naturally, neighboring zones are physically interconnected via walls and doors. The 3D building model of the school building is presented in Fig. 2.6. The HVAC equip- ment of the building is ideal and is designed so supply air temperature may be controlled, while a constant value for the supply air ow rate is passed. Supply air ow rates are chosen based on zone dimensions. The modied EnergyPlus input data le (.idf) that describes the building and HVAC system is presented in the following Github link [154]. All schedules for occupancy levels, electrical equipment operation, lighting, door opening as well as weather conditions are included in the idf le. 2.5.2.2 Simulationresults The HVAC system of the school building is operating only on weekdays from 7am to 5pm. All zones have their individual occupancy schedule that corresponds to their use and corresponding schedules for their equipment and lighting. Doors are scheduled to open and close at several times to capture changes in the ways neighboring zones are connected. We have selected the temperature target for all zones to be T m i = 23 o C. The control system is implemented using sampling timeT s = 60s. The design constants correspond to the desired closed loop poles for each zone and the learning rates of the adaptive laws. The closed loop poles are chosen so that stability is preserved and the system achieves temperature regulation fast and avoid unwanted oscillations or overshoot. Learning rates reect the system disturbances and possible parameter changes. The projection bounds of the adaptive laws are conservative bounds for the controller gains that are estimated on-line. For comparison, we have implemented the distributed control scheme with adaptive learning and a distributed control scheme that does not include adaptation. The performance of the proposed algorithm is examined under dierent scenarios. Firstly, we show the temperature tracking for the school’s thermal zones during one day of operation during January. During 34 Figure 2.7: Temperature tracking of 10 dierent zones one normal day of operation, the system has to deal with varying internal heat loads, as well as intercon- nection changes due to doors. The tracking performance of ten dierent thermal zones during one day 35 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time 22 24 26 28 30 32 34 Supply air Temperature Supply air temperature of Bath Constant Gains Adaptation 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time 10 15 20 25 30 35 Supply air Temperature Supply air temperature of Cafeteria Constant Gains Adaptation 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time 20 22 24 26 28 30 32 Supply air Temperature Supply air temperature of ComputerClass Constant Gains Adaptation 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time 20 22 24 26 28 30 32 Supply air Temperature Supply air temperature of CornerClass1Pod1 Constant Gains Adaptation 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time 22 24 26 28 30 32 34 Supply air Temperature Supply air temperature of CorridorPod1 Constant Gains Adaptation 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time 20 22 24 26 28 30 32 Supply air Temperature Supply air temperature of Gym Constant Gains Adaptation 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time 14 16 18 20 22 Supply air Temperature Supply air temperature of Kitchen Constant Gains Adaptation 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time 22 24 26 28 30 32 34 Supply air Temperature Supply air temperature of MainCorridor Constant Gains Adaptation 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time 10 15 20 25 30 35 Supply air Temperature Supply air temperature of MultiClass1Pod1 Constant Gains Adaptation 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time 20 25 30 35 Supply air Temperature Supply air temperature of Offices Constant Gains Adaptation Figure 2.8: Supply air temperature of 10 zones of operation is presented in Fig. 2.7. By simulating and evaluating the performance for the full month of January, we measured 36.80% improvement in temperature tracking and 13.68% improvement in overall 36 energy consumption, as reported by EnergyPlus software. The corresponding input signals for the same day of operation are presented in Fig. 2.8, and they are all well-behaved. As shown in the gures, the sys- tem reacts whenever changes occur, with the proposed scheme being able to adapt to the changes faster, due to the existence of learning. 0.2 i 0.3 i 0.4 i 0.5 i 0.6 i 0.7 i 0.8 i 0.9 i i 1.1 i 1.2 i 1.3 i 1.4 i 1.5 i 1.6 i 1.7 i 1.8 i 1.9 i 2 i 2.1 i 2.2 i 2.3 i 2.4 i 2.5 i -40 -30 -20 -10 0 10 20 30 40 50 Temperature tracking % -15 -10 -5 0 5 10 15 20 Energy consumption % Performance improvement Temperature tracking Energy consumption Figure 2.9: Improved temperature tracking and energy consumption based on the learning rate i Through on-line learning and improved heat allocation between zones, the proposed control meth- odology has shown to improve both temperature tracking and energy consumption. According to design priorities, the system moderator may choose to give priority to improved temperature tracking or im- proved energy consumption, by changing the learning rate of the controller. We have chosen learning rates i by considering the nominal values of the building parameters, to adapt the estimated parameters according to their scale. In order to illustrate how the choice of learning rate aects the trade-o between improved energy consumption and temperature tracking, we have simulated the system during the month of January, adjusting the value of i . Fig. 2.9 shows how temperature tracking and energy consumption change depending on the learning rate, considering the original choice of i from the previous simulation as the baseline and denoting it as i . Aligning with adaptive control theory, learning rate i should be chosen adequately large to be able to respond fast enough to changes and the varying load demand, but also not too large, in order to avoid undesired oscillations, which may deteriorate system performance and 37 negatively aect HVAC equipment. This implies that learning rates should be selected at the neighbor- hood where temperature tracking and energy consumption become less sensitive to changes in the value of i . It should be noted that the performance corresponds to the same one-month period and, thus, same weather conditions and heat load patterns, and may change for dierent time periods. Therefore, for the choice of i , the controller should be able to learn adequately fast the varying load demand, to provide improved performance throughout the year. Every month of the year corresponds to dierent weather conditions and, thus, dierent loads to be satised by the HVAC system. Therefore, we test the system’s performance during dierent periods of the year to capture the response of the control scheme during dierent weather conditions. The overall system is simulated during an one-year period. During one year, the average improvement is 34:57% in temperature tracking and 9:51% in overall energy consumption. Table 2.2 shows the results for each month and it can be shown that performance is improved, yet dierent for dierent weather conditions. The proposed control scheme is learning the needs of the building for dierent system changes and weather conditions, and is also able to continuously adapt to new conditions, since its learning ability is dynamic. Fig. 2.10 presents the controller gains for the Corner Class 1, Pod 1, during a one-year simulation, where it is shown how controller gains adjust for the dierent needs of each time period. Table 2.2: Performance improvement when learning is introduced Tracking error Energy consumption January 36.80% 13.68% February 36.52% 14.03% March 35.65% 11.79% April 35.35% 12.39% May 35.54% 6.54% June 37.35% 5.19% July 37.90% 7.28% August 36.55% 7.04% September 34.22% 4.52% October 31.77% 9.01% November 30.63% 10.18% December 30.62% 12.81% Full Year 34.57% 9.51% 38 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Time -10 -8 -6 -4 -2 0 2 4 Controller gains Controller gains of CornerClass1Pod1 Figure 2.10: Adaptation of controller gains of Corner Class 1, Pod 1 during 1 year However, parameter changes may happen gradually in a building’s life. Thermal and insulation prop- erties of materials and equipment may degrade signicantly after a few years [54]. Therefore, we simulate system performance during January, but after a few years, considering insulation degradation of 15%, ac- cording to [54]. Since the impact of material degradation is hard to measure and quantify, the gains of the baseline controller without adaptation are calculated using the nominal values of the material properties. If insulation properties degrade, we measured 22.76% improvement in temperature tracking and 7.79% im- provement in overall energy consumption when using the proposed control scheme compared to a scheme that calculates control gains based on the nominal values of the materials. This result implies that not only performance is improved using the proposed scheme, but also the need for maintenance and re-calibration of the control mechanism is reduced. Finally, we examined the impact of considering interconnections in the control design. In this case, we simulated the system performance during one full month of operation keeping the learning proper- ties of the algorithm, but removing the controller terms that correspond to the direct and indirect heat transfer from neighboring zones. We measured that taking into account the physical interconnections of neighboring zones into the control design results in 30:14% improvement in temperature tracking and 10:61% improvement in overall energy consumption. While information on the indirect and direct impact 39 of neighboring zones is not necessary for stability, when local controllers utilize such information, heat is allocated through all thermal zones more eciently. 40 Chapter3 EectofStrongInterconnectionsandCommunicationDelays 3.1 Introduction As technology evolves, science extends its range in examining even more complicated and large-scaled systems. Hence, the size of the control systems is continuously expanding, increasing their complexity and making it more challenging to achieve control objectives, such as stabilization, optimization and con- sensus. In order for such goals to be achieved, a great deal of research has been conducted on Networked Distributed Control Systems (NDCS). A NDCS consists of a plethora of coupled agents, which operate individually, but also interact with each other and exchange information. Instead of a centralized control scheme or local controllers that ignore the eect of interconnections, a distributed scheme involves local controllers which regulate the performance of each agent, but also handle the interconnections in an eort to match the performance of a centralized scheme. Networked systems are found in many dierent forms across multiple elds of science and technology, including energy systems, transportation networks, power grids, nancial networks, ecosystems, medical applications, chemical reactors, Unmanned Aerial Vehicles, buildings and HVAC systems, among others [109], [155]–[160]. With such a wide range of applications, NDCS nd use in helping to achieve several control objectives, such as stabilization, consensus, load balancing, rendezvous, and formation control [161], [162]. 41 Since NDCSs are substantially larger than classical control systems and are analyzed in a signicantly dierent manner, it becomes more challenging for control engineers to design such systems that are stable, satisfy specications and overcome various disturbances that occur in a real, non-ideal system. As a res- ult of the existence of this scientic challenge, there have been numerous eorts to describe and propose solutions for several problems that occur, such as stabilization of NDCSs and examination of the impact of communication delays, disturbances, noise and lossy communication channels, non-linearities, and para- metric uncertainties [42], [106], [163]–[168]. This great deal of research indicates the signicant interest from the scientic community to address the issues that occur from the practical application of a NDCS in any of the mentioned elds or in other elds. Large-scaled systems whose interconnections cannot be measured often occur in the real world, such as multi-zone HVAC systems with heat interactions, economic systems that get aected by other economic systems in an unknown manner, decentralized multi-machine power systems that aect each other in a complicated way or even robots/agents that need to fulll their operations, but aect each other during the process. Subsystems aect each other, but, since they may operate independently, the size of their inuence should be controlled. This chapter considers the case when there exists delays in the communication network of a distrib- uted adaptive control system for HVAC systems, as well as strong interconnections between the network’s agents. The objective is to provide delay-dependent conditions and robustness to communication delays, so that each agent can track eciently a reference model by attenuating the eect of strong intercon- nections via feedback based on the delayed information. First, it is assumed that each agent knows its own dynamics, as well as the interconnection parameters, but receives information about the states of its neighbors with some communication delay. A distributed control scheme is proposed and it is proven that if the communication delays and the weakened interconnections satisfy some Linear Matrix Inequality (LMI) conditions, then the proposed scheme guarantees that the tracking error of each agent is bounded with a bound that depends on the size of the delays and weakened interconnections and reduces to zero as 42 these uncertainties reduce to zero. Then, a more realistic situation is considered where each agent knows neither its dynamics nor the interconnection matrices despite the cooperation and sharing of state inform- ation. For this case a distributed adaptive control scheme is proposed and it is proven that the proposed scheme guarantees that the tracking errors are bounded and small in the mean square sense with respect to size of the delays and weakened interconnections, provided the weakened interconnections and time delays satisfy a set of LMI conditions. An illustrative example is presented, in order to demonstrate the applicability and eectiveness of the proposed schemes. 3.2 ControlObjective Let’s consider an a Linear Time Invariant (LTI) system that can be described by N subsystems (agents) of the following form: _ x i (t) =A i x i (t) +B i u i (t) + N X j=1 A ij x j (t) (3.1) wherex i 2R m i denotes the state of thei th agent,A i 2R m i m i ,u i 2R p i represents the control input of thei th agent,B i 2R m i p i ,x j 2R m j denote the states of the neighboring agents andA ij 2R m i m j are the constant interconnection matrices. The pair (A i ;B i ) is controllable. The control objective is to design an appropriate control law, so that all signals in the overall closed- loop system are bounded andx i tracks the state of a stable reference model given by: _ x ref;i (t) =A m;i x ref;i (t) +B m;i r i (t) (3.2) where x ref;i 2 R m i denotes state of the i th reference model, A m;i 2 R m i m i is known, r i 2 R q i is any known bounded piece-wise continuous signal andB m;i 2R m i q i is known. Each agenti knows its current state exactly at each time t, but the information about neighboring agent j is considered to be 43 received with some communication delay ij 2 R, due to the communication network. Thus, for each agenti the signals that are available for measurement arex i (t),r i (t) andx ij (t ij ). 3.3 DistributedControlofInterconnectedSystemswithDelays 3.3.1 Controllerstructure This section considers that the local parameters as well as the interconnection coecientsA ij are known. A distributed control scheme that guarantees stability and eective tracking when the communication delays satisfy some bounds is proposed and analyzed. Based on model reference control, the proposed control input for each agent is the following: u i (t) =K i x i (t) +L i r i (t) N X j=1 K ij x j (t ij ) (3.3) whereK i 2 R p i m i , L i 2 R p i q i andK ij 2 R p i m j represent the constant gain matrices. Since the structure ofA i andB i is considered to be known, the parametersA m;i andB m;i can be designed, so that there exist gainsK i andL i , such thatA m;i = A i B i K i andB i L i = B m;i , whileK ij are chosen to minimize the eect of the closed loop interconnectionskA ij B i K ij k, and always exist based on Lemma 1 of [42]. The rst two terms of the proposed controller are used for controlling the internal dynamics of each agent, while the remaining part is used for attenuating the eect of strong interconnections with the neighboring agents. By applying the proposed controller thei th closed-loop subsystem becomes: _ x i (t) =A m;i x i (t) +B m;i r i (t) + N X j=1 A ij x j (t) N X j=1 B i K ij x j (t ij ) (3.4) 44 The tracking error of thei th agent is dened as: e i (t) =x i (t)x ref;i (t) (3.5) wheree i 2R m i . Hence, the tracking error should satisfy the following dierential equation: _ e i (t) =A m;i e i (t) + N X j=1 A ij x j (t) N X j=1 B i K ij x j (t ij ) (3.6) which can be written as: _ e i (t) = A m;i e i (t) + N X j=1 (A ij B i K ij )e j (t) + N X j=1 (A ij B i K ij )x ref;j (t) + N X j=1 B i K ij [e j (t)e j (t ij ) +x ref;j (t)x ref;j (t ij )] (3.7) or _ e i (t) =A m;i e i (t) + N X j=1 W ij e j (t) + N X j=1 ij Z t t ij _ e j (r)dr +d i (t) (3.8) where d i (t) = N X j=1 [W ij x ref;j (t) + ij [x ref;j (t)x ref;j (t ij )]] (3.9) andW ij = A ij B i K ij and ij = B i K ij . Since,x ref;i is bounded, thend i (t) is also bounded by some constantd i . In addition, sinceA m;i is considered to have all eigenvalues with negative real parts, then there exist positive denite matricesP i ;Q i , so that: A > m;i P i +P i A m;i =Q i (3.10) 45 Our goal is to show that in this case, stability and tracking can be guaranteed, as long as delays ij satisfy some conditions. Theorem3.1. Consider the system composed of the N subsystems (3.8). Then,8ie i (t)2L 1 and lim t!1 sup st je i (s)j P N j=1 c(kW ij k + ij )forsomec> 0,providedallkW ij kand ij satisfythefollowing set of matrix inequalities: ij min (Q j ) N I + (kW ij kkP i k +kW ji kkP j k)I + ji P j ji > ji P j + ij (1 + 2N)A > m;j A m;j +(1 + 2N) N X k=1 [[A > kj A kj + 1 ij > kj kj ] ik ] 0; 8i;j (3.11) where ij ; ij 2 (0; 1). Proof. The proof is given in Appendix B. Theorem 3.1 states that if the interconnections of the overall system can be weakened enough and if the communication delays are small enough, then each subsystem will be able to track properly its corresponding reference system and the overall tracking error will be bounded. Furthermore, the upper bound of the tracking error for each agent depends on the weakened interconnections, as well as the communication delays. For every link, if the interconnection strength and the communication delay can be reduced to zero, then the behavior of the respective neighboring agent will not aect the performance of the local subsystem. Due to the form of the inequalities of Theorem 3.1, for zero delays, it is guaranteed that there exists some striclty positive margin for the norm of the interconnections. Thus, interconnections do not need to be completely canceled. The inequalities dene a set in the dimensional space of the interconnection strength of the linkskW ij k, within which the system remains stable. As comminication delays increase, this feasible region decreases and eventually vanishes, as the inequalities suggest. 46 On the other hand, if the interconnection strengthkW ij k can be set to zero, then there is some strictly positive margin for the communication delays. Thus, the system is robust to delays. The inequalities dene a polyhedron that has the communication delays as dimensions and within which the system remains stable. As the minimized interconnection strength increases, this feasible region decreases and eventually vanishes, as the inequalities suggest. By constructing and solving the following optimization problem using the matrix inequalities of Theorem 3.1, the upper bounds ij for the communication delays can be obtained as follows: maximize [ ij ] NN subject to ij min (Q j ) N I + (kW ij kkP i k +kW ji kkP j k)I + ji P j ji > ji P j + ij (1 + 2N)A > m;j A m;j + (1 + 2N) N X k=1 [[A > kj A kj + 1 ij > kj kj ] ik ] 0; 8i;j These bounds ij for the delays are sucient. The aforementioned problem is a multi-objective op- timization problem. Thus, if it is feasible, the tolerance on delays for some links may be increased by choosing the corresponding solutions from the Pareto optimal set of solutions with the appropriate choice of weighting coecients [169]. Considering interconnections and delays together, the inequalities of Theorem 3.1 suggest that if all of them can be set to zero, then the overall system is stable. In addition, due to continuity, there exists a feasible region within which the system can tolerate strictly positive interconnection strength and com- munication delays. 3.3.2 Unknownsystemparameters In this section, the case where all the parameters of the systemA i , B i , A ij are unknown is considered and a distributed adaptive control scheme is proposed and analyzed. The proposed scheme guarantees 47 stability and eective tracking, as long as the communication delays satisfy certain upper bounds and the interconnections can be weakened to be within certain upper bounds. The proposed control law has the same form as in (3.3), but in this case all gainsK i ,L i ,K ij cannot be calculated and are replaced by their on line estimatesK i (t),L i (t),K ij (t): u i (t) =K i (t)x i (t) +L i (t)r i (t) N X j=1 K ij (t)x j (t ij ) (3.12) Then, by applying the proposed controller, thei th closed-loop subsystem becomes: _ x i (t) = A m;i x i (t) +B m;i r i (t) + N X j=1 (A ij B i K ij )x j (t) + N X j=1 B i K ij [x j (t)x j (t)] B i [ ~ K i x i (t) ~ L i (t)r i (t) + N X j=1 ~ K ij (t)x j (t ij )] (3.13) where ~ K i (t) = K i (t)K i , ~ L i (t) = L i (t)L i and ~ K ij (t) = K ij (t)K ij . It is considered thatL i is unknown but it is either positive denite or negative denite, wherel i = 1 ifL i is positive denite and l i =1 if it is negative denite. Then, the tracking error of thei th agent satises the following equation: _ e i (t) = A m;i e i (t) + N X j=1 (A ij B i K ij )e j (t) + N X j=1 B i K ij [e j (t)e j (t)] B i [ ~ K i x i (t) ~ L i (t)r i (t) + N X j=1 ~ K ij (t)x j (t ij )] + N X j=1 [(A ij B i K ij )x ref;j (t) +B i K ij (x ref;j (t)x ref;j (t ij )] = A m;i e i (t) + N X j=1 W ij e j (t) + N X j=1 ij Z t t ij _ e j (r)dr +d i (t) B i [ ~ K i x i (t) ~ L i (t)r i (t) + N X j=1 ~ K ij (t)x j (t ij )] (3.14) 48 whereW ij =A ij B i K ij , ij =B i K ij ,A i B i K i =A m;i andB i L i =B m;i . The unknown controller parameters are generated by the following adaptive laws: _ K i (t) = Proj[B > m;i P i e i (t)x > i (t) sgn(l i )] (3.15) _ L i (t) = Proj[B > m;i P i e i (t)r > i (t) sgn(l i )] (3.16) _ K ij (t) = Proj[B > m;i P i e i (t)x > j (t ij ) sgn(l i )] (3.17) The adaptive law guarantees that ~ K j (t);K j (t); ~ L j (t);L j (t) ~ K ij (t);K ij (t)2L 1 , with bounds dened by the projection operator. Theorem 3.2. Consider the system composed of the N subsystems (3.14), (3.15), (3.16) and (3.17). Then,8i e i (t)2L 1 ande i (t)2S([ P N j=1 (kW ij k + ij )] 2 ), provided allkW ij k and ij satisfy the following set of matrix inequalities: ij min (Q j ) N I + (kW ij kkP i k +kW ji kkP j k)I + ji P j ji > ji P j + ij (2 + 3N)A > m;j A m;j +2 ij (2 + 3N)kB j k 2 (kK j;max k +kK > j k) 2 I + (2 + 3N) N X k=1 [A > kj A kj ik ] +(2 + 3N)[ N X k=1 [ 1 ij > kj kj ik ] + 2 ij kB j k 2 (kK ij;max k +kK ij k) 2 I N X k=1 ki ] 0 8i;j (3.18) where ij ; ij 2 (0; 1). Proof. The proof is given in Appendix B. Theorem 3.2 states that if the interconnections of the overall system can be weakened enough and if the communication delays are small enough, then each subsystem will be able to track properly its cor- responding reference system and the overall tracking error will be bounded, even though each agent’s 49 parameters as well as the interconnections are completely unknown. From the form of the matrix in- equalities of Theorems 3.1 and 3.2, it is clear that if the problem is feasible in the case of known system parameters, there always exist nonzero positive boundskW ij k for the interconnection strength and ij for the delays to guarantee stability. However, as in any robust adaptive control system, these bounds depend on the unknown parameters and cannot be calculated. 3.4 SimulationExamples In this section, a numerical example is presented, in order to demonstrate the eectiveness of the proposed control schemes. The system consists of three agents, which are described by the following systems _ x 1 (t) = 2 6 6 4 1 3 4 4:5 3 7 7 5 x 1 (t) + 2 6 6 4 1 2 3 7 7 5 u 1 (t) +A 12 x 2 (t) (3.19) _ x 2 (t) = 2 6 6 4 1 8 1 1 3 7 7 5 x 2 (t) + 2 6 6 4 2 1 3 7 7 5 u 2 (t) +A 21 x 1 (t) +A 23 x 3 (t) (3.20) _ x 3 (t) = 2 6 6 6 6 6 6 4 3 5 5 2 3 2:5 2 2:5 1 3 7 7 7 7 7 7 5 x 3 (t) + 2 6 6 6 6 6 6 4 2 1 1 3 7 7 7 7 7 7 5 u 3 (t) +A 32 x 2 (t) (3.21) wherex i 2 R m i denotes the state of thei th -subsystem,u i 2 R represents the control input of thei th - subsystem andx j 2R m j denotes the state of the neighboringj th -subsystem. Suppose that it is desired for these subsystems to track some reference models, which are described by the following LTI models: _ x ref;1 (t) = 2 6 6 4 1 0 0 1:5 3 7 7 5 x ref;1 (t) + 2 6 6 4 0:5 1 3 7 7 5 r 1 (t) (3.22) 50 _ x ref;2 (t) = 2 6 6 4 1 0 0 3 3 7 7 5 x ref;2 (t) + 2 6 6 4 1 0:5 3 7 7 5 r 2 (t) (3.23) _ x ref;3 (t) = 2 6 6 6 6 6 6 4 1 0 0 0 0:5 0 0 0 1:5 3 7 7 7 7 7 7 5 x ref;3 (t) + 2 6 6 6 6 6 6 4 1 0:5 0:5 3 7 7 7 7 7 7 5 r 3 (t) (3.24) for any bounded piece-wise continuous signalsr i (t)2R, wherex ref;i 2R m i denotes the state of thei th reference system. First, the case where interconnections can be weakened enough is considered. In this case, the inter- connection matrices are the following: A 12 = 2 6 6 4 1:05 0:35 1:85 0:5 3 7 7 5 ;A 21 = 2 6 6 4 3 1 1:5 0:4 3 7 7 5 ;A 23 = 2 6 6 4 3 1 2 1:7 0:3 1:1 3 7 7 5 ;A 32 = 2 6 6 6 6 6 6 4 2 1:85 0:9 1 1 0:9 3 7 7 7 7 7 7 5 (3.25) and communication delay is = 0:15. As shown in Fig. 3.1, if only local controllers that ignore the in- terconnections are used, then the overall tracking error is unbounded and the system becomes unstable. According to the results, this instability can be avoided if local feedback and communicated state inform- ation from other subsystems to weaken the eect of the interconnections are used. For the case of known system parameters, Fig. 3.2(a) shows that as long as the interconnections can be weakened enough and communication delay is small, the overall tracking error converges to a set close to zero and, thus, each subsystem tracks eciently the corresponding reference model. 51 Figure 3.1: Performance of the system with controllers that ignore the interconnections However, if interconnections cannot be weakened enough, then there is no guarantee that the overall system will be stable. For this case we consider A 12 = 2 6 6 4 2:95 2:15 3:95 1:9 3 7 7 5 ;A 21 = 2 6 6 4 3:3 1 1:9 2:2 3 7 7 5 ;A 23 = 2 6 6 4 2:6 1:8 2:5 1:9 2:3 1 3 7 7 5 ;A 32 = 2 6 6 6 6 6 6 4 2:9 1:85 2:5 3:8 1:8 2:6 3 7 7 7 7 7 7 5 (3.26) As shown in Fig. 3.2(b), a distributed control scheme may not be able to provide stability. In addition, since the communicated state information from other subsystems is received with some delay, the size of the delay is also crucial for stability. Thus, even in the rst case where interconnections were initially weakened enough, an increase of the communication delay may cause instability, as shown in Fig. 3.2(c) for communication delay = 0:3. Next, it is considered that each agent knows its own dynamicsA i ;B i , but does not know the inter- connection matrices A ij . In this case, the proposed adaptive control scheme can provide stability and performance, as long as interconnections can be weakened enough and communication delays are small enough, as shown in Fig. 3.3(a). However, if interconnections cannot be weakened enough, tracking error 52 Figure 3.2: Performance of the system with distributed controllers in the known parameters case: (a) Weakened interconnections and small delays, (b) Strong interconnections and small delays, (c) Weakened interconnections and large delays 53 Figure 3.3: Performance of the system with distributed controllers in the unknown interconnections case: (a) Weakened interconnections and small delays, (b) Strong interconnections and small delays, (c) Weakened interconnections and large delays 54 Figure 3.4: Performance of the system with distributed controllers in the unknown parameters case: (a) Weakened interconnections and small delays, (b) Strong interconnections and small delays, (c) Weakened interconnections and large delays 55 may not converge, as shown in Fig. 3.3(b). Moreover, if communication delay is too large, the system may become unstable even in the rst case where interconnections were initially weakened enough. Finally, the case where all system parameters are unknown is analyzed. In this case, the proposed ad- aptive control scheme can provide stability and performance, as long as interconnections can be weakened enough and the delay is small enough, as shown in Fig. 3.4(a). Similarly to the known parameters case, if the system interconnections cannot be weakened enough, the tracking error may not converge, as shown in Fig. 3.4(b). In addition, an increase in communication delay may again cause instability, as shown in Fig. 3.4(c). 56 Chapter4 ActuatorDynamicsofAirHandlingUnits 4.1 Introduction In this chapter we propose a design of a distributed adaptive control scheme which can eectively regulate temperature in multi-zone buildings with AHUs, while taking into account the actuator dynamics and the interconnections between zones and also being able to overcome parameter changes, uncertainties and unknown disturbances. The design of the control scheme depends on the modeling of the underlying components and features a backstepping scheme for each AHU and its underlying zone, as well as exchange of information between neighboring zones. The local backstepping transformation and the controller architecture are based on a PI structure, with communication between neighboring thermal zones. On- line estimation of controller parameters guarantees that the system can adapt to parameter changes as well as unknown heat gains, and distribute heat across zones in an ecient manner. The use of a distributed scheme alleviates the drawbacks of centralization, such as high computational complexity and lack of generalization to dierent systems and buildings [77], [88]. We present an illustrative simulation example, where the proposed algorithm is applied to a multi-zone school building and we show how the proposed scheme may have better performance in such a dynamic environment when compared to a control scheme that considers all parameters to be known and constant. As analyzed in [20], thermal comfort may have a direct impact on learning eectiveness in school. 57 4.2 DescriptionofaMulti-ZoneHVACSystemwithAirHandlingUnits Zone 1 AHU 1 Heat Pump Storage Tank Mixing box Fan Cooling Coil Heating Coil Zone 2 Window Temperature Sensor Storage Tank AHU 2 Mixing box Fan Cooling Coil Heating Coil Window Figure 4.1: Schematic diagram of a multi-zone HVAC system. Gray boxes indicate components with dy- namic behavior. This section provides a detailed description of the structure and modeling of multi-zone HVAC systems. Such systems are composed of building zones, AHUs and thermal storage units, which are analyzed in the following subsections. The basic structure of such systems is demonstrated in Fig. 4.1 and is described by the following equations (4.1), (4.2), (4.3) where the control input is the mass ow rate of the water that passes through each coil _ m c i 2 [0; _ m c max;i ]8i2N =f1;:::;Ng. 4.2.1 Zonemodel A thermal zone is dened as a building area, the climate of which is controlled by an AHU [41]. A typical building may consist of multiple interconnected thermal zones. We consider the following sampled-data dynamical model of the dry-bulb air temperature of thei th zone of a building withN thermal zones, with i2N =f1;:::;Ng [39], [40], [170] using sampling timeT s [151]: T z i (k + 1) =a z i T z i (k) +b z i h T sa i (k) + z i _ m sa i C pa T amb (k) + X j2N i i;j _ m sa i C pa T z j (k) + Q i (k) _ m sa i C pa i (4.1) 58 Table 4.1: Nomenclature of Chapter 4 Symbol Denition a Air density ( kg =m 3 ) i;j Inter-zone coecient ( W = o C) z i External wall coecient ( W = o C) f Fan’s power fraction _ m Mass ow rate ( kg =s) C Specic heat capacity ( J =kg o C) E Energy (J) N Total number of zones N i Set of neighboring zones ofi th zone Q Heat gain (W) T Temperature ( o C) T s Sampling time (s) UA Overall conduction heat transfer coecient of coil ( W = o C) V Volume (m 3 ) W Power (W ) Subscript Denition amb Ambient c Coil f Fan i;j Zone number pa Constant pressure air pw Constant pressure water sa Supply air st Storage tank wm Water and metal z Zone Superscript Denition c Cooling h Heating wherea z i = e _ msa i Cpa+ P j2N i i;j +z i aVz i Cz i Ts ,b z i = (1az i ) _ msa i Cpa _ msa i Cpa+ P j2N i i;j +z i . T z i ( o C) is the air temperature of thei th zone,T sa i ( o C) is the supply air temperature in thei th zone,T amb ( o C) is the outdoor ambient air temperature andT z j ( o C) is the air temperature ofj th neighboring zone for allj2N i whereN i contains the indices of the neighboring zones of thei th zone. The mass ow rate of the air supplied into the zone from the air handling unit is represented by _ m sa i ( kg =s). For a constant air mass ow rate _ m sa i , the air temperature of a zoneT z i can be regulated by the supply air temperatureT sa i and is aected by the temperature of 59 neighboring zonesT z j for allj2N i , the ambient temperatureT amb , and the heat gainQ i that may be a result of human activity, electrical equipment, lights, radiation, or other heat sources. Parameter a ( kg =m 3 ) represents air density,V z i (m 3 ) is the zone volume,C z i ( J =kg o C) represents the zone thermal capacitance, C pa ( J =kg o C) is the air specic heat capacity in constant pressure, z i ( W = o C) is the external wall heat transfer coecient and z i;j ( W = o C) corresponds to the inter-zone heat transfer coecient. Zone temperature can be aected by neighboring zones directly due to convection if there are internal openings, such as open doors, or indirectly due to conduction through walls. The interaction between zones may change drastically due to human activity. Opening of a door between two zones of dierent temperature may cause airow from the zone that has higher temperature to the other one and, sub- sequently, heat exchange due to convection [44]. Heat gainQ i is typically unknown, time-varying and dicult to measure. In addition, degradation aects heat transfer properties of materials signicantly, changing heat transfer coecients as well as their thermal capacities [54]. As temperature changes, air density uctuates. It should be noted that the presented model considers constant ux, air is assumed to be fully mixed, air distribution is uniform and there are no pressure losses across the zones and AHUs. 4.2.2 AirHandlingUnitmodel A typical Air Handling Unit is used for controlling the climate conditions of a thermal zone and consists of a mixing box, a fan, a cooling coil and a heating coil [39], as shown in Fig. 4.1. The mixing box and the fan have a static behavior and hence can be modeled with algebraic equations. The mixing box combines return air from the zone with outside air in order to guarantee circulation of fresh air in the zone and avoid the concentration of contaminants. The fan regulates the air ow rate inside the AHU, receiving air from the mixing box and passing it to the coils. The coils inside the AHU regulate the temperature of air that is supplied to the zone. Coils have a dynamic behavior which is characterized by the temperature change of 60 the water and air that pass through them and is described by the following dynamical equation [39], [40], [170]: T sa i (k + 1) =a sa i T sa i (k) +b sa i h T c i (k) + W f i f (UA) c i + C pa _ m amb i (UA) c i T amb (k) + C pa ( _ m sa i _ m amb i ) (UA) c i T z i (k) i (4.2) wherea sa i = e (UA)c i + _ msa i Cpa Csa i Ts ; b sa i = (1asa i )(UA)c i (UA)c i + _ msa i Cpa . C sa i ( J =kg o C) is the thermal capacitance of the coil and (UA) c i ( W = o C) is the overall conduction heat transfer coecient of the coil. The value of (UA) c i is typically hard to measure accurately and changes drastically over time [53]. The mass ow rate of fresh outside air in the mixing box is denoted by _ m amb i ( kg =s). Due to the fan operation, the temperature of the air that leaves the fan increases, withW f i (W) andf denoting the maximum power of the fan and the fan’s power fraction, respectively. The process presented in (4.2) aects the temperature of the water that passes through the coilT c i ( o C) as follows [39], [40], [170]: T c i (k + 1) =a c i T c i (k) +b c i h _ m c i (k) (T st (k)T c i (k)) + (UA) c i C pw T sa i (k) i (4.3) wherea c i =e (UA)c i Cwm i Ts ; b c i = (1Ac i )Cpw (UA)c i .T st ( o C) represents the temperature of the water that arrives to the coil from the thermal storage tank,C pw ( J =kg o C) is the water specic heat capacity in constant pressure andC wm i ( J =kg o C) is the thermal capacitance in the water metal point of the coil. Thermal storages provide cold or hot water to the coils of all AHUs, and their water temperature can be regulated by a heat pump. Specically, four-pipe HVAC systems facilitate heating and cooling simultaneously to increase control performance [171]. Considering processes (4.2) and (4.3), the goal is to regulate the ow rate of water _ m c i that enters the coil via a valve , as illustrated in Fig. 4.1, in order to achieve supply air temperature that can satisfy the zone needs. 61 4.3 ControlObjective The control objective is to choose the mass ow rate of the water that passes through each coil _ m c i 2 [0; _ m c max;i ]8i2N =f1;:::;Ng at each zone, so that the air temperatureT z i of each zone of the building reaches a desired temperatureT r i . The measured signal at each zonei are zone air temperatureT z i , air temperature of neighboring zonesT z j for allj2 N i , supply air temperatureT sa i , temperature of water in coilT c i , water temperature in the storage tankT st and ambient temperatureT amb . The temperature sensors are denoted in Fig. 4.1. Local Controller Local AHU Cooling Coil Heating Coil Supply Air Zone 1 Zone 2 Figure 4.2: Architecture of the proposed distributed control scheme. The cooling and heating coil valves of each local AHU are regulated by a backstepping control scheme. The designed control mechanism should be distributed in order to reduce computational complexity and avoid single points of failure. In addition, the control algorithm should only utilize temperature meas- urements and not depend on knowledge of the exact values of building parameters, in order to be able to overcome all uncertainties and disturbances in the system and at the same time avoid cumbersome cal- ibrations. Disturbances may include heat loads due to weather conditions, lighting, equipment or human activity, while uncertainties may be a result of system changes, such as material degradation or creation 62 of internal openings between zones, and can be reected in several parameters, such as inter-zone and ex- ternal wall heat transfer coecients z i;j and z i , coil overall conduction heat transfer coecient (UA) c i , air density a , etc. 4.4 DesignofValveFlowController This section presents the design of a distributed adaptive control approach, where the control gains are estimated on-line to respond to possible parameter changes, system uncertainties and disturbances. In the rst part of this section, the network structure of the multi-zone HVAC system is exploited in order to establish the form of the proposed distributed control scheme, which is illustrated in Fig. 4.2. In the second part of the section, the controller is augmented with an on-line parameter estimator which updates the controller parameters in real time in order to accommodate for parametric uncertainties. 4.4.1 Controllerarchitecturebasedonbackstepping The dynamics of the system described by eq. (4.1), (4.2) and (4.3) can be re-written as follows: e z i (k + 1) = > z i z i (k) +d z i (k) (4.4) T sa i (k + 1) = > sa i sa i (k) (4.5) T c i (k + 1) = > c i c i (k) (4.6) wheree z i =T z i T r i ,d z i (k) =b z i Q i (k) _ msa i Cpa and z i = h e z i ;T sa i ; T z j 8j2N i ;T amb ;T r i i > (4.7a) z i = " a z i ;b z i ; b z i i;j _ m sa i C pa j2N i ;b z i z i _ m sa i C pa ;a z i 1 # > = 63 = z 0;i ; z 1;i ; h z 2;i;j i 8j2N i ; z 3;i ; z 4;i > (4.7b) sa i = [T sa i ;T c i ;T z i ;T amb ; 1] > (4.7c) sa i = " a sa i ;b sa i ;b sa i C pa ( _ m sa i _ m amb i ) (UA) c i ;b sa i C pa _ m amb i (UA) c i ;b sa i W f i f (UA) c i # > = h sa 0;i ; sa 1;i ; sa 2;i ; sa 3;i ; sa 4;i i > (4.7d) c i = [T c i ;u c i ;T sa i ] > (4.7e) c i = a c i ;b c i ; b c i (UA) c i C pw ; > = h c 0;i ; c 1;i ; c 2;i i > (4.7f) and u c i (k) = _ m c i (k) (T st (k)T c i (k)) (4.8) Each local controller receives information from neighboring zones for allj2N i , in order to obtain the local control decisions. We propose the following control input, based on a backstepping formulation: _ m c i (k) = 1 b z i b sa i b c i (T st (k)T c i (k)) K > i i (k) (4.9) where y 1 i = z i e z i +K > 1 i 1 i (4.10) y 2 i = sa i y 1 i +K > 2 i 2 i (4.11) 1 i = [ z i ;I i ] > (4.12) K 1 i = z i ;h i > (4.13) 2 i = h > z i ; > sa i ; z j e z j h j I j +y 1 j j2N i ;T amb ;T r i ;I i i > (4.14) 64 K 2 i = h z 0;i +h i z i > z i ; z 1;i > sa i ; h z 2;i;j i j2N i ; z 3;i ; z 4;i ;h i i > (4.15) i = h > z i ; > sa i ;T sa i ;T c i ;y 2 i ; z j e z j h j I j +y 1 j j2N i ; sa j y 1 j +y 2 j h j I j j2N i ;T r i ;T amb ;I i ; 1 i (4.16) K i = h K > z i ;K > sa i ;l sa i ;l c i ; c i ; h l z i;j i j2N i ; h z 2;i;j i j2N i ;l r i ;l amb ;l I i ;l f i i (4.17) I i (k) =I i (k 1) +e z i (k) (4.18) and K z i = h z 0;i +h i z i sa i z 0;i + z 1;i sa 2;i + sa i ( z i h i ) +h i i z i ; K sa i = z 1;i z 0;i +h i z i + sa 0;i sa i sa i ; l sa i = z 1;i sa 1;i c 2;i ; l c i = z 1;i sa 1;i c 0;i ; l z i;j = z 2;i;j z 0;i +h i z i + z j h j sa i ; l amb = z 3;i z 0;i +h i z i + 1 sa i + z 1;i sa 3;i l r i = z 4;i z 0;i +h i z i + 1 sa i + z 1;i sa 2;i ; l I i =h i (1 sa i ); l f i = z 1;i sa 4;i (4.19) The design constants z i ; sa i ; c i ;h i are chosen to provide the system with the desired closed-loop dynamics. Signaly 1 i is based on the current values of z i and desired temperatureT r i . Consequently, signal y 2 i is based on the previous signal y 1 i computed online as well as the current values of sa i , to determine the eect of coil water temperature T c i . Design constants z i and h i are chosen so that the polynomialz 2 +(h i z i 1)z+ z i has roots inside the unit circle and ideally provide zone air temperature dynamics with certain desired performance characteristics. Similarly, sa i is chosen inside the unit circle to ideally provide the supply air temperature dynamics with certain desired characteristics. Design parameter c i is chosen inside the unit circle to ideally provide the coil water temperature dynamics with certain desired characteristics.I i (k) is a backward Euler integrator. 65 In (4.9), we assume thatjT st (k)T c i (k)j > l lb i > 08k when the plant is operating, withl lb i > 0 being a lower bound. This assumption is valid and reects the temperature gradient of the coil, as according to ASHRAE Standard 90.1, the minimum temperature dierence for operating coils should have a strictly positive lower bound [172]. In some works [39], [119], temperature gradients of coils are approximated as known and constant, even though they change during operation, or the model gets linearized around an operating point [173]. WhenT st (k)T c i (k) = 0, then _ m c i (k) = 0. Applying the control law (4.9) - (4.19) to the plant model (4.4), (4.5) and (4.6), we obtain the following closed-loop system for each zonei: e z i = z 1 z 2 + (h i 1 z i )z + z i [d z i ] z 1 (z sa i ) (z 2 + (h i 1 z i )z + z i ) [d 1 i ] + z 1 (z c i ) (z sa i ) (z 2 + (h i 1 z i )z + z i ) [d 2 i ] (4.20) where d z i (k) =b z i Q i (k) _ m sa i C pa (4.21) d 1 i (k) = z 0;i z i +h i d z i (k) + X j2N i z 2;i;j d z j (k) (4.22) d 2 i (k) = h z 0;i z i +h i z 0;i sa i + z 1;i sa 2;i +h i i d z i (k) + X j2N i " z 2;i;j z 0;i +h i z i + z j h j sa i + 1 d z j (k) + z 2;i;j d 1 j (k) # (4.23) The proposed controller in one zone and AHU is based on backstepping, while the overall HVAC system of a building is a distributed network of such interconnected backstepping control systems. The proposed control scheme uses the calculated control inputs for all units and guarantees that temperature will converge to the desired one for all zones, as summarized in the following theorem: 66 Theorem4.1. Consider the overall closed-loop system composed ofN zones and AHUs (4.4), (4.5) and (4.6) usingthecontrolschemedescribedby (4.9)- (4.19). Then,zoneairtemperatureisbounded,i.e.,T z i 2l 1 ,and converges to the desired referenceT r i ask!1, for all zonesi2N =f1;:::;Ng and for constantQ i and T amb . Proof. The proof of Theorem 4.1 is presented in Appendix C. Theorem 4.1 guarantees the zone air temperature in each zone will converge to the desired one, overcoming unknown disturbances as well as heat exchange interactions between zones for the time period when unknown heat gains Q i and ambient temperature remain constant. x 2 l 1 means that kxk 1 = sup k0 jx(k)j exists for signalx. Remark 4.1. It has to be noted that the values of z i , sa i , c i andh i should be selected considering the actual system, in order to avoid unwanted oscillations or overshooting. Appropriate selection of the design constants determines the response properties of the control scheme. 4.4.2 Estimationofcontrollergains In this section, the adaptive laws for on-line estimation of the controller gains are introduced. HVAC sys- tems suer from several uncertainties and parameter changes that are caused by human activity, weather conditions, lighting, electrical equipment, material degradation, or inaccurate measurement and approx- imation of system parameters, such as heat transfer coecients and solar gains. Thus, controller gain adaptation is meant to accommodate for all such deviations of the system parameters from their nominal values. An adaptive algorithm based on gradient descent with projection, as presented and analyzed in [102], is used in order to generate on-line the controller gains in (4.9) - (4.19). The algorithm is applied to every element of each subsystem to calculate the corresponding parameters i on-line. The internal states 67 (i.e., coil water temperature, supply air temperature and zone air temperature) are regulated through the backstepping control scheme presented in Fig. 4.2, by creating the articial states (4.27), (4.28) and applying the proposed control input (4.26) to each subsystem. In order to adaptively compute the parameters, we express the dynamics of each state in eq. (4.4), (4.5), (4.6) in the form of the following linear parametric model: z i (k) = > i i (k) (4.24) where i is a vector with all the unknown parameters to be estimated, and i (k) and z i (k) are some known signals. i (k) corresponds to the on-line estimate of the ideal i and is estimated by the following algorithm, as analyzed in [102]: ^ i (k) = i (k 1) +T s i i (k) i (k) [ i (k)] s = 8 > > > > > > > > > < > > > > > > > > > : [ ^ i (k)] s ifL i;s [ ^ i (k)] s U i;s L i;s if [ ^ i (k)] s <L i;s U i;s if [ ^ i (k)] s >U i;s ; 8s 2 i (k) = 1 +T s > i (k) i i (k) (4.25) where the subscripts corresponds to the elements of i (k) for each equipment part, i = > i 0 is a diagonal positive denite gain matrix that represents the learning rate for each element, i (k) is the nor- malized estimation error, andU i;s andL i;s are conservative bounds for the estimated parameters.L i;1 > 0 has to be strictly positive, a restriction that is imposed by the structure of all equipment dynamics and is important for the stability of the overall control scheme. 68 After replacing the controller gains with their online estimates, the proposed control input is the fol- lowing: _ m c i (k) = 1 l u 1;i (k) (T st (k)T c i (k)) K > i (k) i (k) (4.26) where y 1 i (k) = z i e z i (k) + > z;I i (k) z;I i (k) (4.27) y 2 i (k) = sa i y 1 i (k) +K > 2 i (k) 2 i (k) (4.28) z;I i = [ 1 i ; 1] > (4.29) z;I i (k) = [ z i (k);I z i (k)] > (4.30) 2 i = h > z;I i ; > sa i ; z j e z j +y 1 j +T r j j2N i ;T amb ;T r i ; 1 i > (4.31) K 2 i (k) = h z 0;i (k) z i > z;I i (k); z 1;i (k) > sa i (k); z 2;i;j (k) j2N i ; z 3;i (k); z 4;i (k);I z i (k) i > (4.32) i = h > z;I i ; > sa i ;T sa i ;T c i ;y 2 i ; z j e z j +y 1 j j2N i ; sa j y 1 j +y 2 j j2N i ; T r j j2N i ;T r i ;T amb ; 1 i (4.33) K i (k) = h K > z i (k);K > sa i (k);l sa i (k);l c i (k); c i ; l z i;j (k) j2N i ; z 2;i;j (k) j2N i ; l r i;j (k) j2N i ;l r i (k); l amb (k);l I i (k) i (4.34) 69 and K z i (k) = h z 0;i (k) z i z 0;i (k) sa i + z 1;i (k) sa 2;i (k) i z i (k); K sa i (k) = z 1;i (k) z 0;i (k) z i + sa 0;i (k) sa i sa i (k); l sa i (k) = z 1;i (k) sa 1;i (k) c 2;i (k); l c i (k) = z 1;i (k) sa 1;i (k) c 0;i (k); l z i;j (k) = z 2;i;j (k) z 0;i (k) z i sa i + z j ; l r i;j (k) = z 2;i;j (k) z 0;i (k) z i sa i + 1 l amb (k) = z 3;i (k) z 0;i (k) z i sa i + 1 + z 1;i (k) sa 3;i (k) l r i (k) = z 4;i (k) z 0;i (k) z i sa i + 1 + z 1;i (k) sa 2;i (k); l I i (k) = z 0;i (k) z i sa i + 1 I z i (k) + z 1;i (k) sa 4;i (k) (4.35) andl u 1;i (k) = z 1;i (k) sa 1;i (k) c 1;i (k)>L z 1;i L sa 1;i L c 1;i > 0.I z i (k) will be calculated by the zone para- meters adaptive law. We assume that the temperature gradient of the coil is positivejT st (k)T c i (k)j> l lb i > 0 withl lb i > 0 being a lower bound when the plant is operating [172]. The gainsK i ;K 2 i ; z;I i are estimated by using (4.15) - (4.19) and the on-line estimated parameters i , as described by the adaptive law (4.25). By adding and subtracting (4.10) in (4.4), zone air temperature tracking error dynamics can be expressed as the linear parametric model given in (4.24), with: z z i (k) =e z i (k) z i e z i (k 1) (4.36a) z i (k) = z i (k 1) (4.36b) z i = z 0;i z i ; z 1;i ; h z 2;i;j i 8j2N i ; z 3;i ; z 4;i ;I z i > = z 0;i ; z 1;i ; h z 2;i;j i 8j2N i ; z 3;i ; z 4;i ; z 5;i > (4.36c) z i (k) = z z i (k) 2 z i (k) (4.36d) 70 whereI z i is a gain to compensate for the unknown disturbanced z i . Using the adaptive algorithm that is described in (4.25) on the aforementioned model, we extract the on-line estimates z i (k) of the zone parameters z i as expressed in (4.7b), and thus z i (k), and the value ofI z i (k). Supply air temperature dynamics (4.5) can be expressed as the linear parametric model presented in (4.24) by adding and subtracting sa i T sa i (k), with: z sa i (k) =T sa i (k) sa i T sa i (k 1) (4.37a) sa i (k) = sa i (k 1) (4.37b) sa i = h sa 0;i sa i ; sa 1;i ; sa 2;i ; sa 3;i ; sa 4;i i > = h sa 0;i ; sa 1;i ; sa 2;i ; sa 3;i ; sa 4;i i > (4.37c) sa i (k) = z sa i (k) > sa i (k 1) sa i (k) 2 sa i (k) (4.37d) Using the adaptive algorithm that is described in (4.25) on the aforementioned model, we extract the on-line estimates sa i (k) of the supply air parameters sa i (4.7d), and thus sa i (k). By adding and subtracting c i T c i (k) in (4.6), we can express the coil water temperature dynamics as the linear parametric model given in (4.24), with: z c i (k) =T c i (k) c i T c i (k 1) (4.38a) c i (k) = c i (k 1) (4.38b) c i = h c 0;i c i ; c 1;i ; c 2;i i > = h c 0;i ; c 1;i ; c 2;i i > (4.38c) c i (k) = z c i (k) > c i (k 1) c i (k) 2 c i (k) (4.38d) Using the adaptive algorithm that is described in (4.25) on the aforementioned model, we extract the on-line estimates c i (k) of the coil water parameters c i as expressed in (4.7f), and thus c i (k). 71 The combination of the AHU’s components creates a backstepping control system for each thermal zone with all such systems forming a network. The proposed control input (4.26) of each subsystem (4.4), (4.5) and (4.6), where all controller gains are estimated using the adaptive algorithm (4.25), guarantees that air temperature tracking error for all zones will converge to zero, as summarized in the following theorem: Theorem4.2. Consider the overall closed-loop system composed ofN zones and AHUs (4.4), (4.5) and (4.6) using the control scheme described by (4.26) - (4.35). Then, zone air temperature is bounded, i.e.,T z i 2 l 1 , and converges to the desired referenceT r i ask!1, for all zonesi2N =f1;:::;Ng and for constantQ i andT amb . Proof. The proof of Theorem 4.2 is presented in Appendix C. Theorem 4.2 guarantees that zone air temperature can be appropriately regulated for each thermal zone even in the case of system uncertainties, unknown disturbances or changes in system parameters. Heat exchange is taken into account and the control mechanism is able to adapt when interactions between zones change (i.e., opening/closing of doors). For the analysis, internal gainsQ i and ambient temperature T amb are considered constant. It should be noted that the stability and convergence properties of the proposed control scheme do not depend on the size of the system (i.e., number of zones or AHUs). Thus, the proposed control scheme is scalable, i.e., when zones are added or deleted, only the local and neighboring control algorithm parts may require redesigning and not the overall system. Remark4.2. Properselectionoflearningrates i denesatrade-obetweenenergysavingsandtemperature tracking[109]. Fastlearningratesallowthesystemtolearnuncertaintiesandchanges,andadapttothemap- propriately. However,increasingthelearningratestoomuchmaycauseoscillationsandresultindeteriorated performance and increased energy consumption. 72 Figure 4.3: 3D plan of the ANSI/ASHRAE/IES Standard 90.1-2016 Primary School. 4.5 SimulationResultsandDiscussion In this section, the application of the proposed distributed adaptive control scheme to a multi-zone HVAC system is presented and its performance is analyzed and compared to a baseline non-adaptive scheme with nominal control gains by considering temperature tracking and energy performance. The control structure is based on a PI control design, to align with the majority of control algorithms used by the HVAC control industry that are PID and PI control designs [174]. 4.5.1 Buildingdescription The control algorithm is implemented and used to regulate temperature in a prototype primary school building model. The building model is chosen among the ones oered in the suite of ASHRAE Standard 90.1 prototype buildings, which was developed by Pacic Northwest National Laboratory [153]. A 3D 73 Table 4.2: Building zones No. Zone Name No. Zone Name 1 Bath 14 Kitchen 2 Cafeteria 15 Library Media Center 3 Computer Class 16 Lobby 4 CornerClass 1 Pod 1 17 Main Corridor 5 CornerClass 1 Pod 2 18 Mech 6 CornerClass 1 Pod 3 19 MultiClass 1 Pod 1 7 CornerClass 2 Pod 1 20 MultiClass 1 Pod 2 8 CornerClass 2 Pod 2 21 MultiClass 1 Pod 3 9 CornerClass 2 Pod 3 22 MultiClass 2 Pod 1 10 Corridor Pod 1 23 MultiClass 2 Pod 2 11 Corridor Pod 2 24 MultiClass 2 Pod 3 12 Corridor Pod 3 25 Oces 13 Gym plan of the building is presented in Fig. 4.3 and corresponds to the ANSI/ASHRAE/IES Standard 90.1-2016 Primary School model, located in Denver. The building consists of 25 thermal zones that are presented in Table 4.2. The zones have dierent sizes, their use varies and they are physically interconnected via walls and doors. Table 4.3 shows the set of neighboring zones of each building zone. There exist several classrooms, corridors and activity areas, such as the gym or the cafeteria, which correspond to dierent occupancy patterns, heat transfer from one zone to other due to opening/closing of doors and heat loads from equipment and lighting. This implies that each zone may need specialized HVAC equipment to satisfy its needs. Each zone has an AHU to provide proper temperature regulation. The AHUs are customized to allow regulation of water mass ow rate _ m c i through the coils by the proposed controller. There is also uncer- tainty in the overall conduction heat transfer coecient of the coils (UA) c i of the AHUs [53]. The control algorithm is implemented using Matlab/Simulink. It should be noted that the control algorithm is based only on the building and AHU structure, as illustrated in Fig. 4.1, and not on the EnergyPlus model. The overall system with the EnergyPlus building and HVAC model and the Matlab/Simulink control scheme is co-simulated using the Buildings Control Virtual Test Bed (BCVTB). 74 Table 4.3: List of theN i set for alli2N Set No. of Neighboring Set No. of Neighboring N 1 2,15,17,18, N 14 2,13,18 N 2 1,14,18 N 15 1,3,12,21 N 3 12,15,24 N 16 17,18,19,25 N 4 10,19 N 17 1,10,11,16,18,20,22,23 N 5 11,20 N 18 1,2,13,14,16,17,25 N 6 12,21 N 19 4,10,16, N 7 10,22 N 20 5,11,17, N 8 11,23 N 21 6, 12, 15 N 9 12,24 N 22 7, 10,17 N 10 4,7,17,19,22 N 23 8,11,17 N 11 5,8,17,20,23 N 24 3,9,12 N 12 3,6,9,15,21,24 N 25 13,16,18 N 13 14,18,25 4.5.2 Simulationdetails The performance of the HVAC system under the designed controller is simulated for a 3-months period (rst 90 days of the year) using the prototype Denver weather data from EnergyPlus. The HVAC system operates on weekdays from 7am to 5pm. Occupancy schedules are specied for each zone according to its use. In addition, internal and external doors are scheduled to open and close at several times in order to capture the possible changes in the way thermal zones interact with each other. The simulation les also include heat gains due to lighting and equipment, as well as uncertainty in the UA values of the coils. For all zones the desired temperature is selected asT r = 23 o C. The sampling time is selected asT s = 60s. The implementation of the proposed control scheme requires the choice of several design constants, such as the closed loop poles of each AHU and the learning rates of the adaptive laws. The closed loop poles are chosen to guarantee stability as well as avoid undesired performance behavior, such as system oscillations. In order to nd the appropriate values for the design constants z i , sa i , c i andh i , we match the poles of the following 4 th order polynomial: z 2 + (h i 1 z i )z + z i (z sa i ) (z c i ) (4.39) 75 which corresponds to the characteristic equation of a local closed-loop system of one zone, with the poles of a desired 4 th order polynomial of the following form: P (z) = (z 2 +a 1 z +a 0 )(zp 1 )(zp 2 ): (4.40) The pair of poles (z 2 +a 1 z +a 0 ) is selected to be dominant and is chosen based on desired transient response characteristics for zone air temperature. Following [175], [176], the other two polesp 1 andp 2 should be inside a circle of radiusr m , wherer is the magnitude of the dominant poles andm is usually between 3 and 5. The selected values should also satisfy thath i > 0 and z i , sa i , c i 2 (1; 1). Since the exact values of the system parameters may be unknown, we can use their expected nominal values in the calculations. For example, for zone 13 (Gym), the desired dominant poles are 0:94891i0:08964 and correspond to 20% overshoot and rise timet r 15min [151], whilep 1 =0:2 andp 2 = 0:82. Such a selection of poles corresponds to z i = 0:9085, sa i =0:2, c i = 0:82,h i = 0:0102. In this model, for better performance and in order to avoid oscillations it is preferable that c i and z i are relatively close to the values ofa c i anda z i , respectively, when we match the coecients of (4.40) with the ones of the polynomial of (4.39). Depending on the cooling and heating needs of the zone as well as the capabilities of the respective AHU, dierent selection of specications, such as rise time, may apply. Faster rise time implies bigger eort for the AHU in a shorter amount of time, and thus larger required water ow through the coils, while too fast response due to pole selection may result in oscillatory behavior. It has to be noted that the selected specications are for the ideal case where everything is known and there exist no disturbances, interconnections or parameter changes. Learning rates i are chosen to reect the possible change rates of system parameters and disturb- ances. Thus, the diagonal elements of i for each equipment part may be chosen to be proportional to the corresponding nominal gains to be estimated, and they should also represent the possible changes of the 76 gains. Furthermore, the magnitude of the diagonal elements of gain matrices i should be chosen taking into account the gain adaptation described in (4.25). For example, for the same zone (No.13,Gym) we have that z 1;i = 0:0786. Considering sampling timeT s = 60sec, and elements z i and z i that correspond to temperature values, in order to have z 1;i change by at most 1-2% in each time step, we need z 0;i to be in the order of 10 6 . Conservatism in the selection of i is important; in order to achieve smooth system behavior, the values of i cannot be very high, or else the system may exhibit unwanted oscilla- tions. It should also be noted that the initial values for the controller gains in the adaptive algorithm are chosen to be equal to the nominal values of such gains, in order to make use of any prior knowledge of the system; however, any reasonable initial guess may work. Projection boundsU i;s andL i;s are chosen as conservative bounds for the gains to be estimated, based on the potential change of the parameters they restrict, whileL i;1 is always strictly positive, a restriction that comes naturally from the structure of the system. Following the aforementioned guidelines, an initial selection of design constants may be made, and simulations can be used for ne-tuning. For comparison purposes, two versions of the distributed control scheme have been implemented, one with adaptation of controller gains and one without. Both are implemented in Matlab/Simulink and are connected to the EnergyPlus model using BCVTB. 4.5.3 Simulationresults The proposed distributed adaptive control scheme is able to regulate temperature satisfactorily in all build- ing zones, while overcoming changes in weather conditions, occupancy, internal heat loads and zone in- teractions due to doors, as well as changes in parameters due to material and equipment degradation. The impact of adaptive learning of such changes is illustrated by observing the improvement of the following two metrics,IntegralAbsoluteError(IAE) andE N HVAC , which correspond to temperature tracking accuracy and energy consumption, respectively [174], [177].IAE measures the total absolute temperature tracking 77 error of the HVAC system in all zones, whileE N HVAC is the total energy output of the coils and corres- ponds to the total energy that the coils retrieve from the storage tank in order to regulate the building temperature [147]. The consumed energy is reported byEnergyPlus. For the comparison, we calculate the % improvement of the aforementioned metrics when the proposed adaptive scheme is used. 07:00:00 08:00:00 09:00:00 10:00:00 11:00:00 12:00:00 13:00:00 14:00:00 15:00:00 16:00:00 17:00:00 Time (hours:minutes) 21.5 22 22.5 23 23.5 24 24.5 Zone Temperature ( o C) Constant Gains Adaptive Reference 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 0 0.05 0.1 0.15 Mass flow rate (kg/s) Figure 4.4: Performance of proposed distributed cascade scheme with and without adaptive learning for zone 13 (Gym) during the 4 th day of the year. 78 08:00 10:00 12:00 14:00 16:00 Time (hours:minutes) 0. 07 0.09 0.11 0. 13 08:00 10:00 12:00 14:00 16:00 Time (hours:minutes) -0.6 0 -0. 55 -0.50 -0.45 -0. 40 -0. 35 -0.30 08:00 10:00 12:00 14:00 16:00 Time (hours:minutes) 0. 38 0. 39 0.4 0 0. 41 0. 42 0. 43 0.360 0.365 0.370 0.375 0.380 0.385 0.390 0.395 Cooling Heating 08:00 10:00 12:00 14:00 16:00 Time (hours:minutes) 0. 5581 0. 5582 0. 5583 0. 5584 0. 5585 0. 5586 0. 5587 0. 5588 0.552 0. 553 0.554 0.555 0.556 Cooling Heating θ θ θ θ 1,13 5,13 1,13 1,13 Figure 4.5: Gain adaptation in zone 13 (Gym) during the 4 th day of the year. The supply air and coil gains adapt only when the AHU operates in the respective mode, cooling or heating. First, we tested the performance of the control algorithm by simulating the system for the rst 3 months of the year, including schedules for all aforementioned disturbances and system changes and uncertainties, i.e., doors opening and closing, but without any material or equipment degradation or uncertainty. For this particular selection of design constants, the proposed distributed adaptive control scheme achieves IAE = 4:76% reduction of the overall zone air temperature tracking error and E = 3:31% improve- ment in overall energy consumption of the coils during the 3-month period, when compared to a xed nominal-gains control scheme. This indicates that the introduction of adaptive learning can improve en- ergy consumption without compromising performance; in fact, apart from reduced energy costs, the pro- posed adaptive methodology provides accurate zone air temperature tracking, while removing the need for accurate calibration. The performance of the proposed distributed adaptive control scheme versus the non-adaptive baseline control scheme for one zone (No.13, Gym) during a typical day of operation is shown in Fig. 4.4, when operational disturbances and uncertainties are included. While the baseline controller is able to regulate zone air temperature to the desired setpoint, the proposed distributed adaptive controller, which includes learning of the properties of the AHU elements as well as the zone interactions, achieves similar tracking, while removing the need for accurate calibration. In addition, it can provide better heat allocation and, thus, improved energy performance, with well-behaved control inputs, as implied by the reduced water mass ow rate that passes through the coils. 79 Fig. 4.5 shows the adaptation of parameters z 1;13 , z 5;13 , sa 1;13 , c 1;13 of the underlying adaptive laws (zone, supply air and coil’s water) for the Gym zone (No. 13). As shown in the gure, the gains change as there are system changes and disturbances. If we compare gains sa 1;13 ; c 1;13 in Fig. 4.5 with the control input of the Gym in Fig. 4.4, we can observe that the adaptive gains of the supply air and coil controllers are updated according to the operation of the AHU (i.e., heating or cooling). For the supply air temperature and coil water temperature controllers, the gains are allowed to change only when the system is in the respective cooling or heating mode, which is dened by the controller itself. As internal gains Q i and ambient temperature T amb may change during the day, the zone needs change, putting the ideal AHU operation in cooling or heating mode accordingly. After installation, thermal properties of materials change. As reported in [54], thermal resistance of insulation elements may decrease up to 30% a few years after installation. In addition, the performance of HVAC equipment degrades during operation up to 30%, with the operational value of the overall con- duction heat transfer coecient (UA) c i being from 5% up to even 45% less than its nominal value [53]. In such a case, the proposed distributed adaptive control scheme has shown to overcome such parameter changes without requiring re-tuning and without compromising performance, leading to IAE = 4:67% reduction of the overall zone air temperature tracking error and E = 3:54% improvement in overall en- ergy consumption during the 3-month period, when its performance is compared to a xed nominal-gains control scheme. Thus, the proposed scheme achieves energy savings, while removing the need for exact calibration and re-tuning. Table 4.4: Improvement using the proposed distributed adaptive control scheme vs a non-adaptive con- troller Energy Savings (%) After installation 3.31 After a few years of operation 3.54 80 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 22 22.5 23 23.5 24 Zone Temperature ( o C) Constant Gains Adaptation Reference 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 22 22.5 23 23.5 24 Zone Temperature ( o C) Constant Gains Adaptation Reference 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 22 22.5 23 23.5 24 Zone Temperature ( o C) Constant Gains Adaptation Reference 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 22 22.5 23 23.5 24 Zone Temperature ( o C) Constant Gains Adaptation Reference 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 22 22.5 23 23.5 24 Zone Temperature ( o C) Constant Gains Adaptation Reference 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 22 22.5 23 23.5 24 Zone Temperature ( o C) Constant Gains Adaptation Reference 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 22 22.5 23 23.5 24 Zone Temperature ( o C) Constant Gains Adaptation Reference 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 22 22.5 23 23.5 24 Zone Temperature ( o C) Constant Gains Adaptation Reference Figure 4.6: Zone temperature tracking in Celsius ( o C) using the proposed distributed cascade approach for the 77th day of the year. Fig. 4.6 shows the zone air temperature tracking using the nominal control gains versus using the adaptive control gains, for several thermal zones during an indicative one-day period after a few years of installation. As shown in the gure, the adaptive scheme achieves satisfactory temperature tracking compared to a non-adaptive scheme, without needing to know the exact values of system parameters. The control inputs for the same day are presented in Fig. 4.7. Since the control algorithm allows communication between zones and also reacts when zones interactions or system parameters change, it is able to properly allocate heat energy through the building. The introduction of adaptation results in the valve controllers to pass less water and thus less energy to the coils, reducing energy consumption. It should be noted that 81 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 0 0.01 0.02 0.03 0.04 0.05 Mass flow rate (kg/s) 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 0 0.02 0.04 0.06 0.08 0.1 Mass flow rate (kg/s) 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 0 0.02 0.04 0.06 0.08 0.1 Mass flow rate (kg/s) 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 0 0.05 0.1 0.15 0.2 Mass flow rate (kg/s) 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 0 0.05 0.1 0.15 0.2 Mass flow rate (kg/s) 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 0 0.02 0.04 0.06 0.08 0.1 Mass flow rate (kg/s) 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 0 0.05 0.1 0.15 0.2 Mass flow rate (kg/s) 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 Time (hours:minutes) 0 0.02 0.04 0.06 0.08 0.1 Mass flow rate (kg/s) Figure 4.7: Water mass ow rate of coils using the proposed distributed cascade approach for the 77th day of the year. since the algorithm is demand-response, the AHUs switch from the operation of the cooling coil to the operation of the heating coil and vice versa automatically depending on the zone needs, as indicated by the valve controllers. 82 Chapter5 DistributedFaultDetectioninHVACSystems 5.1 Introduction The reliability of HVAC equipment, such as valves, fans, dampers, and pumps, and sensing devices, such as for measuring temperature and ow of air, water and refrigerants, are crucial for feedback control per- formance. However, faults in actuators and sensors are inevitable, causing unsatisfactory indoor thermal conditions and a waste of energy, estimated between 15% to 30% of a building’s energy use [178], [179]. A large or complete failure (i.e., a permanent interruption of a system’s ability to perform a required func- tion) is more likely to be diagnosed or even be observed by occupants. On the other hand, a small fault (i.e., an undetermined deviation of at least one characteristic property or parameter of the system from the acceptable, usual or standard condition) is dicult or even impossible to be diagnosed without the use of Fault Diagnosis (FD) algorithms. Thus, FD methods, which study the detection, isolation and identication of faults, have gained signicant attention in the area of building systems [180]. The industry of HVAC monitoring systems currently uses ruled-based algorithms to diagnose abnor- mal behaviors and anomalies, due to their simplicity. The rules are formed by comparing sensor data or relations of sensor data with predened constant thresholds obtained by experts (usually also called ex- pert systems). Some examples of such diagnosis schemes include the performance assessment rules that identify the mode of operation using specic relationships of measured information [181], [182] and the 83 cause-eect graphs where the various operation modes of the system (both healthy and faulty modes) are represented as discrete events [183], [184]. However, ruled-based algorithms have several limitations, as they are very specic to the system, they may fail beyond the boundaries of the expertise incorporated in them, and they are dicult to update [180]. State-of-the-art FD algorithms can be divided into data-driven and model-based algorithms. Data- driven methods include mainly computational intelligence algorithms that originate from machine learn- ing and pattern recognition and require large volume of historical data for training. Popular data-driven methods include: Principal Component Analysis (PCA) [185], [186], Support Vector Machines (SVM) [187]–[190], Neural Networks (NN) [191], [192], Genetic Algorithms (GA) [193], [194], and fuzzy logic models [195], [196]. Model-based FD algorithms can be applied without the need for training, and do not need historical data of faulty situations, while they can also introduce modeling uncertainties and require calibration of some modeling parameters. They can be classied according to the type of model, that is, statistical and state-space models. Statistical models use data to identify a simple model such as the autoregressive model with exogenous inputs (ARX) [197], the autoregressive moving average model with exogenous in- puts (ARMAX) [198], [199], and fast Fourier transform (FFT) [200], and attempt to predict the output of the system during operation using techniques such as the average error of residuals. They may require a training interval to obtain the corresponding model parameters and the state of the system (e.g., tem- perature), which is represented as a random variable that is a linear combination of its previous values. In order to obtain a valid prediction of the system’s state using statistical models, adequate training time and knowledge of the initial state of the system are required. State-estimation methods perform online learning of the state based on the real-time data of the system. Some examples are the Kalman ltering method [201], [202] and the observer-based estimation schemes [203], [204]. Such methods require the availability of analytical models that describe the behavior of building environment and HVAC system. 84 This is a challenging task due to the unknown heat gains in buildings, the large number of physically interconnected building zones, and the complexity of the electromechanical equipment. Thus, recently established legislative framework about energy performance and eciency of buildings directives [205], [206] promotes the analytical documentation of the system properties and energy models. In large-scale buildings fault isolation is important in order to isolate faults that aect only part of the building system and to identify the occurrence of multiple homogeneous or heterogeneous faults [141], [204], [207], [208]. Most of the literature on model-based FD addresses the problem of fault diagnosis for single-zone HVAC systems [189], [194], [209], with only few results on fault diagnosis in multi-zone HVAC systems for distributed actuator and sensor fault detection, isolation and accommodation [210]–[214]. The main contribution of this chapter is the design of a distributed fault diagnosis algorithm for detect- ing and isolating faults that can aect the actuators of the air handling units, such as water ow valves, or the sensor devices, such as zone air temperature sensors, supply air temperature sensors, and heating and cooling coil water temperature sensors, in large-scale multi-zone buildings. The presence of actuator and sensor faults within each AHU can result in a similar behavior, making it dicult to perceive the type and the location of single or multiple faults. Moreover, in the case of a distributed control design, sensor faults may propagate to neighboring zones, making it even more dicult to pinpoint the source of a fault. Mod- eling the temperature dynamics of each local AHU and its underlying zone with respect to its neighboring zones, a local monitoring agent using measurements and control inputs is designed to detect and identify the type and location of faults considering bounded modeling uncertainty and measurement noise. Each local diagnosis agent consists of a number of estimators that can estimate on-line the temperature of air in the zone, the temperature of the supply air to the zone from the AHU, and the temperature of water in the cooling and heating coils. For each state (i.e., temperature) the residual dierence between the measured and estimated values is calculated at each time step. Under healthy conditions, the residual is bounded by a corresponding adaptive threshold, which is calculated taking into account modeling uncertainties and 85 measurement noise in order to avoid false alarms. Each pair of residual and adaptive thresholds forms an analytical redundancy relation (ARR). The violation of an ARR indicates the existence of a fault. Exploiting the dependency of each actuator and sensor fault with each ARR, a decision logic is designed to reveal un- der certain conditions the location and type of the fault during the operation of the system. The proposed distributed fault diagnosis algorithm is evaluated through simulating the behavior of the primary school building model under several fault scenarios. Notation R denotes the eld of the real numbers. x 2 R denotes a scalar variable, x 2 R n denotes a vector and A2 R mn a matrix. For any vector x2 R n ,kx(k)k represents the Euclidean (l 2 ) vector norm in R n at each timek, wherek represents time. (A) represents the eigenvalues of matrix A and max (A) represents the maximum eigenvalue of matrix A. 5.2 HVACSystemDynamicsforFaultDiagnosis This section provides a detailed description of the structure and modeling of multi-zone HVAC systems to be used in the fault diagnosis algorithm. Such systems are composed of building zones, AHUs and thermal storage units, which are analyzed in the following subsections. It should be noted that the presented model considers constant ux, air is assumed to be fully mixed, air distribution is uniform and there are no pressure losses across the zones and AHUs. The FD algorithm is implemented in Building Management Systems (BMS). BMS collect measurements and apply control decisions within discrete time steps, and thus the discrete-time representation of the multi-zone HVAC model is presented. The formulation of the state-space equations, i.e., the selection of the state vector of each process, creates an observable set of states in order to design a distributed FD 86 algorithm capable of estimating the state of the local process. The set of measurements used in the design of the underlying estimator can form a unique combination of sensors in order to permit the isolation of faults [207]. 5.2.1 ThermalZonemodel A thermal zone is dened as the building area, the climate of which is controlled by an AHU. A typical building may consist of multiple interconnected thermal zones. We consider the following dynamical model of the air temperature for thei th zone of a building withN thermal zones, withi2N =f1;:::;Ng [39], [40], [170]: T z i (k + 1) =A d z i T z i (k) + z i z i (k) + z i (k) (5.1) wheret =kT s ,A z i = _ msa i Cpa+ P j2N i a i;j +az i aVz i Cz i ;B z i = _ msa i Cpa aVz i Cz i ,A d z i =e Az i Ts ,B d z i = A d z i 1 Az i B z i , z i (k) = B d z i Q i (k) _ msa i Cpa and z i =B d z i 1 h a i;j _ msa i Cpa i j2N i az i _ msa i Cpa (5.2) z i (k) = T sa i (k) T z j (k) j2N i T amb (k) > (5.3) whereT s is the sampling time. The air temperature of thei th zoneT z i can be controlled by the supply air temperature T sa i and is aected by the temperature of neighboring zones T z j 8j 2 N i whereN i contains the indices of the neighboring zones of thei th zone, ambient temperatureT amb , and heat gain Q i , which may be a result of human activity, electrical equipment, lights, radiation, or other heat sources. Zone temperature can be aected by neighboring zones directly due to convection if there are internal openings, such as open doors, or indirectly due to conduction through walls. 87 Table 5.1: Nomenclature of Chapter 5 Symbol Denition a Air density ( kg =m 3 ) W Power (W) a i;j Inter-zone coecient ( W = o C) Q Heat gain (W) a z i External wall coecient ( W = o C) T s Sampling time (s) UA Overall conduction heat transfer coecient of coil ( W = o C) T Temperature ( o C) t time (s) C Specic heat capacity ( J =kg o C) f fault function _ m Mass ow rate ( kg =s) n sensor noise f r Fan’s power fraction N Total number of zones V Volume (m 3 ) N i Indices of neighboring zones ofi th zone Subscript Denition amb Ambient st Storage tank c Cooling h Heating wm Water and metal f Fan hp Heat pump i;j Zone number z Zone m Mixing box o Outside air pa Constant pressure air pw Constant pressure water sa Supply air hc Heating coil cc Cooling coil Superscript Denition d Discrete version ref Reference signal * Nominal value Accent Use b Estimation signal 88 5.2.2 AirHandlingUnitmodel SupplyAir Air Handling Units are used for regulating the climate conditions, i.e., temperature, humidity, and quality of air in a thermal zone. A typical AHU consists of a mixing box, a fan, a cooling coil and a heating coil [39], as shown in Fig. 4.1. The mixing box and the fan have a static behavior and hence can be modeled with algebraic equations. The mixing box combines return air from the zone with outside air in order to guarantee circulation of fresh air in the zone and avoid the concentration of contaminants, while the fan regulates the air ow rate, receiving air from the mixing box and passing it to the coils, and its operation increases the air temperature inside the AHU. The heating and cooling coils regulate the temperature of air that is supplied to the zone. The heating coil receives hot water from a hot water storage tank and transfers thermal energy to the air that passes through the coil, while the cooling coil receives cold water from a cold water storage and absorbs thermal energy from the air that passes through it. Depending on the needs of the zone for heating or cooling, only one of the two coils may be operating at a specic moment. Coils have a dynamic behavior which is characterized by the temperature change of the water and air that pass through them. Whether they are cooling or heating coils, their dynamics follow a similar formulation. Thus, the air that passes through the cooling coil and heating coil is given by: T sa i (k + 1) = A d sa i T sa i (k) + sa i sa i (k) (5.4) wheret =kT s , A d sa i =e Asa i Ts , B d sa i = A 1 sa i A d sa i I B sa i , T sa i = 2 6 6 4 T c;sa i T h;sa i 3 7 7 5 , T c i = 2 6 6 4 T cc i T hc i 3 7 7 5 and sa i = B d sa i I G f;sa i G amb;sa i G ma;sa i (5.5) sa i (k) = T c i (k) > I T amb (k) T z i (k) > (5.6) 89 A sa i = 2 6 6 4 (UA) cc i + _ msa i Cpa Csa i 0 _ msa i Cpa Csa i (UA) hc i + _ msa i Cpa Csa i 3 7 7 5 ; B sa i = 2 6 6 4 (UA) cc i Csa i 0 0 (UA) hc i Csa i 3 7 7 5 (5.7) G f;sa i = W fan i fr (UA) cc i 0 > ; G amb;sa i = Cpa _ mo i (UA) cc i 0 > ; G ma;sa i = Cpa( _ msa i _ mo i ) (UA) cc i 0 > (5.8) CoolingandHeatingCoil Due to the orientation of the heating and the cooling coil, the air that passes through the cooling coil is aected by the air temperature leaving the fanT f i . This process aects the temperature of the water that passes through the cooling coil as follows: T sc i (k + 1) = A d sc i T sc i (k) + sc i sc i (k) + B d sc i C > (T c st (k) CT sc i (k)) _ m cc i (k) (5.9) wheret =kT s , A d sc i =e Asc i Ts , B d sc i = A 1 sc i A d sc i 1 B sc i , T sc i = 2 6 6 4 T c;sa i T cc i 3 7 7 5 and sc i = G f;sc i G amb;sc i G ma;sc i (5.10) sc i (k) = 1 T amb (k) T z i (k) > (5.11) A sc i = 2 6 6 4 (UA) cc i + _ msa i Cpa Csa i (UA) cc i Csa i (UA) cc i Cwm i (UA) cc i Cwm i 3 7 7 5 ; B sc i = 2 6 6 4 1 0 0 Cpw Cwm i 3 7 7 5 ; C = 0 1 (5.12) G f;sc i = W fan i fr Csa i 0 > ; G amb;sc i = Cpa _ mo i Csa i 0 > ; G ma;sc i = Cpa( _ msa i _ mo i ) Csa i 0 > (5.13) The air temperature of the heating coil is aected by the temperature of the air that passes through the cooling coilT c;sa i . This process aects the temperature of the air and water that pass through the heating coil as follows: 90 T sh i (k + 1) = A d sh i T sh i (k) + sh i sh i (k) + B d sh i C > sc T c st (k) C sc i T sh i (k) _ m cc i (k) + B d sh i C > sh T h st (k) C sh i T sh i (k) _ m hc i (k) (5.14) wheret = kT s , T sh i = T c;sa i T cc i T h;sa i T hc i > , A d sh i = e A sh i Ts , B d sh i = A 1 sh i A d sh i 1 B sh i and sh i = G f;sh i G amb;sh i G ma;sh i (5.15) sh i (k) = 1 T amb (k) T z i (k) > (5.16) A sh i = 2 6 6 6 6 6 6 6 6 6 6 4 (UA) cc i + _ msa i Cpa Csa i (UA) cc i Csa i 0 0 (UA) cc i Cwm i (UA) cc i Cwm i 0 0 _ msa i Cpa Csa i 0 (UA) hc i + _ msa i Cpa Csa i (UA) hc i Csa i 0 0 (UA) hc i Cwm i (UA) hc i Cwm i 3 7 7 7 7 7 7 7 7 7 7 5 ;B sh i = 2 6 6 6 6 6 6 6 6 6 6 4 1 0 0 0 0 Cpw Cwm i 0 0 0 0 0 0 0 0 0 Cpw Cwm i 3 7 7 7 7 7 7 7 7 7 7 5 (5.17) C sc = 0 1 0 0 ; C sh = 0 0 0 1 ; G f;sh i = W fan i fr Csa i 0 0 0 > (5.18) G amb;sh i = Cpa _ mo i Csa i 0 0 0 > ; G ma;sh i = Cpa( _ msa i _ mo i ) Csa i 0 0 0 > (5.19) T c st ,T h st represent the temperature of water that arrives to the coil from the chilled water storage tank and the heated water storage tank, respectively. The HVAC model is described by nonlinear dynamics, since the input (i.e., mass ow rate) is multiplied by the state variable (i.e., temperature). However, the AHU operates as a Fan-Coil Unit, which means that the mass ow rate of supply air _ m sa i is predened and constant, while the mass ow rate of water in coils _ m c i acts as the control input and it can vary with time. Therefore, the non-linearity holds only for the coil 91 dynamics, given in equations (5.9), (5.14), while the dynamics of zone air and supply air temperature are linear. 5.3 FaultDiagnosisObjective The objective of this chapter is to design a distributed model-based FD algorithm for detecting and isolating both actuator and sensor faults in multi-zone HVAC systems. The proposed distributed FD algorithm consists of dedicated FD Agents that collect control inputs and sensor measurements from each AHU and underline zone and give alarms in the presence of sensor and actuator faults using the available modeling parameters, while taking into account modeling uncertainties (e.g., internal gains and measurement noise). A fault isolation logic is designed to reveal the location and type of faults. Each agent consists of four estimators that monitor the zone air temperatureT z i , the supply air tem- peratureT sa i and the water temperature in the cooling coilT cc i and heating coilT hc i , respectively. The available sensor measurements for each AHU are: y z i (k) = T z i (k) +n z i (k) +f z i (k) (5.20) y sa i (k) = C 2 6 6 4 T c;sa i (k) T h;sa i (k) 3 7 7 5 +n sa i (k) +f sa i (k) (5.21) y cc i (k) = T cc i (k) +n cc i (k) +f cc i (k) (5.22) y hc i (k) = T hc i (k) +n hc i (k) +f hc i (k) (5.23) with C = 0 1 , wherey z i is the measurement of theith zone air temperatureT z i ,y sa i is the meas- urement of the supply air temperatureT sa i andy cc i ,y hc i are the measurements of water temperature in cooling coil T cc i and heating coil T cc i , respectively. The terms n z i , n sa i , n cc i , n hc i represent the noise corrupting the measurements andf z i ,f sa i ,f cc i ,f hc i are the corresponding sensor faults. 92 Faults can aect the actuation devices of the AHU, i.e., the mechanical valves that regulate the water mass ow rate of the cold and hot water that pass through the cooling and heating coils, respectively. The water mass ow rate in cooling and heating coil valves can be represented by: _ m cc i (k) = u cc i (k) +f m cc i (k) (5.24) _ m hc i (k) = u hc i (k) +f m hc i (k) (5.25) where _ m cc i and _ m hc i are the actual water ow rates of the cooling and heating coils, respectively, u cc i andu hc i are the control inputs, andf m cc i andf m hc i are the actuator faults that can aect the valves of the cooling and heating coil, respectively. Note that due to the physical limitations of valves, the actual water mass ow rate of both coils is bounded, i.e., _ m cc i 2 [0; _ m cc max;i ] and _ m hc i 2 [0; _ m hc max;i ] for alli2N . The signals denoted byf in (5.20)–(5.25) represent the unknown faults and can be described as: f(k) = (kk f )(kk f ) (5.26) wherek f is the rst time instant of fault occurrence, is the fault function and denotes the fault’s time prole where (i) (k), o (k;), with o (k;) = 8 > > < > > : 0; k< 0 1e k ; k 0 (5.27) where being the time evolution rate of faultf. Note that!1 models an abrupt fault, while! 0 describes a fault that evolves gradually. 93 Remark 5.1. For this work the measurements of heated and chilled water temperature in the storage tank denoted byT h st andT c st , respectively, and the measurements of ambient air temperatureT amb are considered available and healthy. 5.4 DistributedFaultDiagnosisAlgorithm This section presents the design of the proposed distributed fault diagnosis algorithm. As illustrated in Fig. 5.1, an FD Agent is designed for each zone-AHU that uses: • the local control inputs for the water mass ow rates for the heating coilu hc i and for the cooling coilu hc i determined by the controller, • the local measurements from the sensors installed in the zone and AHU, i.e., zone air temperature y z i , supply air temperaturey sa i , water temperature in the heating coily hc i , water temperature in the cooling coily cc i , • the air temperature measurements from the sensors located in the neighboring zones y z j , for all j2N i , and • the detection decision signals D z j from theN i neighboring Fault Diagnosis Agents. Fig. 5.1 illustrates the architecture of the local Fault Diagnosis Agent of zone 1, where zone 1 is physic- ally interconnected with zone 2. The diagnosis procedure of each agent consists of the following on-line processes: 1. StateEstimation. Estimation of zone air temperature, supply air temperature, heating coil and cooling coil water temperature. 2. Fault Detection. Fault detection process is based on the development of analytical redundancy re- lations (ARRs) for each estimated quantity. Each ARR corresponds to the comparison of a residual 94 (i.e., the dierence between the measured and estimated quantity) with an adaptive threshold calcu- lated considering a healthy system (i.e., absence of faults). Thus, the residual should be maintained below the adaptive threshold under healthy conditions for all time instances. Violation of the ARRs triggers an alarm and indicates the detection of a fault or faults. 3. DistributedFaultIsolation is responsible for revealing the location of the fault. Local and neighboring detection signals form an observed decision patternD (i) = D cc i ; D hc i ; D sa j ; D z i ; D z j that is com- pared online with a number of theoretical decision patterns obtained by the Fault Signature Matrix illustrated in Fig. 5.1 and in Table 5.3. The matrix uses binary logic and assigns “1” if a fault pattern can directly aect an ARR, “*” if a fault pattern can indirectly (through shared measurements) aect an ARR, and “0” if a fault pattern cannot aect an ARR. If the outcome of the comparison is unique, then we can reach to a decision about the location of the fault(s) or if is not unique we can reduce the number of candidate faults that trigger the alarm. Assumption 5.1. The modeling uncertainty termQ i , produced by the internal heat gains, is not available and is considered to be bounded by known boundQ i such asjQ i (k)jQ i (k), for allk andi2N. Assumption5.2. Thenoiseinmeasurementsn z i n sa i ,n hc i n cc i isunknownbutusingsensorsaccuracygiven bymanufactures’technicalspecicationsweareabletoobtainupperboundsonthemeasurementsnoisesuch that:jn z i (k)jn z i ,jn sa i (k)jn sa i ,jn hc i (k)jn hc i andjn cc i (k)jn cc i , for allk andi2N. 95 Fault Diagnosis Agent 1 Zone Estimator Supply Air Estimator Heating Coil Estimator Cooling Coil Estimator Fault Detection Local Fault Signature Matrix Fault Detection Fault Detection Fault Detection Figure 5.1: The design of the FD Agent 1 of zone 1. Since zone 1 is interconnected with zone 2, the air temperature measurements of zone 2 (y z 2 ) and the detection decision of FD Agent 2 (D z 2 ) are used in the fault diagnosis process of FD Agent 1. 5.4.1 Temperatureestimation ZoneAirTemperatureEstimation This section illustrates the estimators design. A dedicated Luenberger observer is designed to estimate the air temperatureT z i in theith zone as follows: ^ T z i (k + 1) =A d z i ^ T z i (k) + z i Y z i (k) +L z i y z i (k) ^ T z i (k) (5.28) 96 Y z i (k) = y sa i (k) y z j (k) j2N i T amb (k) > (5.29) where ^ T z i is the estimation of theith zone air temperature with ^ T z i (0) = 0 andL z i is the observer gain selected so thatj(A d z i L z i )j < 1, in order to ensure the asymptotic stability of the observer. The estimator is designed using the available sensor measurements of the monitored subsystem, i.e.,y z i , and sensor measurements from physically interconnected subsystems that are collected in vector Y z i . Then, let’s dene the zone air temperature estimation error " z i (k) = T z i (k) ^ T z i (k), where zone state-estimation error dynamics are the following: " z i (k + 1) = A d z i L z i " z i (k) + z i N z i (k) + z i F z i (k) + z i (k)L z i (n z i (k) +f z i (k)) (5.30) N z i (k) = n sa i (k) n z j (k) j2N i 0 > (5.31) F z i (k) = f sa i (k) f z j (k) j2N i 0 > (5.32) where N z i (k) and F z i (k) are the noise and fault vectors aecting the zone estimation air dynamics, that originate from the physically interconnected subsystems. SupplyAirTemperatureEstimation For supply air temperature, a dedicated Luenberger observer is designed to estimate T sa i , which is a vector containing the temperature of the air that passes through the cooling coilT c;sa i and through the heating coilT h;sa i in theith AHU as follows: ^ T sa i (k + 1) = A d sa i ^ T sa i (k) + sa i Y sa i (k) + L sa i y sa i (k) C ^ T sa i (k) (5.33) Y sa i (k) = " y cc i (k) y hc i (k) 1 T amb (k) y z i (k) # > (5.34) 97 where ^ T sa i = ^ T c;sa i ^ T h;sa i > is a vector that contains the estimation of the air temperature in the cooling and heating coil with ^ T sa i (0) = 0 21 and L sa i 2 R 21 is a vector that consists of the observer gains selected such thatj max (A d sa i L sa i C)j < 1, in order to ensure the asymptotic stability of the observer. Then, let’s dene the supply air temperature estimation error " sa i (k) = T sa i (k) ^ T sa i (k), where zone state-estimation error dynamics are the following: " sa i (k + 1) = A d sa i L sa i C " sa i (k) + sa i N sa i (k) + sa i F sa i (k) L sa i n sa i (k) +f sa i (k) (5.35) N sa i (k) = " n cc i (k) n hc i (k) 1 0 n z i (k) # > (5.36) F sa i (k) = " f cc i (k) f hc i (k) 1 0 f z i (k) # > (5.37) where N sa i (k) and F sa i (k) are the noise and fault vectors aecting the zone estimation air dynamics, that originate from the physically interconnected subsystems. CoolingCoilWaterTemperatureEstimation Next, the design of the estimator of cooling coil’s water temperature is presented. A dedicated Luenberger observer is designed to estimate the temperature of the supply air that passes through the cooling coil T c;sa i and the water temperature in the cooling coilT cc i of theith AHU as follows: ^ T sc i (k + 1) = A d sc i ^ T sc i (k) + B d sc i C > T c st (k)y cc i (k) u cc i (k) + sc i Y sc i (k) + L sc i y cc i (k) C ^ T sc i (k) (5.38) Y sc i (k) = 1 T amb (k) y z i (k) > (5.39) where ^ T sc i = ^ T c;sa i ^ T cc i > is a vector that contains the estimation of the air and water temperature in the cooling coil with ^ T sc i (0) = 0 21 and L sc i 2 R 21 is a vector that consists of the observer gains 98 selected such thatj max (A d sc i L sc i C)j < 1 in order to ensure the asymptotic stability of the observer. Then, let’s dene the cooling coil’s air and water temperature estimation error" sc i (k) = T sc i (k) ^ T sc i (k), where zone state-estimation error dynamics are the following: " sc i (k + 1) = A d sc i L sc i C " sc i (k) + B d sc i C > (n cc i (k) +f cc i (k))u cc i (k) + B d sc i C > (T c st (k)T cc i (k))f m cc i (k) + sc i N sc i (k) + sc i F sc i (k) L sc i n cc i (k) +f cc i (k) (5.40) N sc i (k) = 0 0 n z i (k) > (5.41) F sa i (k) = 0 0 f z i (k) > (5.42) where N sc i (k) and F sc i (k) are the noise and fault vectors aecting the cooling coil estimation dynamics, that originate from the physically interconnected subsystems. HeatingCoilWaterTemperatureEstimation A dedicated Luenberger observer is designed to estimate the temperature of the supply air that passes through the heating coil T h;sa i and the water temperature in the heating coil T hc i of the ith AHU as follows: ^ T sh i (k + 1) = A d sh i ^ T sh i (k) + sc i Y sc i (k) + B d sh i C > sc T c st (k) C sc i T sh i (k) u cc i (k) + B d sh i C > sh T h st (k) C sh i T sh i (k) u hc i (k) + L sh i y hc i (k) C ^ T sh i (k) (5.43) Y sh i (k) = 1 T amb (k) y z i (k) > (5.44) where ^ T sh i = ^ T c;sa i ^ T cc i ^ T h;sa i ^ T hc i > is a vector that contains the estimation of the air temper- ature in the cooling and heating coil with ^ T sh i (0) = 0 41 and L sh i 2R 41 is a vector that consists of the 99 observer gains selected such thatj max (A d sh i L sh i C sh )j< 1 in order to ensure the asymptotic stability of the observer. Then, let’s dene zone air temperature estimation error " sh i (k) = T sh i (k) ^ T sh i (k), where zone state-estimation error dynamics are the following: " sh i (k + 1) = A d sh i L sh i C sh " sh i (k) + B d sh i C > sc (n cc i (k) +f cc i (k))u cc i (k) + B d sh i C > sh (n hc i (k) +f hc i (k))u hc i (k) + B d sh i C > sc (T c st (k)T cc i (k))f m cc i (k) + B d sh i C > sh T h st (k)T hc i (k) f m hc i (k) + sh i N sh i (k) + sh i F sh i (k) L sh i n hc i (k) +f hc i (k) (5.45) N sh i (k) = 0 0 n z i (k) > (5.46) F sh i (k) = 0 0 f z i (k) > (5.47) where N sh i (k) and F sh i (k) are the noise and fault vectors aecting the heating coil estimation dynamics, that originate from the physically interconnected subsystems. 5.4.2 Adaptivethresholddesignforfaultdetection This section presents the design of the fault detection algorithm that involves the creation of ARRs com- prised of residuals and adaptive thresholds assuming healthy system, i.e., all fault signals set to zero, f z i (k)= f sa i (k)= f cc i (k) = f hc i (k)= f m cc i (k)= f m hc i (k) = 0 for all i2 N. The ARRs should be valid in the absence of faults and violated in their presence. The residuals are formed as follows: z i (k) = y z i (k) ^ T z i (k) (5.48) sa i (k) = y sa i (k) C ^ T sa i (k) (5.49) cc i (k) = y cc i (k) C ^ T sc i (k) (5.50) hc i (k) = y hc i (k) C sh ^ T sh i (k) (5.51) 100 where the estimations ^ T z i (k), ^ T sa i (k), ^ T sc i (k), ^ T sh i (k) are calculated by the observers presented in (5.28), (5.33), (5.38) and (5.43). Thus, the fault detection process involves the computation of an adaptive threshold for each one of the above residuals such that under healthy conditions the following ARRs must be satised for allk 0: E z i : j z i (k)j z i (k) (5.52) E sa i : j sa i (k)j sa i (k) (5.53) E cc i : j cc i (k)j cc i (k) (5.54) E hc i : j hc i (k)j hc i (k) (5.55) where denotes the corresponding adaptive threshold. The computation of the adaptive thresholds is presented next. ComputationofZone’sAdaptiveThreshold The rst step in the computation of an adaptive threshold is to express the residual with respect to the corresponding estimation error. Thus, the residual z i (k) can be dened using the zone’s air temperature estimation error" z i (k) given as follows: z i (k) =y z i (k) ^ T z i (k) = " z i (k) +n z i (k) +f z i (k) (5.56) From (5.56) it can be concluded that the residual z i (k) can be aected by the sensor faultsf z i ,f sa i and f z j , for allj2 N i . Considering healthy conditions (i.e.,f z i =0,f sa i =0 andf z j =0, for allj2 N i ) and by 101 applying the Minkowski inequality on (5.56) we can obtain the adaptive threshold z i (k) that corresponds to the residual z i (k): j z i (k)j z i (k) = k z i o z i +n z i +B d z i k1 X z=1 ( z i ) k1z h n sa i + X j2N i a i;j _ m sa i C pa n z j +L z i (n z i ) + Q i _ m sa i C pa i (5.57) with A d z i B d z i L z i z i < 1 andj" z i (0)j o z i , where z i and o z i are design parameters. ComputationofSupplyAirAdaptiveThreshold The residual sa i (k) can be dened using the supply air temperature estimation error" sa i (k) given in (5.35) as follows: sa i (k) =y sa i (k) C ^ T sa i (k) = C" sa i (k) +n sa i (k) +f sa i (k) (5.58) From (5.58), the residual sa i (k) can be aected by the sensor faultsf z i ,f sa i ,f cc i ,f hc i . Considering healthy conditions (i.e., f z i =0, f sa i =0, f cc i =0, f hc i =0), the computation of the adaptive threshold sa i (k) on the residual sa i (k) is calculated by applying the Schwartz inequality on (5.58) and the result is the following: j sa i (k)j sa i (k) = ( sa i ) k o sa i +n sa i + k1 X z=1 ( sa i ) k1z B d sa i n cc i +n hc i + k1 X z=1 ( sa i ) k1z B d sa i G ma;sa i n z i + k1 X z=1 ( sa i ) k1z kL sa i kn sa i (5.59) where C ~ A sa i k ( sa i ) k andk" sa i (0)k o sa i for allk 0. Note that sa i < 1 and o sa i are design parameters. 102 ComputationofCoolingCoil’sAdaptiveThreshold The residual cc i (k) can be dened using the cooling coil’s air and water temperature estimation error " sc i (k) given in (5.40) as follows: cci (k) =y cci (k) C ^ T sci (k) = C" sci (k) +n cci (k) +f cci (k) (5.60) From (5.60), it is concluded that the residual cc i (k) can be aected by the sensor faultsf z i andf cc i , as well as the actuator faultf m cc i . Considering healthy conditions (i.e.,f z i =0,f cc i =0,f m cc i =0), the computation of the adaptive threshold cc i (k) on the residual cc i (k) is calculated by applying the Schwartz inequality on (5.60) and the result is the following: j cc i (k)j cc i (k) = ( sc i ) k o sc i +n cc i + k1 X z=1 ( sc i ) k1z B d sc i n cc i ju cc i (z)j + k1 X z=1 ( sc i ) k1z B d sc i G ma;sc i n z i + k1 X z=1 ( sc i ) k1z kL sc i kn cc i (5.61) where C ~ A sc i k ( sc i ) k andk" sc i (0)k o sc i for allk 0. Note that sc i < 1 and o sc i are design parameters. ComputationofHeatingCoil’sAdaptiveThreshold The residual hc i (k) can be dened using the heating coil’s water temperature estimation error" sh i (k) given in (5.45) as follows: hci (k) =y hci (k) C sh ^ T shi (k) = C sh " shi (k) +n hci (k) +f hci (k) (5.62) From (5.62), the residual sh i (k) can be aected by the sensor faultsf z i ,f cc i andf hc i , as well as the actuator faultsf m cc i ,f m hc i . Considering healthy conditions (i.e.,f z i =0,f cc i =0,f hc i =0,f m cc i =0,f m hc i =0 ), the computation 103 of the adaptive threshold sh i (k) on the residual sh i (k) is calculated by applying the Schwartz inequality on (5.62) and the result is the following: j hc i (k)j hc i (k) = ( sh i ) k o sh i +n hc i + k1 X z=1 ( sh i ) k1z B d sh i n cc i ju cc i (z)j + k1 X z=1 ( sh i ) k1z B d sh i n hc i ju hc i (z)j + k1 X z=1 ( sh i ) k1z B d sh i G ma;sh i n z i + k1 X z=1 ( sh i ) k1z kL sh i kn hc i (5.63) where C sh ~ A sh i k ( sh i ) k andk" sh i (0)k o sh i for allk 0. Note that sh i < 1 and o sh i are design parameters. 5.4.3 Distributedfaultdetectionandisolationlogic The fault detection logic is based on a detection signal D, designed for each pair of residual and adaptive threshold as follows: D ? (k) = 8 > > < > > : 1 kk D ? 0 otherwise (5.64) k D ? =fk :j ? (k)j> ? (k)g (5.65) wherek D ? represents the detection time step of the corresponding ARR and? denotes the subscripts of the corresponding subsystem such as?2fz i ; sa i ; cc i ; hc i g. The Distributed Fault Isolation Logic is based on binary logic using the ARRs given in (5.52)-(5.55). Table 5.2 is the dependency matrix that summarizes how each fault can aect the ARRs using the resid- uals included in the fault detection algorithm of Section 5.4.2. If a fault can directly or indirectly aect 104 the corresponding ARR. The indirect eect results if the observer of the estimated quantity, e.g., the ob- server estimating the zone air temperature, uses measurements of another quantity except the one that is estimated, e.g., measurements of the supply air temperature or measurements of a neighboring zone’s air temperature. Moreover, faults in the actuators that inuence the estimated quantity are also classied as direct. Based on the dependency matrix, the local fault signature matrix shows the logic on how the possible combinations of faults can aect the corresponding ARRs, where “1" refers to the case that the corresponding ARR is mainly aected by the corresponding fault, “" refers to the case that the corres- ponding ARR is aected by the corresponding fault due to the sensor measurement exchange between the estimators and “0" denotes the case that the corresponding ARR is not aected by the corresponding fault. Table 5.2: Dependency Matrix ofi-th Fault Diagnosis Agent 1: Direct : Indirect E cc i f m cc i ,f cc i f z i E hc i f m hc i ,f hc i f m cc i ,f cc i ,f z i E sa i f sa i f cc i ,f hc i ,f z i E z i f z i f sa i ,f z j E z j f z j f z i Table 5.3: Local Fault Signature Matrix ofi-th Fault Diagnosis Agent f m cc i f m hc i f cc i f hc i f sa i f z i f z j E cc i 1 0 1 0 0 0 E hc i 1 1 0 0 E sa i 0 0 1 0 E z i 0 0 0 0 1 E z j 0 0 0 0 0 1 In order to reach into a decision about the possible location of a fault in each AHU, we compare at each time step the observed diagnosis setD (i) designed as D (i) (k) = D cc i (k); D hc i (k); D sa i (k); D z i (k); D z j (k) > (5.66) 105 with a set of theoretical patternsF (i) obtained based on the local fault signature matrix, given in Table 5.3. The outcome of the online comparison of the observed diagnosis setD (i) to theN (i) c theoretical pat- ternsF (i) q ,q2f1;:::;N (i) c g is the diagnosis set (i) (t), which is determined as: (i) (t) = n F (i) c i :i2I (i) (t) o (5.67) withI (i) (t) = n k :F (i) k =D (i) (t); k2f1;:::;N (i) c g o . For each fault diagnosis agent, the diagnosis sets contain all the possible fault combinations. In the case the observed fault pattern has a unique match with one of the theoretical fault patterns, then the isolation algorithm can identify the location and type of the fault, otherwise it can reduce the number of candidate fault combinations. 5.5 SimulationExample In this section the application of the proposed distributed fault diagnosis scheme to a multi-zone HVAC system is presented and its performance is analyzed in the presence of sensor and actuator faults. 5.5.1 Buildingdescription The distributed fault diagnosis algorithm is implemented in the prototype primary school building model, which consists of 25 thermal zones. The building model is chosen among the ones oered in the suite of ASHRAE Standard 90.1 prototype buildings, which was developed by Pacic Northwest National Laborat- ory [153], and it is modied for the purposes of this work [126], [154]. The zones have dierent sizes, their use varies and they are physically interconnected via walls and doors. For example, there exist several classrooms, corridors and activity areas, such as the gym or the cafeteria, which correspond to dierent 106 occupancy patterns and heat loads from equipment and lighting. This implies that the temperature in each zone can be aected by various sources of heat that can not be available or modeled. Each zone has an AHU to provide proper temperature regulation. The AHUs are customized to allow regulation of water mass ow rate _ m cc i ; _ m hc i through the coils by a controller, and the system uses a distributed control design. The fault diagnosis algorithm is implemented using Matlab/Simulink. The overall system with the EnergyPlus building and HVAC model and the Matlab/Simulink fault diagnosis scheme is co-simulated using the Buildings Control Virtual Test Bed (BCVTB) [215]. 5.5.2 Simulationdetails The test-bed of the multi-zone HVAC system presented in the previous section is simulated for a 2-days period (1st of April to 2nd of April) using the prototype Denver weather data fromEnergyPlus. The HVAC system operates on weekdays, from 6am to 6pm during the winter period and from 7am to 7pm during the summer period. Occupancy schedules are specied for each zone according to its use. In addition, internal and external doors are scheduled to open and close at several times in order to capture the possible changes in the way thermal zones interact with each other. For all zones the desired temperature is selected as T ref z = 23 o C. The sampling time is selected asT s = 60s. Table 5.4: Design constants for fault diagnosis Variable Value L z i place(A z i ; 1; 0:1(A z i )) L sa i place(A sa i ; C; [1 10 13 2 10 10 ](A sa i )) L sc i place(A sc i ; C; [0:8 0:0087](A sc i )) L sh i place(A sh i ; C sh ; [0:1 0:2 0:5 0:7](A sh i )) Q i 2 W n z i ,n sa i ,n cc i ,n hc i 0.3 o C z i ; sa i ; sc i ; sh i < 1 0.7 o z i ; o sa i ; o sc i ; o sh i 100 o C The implementation of the proposed distributed fault diagnosis algorithm requires the choice of several design parameters, such as zonal modeling uncertainty bounds, sensor noise measurement bounds and 107 observer gains, which are presented in Table 5.4. The observer gains for each observer L z i , L sa i , L sc i , L sh i ,i2N are chosen to guarantee stability of the estimators for each AHU. It is noted that the observers’ initial condition is chosen to be equal to zero and the estimators reset when the HVAC system does not operate in order to avoid fault alarms while the AHUs are inactive. The design bound on the modeling uncertainty for each zoneQ i is selected to be 2 W and represents the maximum unknown heat gain. The design bounds of the sensor measurement noise can be obtained by the technical specications of the manufacturer, and for this simulation example they were selected to be 0:3 o C. 5.5.3 Simulationresults The performance of the proposed distributed FD algorithm is analyzed in this section. Its performance is evaluated with respect to the three fault models that follow the model given in (5.26) and are presented below: • Fault Model 1: Abrupt additive fault with a fault function s =10 o C for the sensor faults and a =0.1 kg/sec for the actuator (valve) faults, where=110 4 determines the fault’s time prole. Denoted by red color in the plots. • Fault Model 2: Abrupt multiplicative fault that its function is selected to increase 10% from the corresponding temperature for the sensor faults and 50% from the nominal control input for the actuator faults, with=110 4 . Denoted by blue color in the plots. • Fault Model 3: Incipient additive fault that has the same fault function as Fault Model 1, where the time prole of the faults is selected such that=0.97. Denoted by green color in the plots. For simplicity, the time of fault occurrence for all faulty cases is selected ask f =129600 sec, which corres- ponds to day 2 of the simulation at 12:00 p.m., i.e., 01:12:00:00. To evaluate the performance of the proposed FD algorithm we run the following simulation scenarios: 108 • Scenario 1: Sensor faultf z 10 with Fault Model 1, • Scenario 2: Sensor faultsf sa 4 with Fault Model 1,f sa 6 with Fault Model 3, • Scenario 3: Sensor faultsf cc 7 with Fault Model 1,f cc 8 with Fault Model 2, • Scenario 4: Actuator faultsf m cc 10 with Fault Model 1,f m cc 12 with Fault Model 3. In order to indicate the impact of faults on energy consumption and thermal comfort, the following simulation result is provided. A fault in the temperature sensor located in Zone 10 (Corridor Pod 1) with Fault Model 1 occurs at 01:12:00:00. The outcome with respect to energy waste and uncomfortable indoor conditions is presented in Fig. 5.2. Specically, the left-hand side plot of Fig. 5.2 shows the dierence between the energy (in kWh) consumed under a faulty scenario and under healthy conditions. Since both simulations run under the same conditions and schedules, we expect energy usage to be maintained close to 0 (light green color), as prior to the fault occurrence. However, when the sensor fault occurs, the energy waste of the AHU in Zone 10 is signicantly increased. Furthermore, if this fault remains undetected for a long time period, then it is projected to cause a signicant waste of energy. 01:00:00:00 01:01:00:00 01:02:00:00 01:03:00:00 01:04:00:00 01:05:00:00 01:06:00:00 01:07:00:00 01:08:00:00 01:09:00:00 01:10:00:00 01:11:00:00 01:12:00:00 01:13:00:00 01:14:00:00 01:15:00:00 01:16:00:00 01:17:00:00 01:18:00:00 01:19:00:00 01:20:00:00 01:21:00:00 01:22:00:00 01:23:00:00 02:00:00:00 Time (dd:hh:mm:ss) Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 8 Zone 9 Zone 10 Zone 11 Zone 12 Zone 13 Zone 14 Zone 15 Zone 16 Zone 17 Zone 18 Zone 19 Zone 20 Zone 21 Zone 22 Zone 23 Zone 24 Zone 25 Energy use (Faulty-Healthy) (kWh) -20 -15 -10 -5 0 5 10 15 20 01:06:01:00 01:06:25:00 01:06:49:00 01:07:13:00 01:07:37:00 01:08:01:00 01:08:25:00 01:08:49:00 01:09:13:00 01:09:37:00 01:10:01:00 01:10:25:00 01:10:49:00 01:11:13:00 01:11:37:00 01:12:01:00 01:12:25:00 01:12:49:00 01:13:13:00 01:13:37:00 01:14:01:00 01:14:25:00 01:14:49:00 01:15:13:00 01:15:37:00 Time (dd:hh:mm:ss) Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 8 Zone 9 Zone 10 Zone 11 Zone 12 Zone 13 Zone 14 Zone 15 Zone 16 Zone 17 Zone 18 Zone 19 Zone 20 Zone 21 Zone 22 Zone 23 Zone 24 Zone 25 Predicted Percentage of Dissatisfied (PPD) (%) 0 5 10 15 20 25 30 Figure 5.2: The impact of a zone temperature sensor fault to AHU’s energy consumption and thermal comfort of occupants. 109 Besides energy waste, the presence of a fault can aect the thermal comfort of occupants. The human body can adapt to the external environment up to a certain range, but as soon as the limits are reached, the body’s responses are perceived as uncomfortable. The Predicted Percentage of Dissatised (PPD) (%), developed by P. O. Fanger [216], gives an estimation of the percentage of people predicted to experience local discomfort. According to the ASHRAE 55 standard for thermal comfort, PPD is recommended to be below 20%. In the right-hand side plot of Fig. 5.2 the PPD is presented considering an oce example where metabolic rate, air speed, clothing insulation, humidity and radiant temperature remain constant and only indoor air temperature varies. We can notice that the ASHRAE 55 criterion for thermal comfort (i.e., less than 20% PPD) is violated when the HVAC system is scheduled to turn on at 01:06:00:00. The controller implemented is able to keep PPD below the thermal comfort threshold in all zones besides the eect of disturbances rising from solar radiance, occupancy, heat transfer from devices, etc. However, the presence of the sensor fault in Zone 10 causes violation of the thermal comfort criterion not only in Zone 10 but also to the majority of its neighboring zones, i.e., Zones 4, 7, 19, and 22. This can be an outcome of the physical interconnection and/or the implementation of the distributed control algorithm [58], [126]. This kind of faults may go unnoticed causing a signicant waste of energy and uncomfortable condi- tions for the occupants, hence the proposed distributed diagnosis algorithm can provide knowledge about the presence and location of faults and inform the sta or building operators to take remedial actions. The results presented next show the eectiveness of the proposed diagnosis algorithm. Fig. 5.3–5.6 show the ARRs of the aected AHUs for the aforementioned fault scenarios. Specically, in each box the absolute values of the residualsj z i j,j sa i j,j cc i j,j hc i j are denoted by black dots, their corresponding thresholds z i , sa i , cc i , hc i are denoted by purple dots and the detection decision signals D z i , D sa i , D cc i , D hc i are denoted by the yellow dashed line. The red dot and the red star represent the location of a sensor and an actuator fault with fault Model 1, respectively, the blue dot shows the location of a sensor fault with fault Model 2, and the green dot and green star show the location of a sensor and 110 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Air Temperature for zone CorridorPod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Supply Air for zone CorridorPod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Cooling Coil for zone CorridorPod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 Time (dd:hh:mm:ss) 0 10 20 0 1 ARR: Heating Coil for zone CorridorPod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Air Temperature for zone MultiClass1Pod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Supply Air for zone MultiClass1Pod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Cooling Coil for zone MultiClass1Pod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 Time (dd:hh:mm:ss) 0 10 20 0 1 ARR: Heating Coil for zone MultiClass1Pod1 Figure 5.3: ARRs of the AHU 10 and 19 for Scenario 1. an actuator fault with fault Model 3, respectively. Note that background light blue and red color on each plot indicate the cooling and heating mode of each AHU, respectively, where cooling mode means that the cooling coil valve is open and heating mode means that the heating coil valve is open. Fig. 5.3 shows the ARRs for AHUs 10 and 19 in fault Scenario 1. According to Table 4.3, zone 10 is connected to zone 19. The purpose of this result to is show the eect of a sensor fault to the ARRs of the local FD Agent and also the eect to one of its neighboring FD Agents. Specically, the sensor fault f z 10 with Fault Model 1 is applied, which is a fault of 10 o C magnitude that can be consider a large one. According to Table 5.3, f z 10 can cause the violation of ARRE z 10 and may cause the violation of ARRs E sa 10 ,E hc 10 ,E cc 10 of the local FD Agent and may also cause the violation of ARRsE z 4 ,E z 7 ,E z 17 ,E z 19 , E z 22 due to the exchange of the sensor information from the neighboring zones included inN 10 for the zone air temperature estimation. From the results in Fig. 5.3, the observed diagnosis sets areD (10) (k) = [0; 0; 1; 1; 0] > andD (19) (k) = [0; 0; 0; 0; 0] > . This means that this fault Scenario caused the violation of 111 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Air Temperature for zone CornerClass1Pod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Supply Air for zone CornerClass1Pod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Cooling Coil for zone CornerClass1Pod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 Time (dd:hh:mm:ss) 0 10 20 0 1 ARR: Heating Coil for zone CornerClass1Pod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Air Temperature for zone CornerClass1Pod3 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Supply Air for zone CornerClass1Pod3 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Cooling Coil for zone CornerClass1Pod3 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 Time (dd:hh:mm:ss) 0 10 20 0 1 ARR: Heating Coil for zone CornerClass1Pod3 Figure 5.4: ARRs of the AHU 4 and 6 for Scenario 2. ARRsE z 10 andE sa 10 , while the remaining ARRs were not violated. The outcome of the Distributed Fault Isolation logic is that this observed pattern may be caused by the following fault combinations:ff z 10 g, ff sa 10 g, orff z 10 ;f sa 10 g, based on which we can exclude the occurrence of faultsf hc 10 ,f cc 10 ,f m hc 10 ,f m cc 10 , f z 4 ,f z 7 ,f z 17 ,f z 19 ,f z 22 and their combinations. Indicatively, we refer only the ARRs of FD Agent 19, but the ARRs in FD Agents 4, 7, 17 and 22 were not violated either. Fig. 5.4 shows the ARRs of AHUs 4 and 6 in fault Scenario 2. Specically, the sensor faultf sa 4 with Fault Model 1 and the sensor faultf sa 6 with Fault Model 3 are applied. According to Table 5.3,f sa 4 andf sa 6 can cause the violation of ARRsE sa 4 andE sa 6 , and may cause the violation of ARRsE z 4 andE z 6 , respectively. From Fig. 5.4, the observed diagnosis sets areD (4) (k) = [0; 0; 1; 0; 0] > andD (6) (k) = [0; 0; 1; 0; 0] > . This means that fault Scenario 2 caused the violation of ARRsE sa 4 andE sa 6 , while the remaining ARRs were not violated. The outcome of the Distributed Fault Isolation logic presented in Section 5.4.3 is that these observed diagnosis sets may only be a result of the following fault combination:ff sa 4 ;f sa 6 g, based on 112 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Air Temperature for zone CornerClass2Pod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Supply Air for zone CornerClass2Pod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Cooling Coil for zone CornerClass2Pod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 Time (dd:hh:mm:ss) 0 10 20 0 1 ARR: Heating Coil for zone CornerClass2Pod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Air Temperature for zone CornerClass2Pod2 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Supply Air for zone CornerClass2Pod2 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Cooling Coil for zone CornerClass2Pod2 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 Time (dd:hh:mm:ss) 0 10 20 0 1 ARR: Heating Coil for zone CornerClass2Pod2 Figure 5.5: ARRs of the AHU 6 and 7 for Scenario 3. which we can exclude the occurrence of the remaining fault combinations. Hence, the proposed distributed fault diagnosis algorithm is able to detect and isolate in real-time the presence of sensor faultsf sa 4 and f sa 6 . Consequently, by using the proposed algorithm we can reduce the maintenance time. Note that fault ff sa 4 g (red dot) corresponds to an abrupt sensor fault that is detected at the occurrence time, whileff sa 6 g (green dot) corresponds to an incipient sensor fault that is detected approximately 1 and 15 minutes after the occurrence time. Fig. 5.5 shows the ARRs of AHUs 7 and 8 in fault Scenario 3. Specically, the abrupt sensor fault f cc 7 with Fault Model 1 and the multiplicative sensor faultf cc 8 with Fault Model 2 are applied. According to Table 5.3,f cc 7 andf cc 8 can cause the violation of ARRsE cc 7 andE cc 8 , and may cause the violation of ARRsE hc 7 ,E sa 7 andE hc 8 ,E sa 8 , respectively. From Fig. 5.5, the observed diagnosis sets areD (7) (k) = [1; 0; 0; 0; 0] > andD (8) (k) = [1; 0; 0; 0; 0] > . This means that fault Scenario 3 causes the violation of ARR E cc 7 andE cc 8 , while the remaining involved ARRs were not violated. The outcome of the Distributed 113 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Air Temperature for zone CorridorPod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Supply Air for zone CorridorPod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Cooling Coil for zone CorridorPod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 Time (dd:hh:mm:ss) 0 10 20 0 1 ARR: Heating Coil for zone CorridorPod1 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Air Temperature for zone CorridorPod3 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Supply Air for zone CorridorPod3 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 0 10 20 0 1 ARR: Cooling Coil for zone CorridorPod3 01:06:00:00 01:09:00:00 01:12:00:00 01:15:00:00 Time (dd:hh:mm:ss) 0 10 20 0 1 ARR: Heating Coil for zone CorridorPod3 Figure 5.6: ARRs of the AHU 10 and 12 for Scenario 4. Fault Isolation logic is that these observed diagnosis sets may only be a result of one of the following fault combinations:ff cc 7 g,ff m cc 7 g, orff cc 7 ;f m cc 7 g for FD Agent 7 andff cc 8 g,ff m cc 8 g, orff cc 8 ;f m cc 8 g for FD Agent 8. Based on the above logic we can exclude the occurrence of the remaining fault combinations. Fig. 5.6 shows the ARRs of AHU 10 and 12 generated by fault Scenario 4. Specically, the actuator fault f m cc 10 with Fault Model 1 and the actuator faultf cc 12 with Fault Model 3 are applied. According to Table 5.3, f m cc 10 andf m cc 12 can cause the violation of ARRsE cc 10 andE cc 12 , and may cause the violation of ARRsE hc 10 andE hc 12 , respectively. From results in Fig. 5.6, the observed diagnosis sets areD (10) (k) = [1; 0; 0; 0; 0] > andD (6) (k) = [0; 0; 0; 0; 0] > . This means that fault Scenario 4 caused only the violation of ARRE cc 10 , while the remaining ARRs includingE cc 12 were not violated. The outcome of the Distributed Fault Isolation logic for the FD Agent 10 is that this observed diagnosis set may be caused by the following fault combinations: ff cc 10 g,ff m cc 10 g, orff cc 10 ;f m cc 10 g, based on which we can exclude the occurrence of the remaining fault combinations. On the other hand, the presence of the actuator fault f cc 12 with Fault Model 3 was not 114 detected by FD Agent 12 due to the selection of design parameters and this is one of the cases of missed faults. 115 Chapter6 Conclusion 6.1 ConcludingRemarks As an integral part of human life, smart buildings are in the center of attention. Understanding the dynamic nature of buildings and uncertainty in HVAC systems, and considering natural heat exchange between neighboring thermal zones in the control design has shown to be an important step in more ecient and cost-eective building control. Taking into account the unique characteristics of the electromechanical equipment of HVAC systems has shown to be crucial when applying distributed control, leading to energy savings and improved climate regulation. Indoor climate conditions have a direct impact in people’s quality of life, aecting most aspects of oc- cupant activity. HVAC systems are used to provide appropriate conditions, regulating indoor temperature, humidity and air quality, and end up being one of the biggest energy consumers in developed countries. In industry, the most common applied control approaches are traditional ones, and especially PID and On/O controllers. However, such controllers are usually unable to optimize system performance and thus research interest has turned to more advanced control algorithms. Several model based control ap- proaches have been proposed, including nonlinear, robust, optimal and model predictive control. Their performance though is heavily dependent on the accuracy of the model, as well as the strength of the dis- turbances, usually requiring the training of accurate prediction models for weather conditions and other 116 kinds of disturbances to operate satisfactorily, thus becoming less eective in the presence of unpredicted internal or external disturbances or time-varying system parameters. Several non-model based control approaches have been proposed, in an eort to use data from the system operation and articial intelli- gence to optimize performance. These approaches include fuzzy logic systems, neural networks, genetic algorithms and reinforcement learning. The application of these techniques has shown promising results, but also raises several challenges, such as requiring prior training. This indicates that a great amount of data and a great amount of learning time are needed, implying improper HVAC control during the initial training time, and thus rendering these approaches dicult to implement. Soft control techniques typically rely on data and may not provide proper stability guarantees in the case of unexpected eects. Adaptive control strategies for HVAC systems are still not wide-spread but they have shown prom- ising results. Buildings climate regulation systems and electromechanical equipment exhibit the presence of strong uncertainties and disturbances, which can be overcome using adaptive control techniques and thus provide the framework for improved performance when compared to methods that do not consider uncertainty. In addition, the need for accurate model and system parameters knowledge is lifted, removing the need for precise calibrations and making the control mechanisms adjustable to time-varying parameter values. Research results have shown the potential for energy savings as well as the improved response properties of adaptive control systems. When compared to non-model based techniques such as neural networks, the adaptive ones can guarantee good stability properties and at the same time carry smaller computational burden without the need of large sets of historical data for training. In this work we have proposed a distributed adaptive control scheme for temperature regulation in multi-zone HVAC systems. The proposed scheme allocates energy between thermal zones eciently and does not require accurate knowledge of model parameters, thus adjusting to parameter changes and un- known disturbances. We have shown that the proposed scheme guarantees bounded signals and conver- gence of zone temperature to the desired one for all building zones, while convergence properties do not 117 depend on the size of the system, thus making the control scheme scalable. The introduction of adapt- ive learning and exchange of information between zones is shown to provide faster response to changes and more accurate temperature control while consuming less energy in realistic scenarios, when human activity, lighting, electrical equipment, door opening and closing, varying weather conditions, and ma- terial degradation are considered. The robustness and stability properties of distributed adaptive control schemes have been studied, showing that such schemes are robust to interconnections eects and com- munication delays. The actuator dynamics of Air Handling Units have been analyzed and local controllers that are designed based on their structure have been proven to be able to regulate temperature satisfact- orily in every thermal zone, with backstepping states that reect the cascade structure of the system. The stability and performance of the proposed control methodology do not depend on the size of the building and HVAC system. Simulated application of the proposed distributed adaptive control schemes to large building has illustrated the satisfactory performance of the proposed control architecture. The simulations have also shown that the introduction of adaptive learning and the consideration of zone interactions im- prove temperature tracking and provide energy savings through improved heat allocation, in realistic scenarios with system uncertainties and unknown heat loads, implying prolonged ecient operation of the HVAC equipment. Finally, a distributed model-based fault diagnosis algorithm that can, in real-time, detect and isolate sensor and actuator faults that can aect the operation of the AHUs in a large-scale, multi-zone building has been proposed. Faults, in opposition to failures, can not be easily diagnosed from the ruled-based algorithms integrated in the existing Buildings Management Systems, causing uncomfort- able thermal conditions and in some cases a huge waste of energy in a long term duration. By combining local and neighboring detection signals, an isolation logic is designed to decrease the number of candidate faults and consequently the maintenance time by the building’s operators. The complexity of HVAC systems and their high energy consumption have rendered the development of more sophisticated control algorithms an imperative need, providing a lot of opportunities for future 118 research. Controlling indoor climate conditions includes regulating humidity and providing high quality air, minimizing the concentration of contaminants. Detecting and identifying operational faults is such systems can prevent unnecessary energy consumption and at the same time facilitate proper climate reg- ulation. Adaptive control has shown to have the potential to oer the way to handle the dynamic and uncertain behavior of these systems while providing stable and ecient control solutions. 6.2 FutureDirections Among the plethora of ideas, starting from theoretical analysis of the mathematical models of intercon- nected systems to proposing practical solutions to HVAC systems, there exist several directions for further investigation. However, it appears crucial to focus on solving current problems and directly improve life quality and energy eciency. Thus, a future direction would be the development of a control algorithm that can appropriately regulate all aspects of indoor environment, including temperature, but also humid- ity and contaminant concentration, while dealing with parameter changes, modeling uncertainties and unknown disturbances. Temperature is not the only important factor to dene thermal comfort. Thermal comfort represents the satisfaction with the thermal environment and is a complex function of temperature and humidity [217]–[219]. If air becomes too dry, people suer from sensory irritation in eyes and upper airways, as well as deteriorated sleep quality and work performance [220]. At the same time, indoor humidity has been shown to be connected to the survival and infectivity of bacteria and viruses, such as inuenza virus [221], [222], and apparently humidity is linked to occupant health [223]–[225]. Apart from the direct health implications of humidity level, it also aects thermal comfort - along with temperature. The existence of thermal discomfort has a direct impact in productivity and well-being. 119 Humidity in a thermal zone is described by the following equation [39]: V z dW z dt =f sa (W sa W z ) + P (t) a (6.1) whereW z described the humidity ratio of the zone inkg=kg (dry air),W s humidity ratio of the supply air in kg/kg (dry air),V z is the volume of the zone inm 3 ,f sa is the volume ow rate of the supply air inm 3 =s, a is th e density of air inkg=m 3 andP (t) is evaporation rate of the occupants and the impact of any other humidity loads. Adding humidity in the control design directly aects the controller structure as it introduces challenging non-linearities. Similarly to the zone, several components of the HVAC equipment aect humidity directly or indirectly and their eect should be considered. Characteristic examples of equipment that aects humidity are the cooling coils (indirect dehumidication), the humidiers and the dehumidiers. Another important aspect of Indoor Air Quality (IAQ), which became apparent during the Covid-19 pandemic, is the concentration of contaminants in the air. An often overlooked part of HVAC systems, proper Ventilation is responsible for providing low levels of contaminants, that may range from microor- ganisms to particle and CO 2 . As ventilation mainly includes providing fresh air from outside into the building, it integrally aects temperature and humidity levels. Exploiting the structure of the HVAC equip- ment can lead to developing control algorithms that regulate all aspects of IAQ in an optimized and energy ecient way. Despite the work on adaptive control mechanisms for HVAC systems, there are still several areas for future research and investigation. As energy consumption is a critical factor, it should be analyzed how adaptive control can aect and improve it, and how it can be incorporated as a control objective in a standardized way. The denition of satisfactory performance can be rened to better incorporate thermal 120 comfort. Research has indicated that thermal comfort itself is adaptive, meaning that the level of satis- faction from indoor climate conditions varies depending on temperature, humidity, clothing, lighting, air currents, etc, and is still a challenge for tracking using adaptive control techniques. This also suggests the need to dene the trade-o between thermal comfort and energy usage in systems with uncertainties and disturbances, and to be able to recognize online the optimal balance. In addition, the increased computation costs in large-scale buildings dictate the need to investigate the application of adaptive control in a dis- tributed way. One of the most promising future directions is the successful fusion of adaptive control with other control techniques in an eort to combine the advantageous aspects of each methodology. Methods that are not robust to disturbances and cannot react appropriately to parameter changes can benet from the learning abilities of adaptive control. Especially MPC has shown to get enhanced, as adaptive meth- ods can be used for disturbance recognition, such as unpredicted occupant activity or weather conditions. Finally, the factors that aect adaptive control performance in HVAC systems need to be analyzed, and controllers for the dierent electromechanical equipment to be designed, taking into consideration how the models of such systems may change during operation. Research on adaptive HVAC control and practical implementations are still in early stages, but have already shown the potential benets of including stable adaptive learning techniques in the eort to ad- vance the development of smart buildings. 121 Bibliography [1] U.S. Environmental Protection Agency, ‘Report to congress on indoor air quality: Volume 2,’ Wash- ington, DC, 1989. [2] K. W. Tham, ‘Indoor air quality and its eects on humans—a review of challenges and developments in the last 30 years,’EnergyandBuildings, vol. 130, pp. 637–650, 2016.doi: 10.1016/j.enbuild.2016. 08.071. [3] M. Frontczak and P. Wargocki, ‘Literature survey on how dierent factors inuence human comfort in indoor environments,’ Building and Environment, vol. 46, no. 4, pp. 922–937, 2011.doi: 10.1016/ j.buildenv.2010.10.021. [4] A. Kaushik, M. Arif, P. Tumula and O. J. Ebohon, ‘Eect of thermal comfort on occupant productiv- ity in oce buildings: Response surface analysis,’ Building and Environment, vol. 180, p. 107 021, 2020.doi: 10.1016/j.buildenv.2020.107021. [5] Y. A. Horr, M. Arif, A. Kaushik, A. Mazroei, M. Katafygiotou and E. Elsarrag, ‘Occupant productiv- ity and oce indoor environment quality: A review of the literature,’ Building and Environment, vol. 105, pp. 369–389, 2016.doi: 10.1016/j.buildenv.2016.06.001. [6] W. O. Collinge, A. E. Landis, A. K. Jones, L. A. Schaefer and M. M. Bilec, ‘Productivity metrics in dynamic LCA for whole buildings: Using a post-occupancy evaluation of energy and indoor environmental quality tradeos,’ Building and Environment, vol. 82, pp. 339–348, 2014. doi: 10 . 1016/j.buildenv.2014.08.032. [7] L. Lan, P. Wargocki, D. P. Wyon and Z. Lian, ‘Eects of thermal discomfort in an oce on perceived air quality, SBS symptoms, physiological responses, and human performance,’ Indoor Air, vol. 21, no. 5, pp. 376–390, 2011.doi: 10.1111/j.1600- 0668.2011.00714.x. [8] W. J. Fisk, D. Black and G. Brunner, ‘Changing ventilation rates in u.s. oces: Implications for health, work performance, energy, and associated economics,’ Building and Environment, vol. 47, pp. 368–372, 2012.doi: 10.1016/j.buildenv.2011.07.001. [9] O. Seppanen and W. Fisk, ‘Some quantitative relations between indoor environmental quality and work performance or health,’ HVAC&R Research, vol. 12, no. 4, pp. 957–973, 2006. doi: 10 . 1080 / 10789669.2006.10391446. 122 [10] Y. Ma, Y. Zhao, J. Liu, X. He, B. Wang, S. Fu, J. Yan, J. Niu, J. Zhou and B. Luo, ‘Eects of tem- perature variation and humidity on the death of COVID-19 in Wuhan, China,’ Science of The Total Environment, vol. 724, p. 138 226, 2020.doi: 10.1016/j.scitotenv.2020.138226. [11] J. Khodakarami and N. Nasrollahi, ‘Thermal comfort in hospitals – a literature review,’ Renewable andSustainableEnergyReviews, vol. 16, no. 6, pp. 4071–4077, 2012.doi: 10.1016/j.rser.2012.03.054. [12] K. J. Collins, ‘Low indoor temperatures and morbidity in the elderly,’AgeandAgeing, vol. 15, no. 4, pp. 212–220, 1986.doi: 10.1093/ageing/15.4.212. [13] ASHRAE, ‘ASHRAE Handbook—HVAC Applications,’ in 2019, ch. 9 : Health Care Facilities. [14] K. Okamoto-Mizuno and K. Mizuno, ‘Eects of thermal environment on sleep and circadian rhythm,’ Journal of Physiological Anthropology, vol. 31, no. 1, 2012.doi: 10.1186/1880- 6805- 31- 14. [15] G. Zheng, K. Li and Y. Wang, ‘The eects of high-temperature weather on human sleep quality and appetite,’ International Journal of Environmental Research and Public Health, vol. 16, no. 2, p. 270, 2019.doi: 10.3390/ijerph16020270. [16] S. S. Joshi, T. J. Lesser, J. W. Olsen and B. F. O’Hara, ‘The importance of temperature and ther- moregulation for optimal human sleep,’ Energy and Buildings, vol. 131, pp. 153–157, 2016. doi: 10.1016/j.enbuild.2016.09.020. [17] P. Wargocki, J. A. Porras-Salazar and S. Contreras-Espinoza, ‘The relationship between classroom temperature and children’s performance in school,’ Building and Environment, vol. 157, pp. 197– 204, 2019.doi: 10.1016/j.buildenv.2019.04.046. [18] P. Wargocki, J. A. Porras-Salazar, S. Contreras-Espinoza and W. Bahneth, ‘The relationships between classroom air quality and children’s performance in school,’ Building and Environment, vol. 173, p. 106 749, 2020.doi: 10.1016/j.buildenv.2020.106749. [19] M. Lee, K. Mui, L. Wong, W. Chan, E. Lee and C. Cheung, ‘Student learning performance and Indoor Environmental Quality (IEQ) in air-conditioned university teaching rooms,’ Building and Environment, vol. 49, pp. 238–244, 2012.doi: 10.1016/j.buildenv.2011.10.001. [20] M. Puteh, M. Ibrahim, M. Adnan, C. N. Che’Ahmad and N. M. Noh, ‘Thermal comfort in classroom: Constraints and issues,’ Procedia - Social and Behavioral Sciences, vol. 46, pp. 1834–1838, 2012.doi: 10.1016/j.sbspro.2012.05.388. [21] Y. Wang, J. Kuckelkorn, F.-Y. Zhao, D. Liu, A. Kirschbaum and J.-L. Zhang, ‘Evaluation on classroom thermal comfort and energy performance of passive school building by optimizing HVAC control systems,’BuildingandEnvironment, vol. 89, pp. 86–106, 2015.doi: 10.1016/j.buildenv.2015.02.023. [22] L. Pérez-Lombard, J. Ortiz and C. Pout, ‘A review on buildings energy consumption information,’ Energy and Buildings, vol. 40, no. 3, pp. 394–398, 2008.doi: 10.1016/j.enbuild.2007.03.007. [23] 2011 Buildings Energy Data Book. U.S. Department of Energy, 2011. [Online]. Available: https : //ieer.org/wp/wp- content/uploads/2012/03/DOE- 2011- Buildings- Energy- DataBook- BEDB.pdf . 123 [24] Council of the European Commission, ‘Directive 2010/31/EU of the European Parliament and of the council of 19 May 2010 on the energy performance of buildings,’ Ocial Journal of the European Union, vol. 153, pp. 13–35, 2010. [25] J. Ngarambe, G. Y. Yun and M. Santamouris, ‘The use of articial intelligence (AI) methods in the prediction of thermal comfort in buildings: Energy implications of AI-based thermal comfort controls,’ Energy and Buildings, vol. 211, p. 109 807, 2020.doi: 10.1016/j.enbuild.2020.109807. [26] M. Rahman, M. Rasul and M. Khan, ‘Energy conservation measures in an institutional building in sub-tropical climate in Australia,’AppliedEnergy, vol. 87, no. 10, pp. 2994–3004, 2010.doi: 10.1016/ j.apenergy.2010.04.005. [27] Y. Himeur, A. Alsalemi, A. Al-Kababji, F. Bensaali and A. Amira, ‘Data fusion strategies for energy eciency in buildings: Overview, challenges and novel orientations,’ Information Fusion, vol. 64, pp. 99–120, 2020.doi: 10.1016/j.inffus.2020.07.003. [28] A.-D. Pham, N.-T. Ngo, T. T. H. Truong, N.-T. Huynh and N.-S. Truong, ‘Predicting energy consump- tion in multiple buildings using machine learning for improving energy eciency and sustainab- ility,’ Journal of Cleaner Production, vol. 260, p. 121 082, 2020.doi: 10.1016/j.jclepro.2020.121082. [29] H. Farzaneh, L. Malehmirchegini, A. Bejan, T. Afolabi, A. Mulumba and P. P. Daka, ‘Articial in- telligence evolution in smart buildings for energy eciency,’AppliedSciences, vol. 11, no. 2, p. 763, 2021.doi: 10.3390/app11020763. [30] H. Mirinejad, K. C. Welch and L. Spicer, ‘A review of intelligent control techniques in HVAC sys- tems,’ in 2012 IEEE Energytech, IEEE, 2012.doi: 10.1109/energytech.2012.6304679. [31] D. Enescu, ‘A review of thermal comfort models and indicators for indoor environments,’Renewable and Sustainable Energy Reviews, vol. 79, pp. 1353–1379, 2017.doi: 10.1016/j.rser.2017.05.175. [32] R. F. Rupp, N. G. Vásquez and R. Lamberts, ‘A review of human thermal comfort in the built envir- onment,’ Energy and Buildings, vol. 105, pp. 178–205, 2015.doi: 10.1016/j.enbuild.2015.07.047. [33] W. Guo and M. Zhou, ‘Technologies toward thermal comfort-based and energy-ecient HVAC systems: A review,’ in 2009 IEEE International Conference on Systems, Man and Cybernetics, IEEE, 2009.doi: 10.1109/icsmc.2009.5346631. [34] P. Mazzei, F. Minichiello and D. Palma, ‘HVAC dehumidication systems for thermal comfort: A critical review,’ Applied Thermal Engineering, vol. 25, no. 5-6, pp. 677–707, 2005. doi: 10 . 1016 / j . applthermaleng.2004.07.014. [35] V. V. Tran, D. Park and Y.-C. Lee, ‘Indoor air pollution, related human diseases, and recent trends in the control and improvement of indoor air quality,’InternationalJournalofEnvironmentalResearch and Public Health, vol. 17, no. 8, p. 2927, 2020.doi: 10.3390/ijerph17082927. [36] A. Cincinelli and T. Martellini, ‘Indoor air quality and health,’InternationalJournalofEnvironmental Research and Public Health, vol. 14, no. 11, p. 1286, 2017.doi: 10.3390/ijerph14111286. 124 [37] W. L. Angel,HVACDesignSourcebook,SecondEdition. McGraw-Hill Education, 1st Jul. 2020, 480 pp., isbn: 1260457249. [Online]. Available: https://www.ebook.de/de/product/38494349/w_larsen_angel_ hvac_design_sourcebook_second_edition.html. [38] S. Seyam, ‘Types of HVAC systems,’ inHVACSystem, InTech, 2018.doi: 10.5772/intechopen.78942. [39] B. Tashtoush, M. Molhim and M. Al-Rousan, ‘Dynamic model of an HVAC system for control ana- lysis,’ Energy, vol. 30, no. 10, pp. 1729–1745, 2005.doi: 10.1016/j.energy.2004.10.004. [40] Z. Afroz, G. Shaullah, T. Urmee and G. Higgins, ‘Modeling techniques used in building HVAC control systems: A review,’RenewableandSustainableEnergyReviews, vol. 83, pp. 64–84, 2018.doi: 10.1016/j.rser.2017.10.044. [41] G. Platt, J. Li, R. Li, G. Poulton, G. James and J. Wall, ‘Adaptive HVAC zone modeling for sustainable buildings,’ Energy and Buildings, vol. 42, no. 4, pp. 412–421, 2010. doi: 10.1016/j.enbuild.2009.10. 009. [42] G. Lymperopoulos and P. Ioannou, ‘Adaptive control of networked distributed systems with un- known interconnections,’ in 2016 IEEE 55th Conference on Decision and Control (CDC), IEEE, 2016. doi: 10.1109/cdc.2016.7798787. [43] S. A. Barakat, ‘Inter-zone convective heat transfer in buildings: A review,’ Journal of Solar Energy Engineering, vol. 109, no. 2, pp. 71–78, 1987.doi: 10.1115/1.3268195. [44] O. M. Lidwell, ‘Air exchange through doorways. The eect of temperature dierence, turbulence and ventilation ow,’ Journal of Hygiene, vol. 79, no. 1, 1977.doi: 10.1017/S0022172400052931. [45] G. Lymperopoulos and P. Ioannou, ‘Distributed adaptive HVAC control for multi-zone buildings,’ in Proc. of the 58th Conference on Decision and Control (CDC), IEEE, 2019. doi: 10 . 1109 / cdc40024 . 2019.9029952. [46] M. Elnour and N. Meskin, ‘Multi-zone HVAC control system design using feedback linearization,’ in Proc. of the 5th International Conference on Control, Instrumentation, and Automation (ICCIA), IEEE, 2017.doi: 10.1109/icciautom.2017.8258687. [47] S. Goyal, H. A. Ingley and P. Barooah, ‘Eect of various uncertainties on the performance of occupancy-based optimal control of HVAC zones,’ in Proc. of the 51st IEEE Conference on Decision and Control (CDC), IEEE, 2012.doi: 10.1109/cdc.2012.6426111. [48] A. Mirakhorli and B. Dong, ‘Occupancy behavior based model predictive control for building indoor climate—a critical review,’EnergyandBuildings, vol. 129, pp. 499–513, 2016.doi: 10.1016/j.enbuild. 2016.07.036. [49] S. Salimi and A. Hammad, ‘Critical review and research roadmap of oce building energy man- agement based on occupancy monitoring,’ Energy and Buildings, vol. 182, pp. 214–241, 2019. doi: 10.1016/j.enbuild.2018.10.007. 125 [50] Z. Yang and B. Becerik-Gerber, ‘Assessing the impacts of real-time occupancy state transitions on building heating/cooling loads,’ Energy and Buildings, vol. 135, pp. 201–211, 2017. doi: 10 . 1016 / j . enbuild.2016.11.038. [51] M. W. Khan, M. A. Choudhry, M. Zeeshan and A. Ali, ‘Adaptive fuzzy multivariable controller design based on genetic algorithm for an air handling unit,’Energy, vol. 81, pp. 477–488, 2015.doi: 10.1016/j.energy.2014.12.061. [52] H. Setayesh, H. Moradi and A. Alasty, ‘A comparison between the minimum-order & full-order observers in robust control of the air handling units in the presence of uncertainty,’ Energy and Buildings, vol. 91, pp. 115–130, 2015.doi: 10.1016/j.enbuild.2015.01.016. [53] J. Firrantello, W. Bahneth and P. Kremer, ‘Field measurement and modeling of UVC cooling coil irradiation for heating, ventilating, and air conditioning energy use reduction (RP-1738)—part 1: Field measurements,’ Science and Technology for the Built Environment, vol. 24, no. 6, pp. 588–599, 2017.doi: 10.1080/23744731.2017.1402662. [54] G. Eleftheriadis and M. Hamdy, ‘Impact of building envelope and mechanical component degrad- ation on the whole building performance: A review paper,’ Energy Procedia, vol. 132, pp. 321–326, 2017.doi: 10.1016/j.egypro.2017.09.739. [55] Y. Ma, J. Matusko and F. Borrelli, ‘Stochastic model predictive control for building HVAC sys- tems: Complexity and conservatism,’IEEETransactionsonControlSystemsTechnology, vol. 23, no. 1, pp. 101–116, 2014.doi: 10.1109/TCST.2014.2313736. [56] Z. Wu, Q.-S. Jia and X. Guan, ‘Optimal control of multiroom HVAC system: An event-based ap- proach,’ IEEE Transactions on Control Systems Technology, 2015.doi: 10.1109/tcst.2015.2446955. [57] F. Smarra, A. Jain, T. de Rubeis, D. Ambrosini, A. D’Innocenzo and R. Mangharam, ‘Data-driven model predictive control using random forests for building energy optimization and climate con- trol,’ Applied Energy, vol. 226, pp. 1252–1272, 2018.doi: 10.1016/j.apenergy.2018.02.126. [58] P. D. Moroşan, R. Bourdais, D. Dumur and J. Buisson, ‘Building temperature regulation using a distributed model predictive control,’EnergyandBuildings, vol. 42, no. 9, pp. 1445–1452, 2010.doi: 10.1016/j.enbuild.2010.03.014. [59] G. Lymperopoulos and P. Ioannou, ‘Distributed adaptive control of multi-zone HVAC systems,’ in Proc. of the 27th Mediterranean Conference on Control and Automation (MED), IEEE, 2019, pp. 553– 558.doi: 10.1109/med.2019.8798509. [60] L. Ferrarini and G. Mantovani, ‘Modeling and control of thermal energy of a large commercial building,’ in Proc. of the IEEE International Workshop on Intelligent Energy Systems (IWIES), 2013, pp. 149–154. [61] M. R. Kulkarni and F. Hong, ‘Energy optimal control of a residential space-conditioning system based on sensible heat transfer modeling,’BuildingandEnvironment, vol. 39, no. 1, pp. 31–38, 2004. doi: 10.1016/j.buildenv.2003.07.003. 126 [62] M. Imal, ‘Design and implementation of energy eciency in HVAC systems based on robust PID control for industrial applications,’ Journal of Sensors, vol. 2015, pp. 1–15, 2015.doi: 10.1155/2015/ 954159. [63] D. Lim, B. Rasmussen and D. Swaroop, ‘Selecting PID control gains for nonlinear HVAC&r systems,’ HVAC&R Research, vol. 15, no. 6, pp. 991–1019, 2009.doi: 10.1080/10789669.2009.10390876. [64] A. Afram and F. Janabi-Shari, ‘Theory and applications of HVAC control systems – a review of model predictive control (MPC),’BuildingandEnvironment, vol. 72, pp. 343–355, 2014.doi: 10.1016/ j.buildenv.2013.11.016. [65] F. Behrooz, N. Mariun, M. Marhaban, M. Mohd Radzi and A. Ramli, ‘Review of Control Techniques for HVAC Systems—Nonlinearity Approaches Based on Fuzzy Cognitive Maps,’ Energies, vol. 11, no. 3, pp. 1–41, 2018.doi: 10.3390/en11030495. [66] F. Belic, Z. Hocenski and D. Sliskovic, ‘HVAC control methods - a review,’ in201519thInternational Conference on System Theory, Control and Computing (ICSTCC), IEEE, 2015. doi: 10 . 1109 / icstcc . 2015.7321372. [67] D. S. Naidu and C. Rieger, ‘Advanced control strategies for heating, ventilation, air-conditioning, and refrigeration systems—an overview: Part i: Hard control,’ HVAC&R Research, vol. 17, no. 1, pp. 2–21, 2011.doi: 10.1080/10789669.2011.540942. [68] M. Gholamzadehmir, C. D. Pero, S. Bua, R. Fedrizzi and N. Aste, ‘Adaptive-predictive control strategy for HVAC systems in smart buildings – a review,’ Sustainable Cities and Society, vol. 63, p. 102 480, 2020.doi: 10.1016/j.scs.2020.102480. [69] T. I. Salsbury, ‘A survey of control technologies in the building automation industry,’IFACProceed- ings Volumes, vol. 38, no. 1, pp. 90–100, 2005.doi: 10.3182/20050703- 6- cz- 1902.01397. [70] D. W. U. Perera, C. Pfeier and N.-O. Skeie, ‘Control oftemperature and energy consumption in buildings - a review,’ The International Journal of Energy and Environment, 2014,issn: 2076-2909. [71] Z. ( Yu, G. Huang, F. Haghighat, H. Li and G. Zhang, ‘Control strategies for integration of thermal energy storage into buildings: State-of-the-art review,’EnergyandBuildings, vol. 106, pp. 203–215, 2015.doi: 10.1016/j.enbuild.2015.05.038. [72] M. Royapoor, A. Antony and T. Roskilly, ‘A review of building climate and plant controls, and a survey of industry perspectives,’ Energy and Buildings, vol. 158, pp. 453–465, 2018.doi: 10.1016/j. enbuild.2017.10.022. [73] X. D. He and H. H. Asada, ‘A New Feedback Linearization Approach to Advanced Control of Multi- Unit HVAC Systems,’ in Proc. of the American Control Conference (ACC), 2003, pp. 2311–2316. [74] H. Moradi, M. Saar-Avval and F. Bakhtiari-Nejad, ‘Nonlinear multivariable control and perform- ance analysis of an air-handling unit,’ Energy and Buildings, vol. 43, no. 4, pp. 805–813, 2011. doi: 10.1016/j.enbuild.2010.11.022. 127 [75] S. Wang and Z. Ma, ‘Supervisory and optimal control of building HVAC systems:A review,’HVAC&R Research Research, vol. 14, no. 1, pp. 3–32, 2008.doi: 10.1080/10789669.2008.10390991. [76] R. He and H. Gonzalez, ‘Zoned HVAC control via PDE-constrained optimization,’ in2016American Control Conference (ACC), IEEE, 2016.doi: 10.1109/acc.2016.7524977. [77] B. Su and S. Wang, ‘An agent-based distributed real-time optimal control strategy for building HVAC systems for applications in the context of future IoT-based smart sensor networks,’ Applied Energy, vol. 274, p. 115 322, 2020.doi: 10.1016/j.apenergy.2020.115322. [78] M. Anderson, M. Buehner, P. Young, D. Hittle, C. Anderson, J. Tu and D. Hodgson, ‘MIMO robust control for HVAC systems,’IEEETransactionsonControlSystemsTechnology, vol. 16, no. 3, pp. 475– 483, 2008.doi: 10.1109/TCST.2007.903392. [79] C. P. Underwood, ‘Robust control of HVAC plant I: modelling,’ Building Services Engineering Re- search and Technology, vol. 21, no. 1, pp. 53–61, 2000. [80] Z. Wang, G. Hu and C. J. Spanos, ‘Distributed model predictive control of bilinear HVAC systems using a convexication method,’ in201711thAsianControlConference(ASCC), IEEE, 2017, pp. 878– 883.doi: 10.1109/ASCC.2017.8287414. [81] Y. Ma, G. Anderson and F. Borrelli, ‘A distributed predictive control approach to building temper- ature regulation,’ in Proceedings of the 2011 American Control Conference, IEEE, 2011.doi: 10.1109/ acc.2011.5991549. [82] P.-D. Moroşan, R. Bourdais, D. Dumur and J. Buisson, ‘A distributed MPC strategy based on bend- ers’ decomposition applied to multi-source multi-zone temperature regulation,’ Journal of Process Control, vol. 21, no. 5, pp. 729–737, 2011.doi: 10.1016/j.jprocont.2010.12.002. [83] Y. Long, S. Liu, L. Xie and K. H. Johansson, ‘A Hierarchical Distributed MPC for HVAC systems,’ in Proc.oftheAmericanControlConference(ACC), 2016, pp. 2385–2390.doi: 10.1109/ACC.2016.7525274. [84] M. Razmara, M. Maasoumy, M. Shahbakhti and R. D. Robinett, ‘Optimal exergy control of building HVAC system,’AppliedEnergy, vol. 156, no. 1, pp. 555–565, 2015.doi: 10.1016/j.apenergy.2015.07. 051. [85] J. Álvarez, J. Redondo, E. Camponogara, J. Normey-Rico, M. Berenguel and P. Ortigosa, ‘Optimizing building comfort temperature regulation via model predictive control,’EnergyandBuildings, vol. 57, pp. 361–372, 2013.doi: 10.1016/j.enbuild.2012.10.044. [86] S. Fielsch, T. Grunert, M. Stursberg and A. Kummert, ‘Model Predictive Control for Hydronic Heat- ing Systems in Residential Buildings,’ inProc.ofthe20thWorldCongressTheInternationalFederation ofAutomaticControl, vol. 50, Elsevier BV, 2017, pp. 4216–4221.doi: 10.1016/j.ifacol.2017.08.817. [87] A. Parisio and S. P. Gutierrez, ‘Distributed model predictive control for building demand-side man- agement,’ in 2018 European Control Conference (ECC), IEEE, 2018, pp. 2549–2554. doi: 10 . 23919 / ECC.2018.8550233. 128 [88] N. R. Patel, M. J. Risbeck, J. B. Rawlings, M. J. Wenzel and R. D. Turney, ‘Distributed economic model predictive control for large-scale building temperature regulation,’ in2016AmericanControl Conference (ACC), IEEE, 2016, pp. 895–900.doi: 10.1109/ACC.2016.7525028. [89] S. Koehler and F. Borrelli, ‘Building temperature distributed control via explicit MPC and ’Trim and Respond’ methods,’ inEuropeanControlConference(ECC), 2013, pp. 4334–4339.doi: 10.23919/ECC. 2013.6669781. [90] M. Baranski, L. Meyer, J. Fütterer and D. Müller, ‘Comparative study of neighbor communication approaches for distributed model predictive control in building energy systems,’ Energy, vol. 182, no. 1, pp. 840–851, 2019.doi: 10.1016/j.energy.2019.06.037. [91] S. M. Koehler, F. Chuang, Y. Ma, A. Daly and F. Borrelli, ‘Distributed model predictive control for forced-air systems,’ inIntelligentBuildingControlSystems:ASurveyofModernBuildingControland Sensing Strategies, J. T. Wen and S. Mishra, Eds. Springer International Publishing, 2018, pp. 167– 189.doi: 10.1007/978- 3- 319- 68462- 8_7. [92] F. Bisegna, C. Burattini, M. Manganelli, L. Martirano, B. Mattoni and L. Parise, ‘Adaptive control for lighting, shading and HVAC systems in near zero energy buildings,’ in2016IEEE16thInternational ConferenceonEnvironmentandElectricalEngineering(EEEIC), IEEE, 2016.doi: 10.1109/eeeic.2016. 7555768. [93] S. Soyguder, M. Karakose and H. Alli, ‘Design and simulation of self-tuning PID-type fuzzy adaptive control for an expert HVAC system,’ExpertSystemswithApplications, vol. 36, no. 3, pp. 4566–4573, 2009.doi: 10.1016/j.eswa.2008.05.031. [94] C.-M. Lin, H.-Y. Liu, K.-Y. Tseng and S.-F. Lin, ‘Heating, ventilation, and air conditioning system optimization control strategy involving fan coil unit temperature control,’ Applied Sciences, vol. 9, no. 11, p. 2391, 2019.doi: 10.3390/app9112391. [95] J. Liang and R. Du, ‘Thermal comfort control based on neural network for HVAC application,’ in Proc. of the IEEE Conference on Control Applications (CCA), IEEE, 2005, pp. 819–824. doi: 10 . 1109 / cca.2005.1507230. [96] C. Zhang, S. R. Kuppannagari, R. Kannan and V. K. Prasanna, ‘Building HVAC scheduling using reinforcement learning via neural network based model approximation,’ in Proceedings of the 6th ACM International Conference on Systems for Energy-Ecient Buildings, Cities, and Transportation, ACM, 2019.doi: 10.1145/3360322.3360861. [97] H. Moradi, H. Setayesh and A. Alasty, ‘PID-fuzzy control of air handling units in the presence of uncertainty,’ International Journal of Thermal Sciences, vol. 109, pp. 123–135, 2016. doi: 10.1016/j. ijthermalsci.2016.05.024. [98] S. Soyguder and H. Alli, ‘Fuzzy adaptive control for the actuators position control and modeling of an expert system,’ Expert Systems with Applications, vol. 37, no. 3, pp. 2072–2080, 2010. doi: 10.1016/j.eswa.2009.06.071. 129 [99] S. M. Attaran, R. Yusof and H. Selamat, ‘A novel optimization algorithm based on epsilon constraint- RBF neural network for tuning PID controller in decoupled HVAC system,’AppliedThermalEngin- eering, vol. 99, no. 1, pp. 613–624, 2016.doi: 10.1016/j.applthermaleng.2016.01.025. [100] M. Zaheer-Uddin and N. Tudoroiu, ‘Neuro-PID tracking control of a discharge air temperature system,’ Energy Conversion and Management, vol. 45, no. 1, pp. 2405–2415, 2004. [101] P. Ioannou and J. Sun, Robust Adaptive Control. Dover Publications, 19th Dec. 2012, 821 pp., isbn: 9780486498171. [102] P. Ioannou and B. Fidan,AdaptiveControlTutorial(AdvancesinDesignandControl). SIAM, Society for Industrial and Applied Mathematics, 2006,isbn: 9780898716153. [103] Y. Landau, Adaptive control : the model reference approach. New York: Dekker, 1979. [104] G. Lymperopoulos and P. Ioannou, ‘Adaptive aircraft control in the presence of unstructured dy- namic uncertainties,’ Journal of Guidance, Control, and Dynamics, vol. 42, no. 1, pp. 153–162, 2019. doi: 10.2514/1.g003693. [105] K. S. Tsakalis and P. A. Ioannou, Linear Time-Varying Systems: Control and Adaptation. Prentice Hall, 1992,isbn: 0130129232. [106] G. Lymperopoulos and P. Ioannou, ‘Model reference adaptive control for networked distributed systems with strong interconnections and communication delays,’ Journal of Systems Science and Complexity, vol. 31, no. 1, pp. 38–68, 2018.doi: 10.1007/s11424- 018- 7172- 2. [107] H. N. Nounou and M. N. Nounou, ‘Resilient adaptive control of uncertain time-delay systems,’ in Time-Delay Systems, InTech, 2011.doi: 10.5772/15569. [108] A. Buonomano, U. Montanaro, A. Palombo and S. Santini, ‘Adaptive control for building thermo- hygrometric analysis: A novel dynamic simulation code for indoor spaces with multi-enclosed thermal zones,’ Energy Procedia, vol. 78, pp. 2190–2195, 2015.doi: 10.1016/j.egypro.2015.11.316. [109] G. Lymperopoulos and P. Ioannou, ‘Building temperature regulation in a multi-zone HVAC system using distributed adaptive control,’ Energy and Buildings, vol. 215, 2020. doi: 10 . 1016 / j . enbuild . 2020.109825. [110] Z. Huaguang and L. Cai, ‘Decentralized nonlinear adaptive control of an HVAC system,’IEEETrans- actions on Systems, Man and Cybernetics, Part C (Applications and Reviews), vol. 32, no. 4, pp. 493– 498, 2002.doi: 10.1109/TSMCC.2002.807271. [111] A. Dounis and C. Caraiscos, ‘Advanced control systems engineering for energy and comfort man- agement in a building environment—a review,’ Renewable and Sustainable Energy Reviews, vol. 13, no. 6-7, pp. 1246–1261, 2009.doi: 10.1016/j.rser.2008.09.015. [112] G. Singh, M. Zaheer-uddin and R. Patel, ‘Adaptive control of multivariable thermal processes in HVAC systems,’ Energy Conversion and Management, vol. 41, no. 15, pp. 1671–1685, 2000. doi: 10. 1016/s0196- 8904(99)00182- x. 130 [113] G. Lymperopoulos and P. Ioannou, ‘Adaptive networked distributed model reference control sys- tems with strong interconnections and delays,’ in2017IEEE56thAnnualConferenceonDecisionand Control (CDC), IEEE, 2017.doi: 10.1109/cdc.2017.8264366. [114] Q. Zhou and M. Liu, ‘An on-line self-tuning algorithm of pi controller for the heating and cooling coil in buildings,’ Energy Systems Laboratory; Texas A&M University, 1998. [115] J. Bai and X. Zhang, ‘A new adaptive PI controller and its application in HVAC systems,’ Energy Conversion and Management, vol. 48, no. 4, pp. 1043–1054, 2007. doi: 10.1016/j.enconman.2006.10. 023. [116] J. Bai, S. Wang and X. Zhang, ‘Development of an adaptive smith predictor-based self-tuning PI controller for an HVAC system in a test room,’EnergyandBuildings, vol. 40, no. 12, pp. 2244–2252, 2008.doi: 10.1016/j.enbuild.2008.07.002. [117] S. I. Chaudhry and M. Das, ‘Adaptive control of indoor temperature in a building using a desirable reference temperature prole,’ in 2013 IEEE 56th International Midwest Symposium on Circuits and Systems (MWSCAS), IEEE, 2013.doi: 10.1109/mwscas.2013.6674899. [118] M.-L. Chiang and L.-C. Fu, ‘Adaptive control of switched systems with application to HVAC system,’ in 2007 IEEE International Conference on Control Applications, IEEE, 2007. doi: 10 . 1109 / cca . 2007 . 4389258. [119] S. Yuan, L. Zhang, O. Holub and S. Baldi, ‘Switched adaptive control of air handling units with discrete and saturated actuators,’ IEEE Control Systems Letters, vol. 2, no. 3, pp. 417–422, 2018.doi: 10.1109/lcsys.2018.2840041. [120] J. T. Wen, S. Mishra, S. Mukherjee, N. Tantisujjatham and M. Minakais, ‘Building temperature con- trol with adaptive feedforward,’ in 52nd IEEE Conference on Decision and Control, IEEE, 2013. doi: 10.1109/cdc.2013.6760646. [121] W. Tumin, M. M. Olama and S. M. Djouadi, ‘Adaptive control for residential HVAC systems to support grid services,’ in2021IEEEPower&EnergySocietyInnovativeSmartGridTechnologiesCon- ference (ISGT), IEEE, 2021.doi: 10.1109/isgt49243.2021.9372229. [122] S. Baldi, F. Zhang, T. L. Quang, P. Endel and O. Holub, ‘Passive versus active learning in operation and adaptive maintenance of Heating, Ventilation, and Air Conditioning,’AppliedEnergy, vol. 252, p. 113 478, 2019.doi: 10.1016/j.apenergy.2019.113478. [123] A. Beghi and L. Cecchinato, ‘Modelling and adaptive control of small capacity chillers for HVAC applications,’ Applied Thermal Engineering, vol. 31, no. 6-7, pp. 1125–1134, 2011. doi: 10 . 1016 / j . applthermaleng.2010.12.007. [124] J. Bai, Y. Li and J. Chen, ‘Real-time performance assessment and adaptive control for a water chiller unit in an HVAC system,’IOPConferenceSeries:EarthandEnvironmentalScience, vol. 121, p. 052 005, 2018.doi: 10.1088/1755- 1315/121/5/052005. [125] D. Jie, ‘Modeling and simulation of temperature control system of coating plant air conditioner,’ Procedia Computer Science, vol. 107, pp. 196–201, 2017.doi: 10.1016/j.procs.2017.03.078. 131 [126] G. Lymperopoulos, P. M. Papadopoulos, P. Ioannou and M. M. Polycarpou, ‘Distributed adaptive control of air handling units for interconnected building zones,’ inThe2020AmericanControlCon- ference, 2020.doi: 10.23919/ACC45564.2020.9147454. [127] A. Buonomano, U. Montanaro, A. Palombo and S. Santini, ‘Temperature and humidity adaptive con- trol in multi-enclosed thermal zones under unexpected external disturbances,’EnergyandBuildings, vol. 135, pp. 263–285, 2017.doi: 10.1016/j.enbuild.2016.11.015. [128] N. K. Dhar, N. K. Verma and L. Behera, ‘Adaptive critic-based event-triggered control for HVAC system,’ IEEE Transactions on Industrial Informatics, vol. 14, no. 1, pp. 178–188, 2018. doi: 10.1109/ tii.2017.2725899. [129] A. Adegbenro, M. Short and C. Angione, ‘An integrated approach to adaptive control and super- visory optimisation of HVAC control systems for demand response applications,’ Energies, vol. 14, no. 8, p. 2078, 2021.doi: 10.3390/en14082078. [130] M. Short, ‘Real-time innite horizon adaptive/predictive control for smart home HVAC applica- tions,’ inProceedingsof2012IEEE17thInternationalConferenceonEmergingTechnologies&Factory Automation (ETFA 2012), IEEE, 2012.doi: 10.1109/etfa.2012.6489677. [131] Y. Stauer, E. Olivero, E. Onillon, C. Mahmed and D. Lindelof, ‘NeuroCool: Field tests of an adaptive, model-predictive controller for HVAC systems,’ Energy Procedia, vol. 122, pp. 127–132, 2017. doi: 10.1016/j.egypro.2017.07.316. [132] M. Schmelas, T. Feldmann and E. Bollin, ‘Adaptive predictive control of thermo-active building systems (TABS) based on a multiple regression algorithm,’ Energy and Buildings, vol. 103, pp. 14– 28, 2015.doi: 10.1016/j.enbuild.2015.06.012. [133] S. Yang, M. P. Wan, W. Chen, B. F. Ng and D. Zhai, ‘An adaptive robust model predictive control for indoor climate optimization and uncertainties handling in buildings,’ Building and Environment, vol. 163, p. 106 326, 2019.doi: 10.1016/j.buildenv.2019.106326. [134] D. Lindelof, H. Afshari, M. Alisafaee, J. Biswas, M. Caban, X. Mocellin and J. Viaene, ‘Field tests of an adaptive, model-predictive heating controller for residential buildings,’EnergyandBuildings, vol. 99, pp. 292–302, 2015.doi: 10.1016/j.enbuild.2015.04.029. [135] T. Chen, ‘Application of adaptive predictive control to a oor heating system with a large thermal lag,’ Energy and Buildings, vol. 34, no. 1, pp. 45–51, 2002.doi: 10.1016/s0378- 7788(01)00076- 7. [136] X. Zhang, W. Shi, B. Yan, A. Malkawi and N. Li, ‘Decentralized and Distributed Temperature Control via HVAC Systems in Energy Ecient Buildings,’ in Proc. of the IEEE Global Conference on Signal and Information Processing, 2017, pp. 1–10. [137] M. Baranski, J. Fütterer and D. Müller, ‘Development of a generic model-Assisted control algorithm for building HVAC systems,’ Energy Procedia, vol. 122, no. 1, pp. 1003–1008, 2017. doi: 10 . 1016 / j . egypro.2017.07.465. [138] ——, ‘Distributed exergy-based simulation-assisted control of HVAC supply chains,’ Energy and Buildings, vol. 175, no. 1, pp. 131–140, 2018.doi: 10.1016/j.enbuild.2018.07.006. 132 [139] T. Hatanaka, X. Zhang, W. Shi, M. Zhu and N. Li, ‘Physics-integrated hierarchical/distributed HVAC optimization for multiple buildings with robustness against time delays,’ in 2017 IEEE 56th Annual Conference on Decision and Control (CDC), IEEE, 2017, pp. 6573–6579. doi: 10 . 1109 / cdc . 2017 . 8264650. [140] N. Radhakrishnan, S. Srinivasan, R. Su and K. Poolla, ‘Learning-based hierarchical distributed HVAC scheduling with operational constraints,’ IEEE Transactions on Control Systems Technology, vol. 26, no. 5, pp. 1892–1900, 2018.doi: 10.1109/TCST.2017.2728004. [141] V. Reppa, P. Papadopoulos, M. M. Polycarpou and C. G. Panayiotou, ‘A distributed architecture for HVAC sensor fault detection and isolation,’ IEEE Transactions on Control Systems Technology, vol. 23, no. 4, pp. 1323–1337, 2015.doi: 10.1109/tcst.2014.2363629. [142] P. M. Papadopoulos, V. Reppa, M. M. Polycarpou and C. G. Panayiotou, ‘Distributed sensor fault accommodation of multi-zone hvac systems,’ inProc.oftheIEEEConferenceonDecisionandControl (CDC), IEEE, IEEE, Dec. 2018, pp. 7296–7301.doi: 10.1109/CDC.2018.8619178. [143] C. Guo, Q. Song and W. Cai, ‘A neural network assisted cascade control system for air handling unit,’IEEETransactionsonIndustrialElectronics, vol. 54, no. 1, pp. 620–628, 2007.doi: 10.1109/TIE. 2006.888809. [144] C. Price and B. P. Rasmussen, ‘Optimal tuning of cascaded control architectures for nonlinear HVAC systems,’ Science and Technology for the Built Environment, vol. 23, no. 8, pp. 1190–1202, 2017.doi: 10.1080/23744731.2016.1262663. [145] J. Zhuang, Y. Chen and J. Wu, ‘Cascade control for supply air temperature in a variable air volume system,’ inProc.ofIOPConferenceSeries:EarthandEnvironmentalScience, vol. 238, IOP Publishing, 2019, pp. 1–7.doi: 10.1088/1755- 1315/238/1/012021. [146] D. Ivanova, N. Valov and M. Deyanov, ‘Application of the genetic algorithm for cascade control of a HVAC system,’MATECWebofConferences, vol. 292, N. Mastorakis, V. Mladenov and A. Bulucea, Eds., pp. 1–5, 2019.doi: 10.1051/matecconf/201929201064. [147] EnergyPlus Development Team, ‘Energyplus engineering reference,’ EnergyPlus Version 9.3., 2020. [148] H. Huang, L. Chen, M. Mohammadzaheri, E. Hu and M. Chen, ‘Multi-zone temperature prediction in a commercial building using articial neural network model,’ in 2013 10th IEEE International Conference on Control and Automation (ICCA), IEEE, 2013.doi: 10.1109/ICCA.2013.6565010. [149] N. Elbaccouche, B. Boussaid, N. Abdelkrim and C. Aubrun, ‘Hierarchical predictive control for mul- tizone building,’ in 2016 14th International Conference on Control, Automation, Robotics and Vision (ICARCV), IEEE, 2016.doi: 10.1109/icarcv.2016.7838702. [150] J. B. Petersen, J. D. Bendtsen and J. Stoustrup, ‘Multi-zone modeling and energy ecient control of shopping center cooling,’ in 2018 IEEE Conference on Control Technology and Applications (CCTA), IEEE, 2018.doi: 10.1109/ccta.2018.8511559. [151] C. Phillips, Digital control system analysis and design. Englewood Clis, N.J: Prentice Hall, 1995, isbn: 9780133098327. 133 [152] K. Ogata, Discrete-Time Control Systems (2nd Edition). Pearson, 1995,isbn: 0-13-034281-5. [153] Pacic Northwest National Laboratory, ‘90.1 Prototype Building Models Primary School,’ Available in https://www.energycodes.gov/901- prototype- building- models- primary- school . [154] ‘Github link for the Input Data Files (IDF).’ Available in https : / / github . com / lymperop / Primary _ School_Building. (2019). [155] R. Huck, J. Havlicek, J. Sluss and A. Stevenson, ‘A low-cost distributed control architecture for intelligent transportation systems deployment in the state of oklahoma,’ in Proceedings. 2005 IEEE Intelligent Transportation Systems, 2005., IEEE.doi: 10.1109/itsc.2005.1520173. [156] S. Battiston, J. B. Glattfelder, D. Garlaschelli, F. Lillo and G. Caldarelli, ‘The structure of nancial networks,’ in Network Science, Springer London, 2010, pp. 131–163.doi: 10.1007/978- 1- 84996- 396- 1_7. [157] C. W. Reynolds, ‘Flocks, herds and schools: A distributed behavioral model,’ in Proceedings of the 14thannualconferenceonComputergraphicsandinteractivetechniques-SIGGRAPH’87, ACM Press, 1987.doi: 10.1145/37401.37406. [158] C. Meng, T. Wang, W. Chou, S. Luan, Y. Zhang and Z. Tian, ‘Remote surgery case: Robot-assisted teleneurosurgery,’ in IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA ’04. 2004, IEEE, 2004.doi: 10.1109/robot.2004.1307250. [159] F. Kazempour and J. Ghaisari, ‘Stability analysis of model-based networked distributed control systems,’JournalofProcessControl, vol. 23, no. 3, pp. 444–452, 2013.doi: 10.1016/j.jprocont.2012. 12.010. [160] R. Olfati-Saber, J. A. Fax and R. M. Murray, ‘Consensus and cooperation in networked multi-agent systems,’ Proceedings of the IEEE, vol. 95, no. 1, pp. 215–233, 2007.doi: 10.1109/jproc.2006.887293. [161] W. Ren, R. W. Beard and E. M. Atkins, ‘Information consensus in multivehicle cooperative control,’ IEEE Control Systems, vol. 27, no. 2, pp. 71–82, 2007.doi: 10.1109/mcs.2007.338264. [162] J. P. Hespanha, P. Naghshtabrizi and Y. Xu, ‘A survey of recent results in networked control sys- tems,’ Proceedings of the IEEE, vol. 95, no. 1, pp. 138–162, 2007.doi: 10.1109/jproc.2006.887288. [163] M. Razeghi-Jahromi and A. Seyedi, ‘Stabilization of distributed networked control systems with constant feedback delay,’ in 52nd IEEE Conference on Decision and Control, IEEE, 2013. doi: 10 . 1109/cdc.2013.6760612. [164] P. Naghshtabrizi and J. P. Hespanha, ‘Analysis of distributed control systems with shared com- munication and computation resources,’ in 2009 American Control Conference, IEEE, 2009. doi: 10.1109/acc.2009.5160461. [165] G. Walsh, H. Ye and L. Bushnell, ‘Stability analysis of networked control systems,’IEEETransactions on Control Systems Technology, vol. 10, no. 3, pp. 438–446, 2002.doi: 10.1109/87.998034. 134 [166] X. Wang and N. Hovakimyan, ‘Distributed control of uncertain networked systems: A decoupled design,’ IEEE Transactions on Automatic Control, vol. 58, no. 10, pp. 2536–2549, 2013. doi: 10.1109/ tac.2013.2264554. [167] S. Oh and S. Sastry, ‘Distributed networked control system with lossy links: State estimation and stabilizing communication control,’ inProceedingsofthe45thIEEEConferenceonDecisionandCon- trol, IEEE, 2006.doi: 10.1109/cdc.2006.377338. [168] J. Nilsson, B. Bernhardsson and B. Wittenmark, ‘Stochastic analysis and control of real-time sys- tems with random time delays,’ Automatica, vol. 34, no. 1, pp. 57–64, 1998. doi: 10 . 1016 / s0005 - 1098(97)00170- 2. [169] X.-S. Yang,Engineeringoptimization:anintroductionwithmetaheuristicapplications. Hoboken, N.J: John Wiley, 2010,isbn: 9780470582466. [170] A. Afram and F. Janabi-Shari, ‘Review of modeling methods for HVAC systems,’AppliedThermal Engineering, vol. 67, no. 1-2, pp. 507–519, 2014.doi: 10.1016/j.applthermaleng.2014.03.055. [171] J. Ji, G. Pei, T.-t. Chow, W. He, A. Zhang, J. Dong and H. Yi, ‘Performance of multi-functional domestic heat-pump system,’ Applied Energy, vol. 80, no. 3, pp. 307–326, 2005. doi: 10 . 1016 / j . apenergy.2004.04.005. [172] ‘ANSI/ASHRAE/IES Standard 90.1-2019 – Energy Standard for Buildings Except Low-Rise Resid- ential Buildings,’ 2019,issn: 1041-2336. [173] G.-Y. Jin, P.-Y. Tan, X.-D. Ding and T.-M. Koh, ‘Cooling coil unit dynamic control of in HVAC system,’ in 2011 6th IEEE Conference on Industrial Electronics and Applications, IEEE, 2011. doi: 10. 1109/iciea.2011.5975722. [174] Z. O’Neill, Y. Li and K. Williams, ‘HVAC control loop performance assessment: A critical review (1587-RP),’ Science and Technology for the Built Environment, vol. 23, no. 4, pp. 619–636, 2016. doi: 10.1080/23744731.2016.1239466. [175] Q.-G. Wang, Z. Zhang, K. J. Astrom and L. S. Chek, ‘Guaranteed dominant pole placement with PID controllers,’ Journal of Process Control, vol. 19, no. 2, pp. 349–352, 2009. doi: 10 . 1016 / j . jprocont . 2008.04.012. [176] E. Dincel and M. T. Söylemez, ‘Guaranteed dominant pole placement with discrete-PID controllers: A modied nyquist plot approach,’ IFAC Proceedings Volumes, vol. 47, no. 3, pp. 3122–3127, 2014. doi: 10.3182/20140824- 6- za- 1003.02442. [177] P. D. Domański, Control Performance Assessment: Theoretical Analyses and Industrial Practice. Springer International Publishing, 2020.doi: 10.1007/978- 3- 030- 23593- 2. [178] K. W. Roth, D. Westphalen, M. Y. Feng, P. Llana and L. Quartararo, ‘Energy Impact of Commer- cial Building Controls and Performance Diagnostics: Market Characterization, Energy Impact of Building Faults and Energy Savings Potential,’ Tech. Rep., 2005. 135 [179] N. E. Fernandez, S. Katipamula, W. Wang, Y. Xie, M. Zhao and C. D. Corbin, ‘Impacts of commer- cial building controls on energy savings and peak load reduction,’ Tech. Rep., 2017. doi: 10 . 2172 / 1400347. [180] S. Katipamula and M. R. Brambley, ‘Review Article: Methods for Fault Detection, Diagnostics, and Prognostics for Building Systems—A Review, Part I,’ HVAC and R Research, vol. 11, no. 1, pp. 3–25, 2005.doi: 10.1080/10789669.2005.10391133. [181] J. Schein, S. T. Bushby, N. S. Castro and J. M. House, ‘A rule-based fault detection method for air handling units,’ Energy and Buildings, vol. 38, no. 12, pp. 1485–1492, 2006.doi: 10.1016/j.enbuild. 2006.04.014. [182] H. Yang, S. Cho, C.-S. Tae and M. Zaheeruddin, ‘Sequential rule based algorithms for temperature sensor fault detection in air handling units,’ Energy Conversion and Management, vol. 49, no. 8, pp. 2291–2306, 2008.doi: 10.1016/j.enconman.2008.01.029. [183] Y. Zhao, J. Wen and S. Wang, ‘Diagnostic Bayesian networks for diagnosing air handling units faults - Part II: Faults in coils and sensors,’AppliedThermalEngineering, vol. 90, pp. 145–157, 2015. doi: 10.1016/j.applthermaleng.2015.07.001. [184] M. Sampath, R. Sengupta, S. Lafortune, K. Sinnamohideen and D. Teneketzis, ‘Failure diagnosis using discrete-event models,’IEEETransactionsonControlSystemsTechnology, vol. 4, no. 2, pp. 105– 124, 1996.doi: 10.1109/87.486338. [185] S. Wang and F. Xiao, ‘AHU sensor fault diagnosis using principal component analysis method,’ Energy and Buildings, vol. 36, no. 2, pp. 147–160, 2004.doi: 10.1016/j.enbuild.2003.10.002. [186] Z. Du and X. Jin, ‘Tolerant control for multiple faults of sensors in VAV systems,’EnergyConversion and Management, vol. 48, no. 3, pp. 764–777, 2007.doi: 10.1016/j.enconman.2006.09.007. [187] M. Kumar and I. Kar, ‘Fault Detection and Diagnosis of Air-Conditioning Systems using Residuals,’ Proc.ofthe10thIFACInternationalSymposiumonDynamicsandControlofProcessSystems, vol. 46, no. 32, pp. 607–612, 2013.doi: 10.3182/20131218- 3- IN- 2045.00113. [188] A. Beghi, L. CecChinato, L. Corso, M. Rampazzo and F. Simmini, ‘Process history-based Fault De- tection and Diagnosis for VAVAC systems,’ in Proc. of the IEEE International Conference on Control Applications (CCA), IEEE, 2013, pp. 1165–1170, isbn: 978-1-4799-1559-0. doi: 10 . 1109 / CCA . 2013 . 6662909. [189] J. Liang and R. Du, ‘Model-based Fault Detection and Diagnosis of HVAC systems using Support Vector Machine method,’ International Journal of Refrigeration, vol. 30, no. 6, pp. 1104–1114, 2007. doi: 10.1016/j.ijrefrig.2006.12.012. [190] T. Mulumba, A. Afshari, K. Yan, W. Shen and L. K. Norford, ‘Robust model-based fault diagnosis for air handling units,’ Energy and Buildings, vol. 86, pp. 698–707, 2015. doi: 10 . 1016 / j . enbuild . 2014.10.069. 136 [191] S. Wang and Y. Chen, ‘Fault-tolerant control for outdoor ventilation air ow rate in buildings based on neural network,’BuildingandEnvironment, vol. 37, no. 7, pp. 691–704, 2002.doi: 10.1016/S0360- 1323(01)00076- 2. [192] Z. Du, X. Jin and Y. Yang, ‘Fault diagnosis for temperature, ow rate and pressure sensors in VAV systems using wavelet neural network,’ Applied Energy, vol. 86, no. 9, pp. 1624–1631, 2009. doi: 10.1016/j.apenergy.2009.01.015. [193] S. Wang and J.-B. Wang, ‘Robust sensor fault diagnosis and validation in HVAC systems,’ Trans- actions of the Institute of Measurement and Control, vol. 24, no. 3, pp. 231–262, 2002. doi: 10 . 1191 / 0142331202tm030oa. [194] M. Padilla, D. Choinière and J. A. Candanedo, ‘A model-based strategy for self-correction of sensor faults in variable air volume air handling units,’ Science and Technology for the Built Environment, vol. 21, no. 7, pp. 1018–1032, 2015.doi: 10.1080/23744731.2015.1025682. [195] X.-F. Liu and A. Dexter, ‘Fault-tolerant supervisory control of VAV air-conditioning systems,’ En- ergy and Buildings, vol. 33, no. 4, pp. 379–389, 2001.doi: 10.1016/S0378- 7788(00)00120- 1. [196] C. Lo, P. Chan, Y. Wong, A. Rad and K. Cheung, ‘Fuzzy-genetic algorithm for automatic fault de- tection in HVAC systems,’ Applied Soft Computing, vol. 7, no. 2, pp. 554–560, 2007.doi: 10.1016/j. asoc.2006.06.003. [197] H. Yoshida, S. Kumar and Y. Morita, ‘Online fault detection and diagnosis in VAV air handling unit by RARX modeling,’ Energy and Buildings, vol. 33, no. 4, pp. 391–401, 2001. doi: 10 . 1016 / S0378 - 7788(00)00121- 3. [198] J. C. M. Yiu and S. Wang, ‘Multiple ARMAX modeling scheme for forecasting air conditioning system performance,’ Energy Conversion and Management, vol. 48, no. 8, pp. 2276–2285, 2007.doi: 10.1016/j.enconman.2007.03.018. [199] W. Turner, A. Staino and B. Basu, ‘Residential HVAC fault detection using a system identication approach,’ Energy and Buildings, vol. 151, pp. 1–17, 2017.doi: 10.1016/j.enbuild.2017.06.008. [200] X.-B. Yang, X.-Q. Jin, Z.-M. Du, Y.-H. Zhu and Y.-B. Guo, ‘A hybrid model-based fault detection strategy for air handling unit sensors,’ Energy and Buildings, vol. 57, pp. 132–143, 2013. doi: 10 . 1016/j.enbuild.2012.10.048. [201] H. Shahnazari, P. Mhaskar, J. M. House and T. I. Salsbury, ‘Distributed fault diagnosis of heating, ventilation, and air conditioning systems,’AmericanInstituteofChemicalEngineers(AIChE)Journal, vol. 65, no. 2, pp. 640–651, 2019.doi: 10.1002/aic.16486. [202] M. Bonvini, M. D. Sohn, J. Granderson, M. Wetter and M. A. Piette, ‘Robust on-line fault detection diagnosis for HVAC components based on nonlinear state estimation techniques,’ Applied Energy, vol. 124, pp. 156–166, 2014.doi: 10.1016/j.apenergy.2014.03.009. 137 [203] B. T. Thumati, M. a. Feinstein, J. W. Fonda, A. Turnbull, F. J. Weaver, M. E. Calkins and S. Jagan- nathan, ‘An online model-based fault diagnosis scheme for HVAC systems,’ in IEEE International Conference on Control Applications, 2011, pp. 70–75, isbn: 9781457710629. doi: 10 . 1109 / CCA . 2011 . 6044486. [204] P. M. Papadopoulos, V. Reppa, M. M. Polycarpou and C. G. Panayiotou, ‘Distributed Diagnosis of Actuator and Sensor Faults in HVAC Systems,’Proceedingsofthe20thIFACWorldCongress, vol. 50, no. 1, pp. 4209–4215, 2017.doi: 10.1016/j.ifacol.2017.08.816. [205] ‘Directive 2010/31/EU of the European Parliament and of the Council of 19 May 2010 on the energy performance of buildings,’ Ocial Journal, vol. L 153, pp. 13–35, 2010. [206] ‘Directive 2012/27/EU of the European Parliament and of the Council of 25 October 2012 on energy eciency, amending Directives 2009/125/EC and 2010/30/EU and repealing Directives 2004/8/EC and 2006/32/EC Text with EEA relevance,’ Ocial Journal, vol. L 315, pp. 1–56, 2012. [207] V. Reppa, M. M. Polycarpou and C. G. Panayiotou, ‘Sensor Fault Diagnosis,’FoundationsandTrends in Systems and Control, vol. 3, no. 1-2, pp. 1–248, 2016,issn: 2325-6818.doi: 10.1561/2600000007. [208] P. M. Papadopoulos, G. Lymperopoulos, M. M. Polycarpou and P. Ioannou, ‘Model-based fault de- tection and localization algorithm for air handling units in large-scale buildings,’ in16thConference oftheInternationalSocietyofIndoorAirQualityandClimate:CreativeandSmartSolutionsforBetter Built Environments, Indoor Air 2020, 2020. [209] S. Wang, Q. Zhou and F. Xiao, ‘A system-level fault detection and diagnosis strategy for HVAC systems involving sensor faults,’ Energy and Buildings, vol. 42, no. 4, pp. 477–490, 2010. doi: 10 . 1016/j.enbuild.2009.10.017. [210] H. Shahnazari, P. Mhaskar, J. M. House and T. I. Salsbury, ‘Modeling and fault diagnosis design for HVAC systems using recurrent neural networks,’ Computers and Chemical Engineering, vol. 126, pp. 189–203, 2019.doi: 10.1016/j.compchemeng.2019.04.011. [211] V. Gunes, S. Peter and T. Givargis, ‘Improving Energy Eciency and Thermal Comfort of Smart Buildings with HVAC Systems in the Presence of Sensor Faults,’ in Proceedings of IEEE 17th Inter- national Conference on High Performance Computing and Communications, 7th International Sym- posium on Cyberspace Safety and Security, and 12th International Conference on Embedded Software and Systems, 2015, pp. 945–950.doi: 10.1109/HPCC- CSS- ICESS.2015.154. [212] V. Reppa, P. Papadopoulos, M. M. Polycarpou and C. G. Panayiotou, ‘Distributed detection and isolation of sensor faults in HVAC systems,’ in21stMediterraneanConferenceonControlandAuto- mation, IEEE, 2013, pp. 401–406,isbn: 978-1-4799-0997-1.doi: 10.1109/MED.2013.6608753. [213] P. M. Papadopoulos, V. Reppa, M. M. Polycarpou and C. G. Panayiotou, ‘Distributed Adaptive Estim- ation Scheme for Isolation of Sensor Faults in Multi-zone HVAC Systems,’ in 9th IFAC Symposium onFaultDetection,SupervisionandSafetyforTechnicalProcesses, Elsevier, 2015, pp. 1146–1151.doi: 10.1016/j.ifacol.2015.09.681. 138 [214] ——, ‘Distributed adaptive sensor fault tolerant control for smart buildings,’ in201554thIEEECon- ference on Decision and Control (CDC), IEEE, 2015, pp. 3143–3148, isbn: 978-1-4799-7886-1. doi: 10.1109/CDC.2015.7402690. [215] M. Wetter, ‘Co-simulation of building energy and control systems with the Building Controls Vir- tual Test Bed,’JournalofBuildingPerformanceSimulation, vol. 4, no. 3, 2011.doi: 10.1080/19401493. 2010.518631. [216] P. O. Fanger,ThermalComfort:analysisandapplicationsinenvironmentalengineering. Copenhagen: Danish Technical Press, 1970, p. 244. [217] H. Djamila, ‘Indoor thermal comfort predictions: Selected issues and trends,’ Renewable and Sus- tainable Energy Reviews, vol. 74, pp. 569–580, 2017.doi: 10.1016/j.rser.2017.02.076. [218] B. Olesen and K. Parsons, ‘Introduction to thermal comfort standards and to the proposed new version of EN ISO 7730,’EnergyandBuildings, vol. 34, no. 6, pp. 537–548, 2002.doi: 10.1016/s0378- 7788(02)00004- x. [219] S. Carlucci, L. Bai, R. de Dear and L. Yang, ‘Review of adaptive thermal comfort models in built environmental regulatory documents,’ Building and Environment, vol. 137, pp. 73–89, 2018. doi: 10.1016/j.buildenv.2018.03.053. [220] P. Wolko, ‘Indoor air humidity, air quality, and health – an overview,’ International Journal of Hygiene and Environmental Health, vol. 221, no. 3, pp. 376–390, 2018. doi: 10 . 1016 / j . ijheh . 2018 . 01.015. [221] A. V. Arundel, E. M. Sterling, J. H. Biggin and T. D. Sterling, ‘Indirect health eects of relative humidity in indoor environments.,’ Environmental Health Perspectives, vol. 65, pp. 351–361, 1986. doi: 10.1289/ehp.8665351. [222] A. C. Lowen, S. Mubareka, J. Steel and P. Palese, ‘Inuenza virus transmission is dependent on relative humidity and temperature,’ PLoS Pathogens, vol. 3, no. 10, R. S. Baric, Ed., e151, 2007. doi: 10.1371/journal.ppat.0030151. [223] S. T. Larsen, P. Wolko, M. Hammer, V. Kofoed-Sørensen, P. A. Clausen and G. D. Nielsen, ‘Acute airway eects of airborne formaldehyde in sensitized and non-sensitized mice housed in a dry or humid environment,’ Toxicology and Applied Pharmacology, vol. 268, no. 3, pp. 294–299, 2013. doi: 10.1016/j.taap.2013.02.006. [224] Y. Sunwoo, C. Chou, J. Takeshita, M. Murakami and Y. Tochihara, ‘Physiological and subjective responses to low relative humidity,’ Journal of physiological anthropology, vol. 25, no. 1, pp. 7–14, 2006.doi: 10.2114/jpa2.25.7. [225] K. A. Angelon-Gaetz, D. B. Richardson, S. W. Marshall and M. L. Hernandez, ‘Exploration of the eects of classroom humidity levels on teachers’ respiratory symptoms,’ International Archives of OccupationalandEnvironmentalHealth, vol. 89, no. 5, pp. 729–737, 2016.doi: 10.1007/s00420- 016- 1111- 0. 139 [226] L. V. Stephen Boyd,ConvexOptimization. Cambridge University Press, 8th Mar. 2019, 732 pp.,isbn: 0521833787. [Online]. Available: https : / / www . ebook . de / de / product / 3677442 / stephen _ boyd _ lieven _ vandenberghe_convex_optimization.html. 140 Appendix AppendixA Notation R denotes the eld of real numbers. For any vectorx2 R n ,jx(k)j represents the Euclidean (l 2 ) vector norm inR n at each timek, and for discrete time signalskx k k 2 , P k i=0 ki x > (i)x(i) 1 2 denes the exponentially weightedl 2 norm of a vector functionx(k), where2 (0; 1] is a positive constant andk represents time.x2l 1 means thatkxk 1 = sup k0 jx(k)j exists. ProofofTheorem2.1 This proof covers the case of the gradient-descent based adaptive law. i (k) is generated by eq. (2.14a) and (2.14b), which can be written in the following compact form: i (k) = Proj [ i (k 1) +T s i i (k) i (k)] (6.2) where we use the projection operator for discrete systems, as described in [102], with box constraints for each element of i . The projection operator restricts adaptation between predened bounds, within which 141 the unknown desired gains are assumed to lie. First, we establish the properties ofm i . From the denition ofm i , we derive the following inequality: m 2 i (k) 1 +T s min ( i )j i (k)j 2 (6.3) which implies: j i (k)j 2 m 2 i (k) j i (k)j 2 1 +T s min ( i )j i (k)j 2 (6.4) which in turn implies that the normalized i vector signal is bounded, i.e., j i j m i 2 l 1 . Therefore, the normalized estimation error i is also bounded, i.e., i 2l 1 , since it can be written as follows: i (k) = > i i (k) m 2 i (k) (6.5) Using equation (6.5) we can derive that the quantity i m i 2l 1 is also bounded, since we can write: i (k)m i (k) = > i i (k) m i (k) = > i (k 1) i (k) m i (k) (6.6) where i (k) = i (k) i is the estimation error of the gain. The projection operator in equation (6.2) connes the estimate i inside the convex setS i , which is a hyperbox that is created by all the lower and upper bound constraints [L i ] s ; [U i ] s . Then, equation (6.2) can be rewritten as follows: i (k) = i (k 1) +T s i i (k) i (k) +g(k) (6.7) 142 where g(k) = 8 > > > > < > > > > : 0 if i (k 1) +T s i i (k) i (k)2S i ^ i (k) ~ i (k) if i (k 1) +T s i i (k) i (k) = 2S i (6.8) where ~ i (k) = i (k 1) +T s i i (k) i (k) and ^ i (k) is the orthogonal projection of ~ i (k) toS i . The projection operator guarantees that i 2l 1 . Next, we consider the following Lyapunov-like function: V i (k) = > i (k) 1 i i (k) (6.9) Then, using equation (6.9), we dene the following dierence: V i (k)V i (k 1) = > i (k 1) 1 i i (k 1) +T 2 s 2 i (k) > i (k) i i (k) +g > i (k) 1 i g i (k) + 2 > i (k 1)T s i (k) i (k) + 2g > i (k)T s i (k) i (k) + 2g > i (k) i (k 1) > i (k 1) 1 i i (k 1) (6.10) The following three terms from (6.10) satisfy: g > i (k) 1 i g i (k) + 2g > i (k) i (k 1) +T s i (k) i (k) max 1 i ^ i (k) ~ i (k) > ^ i (k) ~ i (k) + 2 ^ i (k) ~ i (k) > ~ i (k) i (k) c 2 ^ i (k) i (k) > ~ i (k) ^ i (k) ^ i (k) ~ i (k) 2 0 (6.11) 143 where the last inequality is obtained by using ( ^ i i ) > ( ~ i ^ i ) 0 because of the projection property. Using (6.11) in (6.10) we obtain: V i (k)V i (k 1) =T 2 s 2 i (k) > i (k) i i (k) + 2 > i (k 1)T s i (k) i (k) =T 2 s 2 i (k) > i (k) i i (k) 2T s 2 i (k)m 2 i (k) =2T s 2 i (k)m 2 i (k) 1 T s > i (k) i i (k) 2m 2 i (k) (6.12) Since for the chosenm i we have that Ts > i (k) i i (k) m 2 i (k) < 1, it follows that: V i (k)V i (k 1)T s 2 i (k)m 2 i (k) 0 (6.13) Using the last inequality, we derive the following one: M X k=1 2 i (k)m 2 i (k)c(V i (0)V i (M)) (6.14) for some positive nite constantc> 0 and any positive integerM > 0. In addition, since the gain estimate is bounded, i.e. i 2 l 1 , it follows that the Lyapunov-like function (6.10) is bounded too, i.e. V i 2 l 1 . Therefore, it follows from (6.14) that i (k)m i (k)2 l 2 which implies that i (k)m i (k)! 0 ask!1. Finally, because of the adaptive law (6.2), we have the following relation: i (k) i (k 1)T s i i (k)m i (k) i (k) m i (k) (6.15) which implies thatj i (k) i (k1)j2l 2 and, thus,j i (k) i (k1)j! 0 ask!1. To summarize, the estimation scheme (2.14) has the following properties: 1. i (k); i (k); i (k)m i (k)2l 1 144 2. i (k)m i (k);j i (k) i (k 1)j2l 2 3. i (k)m i (k);j i (k) i (k 1)j! 0, ask!1 In order to prove that temperature tracking is guaranteed even without prior knowledge of system para- meters, we dene the following signal: m 2 f (k), 1 + N X i=1 ke i k1 k 2 2 (6.16) wherek() k k 2 is thel 2 norm for some2 (A 2 d ; 1]8i, so that 1 p z+A d has stable poles8i, whereA d = max 8i;8j2N i a m i ;a w i;j and a w i;j = e 2 Uw i;j Aw i;j Cw i;j Ts . We can rewrite the zone temperature tracking error dynamics as follows: e i = z za m i [ i m 2 i ] (6.17) In addition, from the denition ofm i , we derive the following inequality: m 2 i (k) 1 +T s max ()j i (k)j 2 (6.18) and for the elements of i , we have that for some nite c > 0,jT w i;j (k 1)j ck (T z i ) k1 k 2 + ck T zp k1 k 2 +c from (2.1b),(2.1c),jT z i (k1)jje i (k1)j+c,T m i ;T o 2l 1 and, thus,jT sa i (k1)j cje i (k 1)j +c P j2M i ke j k1 k 2 +c P j2N i je j (k 1)j +c. In addition, using Lemma A.12.33.ii of [102] and consideringH(z) = 1, we have thatje i (k)jke i k k 2 . Therefore,j i (k)j c P N i=1 ke i k1 k 2 2 +c. That means that the signalm i is bounded by the signalm f : m i cm f (6.19) 145 for some nite c > 0. Applying Lemma A.12.33 from [102] and since z zam i is analytic injzj p , equation (6.17) gives the following inequality for some nitec> 0: ke i k k 2 ck i m 2 i k k 2 (6.20) which, combined with inequality (6.19), produces the following inequality: ke i k k 2 ck i m i m f k k 2 (6.21) Combining relations (6.16) and (6.21), we derive the following inequality: m 2 f (k) 1 +ck~ g k1 m f k1 k 2 2 (6.22) where ~ g 2 = N X i=1 j i m i j 2 (6.23) By applying Lemma A.12.31 from [102] on inequality (6.22) and using the fact that the geometric mean of a series is less than the arithmetic mean, for some nitec> 0 we obtain the following inequality: m 2 f (k) 1 +c k1 X p=0 kp ~ g 2 (p)m 2 f (p) 1 +c k1 X p=0 2 4 kp ~ g 2 (p) Y p 0. Therefore, again since ~ g 2 l 2 , for < 1, we conclude that the signal m f is bounded, i.e.,m f 2l 1 , since the series in the last inequality converges. Hence, all temperature tracking errorse i 2l 1 8i are bounded too, due to the denition ofm f (6.16). Hence, signals i ;m i 2l 1 are also bounded. In addition, from (2.14c) we have thatz i = i m 2 i = i m i m i . Since i m i 2l 2 andm i 2l 1 , thenz i 2l 2 . This means that > i i 2l 2 and > i i ! 0. Thus, from: e i = z za m i [ > i i ] (6.26) we have that for all zones the temperature tracking error is inl 2 . This implies that8ie i ! 0 ask!1. AppendixB Notation R denotes the eld of real numbers. For any vectorx2R n ,jx(t)j represents the Euclidean (L 2 ) vector norm inR n at each time t and x 2 L 1 means thatkxk 1 = sup t0 jx(t)j exists. For any matrix A, kAk represents the induced matrix norm corresponding to the vector normjj. max (A) and min (A) represent respectively the maximum and minimum eigenvalue of a matrix A. Finally, x2S(w) means that x is w-small in the mean square sense, i.e. x satises R t+T t x > ()x()d c 0 R t+T t w()d +c 1 , wherec 0 ,c 1 are constants independent of w andT is a non-negative nite time interval. 147 GeneralizedProofforTheorems3.1and3.2 Proof. Choose the Lyapunov-Krasovskii functional: V = N X i=1 e > i (t)P i e i (t) + N X i=1 N X j=1 Z t t ij e > j (r)G j e j (r)dr + N X i=1 N X j=1 Z t t ij Z t s _ e > j (r) _ e j (r)drds + N X i=1 [Tr[ ~ K > i (t) i ~ K i (t) + ~ L > i (t) i ~ L i (t)] + N X j=1 Tr[ ~ K > ij (t) i ~ K ij (t)]] (6.27) whereG j are positive denite matrices. The derivative is: _ V = N X i=1 [e > i (t)(A > m;i P i +P i A m;i )e i (t) + 2 N X j=1 e > j (t)W > ij P i e i (t) + 2d > i (t)P i e i (t) +2 N X j=1 Z t t ij _ e > j (r) > ij P i e i (t)dr 2e > i (t)P i B i [ ~ K i x i (t) ~ L i (t)r i (t)] 2e > i (t)P i B i N X j=1 ~ K ij (t)x j (t ij )] + N X i=1 N X j=1 [e > j (t)G j e j (t) e > j (t ij )G j e j (t ij )] + N X i=1 N X j=1 [ ij _ e > j (t) _ e j (t) Z t t ij _ e > j (r) _ e j (r)dr] + N X i=1 [2 Tr[ ~ K > i (t) i _ ~ K i (t) + ~ L > i (t) i _ ~ L i (t)] + N X j=1 2 Tr[ ~ K > ij (t) i _ ~ K ij (t)]] (6.28) Based (3.10), we have that: N X i=1 e > i (t)(A > m;i P i +P i A m;i )e i (t) N X i=1 N X j=1 [e > i (t)Q i e i (t)] N X i=1 N X j=1 [ min (Q i )je i (t)j 2 ] (6.29) In addition, using (3.14) we have that: _ e > j (t) _ e j (t) = e > j (t)A > m;j A m;j e j (t) + 2e > j (t)A > m;j N X k=1 (W jk + jk )e k (t) + 2e > j (t)A > m;j d j (t) 2e > j (t)A > m;j N X k=1 jk e k (t jk ) + N X k=1 N X q=1 e > k (t jk ) > jk jq e q (t jq ) 148 + N X k=1 N X q=1 e > k (t)(W jk + jk ) > (W jq + jq )e q (t) + 2 N X k=1 e > k (t)(W jk + jk ) > d j (t) 2 N X k=1 N X q=1 e > k (t)(W jk + jk ) > jq e q (t jq ) 2 N X k=1 e > k (t jk ) > jk d j (t) +d > j (t)d j (t) 2e > j (t)A > m;j B j ~ K j (t)x j (t) + 2e > j (t)A > m;j B j ~ L j (t)r j (t) 2 N X k=1 e > k (t)(W jk + jk ) > B j ~ K j (t)x j (t) 2d > j (t)B j ~ K j (t)x j (t) +2 N X k=1 e > k (t)(W jk + jk ) > B j ~ L j (t)r j (t) + 2d > j (t)B j ~ L j (t)r j (t) +2 N X k=1 e > k (t jk ) > jk B j ~ K j (t)x j (t) 2 N X k=1 e > k (t jk ) > jk B j ~ L j (t)r j (t) +x > j (t) ~ K > j (t)B > j B j ~ K j (t)x j (t) 2x > j (t) ~ K > j (t)B > j B j ~ L j (t)r j (t) +2 N X k=1 x > j (t) ~ K > j (t)B > j B j ~ K jk (t)x k (t jk ) +r > j (t) ~ L > j (t)B > j B j ~ L j (t)r j (t) 2 N X k=1 r > j (t) ~ L > j (t)B > j B j ~ K jk (t)x k (t jk ) 2e > j (t)A > m;j N X k=1 B j ~ K jk (t)x k (t jk ) 2 N X k=1 d > j (t)B j ~ K jk (t)x k (t jk ) 2 N X k=1 N X q=1 e > k (t)(W jk + jk ) > B j ~ K jq (t)x q (t jq ) +2 N X k=1 N X q=1 e > k (t jk ) > jk B j ~ K jq (t)x q (t jq ) + N X k=1 N X q=1 x > k (t jk ) ~ K > jk B > j B j ~ K jq (t)x q (t jq ) (6.30) Fenchel’s inequality [226] suggests that: 2x > Ryx > x +y > R > Ry (6.31) 149 Therefore, we have that: 2 Z t t ij _ e j (r) > > ij P i e i (t)dr ij e i (t) > P i ij > ij P i e i (t) + Z t t ij _ e j (r) > _ e j (r)dr (6.32) 2e > j (t)A > m;j N X k=1 (W jk + jk )e k (t) Ne > j (t)A > m;j A m;j e j (t) + N X k=1 e > k (t)(W jk + jk ) > (W jk + jk )e k (t) (6.33) 2e > j (t)A > m;j N X k=1 jk e k (t jk )Ne > j (t)A > m;j A m;j e j (t) + N X k=1 e > k (t jk ) > jk jk e k (t jk ) (6.34) N X k=1 N X q=1 e > k (t)(W jk + jk ) > (W jq + jq )e q (t) N X k=1 Ne > k (t)(W jk + jk ) > (W jk + jk )e k (t) (6.35) 2 N X k=1 N X q=1 e > k (t)(W jk + jk ) > jq e q (t jq ) N X k=1 Ne > k (t)(W jk + jk ) > (W jk + jk )e k (t) + N X k=1 Ne > k (t jk ) > jk jk e k (t jk ) (6.36) N X k=1 N X q=1 e > k (t jk ) > jk jq e q (t jq ) N X k=1 Ne > k (t jk ) > jk jk e k (t jk ) (6.37) 2 N X j=1 e > j (t)W > ij P i e i (t) N X j=1 [kW ij kkP i ke > j (t)e j (t) +kW ij kkP i ke > i (t)e i (t)] (6.38) 150 ProofofTheorem3.1 For the case when all system parameters A i ;B i and A ij are known, K i ;L i and K ij can be calculated and, thus, we do not need to use their online estimates. Hence, ~ K i (t) = 0; ~ L i (t) = 0 and ~ K ij (t) = 0. Therefore, using (6.30) and (6.33)-(6.37), we have that: _ e > j (t) _ e j (t) (1 + 2N)e > j (t)A > m;j A m;j e j (t) +d j (t) > d j (t) + N X k=1 (1 + 2N)e > k (t)(W jk + jk ) > (W jk + jk )e k (t) + N X k=1 (1 + 2N)e > k (t jk ) > jk jk e k (t jk ) + 2e > j (t)A > m;j d j (t) +2 N X k=1 e > k (t)(W jk + jk ) > d j (t) 2 N X k=1 e > k (t jk ) > jk d j (t) (6.39) Thus, using (6.28), (6.29), (6.32), (6.38) and (6.39), we have that: _ V N X i=1 N X j=1 e > j (t) e j (t ij ) > ij 2 6 6 4 e j (t) e j (t ij ) 3 7 7 5 +M where ij = 2 6 6 4 ij;1 0 0 ij;2 3 7 7 5 (6.40) with ij;1 = ij min (Q j ) N I + (kW ij kkP i k +kW ji kkP j k)I + ji P j ji > ji P j + max (G j ) + ij (1 + 2N)A > m;j A m;j + (1 + 2N) N X k=1 [(W kj + kj ) > (W kj + kj ) ik ] (6.41) 151 and ij;2 = ij min (G j )I + (1 + 2N) N X k=1 [ > kj kj ik ] (6.42) with ij ; ij 2 (0; 1), withM being: M = N X i=1 N X j=1 [(1 ij ) min (Q j ) N e > j (t)e j (t) + 2d > j (t) P j N e j (t) + ij d > j (t)d > j (t) (1 ij ) min (G j )e j (t ij ) > e j (t ij ) + 2 ij e > j (t)A > m;j d j (t) +2e > j (t)(W ij + ij ) > d i (t) N X k=1 ki 2e > j (t ij ) > ij d i (t) N X k=1 ki ] (6.43) We need ij to be negative denite. Thus, this corresponds to ij;1 and ij;2 to be negative denite, which, in turn, means that we want: ij min (Q j ) N I + (kW ij kkP i k +kW ji kkP j k)I + ji P j ji > ji P j + ij (1 + 2N)A > m;j A m;j +(1 + 2N) N X k=1 [[A > kj A kj + 1 ij > kj kj ] ik ] 0 (6.44) since min (G) max (G) andW kj + kj =A kj . If ij are negative denite, for sucinetly small> 0 we have that: _ V V +V + N X i=1 N X j=1 [(1 ij ) min (Q j ) N e > j (t)e j (t) + 2d > j (t) P j N e j (t) (1 ij ) min (G j )e j (t ij ) > e j (t ij ) + 2 ij e > j (t)A > m;j d j (t) + ij d > j (t)d > j (t) +2e > j (t)(W ij + ij ) > d i (t) N X k=1 ki 2e > j (t ij ) > ij d i (t) N X k=1 ki ] (6.45) By examining the Lyapunov functional, we get the following inequalities for some constantc> 0: N X i=1 e > i (t)P i e i (t) N X i=1 N X j=1 max (P j ) N e > j (t)e j (t) = N X i=1 N X j=1 cje j (t)j 2 (6.46) 152 N X i=1 N X j=1 Z t t ij e > j (r)G j e j (r)dr N X i=1 N X j=1 max (G j ) Z t t ij e > j (r)e j (r)dr N X i=1 N X j=1 max (G j )(t (t ij )) max t ij rt je j (r)j 2 = N X i=1 N X j=1 max (G j ) ij cje j (t)j 2 = N X i=1 N X j=1 cje j (t)j 2 (6.47) Using (6.39), we have that: N X i=1 N X j=1 Z t t ij Z t s _ e > j (r) _ e j (r)drds N X i=1 N X j=1 [(1 + 2N)kA > m;j A m;j kc 2 ij 2 je j (t)j 2 +(1 + 2N)cje j (t)j 2 kW ij + ij k 2 N X k=1 2 ki 2 +(1 + 2N)cje j (t ij )j 2 k ij k 2 N X k=1 2 ki 2 +d 2 j 2 ij 2 +2kA m;j k 2 d j cje j (t)j 2 ij 2 + 2kW ij + ij kd i cje j (t)j N X k=1 2 ki 2 +2k ij kd i cje j (t ij )j N X k=1 2 ki 2 ] N X i=1 N X j=1 [cje j (t)j 2 +cje j (t ij )j 2 +cje j (t)j +cje j (t ij )j +c] (6.48) and using (6.43) we have that: M N X i=1 N X j=1 [cje j (t)j 2 +cje j (t)j +ccje j (t ij )j 2 +cje j (t ij )j] (6.49) Therefore, using (6.46), (6.47), (6.48) and (6.49), we can write (6.45) as: _ V V + N X i=1 N X j=1 [cje j (t)j 2 +cje j (t)j +ccje j (t ij )j 2 +cje j (t ij )j +cje j (t)j 2 153 +cje j (t ij )j 2 +cje j (t)j +cje j (t ij )j +c] V + N X i=1 N X j=1 [cje j (t)j 2 +cje j (t)j +ccje j (t ij )j 2 +cje j (t ij )j] V + N X i=1 N X j=1 [c(je j (t)jc) 2 c(je j (t ij )jc) 2 +c] V + N X i=1 N X j=1 c =V +C (6.50) whereC > 0 is a constant. By using Lemma 3.2.4 of [101], we have that the Lyapunov function is bounded as follows: V (t)e (tt 0 ) V (t 0 ) + Z t t 0 e (ts) Cdse (tt 0 ) V (t 0 ) + C (6.51) Hence, the Lyapunov functionV 2L 1 , which means that the errore i (t)2L 1 . Then, from (3.7), we have that _ e i (t)2L 1 too. Furthermore, from (3.8) we have that: _ e i (t) = A m;i e i (t) + N X j=1 W ij e j (t) + N X j=1 ij Z t t ij _ e j (r)dr + N X j=1 W ij x ref;j (t) + N X j=1 ij Z t t ij _ x ref;j (r)dr (6.52) The solution of the dierential equation above is: e i (t) = i (t;t 0 )e i (t 0 ) + Z t t 0 i (t;s) N X j=1 W ij e j (s)ds + Z t t 0 i (t;s) N X j=1 ij Z s s ij _ e j (r)drds + Z t t 0 i (t;s) N X j=1 W ij x ref;j (s)ds + Z t t 0 i (t;s) N X j=1 ij Z s ts ij _ x ref;j (r)drds (6.53) 154 where i (t;t 0 ) =e A m;i (tt 0 ) is the state-transition matrix of the system. Thus, for 0;i > 0; 0;i > 0 such thatk i (t;t 0 )k 0;i e 0;i (tt 0 ) , we have that: je i (t)j 0;i e 0;i (tt 0 ) je i (t 0 )j + Z t t 0 0;i e 0;i (ts) N X j=1 kW ij kje j (s)jds + Z t t 0 0;i e 0;i (ts) N X j=1 k ij k Z s s ij j _ e j (r)jdrds + Z t t 0 0;i e 0;i (ts) N X j=1 kW ij kjx ref;j (s)jds + Z t t 0 0;i e 0;i (ts) N X j=1 k ij k Z s ts ij j _ x ref;j (r)jdrds (6.54) and sincee i ; _ e i ;x ref;i ; _ x ref;i 2L 1 and they are also continuous, we have that: je i (t)j 0;i e 0;i (tt 0 ) je i (t 0 )j + N X j=1 ckW ij k Z t t 0 0;i e 0;i (ts) ds + N X j=1 c ij k ij k Z t t 0 0;i e 0;i (ts) ds + N X j=1 ckW ij k Z t t 0 0;i e 0;i (ts) ds + N X j=1 c ij k ij k Z t t 0 0;i e 0;i (ts) ds (6.55) In addition, we have that: Z t t 0 0;i e 0;i (ts) ds = 0;i 1 0;i (1e 0;i (tt 0 ) ) 0;i 0;i (6.56) Hence, we have that: je i (t)j 0;i e 0;i (tt 0 ) je i (t 0 )j + N X j=1 ckW ij k 0;i 0;i + N X j=1 c ij k ij k 0;i 0;i + N X j=1 ckW ij k 0;i 0;i + N X j=1 c ij k ij k 0;i 0;i (6.57) 155 Therefore, we have that: lim t!1 sup st je i (s)j N X j=1 ckW ij k 0;i 0;i + N X j=1 c ij k ij k 0;i 0;i N X j=1 c(kW ij k + ij ) (6.58) ProofofTheorem3.2 For the case when all system parametersA i ;B i andA ij are considered to be unknown,K i ;L i andK ij cannot be calculated and they are replaced by their online estimatesK i (t) andL i (t) andK ij (t). In addition, using Fenchel’s inequality, we have that: 2e > j (t)A > m;j N X k=1 B j ~ K jk (t)x k (t jk ) N X k=1 x > k (t jk ) ~ K > jk (t)B > j B j ~ K jk (t)x k (t jk ) +Ne > j (t)A > m;j A m;j e j (t) (6.59) 2 N X k=1 N X q=1 e > k (t)(W jk + jk ) > B j ~ K jq (t)x q (t jq ) N X k=1 Ne > k (t)(W jk + jk ) > (W jk + jk )e k (t) + N X k=1 Nx > k (t jk ) ~ K > jk (t)B > j B j ~ K jk (t)x k (t jk ) (6.60) 2 N X k=1 N X q=1 e > k (t jk ) jk B j ~ K jq (t)x q (t jq ) N X k=1 Ne > k (t jk ) > jk jk e k (t jk ) + N X k=1 Nx > k (t jk ) ~ K > jk (t)B > j B j ~ K jk (t)x k (t jk ) (6.61) 156 N X k=1 N X q=1 x > k (t jk ) ~ K > jk (t)B > j B j ~ K jq (t)x q (t jq ) N X k=1 Nx > k (t jk ) ~ K > jk (t)B > j B j ~ K jk (t)x k (t jk ) (6.62) 2e > j (t)A > m;j B j ~ K j (t)x j (t)e > j (t)A > m;j A m;j e j (t) +x > j (t) ~ K > j (t)B > j B j ~ K j (t)x j (t) (6.63) 2 N X k=1 e > k (t)(W jk + jk ) > B j ~ K j (t)x j (t) N X k=1 e > k (t)(W jk + jk ) > (W jk + jk )e k (t) +Nx > j (t) ~ K > j (t)B > j B j ~ K j (t)x j (t) (6.64) 2 N X k=1 e > k (t jk ) > jk B j ~ K j x j (t) N X k=1 e > k (t jk ) > jk jk e k (t jk ) +Nx > j (t) ~ K > j (t)B > j B j ~ K j (t)x j (t) (6.65) 2 N X k=1 x > j (t) ~ K > j (t)B > j B j ~ K jk (t)x k (t jk ) N X k=1 x > k (t jk ) ~ K > jk (t)B > j B j ~ K jk (t)x k (t jk ) +Nx > j (t) ~ K > j (t)B > j B j ~ K j (t)x j (t) (6.66) Thus, using (6.33) - (6.37), and (6.59) - (6.66), we have that: _ e > j (t) _ e j (t) (2 + 3N)e > j A > m;j A m;j e j (t) + (2 + 3N) N X k=1 e > k (t)(W jk + ik ) > (W jk + jk )e k (t) +(2 + 3N) N X k=1 e > k (t jk ) > jk jk e k (t jk ) +(4 + 6N)kB j k 2 (kK j;max k +kK j k) 2 e > j (t)e j (t) +(4 + 6N) N X k=1 kB j k 2 (kK jk;max k +kK jk k) 2 e > k (t jk )e k (t jk ) +M 1 (6.67) 157 where M 1 = 2e > j (t)A > m;j d j (t) + 2 N X k=1 e > k (t)(W jk + jk ) > d j (t) 2 N X k=1 e > k (t jk ) > jk d j (t) +d > j (t)d j (t) + 2e > j (t)A > m;j B j ~ L j (t)r j (t) + 2 N X k=1 e > k (t)(W jk + jk ) > B j ~ L j (t)r j (t) 2 N X k=1 e > k (t jk ) > jk B j ~ L j (t)r j (t) 2d > j (t)B j ~ K j (t)e j (t) 2d > j (t)B j ~ K j (t)x ref;j (t) +2d > j (t)B j ~ L j (t)r j (t) 2 N X k=1 d > j (t)B j ~ K jk (t)e k (t jk ) 2 N X k=1 d > j (t)B j ~ K jk (t)x ref;k (t jk ) 2e > j (t) ~ K > j B > j B j ~ L j (t)r j (t) 2x > ref;j (t) ~ K > j (t)B > j B j ~ L j (t)r j (t) +r > j (t) ~ L > j (t)B > j B j ~ L j (t)r j (t) 2 N X k=1 r > j (t) ~ L > j (t)B > j B j ~ K jk (t)e k (t jk ) 2 N X k=1 r > j (t) ~ L > j (t)B > j B j ~ K jk (t)x ref;k (t jk ) +(4 + 6N)kB j k 2 (kK j;max k +kK j k) 2 x > ref;j (t)x ref;j (t) +(4 + 6N) N X k=1 kB j k 2 (kK jk;max k +kK jk k) 2 x > ref;k (t jk )x ref;k (t jk ) (6.68) Thus, using (6.28), (6.29), (6.32), (6.38) and (6.67), we have that: _ V N X i=1 N X j=1 e > j (t) e j (t ij ) > ij 2 6 6 4 e j (t) e j (t ij ) 3 7 7 5 +M where ij = 2 6 6 4 ij;1 0 0 ij;2 3 7 7 5 (6.69) 158 with ij;1 = ij min (Q j ) N I + (kW ij kkP i k +kW ji kkP j k)I + ji P j ji > ji P j + ij (2 + 3N)(A > m;j A m;j + 2kB j k 2 (kK j;max k +kK j k) 2 I) + max (G j ) +(2 + 3N) N X k=1 [(W kj + kj ) > (W kj + kj ) ik ] (6.70) and ij;2 = ij min (G j )I + 2(2 + 3N)kB j k 2 (kK ij;max k +kK ij k) 2 I N X k=1 ki + (2 + 3N) N X k=1 [ > kj kj ik ] (6.71) with ij ; ij 2 (0; 1), withM being: M = N X i=1 N X j=1 [(1 ij ) min (Q j ) N e > j (t)e j (t) (1 ij ) min (G j )e j (t ij ) > e j (t ij ) + ij M 1 ] + 2 N X i=1 [Tr[ ~ K > i (t) i _ ~ K i (t)]e > i (t)P i B i ~ K i (t)x i (t) + Tr[ ~ L > i (t) i _ ~ L i (t)] +e > i (t)P i B i ~ L i (t)r i (t)] + 2 N X i=1 N X j=1 [Tr[ ~ K > ij (t) i _ ~ K ij (t)]e > i (t)P i B i ~ K ij (t)x j (t ij )] (6.72) We need to be negative denite. Thus, this corresponds to ij;1 and ij;2 to be negative denite, which, in turn, means that we want: ij min (Q j ) N I + (kW ij kkP i k +kW ji kkP j k)I + ji P j ji > ji P j + ij (2 + 3N)A > m;j A m;j +2 ij (2 + 3N)kB j k 2 (kK j;max k +kK > j k) 2 I + (2 + 3N) N X k=1 [A > kj A kj ik ] +(2 + 3N)[ N X k=1 [ 1 ij > kj kj ik ] + 2 ij kB j k 2 (kK ij;max k +kK ij k) 2 I N X k=1 ki ] 0 (6.73) 159 since min (G) max (G) andW kj + kj =A kj . If ij are negative denite, then for some suciently small> 0 we have that: _ V M =V +V +M (6.74) By assuming thatL i is either positive denite or negative denite and i 1 =L i sgn(l i ), wherel i = 1 if L i is positive denite andl i =1 if it is negative denite, by choosing: _ ~ K i = _ K i = Proj[B > m;i P i e i (t)x > i (t) sgn(l i )] (6.75) _ ~ L i = _ L i = Proj[B > m;i P i e i (t)r > i (t) sgn(l i )] (6.76) _ ~ K ij = _ K ij = Proj[B > m;i P i e i (t)x > j (t ij ) sgn(l i )] (6.77) we have that: 2 N X i=1 [Tr[ ~ K > i (t) i _ ~ K i (t)]e > i (t)P i B i ~ K i (t)x i (t) + Tr[ ~ L > i (t) i _ ~ L i (t)] +e > i (t)P i B i ~ L i (t)r i (t)] +2 N X i=1 N X j=1 [Tr[ ~ K > ij (t) i _ ~ K ij (t)]e > i (t)P i B i ~ K ij (t)x j (t ij )] 0 (6.78) and N X i=1 [Tr[ ~ K > i (t) i ~ K i (t) + ~ L > i (t) i ~ L i (t)] + N X j=1 Tr[ ~ K > ij (t) i ~ K ij (t)]]c (6.79) In addition, for some constantc> 0, because of (6.68), (6.72) and (6.75) - (6.77), we have that: M N X i=1 N X j=1 [(1 ij ) min (Q j ) N e > j (t)e j (t) (1 ij ) min (G j )e j (t ij ) > e j (t ij ) 160 + ij M 1 ] N X i=1 N X j=1 [cje j (t)j 2 cje j (t ij )j 2 +cje j (t)j +cje j (t ij )j +cd 2 j +c] (6.80) Furthermore, from (6.67), we have that: N X i=1 N X j=1 Z t t ij Z t s _ e > j (r) _ e j (r)drds N X i=1 N X j=1 [cje j (t)j 2 +cje j (t ij )j 2 +cje j (t)j +c +cje j (t ij )j] (6.81) From (6.46), (6.47), (6.79), (6.80) and (6.81), we have that: _ V V + N X i=1 N X j=1 [cje j (t)j 2 +cje j (t)jcje j (t ij )j 2 +cje j (t ij )j +c +cd 2 j +cje j (t)j 2 +cje j (t ij )j 2 +cje j (t)j +c +cje j (t ij )j] V + N X i=1 N X j=1 [cje j (t)j 2 +cje j (t)jcje j (t ij )j 2 +cje j (t ij )j +c +cd 2 j ] V + N X i=1 N X j=1 [c(je j (t)jc) 2 c(je j (t ij )jc) 2 +c +cd 2 j ] V + N X i=1 N X j=1 c =V +C (6.82) whereC > 0 is a constant. By using Lemma 3.2.4 of [101], we have that the Lyapunov function is bounded as follows: V (t)e (tt 0 ) V (t 0 ) + Z t t 0 e (ts) Cdse (tt 0 ) V (t 0 ) + C (6.83) Hence, the Lyapunov functionV 2L 1 , which means that the errore i (t)2L 1 . Then, from (??), we have that _ e i (t)2L 1 too. In addition, from (6.74) and (6.80), we have that: _ V N X i=1 N X j=1 [cje j (t)j 2 +cje j (t)jcje j (t ij )j 2 +cje j (t ij )j +c +cd 2 j ] (6.84) 161 Therefore, from Lemma 6.1, we have thate i (t)2S([ P N j=1 (kW ij k+ ij )] 2 ). Lemma 6.1 is given in the end of the Appendix. Lemma6.1. If for someV 2L 1 and some constantc> 0, we have that _ V N X i=1 N X j=1 [cje j (t)j 2 +cje j (t)jcje j (t ij )j 2 +cje j (t ij )j +c +cd 2 j ] (6.85) whered i is an upper bound of (3.9), thene i (t)2S([ P N j=1 (W ij + ij )] 2 ). Proof. _ V N X i=1 N X j=1 [cje j (t)j 2 c(je j (t)jd j ) 2 c(je j (t ij )jd j ) 2 +cd 2 j +c] (6.86) ) _ V N X i=1 N X j=1 [cje j (t)j 2 +cd 2 j +c] (6.87) ) N X i=1 N X j=1 Z t 0 cje j (r)j 2 drV (0)V (t) + N X i=1 N X j=1 [ Z t 0 (cd 2 j +c)dr] (6.88) However, d i (t) = N X i=1 (A ij B i K ij )x ref;j (t) + N X i=1 B i K ij [x ref;j (t)x ref;j (t ij )] (6.89) and, thus, d i N X i=1 ckW ij k + N X i=1 c ij c N X i=1 (kW ij k + ij ) (6.90) Therefore, N X i=1 N X j=1 Z t 0 cje j (r)j 2 dr N X i=1 N X j=1 [ Z t 0 (c[ N X k=1 (kW jk k + jk )] 2 +c)dr] (6.91) Hence, we have thate i (t)2S([ P N j=1 (kW ij k + ij )] 2 ). 162 AppendixC Notation R denotes the eld of real numbers. For any vectorx2 R n ,jx(k)j represents the Euclidean (l 2 ) vector norm inR n at each timek, and for discrete time signalskx k k 2 , P k i=0 ki x > (i)x(i) 1 2 denes the exponentially weightedl 2 norm of a vector functionx(k), where2 (0; 1] is a positive constant andk represents time.x2l 1 means thatkxk 1 = sup k0 jx(k)j exists. ProofofTheorem4.1 By creating the backstepping states (4.10), (4.11) and applying the proposed controller (4.9), we get the following closed loop system8i : 2 6 6 6 6 6 6 6 6 6 6 4 e z i I i y 1 i y 2 i 3 7 7 7 7 7 7 7 7 7 7 5 (k + 1) = 2 6 6 6 6 6 6 6 6 6 6 4 z i h i 1 0 z i 1h i 1 0 0 0 sa i 1 0 0 0 c i 3 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4 e z i I i y 1 i y 2 i 3 7 7 7 7 7 7 7 7 7 7 5 (k) + 2 6 6 6 6 6 6 6 6 6 6 4 d z i d z i d 1 i d 2 i 3 7 7 7 7 7 7 7 7 7 7 5 (k) (6.92) where d z i (k) =b z i Q i (k) _ m sa i C pa (6.93) d 1 i (k) = z 0;i z i +h i d z i (k) + X j2N i z 2;i;j d z j (k) (6.94) d 2 i (k) = h z 0;i z i +h i z 0;i sa i + z 1;i sa 2;i +h i i d z i (k) + X j2N i " z 2;i;j z 0;i +h i z i + z j h j sa i + 1 d z j (k) + z 2;i;j d 1 j (k) # (6.95) 163 From here, it can be seen that we choosej sa i j< 1;j c i j< 1 and z i ;h i to make the block 2 6 6 4 z i h i z i 1h i 3 7 7 5 have eigenvalues inside the unit circle. Then, we have that zone air temperature tracking error goes to 0 exponentially fast for constant disturbancesd z i , i.e.,e z i ! 0, since e z i = z 1 z 2 + (h i 1 z i )z + z i [d z i ] z 1 (z sa i ) (z 2 + (h i 1 z i )z + z i ) [d 1 i ] + z 1 (z c i ) (z sa i ) (z 2 + (h i 1 z i )z + z i ) [d 2 i ] (6.96) ProofofTheorem4.2 By creating the adaptive backstepping states (4.27), (4.28) and applying the proposed controller (4.26), we get the following closed loop system8i : e z i (k + 1) = z i e z i (k) +y 1 i (k) ~ > z i (k) z;I i (k) (6.97) y 1 i (k + 1) = sa i y 1 i (k) +y 2 i (k) + > z i (k + 1) z;I i (k + 1) z 0;i (k) ~ > z i (k) z;I i (k) z 1;i (k) ~ > sa i (k) sa i (k) X j2N i z 2;i;j (k) ~ > z j (k) z;I j (k) (6.98) y 2 i (k + 1) = c i y 2 i (k) sa i > z i (k + 1) z;I i (k + 1) + K > 2 i (k + 1) 2 i (k + 1) + X j2N i z 2;i;j (k) > z;I j (k + 1) z;I j (k + 1) z i (k) ~ > z i (k) z;I i (k) sa i (k) ~ > sa i (k) sa i (k) c i (k) ~ > c i (k) c i (k) X j2N i z i;j (k) ~ > z j (k) z;I j (k) X j2N i sa i;j (k) ~ > sa j (k) sa j (k) X j2N i X p2N j z 2;j;p (k) ~ > zp (k) z;I p (k) (6.99) 164 where ~ z i (k) = z i (k) z i , z i (k) = z i (k) z i (k1) and the following signals that are bounded by the adaptive law: z i (k) = z 0;i (k) sa i z 0;i (k) + z i + z 1;i (k) sa 2;i (k) + sa i z i sa i (k) = z 1;i (k) z 0;i (k) + sa 0;i (k) c i (k) = z 1;i (k) sa 1;i (k) z i;j (k) = z 2;i;j (k) z 0;i (k) sa i + z 0;j (k) + z j sa i;j (k) = z;1 i (k) z 2;i;j (k) (6.100) Next, we consider the following signal: 2 f (k) = 1 + N X i=1 k(Y i ) k1 k 2 2 (6.101) whereY i = [e z i ;y 1 i ;y 2 i ] > and for some > 0 so that 1 (zc i )(zsa i )(zz i ) is analytic injzj > p for some2 (0; 1] for alli. Signal f has the property of bounding from above most signals in the closed-loop plant. It is used for analysis only and its bounding properties are summarized in the following lemma. Lemma6.2. The signal f guarantees that k z;I i k k 2 ( f ) k ; ( z i ) k ( f ) k ; k( sa i ) k k 2 ( f ) k ; ( sa i ) k ( f ) k ; k( c i ) k k 2 ( f ) k ; ( c i ) k ( f ) k Proof. From the denition of f , we have that k(ez i ) k1 k 2 ( f) k ; k(y 1 i ) k1 k 2 ( f) k ; k(y 2 i ) k1 k 2 ( f) k 2L 1 . Using (4.27), we derive thatk(T sa i ) k1 k 2 ck(e z i ) k1 k 2 + ck(y 1 i ) k1 k 2 + c P j2N i k e z j k1 k 2 + c for some constantc> 0, thus k(Tsa i ) k1 k 2 ( f) k 2L 1 . Hence, k( z;I i ) k1 k 2 ( f) k ; (z i ) k ( f) k ; (z i ) k ( f) k 2L 1 . 165 Using (4.28), we have thatk(T c i ) k1 k 2 c +ck(e z i ) k1 k 2 +ck(y 1 i ) k1 k 2 +ck(y 2 i ) k1 k 2 + ck(T sa i ) k1 k 2 +c P j2N i k e z j k1 k 2 +c P j2N i k y 1 j k1 k 2 which means that k(Tc i ) k1 k 2 ( f) k 2L 1 and thus k( sa i ) k1 k 2 ( f) k 2L 1 . This implies that (sa i ) k ( f) k ; (sa i ) k ( f) k 2L 1 . From (4.8) and (4.26), we get thatk(u c i ) k1 k 2 c+ck(e z i ) k1 k 2 +ck(y 2 i ) k1 k 2 +ck(T sa i ) k1 k 2 + ck(T c i ) k1 k 2 +c P j2N i k e z j k1 k 2 +c P j2N i k y 1 j k1 k 2 +c P j2N i k y 2 j k1 k 2 , which means that k(uc i ) k1 k 2 ( f) k 2L 1 and thus k( c i ) k1 k 2 ( f) k ; (c i ) k ( f) k ; (c i ) k ( f) k 2L 1 . Sincek z;I i k k 2 ck z;I i k1 k 2 + ck( sa i ) k1 k 2 + c P j2N i k z;I j k1 k 2 + c from the denition of z;I i (4.7a), we also have that k( z;I i ) k k 2 ( f) k 2L 1 . Sincek( sa i ) k k 2 ck z;I i k1 k 2 + ck( sa i ) k1 k 2 +ck( c i ) k1 k 2 +c from the denition of sa i (4.7c), we also have that k( sa i ) k k 2 ( f) k 2L 1 . Sincek( c i ) k k 2 ck( sa i ) k1 k 2 + ck( c i ) k1 k 2 + ck(u c i ) k k 2 from the denition of c i (4.7e) andk(u c i ) k k 2 c + ck z;I i k k 2 +ck( sa i ) k k 2 +c P j2N i k z;I j k k 2 + c P j2N i k sa j k k 2 + c P j2N i P p2N j k z;I p k k 2 +c P j2N i P p2N j k sap k k 2 from the denition ofu c i (4.8), (4.26) , we also have that k( c i ) k k 2 ( f) k 2L 1 , From (6.97) - (6.99) and given thatj z i j;j sa i j;j c i j < 1, by applying Lemma A.12.33 from [102] and using the proposed adaptive law we have that: k(Y i ) k1 k 2 ck( z i z i f ) k1 k 2 +ck ~ z i f k1 k 2 +ck( sa i sa i f ) k1 k 2 +ck ~ sa i f k1 k 2 +ck( c i c i f ) k1 k 2 +ck ~ c i f k1 k 2 +c X j2N i " k z j z j f k1 k 2 +k ~ z j f k1 k 2 +k sa j sa j f k1 k 2 +k ~ sa j f k1 k 2 # +c X j2N i X p2N j " k zp zp f k1 k 2 +k ~ zp f k1 k 2 # (6.102) 166 where ( i i f ) k = i (k) i (k) f (k). Based on (6.102), we derive the following inequality: 2 f (k)c +c (~ g f ) k1 2 2 (6.103) where ~ g 2 = N X i=1 h j z i z i j 2 +j sa i sa i j 2 +j c i c i j 2 +j ~ z i j 2 +j ~ sa i j 2 +j ~ c i j 2 i (6.104) By applying Lemma A.12.31 from [102] on inequality (6.103) and using the fact that the geometric mean of a series is less than the arithmetic mean, for some nitec> 0 we obtain the following inequality: 2 f (k) 1 +c k1 X p=0 kp ~ g 2 (p) 2 f (p) 1 +c k1 X p=0 2 4 kp ~ g 2 (p) Y p 0. Therefore, again since ~ g2 l 2 and < 1, we conclude that signal we dened in (6.101) is bounded, i.e., f 2l 1 , since the series in the last inequality converges. Hence, all temperature 167 tracking errorse z i 2l 1 8i are bounded too, due to the denition of f . Hence, all signals i ; i ; i 2l 1 are also bounded for alli2N . Using (6.97), (4.5) and (4.6), we have that (e z i ) k =W z i (z)W sa i (z)W c i (z) " z i ~ > z i z;I i k sa i ~ > sa i sa i k c i ~ > c i c i k X j2N i h z i;j ~ > z j z;I j k + sa i;j ~ > sa j sa j k i X j2N i X p2N j z 2;j;p ~ > zp z;I p k # +W z i (z)W sa i (z)W c i (z)z " sa i > z i z;I i k + K > 2 i 2 i k + X j2N i > z j z;I j k # +W z i (z)W sa i (z) " z;0 i k h ~ > z i z;I i i k h z;1 i ~ > sa i sa i k i X j2N i h z 2;i;j k h ~ > z;I j z;I j i k i # +W z i (z)W sa i (z)z h > z i z;I i i k +W z i (z) h ~ > z i z;I i i k (6.107) Thus, the tracking error for each zone is bounded by signals that converge to zero: j (e z i ) k jc N X i=0 " k z i 2 z i k k 2 +ck sa i 2 sa i k k 2 +ck c i 2 c i k k 2 +ck ~ z;I i k k 2 +ck ~ sa i k k 2 +ck ~ c i k k 2 # (6.108) Thus,e z i ! 08i. 168
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Creator
Lymperopoulos, Georgios
(author)
Core Title
Distributed adaptive control with application to heating, ventilation and air-conditioning systems
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Degree Conferral Date
2021-08
Publication Date
07/16/2023
Defense Date
05/21/2021
Publisher
University of Southern California
(original),
University of Southern California. Libraries
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Tag
actuator dynamics,delay systems,distributed control,fault diagnosis,HVAC systems,OAI-PMH Harvest,robust adaptive control,smart buildings
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application/pdf
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Language
English
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Electronically uploaded by the author
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Advisor
Ioannou, Petros (
committee chair
), Flashner, Henryk (
committee member
), Prasanna, Viktor K. (
committee member
)
Creator Email
geo.lympero@gmail.com,lymperop@usc.edu
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https://doi.org/10.25549/usctheses-oUC15610962
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UC15610962
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etd-Lymperopou-9779
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Dissertation
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Lymperopoulos, Georgios
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University of Southern California Dissertations and Theses
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Tags
actuator dynamics
delay systems
distributed control
fault diagnosis
HVAC systems
robust adaptive control
smart buildings