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Fault -tolerant control in complex systems with real-time applications
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Fault -tolerant control in complex systems with real-time applications
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NOTE TO USERS Page(s) not included in the original m anuscript and are unavailable from the author or university. The m anuscript was scanned as received. ii This reproduction is the best copy available. ® UMI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. FAULT-TOLERANT CONTROL IN COMPLEX SYSTEMS WITH REAL-TIME APPLICATIONS by Ali A. Abdullah A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2005 Copyright 2005 Ali A. Abdullah Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3180413 INFORM ATION TO USERS The quality of this reproduction is dependent upon the quality of the copy subm itted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and im proper alignm ent can adversely affect reproduction. In the unlikely event that the author did not send a com plete m anuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI M icroform 3180413 C opyright 2006 by ProQuest Information and Learning Company. All rights reserved. This m icroform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Inform ation and Learning Com pany 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dedication To my parents, brothers, sisters, wife, and daughter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements I would like to express my appreciation to Professor Petros Ioannou, my advisor, for his guidance, support, and patience throughout my doctoral research. His observations and advice helped me to think deeply about research. I thank him for providing me with the opportunity to work with him in many projects. I would also like to thank the following professors: Edmond Jonckheere, Henryk Flashner, Maj Dean Mirmirani, and Michael Safonov for serving on my guidance com mittee and for their critical comments and suggestions. I am grateful to Kuwait University for the award of the fellowship which has sup ported me during my years of research. I greatly appreciate Professor Helen Boussalis, Dr. Kun, Professor K. Rad, Demetrios Florakis, Salvador Fallorina, Mike Lara, and Rama for their support and discussions through the years I spent in SPACE Lab. Also I wish to thank Dr. Haojian Xu, Dr. Hossein Jula, and Dr. Baris Fidan for sharing information and enjoying discussions with them. I would also like to acknowledge the many close and dear friends I made in USA. Last, but not least, I am very grateful to my parents, brothers, sisters, wife, and daughter, for their love, patience, and understanding. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents Dedication iii Acknowledgements iv List of Tables vii List of Figures viii A bstract xi 1 Introduction 1 2 Decentralized Fault-Tolerant Control System Design 7 2.1 Problem S ta te m e n t....................................................................................... 8 2.2 Structure of a Decentralized Fault-Tolerant Control S y stem ................... 9 2.2.1 Decentralized Control D e s ig n ........................................................ 10 2.2.1.1 Decentralized State Feedback Proportional Plus Inte gral Controller ................................................................. 10 2.2.1.2 Decentralized Output Feedback Proportional Plus In tegral C ontroller............................................................. 11 2.2.1.3 Decentralized Direct Adaptive Output Feedback Con troller .............................................................................. 13 2.2.2 Decentralized Fault Detection and Iso latio n ................................. 19 2.2.2.1 Fault M o d ellin g ............................................................... 20 22.2.2 Residual G eneration........................................................ 21 2.2.2.3 Residual T esting............................................................... 24 2.2.2.4 Fault Iso latio n .................................................................. 28 2.2.3 Supervision S c h e m e ........................................................................ 29 3 Segmented Telescope Test Bed 31 3.1 Introduction.................................................................................................... 31 3.2 Structure of the Test B e d ............................................................................. 33 3.2.1 Primary M irro r.................................................................................. 33 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2.2 Secondary M irror............................................................................... 36 3.2.3 Supporting T ru s s ............................................................................... 36 3.2.4 Isolation Platform ........................................................................... 37 3.3 Data Acquisition S y s te m .............................................................................. 38 3.3.1 Edge S e n so rs..................................................................................... 39 3.3.2 Actuators and Am plifiers.................................................................. 40 3.3.3 A/D and D/A Converters.................................................................. 42 3.3.4 Digital Signal Processors.................................................................. 42 3.4 Primary Mirror System M o d e l..................................................................... 43 4 Implementation of Decentralized Fault-Tolerant Control for the Segmented Telescope Test Bed 45 4.1 Performance Requirem ents........................................................................... 45 4.2 Flowchart of the Fault-Tolerant Control Algorithm ................................. 46 4.3 Decentralized C ontrol..................................................................................... 48 4.4 Decentralized Fault Detection and Iso la tio n .............................................. 58 4.5 Decentralized Fault-Tolerant C ontrol........................................................... 62 5 Fault-Tolerant Control for a Class of Nonlinear Systems 68 5.1 Introduction..................................................................................................... 68 5.2 Problem S ta te m e n t........................................................................................ 70 5.3 Structure of a Fault-Tolerant Control System .............................................. 72 5.3.1 Control D e s ig n .................................................................................. 73 5.3.2 Fault D iagnosis.................................................................................. 77 5.3.3 Fault A ccom m odation..................................................................... 81 5.4 Illustration Example ..................................................................................... 89 5.4.1 Control S y stem .................................................................................. 89 5.4.2 Fault D iagnosis.................................................................................. 90 5.4.3 Fault A ccom m odation..................................................................... 92 6 Application of Fault-Tolerant Control for an F-16 Aircraft Longitudinal Mo tion 93 6.1 Nonlinear Equations ..................................................................................... 93 6.2 Linear Parameter Varying M o d e l.................................................................. 97 6.3 Control S y s te m ............................................................................................... 104 6.4 Fault D iag n o sis............................................................................................... I l l 6.5 Fault Accommodation .................................................................................. 118 7 Conclusions and Future Research Directions 123 7.1 Results............................................................................................................... 123 7.2 Future Research D irections........................................................................... 124 Reference List 125 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables 3.1 Maximum singular values............................................................................. 44 4.1 The shape error values for the three types of controllers........................... 50 4.2 Commonly inductive sensor f a u l t s ............................................................. 58 4.3 Sensor failure codes for the subsystem / ' ................................................... 58 4.4 Fault occurrence and isolation tim e ............................................................. 60 4.5 The steady state shape errors (jU m )............................................................. 63 6.1 The constant coefficient values of the F-16 aircraft longitudinal equations 98 6.2 Parameter b o u n d s.......................................................................................... 104 6.3 Fault occurrence and detection t i m e .......................................................... 112 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures 2.1 General scheme of the DFTC system....................................................... 10 2.2 The structure of the DDAOF control system........................................... 13 2.3 The z'th local unit of the DSFDI scheme................................................... 20 3.1 An example of deployed NGST in space................................................. 32 3.2 The structure of the segmented telescope test bed at SPACE Lab......... 34 3.3 Test bed dimensions.................................................................................... 35 3.4 The location of actuators and edge sensors under the primary mirror. . . 36 3.5 Secondary m irro r....................................................................................... 37 3.6 Isolation platform........................................................................................ 38 3.7 Data flow through the system.................................................................... 39 3.8 The structure of the inductive sensor KDM-8200................................... 40 3.9 Actuator model ML2-3002-10 0 L B ......................................................... 41 4.1 The flowchart of the DFTC algorithm...................................................... 47 4.2 Real-time results of the mirror no. 1, for three cases: (a) with DSFPI control; (b) with DOFPI control; (c) with DDAOF control................... 51 4.3 Real-time results of the mirror no. 2, for three cases: (a) with DSFPI control; (b) with DOFPI control; (c) with DDAOF control.................. 52 4.4 Real-time results of the mirror no. 3, for three cases: (a) with DSFPI control; (b) with DOFPI control; (c) with DDAOF control.................. 53 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 Real-time results of the mirror no. 4, for three cases: (a) with DSFPI control; (b) with DOFPI control; (c) with DDAOF control........ 54 4.6 Real-time results of the mirror no. 5, for three cases: (a) with DSFPI control; (b) with DOFPI control; (c) with DDAOF control........ 55 4.7 Real-time results of the mirror no. 6, for three cases: (a) with DSFPI control; (b) with DOFPI control; (c) with DDAOF control........ 56 4.8 Shape errors, for three cases: (a) with DSFPI control; (b) with DOFPI control; (c) with DDAOF control................................................... 57 4.9 The Residual signals for all 6 cases............................................... 61 4.10 Sensor no. 1 is failed (case no. 3): without using DFTC (left column); with using DFTC (right column)................................................... 64 4.11 Sensor no. 18 is failed (case no. 5): without using DFTC (left column); with using DFTC (right column).................................................... 65 4.12 Sensor no. 4 is failed (case no. 4): without using DFTC (left column); with using DFTC (right column)................................................... 66 4.13 Sensor no. 3 and 8 are failed in the presence of external disturbances (case no. 6): without using DFTC. (left column); with using DFTC (right column)............................................................................................. 67 5.1 Structure of fault-tolerant control system..................................... 72 5.2 Fault diagnostic architecture.......................................................... 77 5.3 Simulation results of nominal control system.............................. 90 5.4 Simulation results of fault diagnostic scheme.............................. 91 5.5 Simulation results with and without fault accommodation........ 92 6.1 Control surfaces and three body axes of rotation of an F-16 aircraft. . . 95 6.2 Input signals (constant) and output responses of nonlinear and LPV mod els at starting speed 400 ft/sec and altitude 5000 ft..................... 100 6.3 Input signals (constant) and output responses of nonlinear and LPV mod els at starting speed 600 ft/sec and altitude 40000 ft................... 101 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4 Input signals (rectangular) and output responses of nonlinear and LPV models at starting speed 400 ft/sec and altitude 5000 ft.............................. 102 6.5 Input signals (rectangular) and output responses of nonlinear and LPV models at starting speed 600 ft/sec and altitude 40000 ft........................... 103 6.6 F-16 aircraft longitudinal control system structure....................................... 106 6.7 Simulation results of nominal control system (scenario no. 1)............... 108 6.8 Simulation results of nominal control system (scenario no. 2)............... 109 6.9 Simulation results of nominal control system (scenario no. 3)............... 110 6.10 Simulation results of fault diagnostic scheme (case no. 1)......................... 113 6.11 Simulation results of fault diagnostic scheme (case no. 2)......................... 114 6.12 Simulation results of fault diagnostic scheme (case no. 3)......................... 115 6.13 Simulation results of fault diagnostic scheme (case no. 4)......................... 116 6.14 Simulation results of fault diagnostic scheme (case no. 5)......................... 117 6.15 Simulation results with and without fault accommodation (case no. 1). . 120 6.16 Simulation results with and without fault accommodation (case no. 2). . 121 6.17 Simulation results with and without fault accommodation (case no. 3). . 122 x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract Designing and implementing a fault-tolerant control (FTC) system for large scale sys tems pose a great challenge due to the high dimensionality, multiple inputs and outputs, and processor limitations. In this study, the problem of designing and implementing a decentralized FTC system is considered for large scale systems. The decentralized FTC system consists of three schemes: decentralized nominal controller, decentralized fault detection and isolation, and decentralized supervision scheme. The proposed approach overcomes the design and implementation complexities by dividing the total task of the FTC system into a number of small tasks. These small tasks can be easily implemented by running a number of commercial processors in parallel. The proposed design is ap plied to a typical large scale system: a segmented telescope test bed which consists of 6 segments giving an overall high order system. The decentralized FTC system consists of 6 local units, each one responsible for controlling one segment, detecting and isolat ing faults, and reconfiguring the controller for fault accommodation. The algorithm of the 6 local units is implemented by running 3 processors in parallel. Real time results demonstrate the capability of the proposed decentralized FTC system to tolerate sensor failure in large scale systems such as in the segmented telescope test bed. In addition, xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. since designing an FTC system to recover the original system performance is impractical in many situations, due to the new physical constraints caused by the fault, designing an FTC system with the ability to comply with the new physical constraints is considered for a class of nonlinear systems subject to actuator saturation fault. The approach is based on the conceptual tools of linear matrix inequalities and on-line fault model estimation. The FTC system consists of nominal control, fault diagnostic, and fault accommodation schemes. These schemes are designed to: 1) achieve stability and tracking requirements; 2) detect, isolate, and estimate a fault; 3) reduce the fault effect on the system. A nonlin ear model of an F-16 aircraft longitudinal motion is used to demonstrate the effectiveness of the proposed FTC system. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction The design of fault-tolerant control systems is one of the most important issues for safety- critical systems such as nuclear power stations, chemical plants, aircraft, and spacecraft. Fault-tolerant control systems are used to improve system reliability, maintainability, and survivability by completing a task after failure, increasing maintenance time, and preventing damages, respectively. Until recently, most controllers have been designed to achieve the desired performances in the presence of modelling errors, noise, and dis turbances. Such controllers are known as robust controllers. These robust controllers include, for example, H„ and adaptive controllers. In general, no robust controllers can guarantee closed-loop system stability or performance in the presence of faults. The inability of the robust controllers to tolerate faults has motivated many researchers to design controllers with a feature of tolerance for specific types of faults. These con trollers are known as reliable controllers or passive fault-tolerant controllers in which fault tolerance is achieved using a single controller with fixed structure and parameters. 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Methods of passive fault-tolerant control system design include for example: LQG con trol [73], observer-based control with Ha „ norm bound [57], decentralized observer-based control with H„ norm bound [68], and H„ control [72], Although the structure of the pas sive fault-tolerant control systems is simple, it has many disadvantages, for instance: i) it is limited to specific types of faults like outage, bias, and gain changing; ii) it can toler ate only a preselected set of sensors/actuators; and iii) it degrades system performance to achieve fault tolerance. Therefore, many researchers have considered another approach of designing the fault-tolerant control systems with the ability to tolerate various types of faults at the same time while maintaining an acceptable level of performance. These fault-tolerant control systems are known as active fault-tolerant control systems. Their structures and/or parameters can be changed to accommodate failures. The key ele ment in achieving active fault tolerance is to design redundant systems. Redundancies can be provided by: equipment (physical redundancies), mathematical functions (ana lytical redundancies), or both equipment and mathematical functions. For the concepts and methods in fault-tolerant control, one can refer to [11] and for a good survey in fault-tolerant control systems see [47]. One method of active fault tolerance using the physical redundancy approach to achieve sensor fault tolerance is to operate a number of redundant sensors in parallel with the system sensors. For fault isolation, voters are used to determine the faulty sensor location. For sensor fault tolerance, the faulty sensor is replaced with a healthy one. Generally speaking, n sensors measuring a quantity can be used to tolerate (« — l)/2 failed sensors where n is an odd number. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. One practical example of using redundant sensors for sensor fault tolerance is an electro mechanical brake pedal system [32], Four position sensors are used to sense the move ment of a brake pedal and two voters are used to detect a sensor fault. For fault tolerance, the faulty sensor is removed from the control loop and replaced with a healthy one. Al though the physical redundancy approach is straightforward and most likely performs better than the analytical redundancy approach, especially for reducing the risk of false alarms, cost and space are the main limitations of the physical redundancy approach. The analytical redundancy approach seems to be an attractive alternative to achieve fault tol erance where redundancies are provided by the mathematical functions. Following this approach, one can reduce the cost of additional equipments and overcome the problem of space limitation, though at the expense of computational complexity and performance degradation. In the last decade, the subject of active fault-tolerant control system has drawn much attention of researchers; see for example [12], [21], [52], [53], [61], and [62], In the analytical redundancy approach, many methods have been proposed, includ ing: • Multiple observers with fault detection and isolation (FDI) scheme [16]: the in tention is to generate a number of estimated state vectors. One estimated state vector not effected by a sensor fault is used for a state feedback controller, assum ing that the observability condition is applied. In this case, n banks of observers are used to estimate n redundant state vectors where n is the number of sensors. To determine the location of a faulty sensor, each bank of observers is driven by one 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sensor output and all actuator inputs. If one of the sensors fails, the location of the faulty sensor is identified by a detection logic that operates on the n state vectors. Then, one healthy redundant state vector is used for the feedback controller. This approach may be unfeasible for a large order system with many sensors because n redundant state vectors must be generated on-line. • Multiple observers with decision by majority [65]: this approach also uses a num ber of observers, but without the FDI scheme. For a state feedback control system, the system outputs are divided into three sets, where each output belongs to only one set. Three banks of observers are used to estimate three redundant state vec tors, where each bank is driven by all actuator inputs and one set of sensor outputs. Based on a decision by the majority, an extension of a scalar case where a middle value of a scalar function is adopted at time t, one estimated state vector is selected at time t for the state feedback controller. In general, this method cannot identify the location of the faulty sensor. Also, the possibility to adopt an estimated state vector generated by a faulty sensor still exists. • Predesigned controllers with FDI scheme ([5], [33], and [41]): the idea is to de sign a number of controllers with a switching logic used to adopt a right controller in the presence of a fault. In this case, a number of controllers are off-line designed for a number of possible faults that could happen in the system. When one fault oc curs, an FDI scheme identifies the location of a fault and a switching logic adapts one controller designed for that instance of fault. For example in [33], throttles 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. are used to compensate for the loss of the control surfaces where collective throttle inputs are used to control the longitudinal motion of an airplane and differential throttle inputs are used to control the lateral motion of an airplane. • Neural networks ([13], [42], and [70]): this design idea is similar to the multiple observers with FDI scheme but with neural network used as a model tool. For example in [70], neural networks are used to tolerate sensor faults in an outdoor ventilation control unit. Three neural networks are trained to predict the measure ments of the three airflow sensors used in a control unit. The difference between a sensor reading and its prediction reading from a neural network is used to de tect a sensor fault and identify its location. For sensor fault tolerance, the faulty sensor reading is replaced with an estimated output from a neural network. Many questions are raised in this approach, including: the selection of the structure of the neural network; the complexity of the neural network structure; and the time needed for estimation. • Fault estimation and control reconfiguration ([44], [48], [49], and [50]): in this method, an FDI scheme is used to detect and locate a fault. Then, an estima tion scheme is used to estimate the fault magnitude and reconfigure the nominal controller for fault accommodation. In [44], the fault magnitude is estimated by representing the fault as one of the closed-loop system states. Then, the state space representation of the closed-loop system with fault is used to determine the fault magnitude. Another method of fault estimation is the employment of a function 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. approximator with adjustable parameters [48]. After fault estimation, a nominal control law is modified by adding a new term (function of fault magnitude) to the original control law for fault accommodation. Most of the above methods of active fault-tolerant control systems are required FDI schemes. The main task of the FDI scheme is to detect a fault and identify its location. Many fault detection and isolation design methods (mostly designed for sensor/actuator faults) have been proposed in the last three decades. Examples of well-known methods are: state estimation ([17], [24], and [71]), parity equations ([27], [28], and [39]), pa rameter estimation [30], and neural network approach ([8], [46], [55], and [60]). For a comprehensive survey of FDI, one can refer to [31], and for comparison between dif ferent methods of FDI, [9]. None of the above FDI methods deal with the problem of modelling errors and disturbances. Much work has been done to enhance the perfor mance of FDI scheme in the presence of modelling errors, disturbances, and noise. The approach has generally attempted to increase the sensitivity of fault indication signals, called residuals, for specific faults need to be detected at the same time to reduce the sensitivity of fault indication signals in the absence of these faults. Methods of robust FDI include, for example: robust observer ([20], [45], and [54]), robust parity equations ([15], [19], and [38]), and frequency-domain optimization ([18], [23], and [51]). For more details regarding methods of robust FDI, see [14] and [22], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Decentralized Fault-Tolerant Control System Design A decentralized fault-tolerant control (DFTC) system is designed and analyzed for a class of large scale systems. The objective of the DFTC system is to meet the perfor mance requirements under normal and failure situations. The objective is accomplished by integrating three schemes: controller; fault detection and isolation; and reconfigurable controllers. These schemes have a decentralized structure for the reason that the imple mentation of a centralized fault-tolerant control system for large scale systems is usually not feasible for existing digital signal processors. However, with a decentralized ap proach the feasibility of real time implementation can be achieved by dividing the total task of the fault-tolerant control system into a number of small tasks implemented by a number of processors running in parallel. The proposed design is implemented in real time on a segmented telescope test bed and demonstrated to meet the performance re quirements in the presence of sensor failures. 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.1 Problem Statement The design and implementation of fault-tolerant control systems for large scale systems pose a great challenge due to the high system dimensionality, multiple inputs and outputs, and processor limitations. In this study, the following problems are considered. Problem 2.1 Design a fault-tolerant control system fo r a class o f large scale systems modelled as: y = G{s)u (2 .1) where G{s) = ( \ G ii(s) ... Gin(s) ^ Gm(s) Gnn{s) y is the overall transfer function matrix, y = (y\ ,... ,y f,... ,yJj)T € R O T is the measured output vector, and u — (w [,..., u f ,..., u^)T G R” is the input vector. Problem 2.2 Implement the proposed fault-tolerant control system to a typical large scale system: a segmented telescope test bed. 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Most of the existing methods for designing fault-tolerant control system cannot be ap plied directly to large scale systems without further consideration. To establish an effec tive and computationally tractable method for fault tolerance for large scale systems, a decentralized approach is suggested here. 2.2 Structure of a Decentralized Fault-Tolerant Control System Designing a DFTC system is carried out by dividing a large scale system into a number of subsystems, where each one consists of a set of actuators and sensors. Next, a local fault-tolerant control system is designed for each of the subsystems to control the corre sponding subsystem under normal and failure situations. In this approach, the total task of a fault-tolerant control system is divided into a number of small tasks. These small tasks can be easily implemented by running a number of processors in parallel. The proposed DFTC system (Fig. 2.1) consists of three schemes: decentralized controller, decentralized fault detection and isolation, and decentralized supervision scheme. Each of the three decentralized schemes consists of a number of local units, equal to the num ber of subsystems. The local control unit is responsible for locally controlling a part of a system under normal operation. The local fault detection and isolation unit is locally responsible for fault detection and isolation. The local supervision unit is locally respon sible for integrating the diagnostic information, provided by the local fault detection and 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference signals |U>calno^] J^calno^^ [^calnoNj Controller | Local no. 1 1 l^ ^ c a ln o jJ | Local no. N| Sensor fault information l^cjljio ^ |^)painaT] I Local n o .^l \ctua[or inpujs Actuators h Large scale system Sensors iL ITT5 Sensor outputs Figure 2.1: General scheme of the DFTC system. isolation unit, with reconfigurable controllers for fault accommodating. In the following sections, the design of the three parts of the DFTC system is presented. 2.2.1 Decentralized Control Design Many types of decentralized controllers have been proposed in the literature. Three types of decentralized controllers are investigated here and used to control the segmented telescope test bed. 2.2.1.1 Decentralized State Feedback Proportional Plus Integral Controller The decentralized state feedback proportional plus integral controller is generated by: Local state estimator: x}(() = A,Xi(t) +B,u,(t) +F}(y,- — C,Xj) _ t Local controller: w,(f) = —KiXi{t) —Li fyi{t)dt i= o 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Xj is the estimate of the /th local state vector Xj, and Gu(s) — Q(sl — Aj)~xBi. The gain matrices Fj, Kj and I , are obtained following the standard LQR plus integral design procedure [1]. The controller gains are varied using different weights in the LQR cost until a desired closed-loop response is obtained. 2.2.1.2 Decentralized O utput Feedback Proportional Plus Integral Controller The decentralized output feedback proportional plus integral (DOFPI) controller is gen erated by: t U j ( t ) = -Kiyi{t)-U J yi{t)dt i = \ , . . . , N ( 2 .2 ) o where the gain matrices Kj and Li are computed as follow. Consider the following state space representation of the system (2.1): x = Ax + Bu (2.3) y = Cx (2.4) where x G R” is the system state vector, and the matrices A, B, and C are of appropriate dimensions. The time derivative of the state equation (2.3) is given by: v = Avv + Bvii y = Cvv 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where v = (xT,yT)T, Av = ( \ A 0 , Bv = (Bt , 0)t , and Cv = (0 ,1). The control law \ c V (2.2) can be written in a compact form as u = - F v with the following structure of F: F = ( . KiQ L\ 0 K2C2 0 L2 0 0 0 Ln where C = (C f ,... ,Cjj)T. The Lyapunov inequality: (2.5) P(AV - BV F) + (Av - BvF)t P < 0 (2.6) is used to compute the constant gain matrix F. The problem is to find a positive definite matrix P and the controller gain F satisfying (2.6). This problem is not easy to solve because the inequality (2.6) is not linear in terms of P and F. However, if we fix P > 0 then the inequality is converted to a linear matrix inequality (LMI), which can be easily solved for F. The design procedure for the DOFPI controller is summarized in the following steps: 1. Add and subtract each of the terms PAe to the first part and A^P to the second part of the inequality (2.6) to get P(AV + Ae — Ae — BV F ) + (Av + Ae —Ae — BV F )TP < 0. This modification is required for the next step because Av has a number of 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. eigenvalues at zero. The matrix Ae is selected such that all eigenvalues of Av+Ae ( \ 0 0 have a negative real part. For example one can select Ae = , where e is a negative scalar 0 el 2. Find the positive definite matrix P satisfying: P(AV + Ae) + (Av + Ae)TP = -Q, where Q is any positive definite matrix. For example select Q — I 3. Solve P{— Ae — BV F ) + (—Ae — BV F)TP < Q iovF using the LMI-Toolbox [25], 2.2.1.3 Decentralized Direct Adaptive O utput Feedback Controller The structure of the decentralized direct adaptive output feedback (DDAOF) control system is shown in Fig. 2.2. In this figure, G(s) is the plant transfer function ma- G(s) Reference u !» Actuators Large scale ^ < MSh i i i system Sensors i y DDAOF control laws: Figure 2.2: The structure of the DDAOF control system. trix, and Gy(s) = diag{Gfi(s),...,Gf(s),...,GfN(s)} — C(sl — A) X B + D with A = diag{A\,...,Ai,...,AN), B = diag{B\,...,Bu ...,BN}, C = diag{Cu ...,Ci,...,CN}, and D = d i a g { D \ • ,£>n } where Aj, Bj, Cj, and Dj are constant design matri ces of appropriate dimensions. The DDAOF control laws are given by Theorem 2.1 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Theorem 2.1 I f there exist a scalar S andfilter Gf(s) such that Gm(s) — G(s)Gf{s) + 8 1 is strictly positive real (SPR), then the following DDAOF control laws: U j = Gf (s)ui i = l,2 ,...,N ~ / \ fe(0 A'(0 / \ ui{t) - “ l + 8(ki(t) + li(t))yi{t) tyt) = a iyf (t)yi(t) hit) = fryf (t)yi(t) where a,- and j3, are design positive constants, can stabilize the system o f Fig. 2.2 and force the output vector y to zero exponentially fast. Lemma 2.1 (Kalman-Yakubovich-Popov Lemma) [35] is used to prove Theorem 2.1. Lemma 2.1 The transferfunction matrix Gm{s) = C {si—A)~l B + 5 1 is SPR if and only if there exist matrices P = P T > 0, L, and W , and a constant e > 0 such that: PA+ArP = - L rL - eP (2.7) P$ = Ct - L t W (2.8) WTW = 2 81 (2.9) 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Proof o f Theorem 2.1: Consider the following state space representation of Gm(s) — G{s)G/(s) + 51 where G (s)= C {sI-A)~lB: x = Ax+ Bu y = C x + 8 u = y + 8 u (2 .10) (2 .11) where x = (x f , xT)T, A Use / - \ A BC , B = (DtBt , Bt )t , and C = (C, 0). V(£) = ^xrPx (2.12) as a Lyapunov function candidate. The derivative of (2.12) along the trajectories of the system (2.10) is given by v(x) = X - x TPic + \ k TPx (2.13) Substituting (2.10) into (2.13), we obtain F(jc) = ^x1 (PA+A1 P )x+ xT PBu (2.14) 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Using (2.7) and (2.8) from Lemma 2.1, yields V(x) = - \ xtLtLx - \ extPx + xt {Ct — LtW)u = — jxtLtLx — j £xtPx + xt (Ct — LtW)u + 8 utu — 8uTii = —jxtLtLx — j £xtPx + (Cx+ 8 u)tu — 8 ut u — xtLtW u Using (2.9) from Lemma 2.1 and (2.11), we obtain V{x) = - ^ extPx - X -{L x + W u)t{Lx + W u) + / u (2.15) N Substituting y T u = y \u \ + . . . + y j uj + ... + y T NU N = X y f «/ int0 (2.15), we have 1=1 1 1 N V(x) = - - E x TP x - - ( L x + Wu)T(Lx+Wu) + '^ iyJiii (2.16) 2 2 i= i Using the control laws given in Theorem 2.1, we have (i + 5 (* ,(0 + M 0 ))« /(0 = -(* /(') Then *2/(0=- ( M + m i y M + s m ) = (2 .17) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Substituting (2.17) into (2.16), we have V (x) = -]-exTPx -)-(L x + W u)T {Lx+W u) - £ ( £ , - + tyyfyt L 1 ,-= i Since (kj + U)y]yi > 0, we have V(x) < -] ^ e x TPx which implies that p (/)|| is bounded and converges to zero exponentially fast. Since x = [xT^ ] T we also have p (f)|| and p (f)|| approaching zero exponentially fast. Since y = Cx we have that \\y(t)\\ approaching zero exponentially fast. The above analysis also implies that x, x, and y are square integrable i.e. they are L2-signals. From (2.7) t we have /,(?) = ft / yj{i)yi(x)dx, since y, £ Lj it follows that /,-(/) is bounded and o o o ______________ _ lim = ft f y [ (r)yi(r) dx = /,• < OO. Hence, all signals are bounded in addition o to ||y,-(/)|| converging to zero exponentially fast □. Lemma 2.2 shows that there always exists a scalar 8 such that Gm(s) is SPR. Lemma 2.2 For any n x n proper transfer function matrix Gm(s) = C(sl — A) lB with all elements analytic in the closed right-half complex plane, there exists a scalar 8 such that Gm(s) = Gm{s) + 81 is SPR. Definition 2.1 [66] is used for the proof of Lemma 2.2. Definition 2.1 Let Gm(s) be an n x n transfer function matrix and also let Gh (jw) = Gm{Jw) + G^i-jw). Then Gm(s) is SPR if: 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1. Gm(s) is analytic in the closed right-half complex plane 2. Gf,(jw) > 0 Vw € (— °°,°°) 3. GhW > 0 4. limn,_,oo vP’Ghijw) > 0 if Gh{°°) is singular. Proof o f Lemma 2.2\ Since we assumed that all elements of Gm(s) are analytic in the closed right-half complex plane, therefore condition 1 of definition 2.1 is satisfied. Now Gm(s) is SPR if Gf,(jw) is positive definite for any real w or the eigenvalues of G/,(yw) are positive for any real w (including w = °°). The characteristic equation of Gh(jw) at each w is: A ( ^ w ) = \ K l - G h{jw)\ = \lw I - { G m{jw) + GT m{-jw) + 28l)\ = \{?iw- 2 8 ) I - { G m(jw)-{-Glt{-jw))\ = 0 where A *, is the eigenvalue of Gh(jw) at each w. Let A m in be the minimum eigenvalue of Gm(jw) + GT m{-jw) Vw G ( — oo? o o ), then for 8 > — jAmm we have A^ > 0 . There fore condition 2 of definition 2.1 is satisfied. For condition 3 of definition 2.1 we have Gh(°°) = 281 > 0 for 5 > 0 . Therefore condition 3 of definition 2.1 is satisfied. Choos ing 8 > max{0, — jAmm} we satisfied conditions 2 and 3 as well as condition 4 □. The following Lemma 2.3 [64] can be used to find the constant matrices of the filter and 8 analytically. 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lemma 2.3 The transfer function matrix Gm(s) = C(sI—A) lB + 8 l is SPR if and only if there exists a positive definite matrix H such that: . * V " T * A A < 7 . ^ AH + H AT £ - H C t \ BT - C H -281 <0 (2.18) The matrix inequality (2.18) is not linear in terms of H and the filter matrices and there fore it is not easy to solve. However, the matrix inequality (2.18) can be converted into a linear matrix inequality by fixing some filter matrices. For example, choose A = a l and C = I, where a is some negative value. In this case the matrix inequality (2.18) can be easily solved for the other design matrices and 8 using the LMI-Toolbox [25]. 2.2.2 Decentralized Fault Detection and Isolation The decentralized fault detection and isolation scheme is designed for a large scale sys tem to indicate any possible fault that occurs locally and to identify its location. In the test bed, the sensor fault is considered. To detect sensor faults in a large scale system, the decentralized sensor fault detection and isolation (DSFDI) scheme is designed with N local units where each local unit is responsible for sensor fault detection and isolation in the corresponding subsystem. Fig. 2.3 shows the z'th local unit of the DSFDI scheme. To identify the location of a faulty sensor, each local unit is designed with a number of residual generators where each generator is driven by all local inputs and one local output. The design steps of the DSFDI scheme are: 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Local inputs Sensor fault * location Res. generator no. 1 Res. generator no. j Res. generator no. ftij i i Subsystem no. i Figure 2.3: The z'th local unit of the DSFDI scheme. 2.2.2.1 Fault Modelling Many types of sensor faults can occur. These faults are usually described based on the fault time behavior and magnitude, for example: sudden, slowly developing, intermit tent, hard, or soft faults. Generally, sudden and hard faults must be detected early to prevent system damages, while for slowly developing faults like drift faults, the detec tion time is longer because of the difficulty of detecting these types of faults early. On the other hand, for soft faults like a small change in a sensor gain, fault detection may be unnecessary, since the nominal control system can usually tolerate this kind of fault. The effects of these different types of sensor faults on a system can be modelled as unknown vector (function of time) adding to the system output vector. To find a suitable model for designing the DSFDI scheme, the equation (2.1) is written as: */(s) = G,/(*)«,(*)+ £ Gu(s)uj(s) I'= 1 , 2 , . . . , A T (2.19) j=lj& 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where yt G R'”i is the output vector of the subsystem i, and « ,• G M”' is the input vector of the subsystem i. Gu(s) € Cm,xni is the transfer function matrix of the subsystem i, and Gij(s) G Cm'xnJ is the interconnection transfer function matrix between subsystems i and j. The state space representation of (2.19) with the interconnections shown separately is given by: *,(/) = AjXj(t)+BiUj(t) + dj(t) (2.20) yi(t) = CiXi{t) (2.21) where Gu(s) = C i(sl-A j)~ lBj, and Gij(s)uj(s) = Ci(sl — Ai)~xdi(s). The ef fects of sensor faults on the system is modelled as unknown function adding to the system output, therefore the state space representation (2.20)-(2.21) is written as: Xi(t) = AiXi(t)+BiUi(t)+di(t) (2.22) y,(t) - CiXi{t)+f{t) (2.23) where fi(t) G Rm i is the /th unknown sensor fault vector affecting the local sensor read ings of the subsystem i. 2.2.22 Residual Generation For each subsystem, m, residual generators where each one is driven by all local inputs and one local output is used to generate m ,- fault indication signals called residuals. These 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. residuals are used to indicate any possible sensor fault. Therefore, the residuals should be close to zero when there is no sensor failure and different than zero in the presence of sensor failure. Each of the residual generators is designed using parity equations method ([15], [19], and [38]). To design a residual generator, the following discrete version of the state space equations (2.22)-(2.23) is considered: Xi(k + 1) = AiXi(k) + BiUi(k) + d i( k ) yij{k) = CijXj(k) + f i j { k ) j = 1 , 2 , . . . , mi where yy is the y'th element of the sensor output vector y,-, and fjj is the yth element of the sensor fault vector fj. From this, the m + 1 samples of the output yy can be written as: Yjj = LijXj(k — m ) MijUj + N jjD i + F y (2.24) where ( \ { _ \ y<j(k-m) < j yij(k — m + 1) QjAj YtJ = i Lij — ^ yu(k) j C-Am \ > J i / 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Mij = CijBi 0 0 Ui ( \ U i ( k — m ) U j ( k — m + 1) NU = Fa = Cu 0 A .> - 1 C-Am~ 2 \ L' r i y i ( . \ M k~ m) f ij( k - m + 1) \ y y «»(*) y 0 0 0 / ° y / ,A = d j ( k — m) d i ( k — m + 1) y di(k) j , and The yth residual of subsystem i is generated by: r i j ( k ) = V,j (Yjj — MijUj) (2.25) where F/y € Rm+1 is a design vector and m is a parity equations order to be selected. The vector F/y is to be computed such that the influence of subsystem interactions on the residual r,y is minimized at the same time that the influence of sensor failure on the 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. residual r\j is maximized. To find such a vector, the internal structure of the equation (2.25) is found by substituting (2.24) into (2.25) to get: rtJ(k) = V Tj (LijX{k - m ) + NjjDj + Fu) = V? {HjjE, + F,j) where Hy = (I,y, Ny), and Et = (x(k - m)T, Df)T. To minimize the effect of Ej and at the same time to maximize the effect of Fjj on the residual r,y, the performance index I I V?H - ■ || ^ Jij — ii'J T i 2 is minimized for Vjj. The singular value decomposition technique [29] I I > j II2 is used to find such an optimal vector. To find Vjj, write Hjj as Ujj'LjjMj- with UjjUfj = U-jUjj = I, MjjMjj = MjjMjj = I, and with singular values of Hjj on the diagonal (descending order). Then Vjj is the last column of Ujj. 2 2 .2.3 Residual Testing Each residual is tested for the likelihood of sensor fault. A decision about the existence of a sensor failure is made by comparing an evaluation function rjj(k) of the residual to a preselected threshold value 7 } y according to the following logic: Tjj (k) < Tjj = > no sensor fa u lt ?jj{k) > Tjj => sensor fa u lt where Tjj(k) = [yq^~ L k ntk rfj(n)r‘j(n)}^2 * s the average energy of the residual over a detection window a. If the evaluation function value exceeds the threshold value Tjj, 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a sensor failure will be considered in the system. The threshold values can be selected from experiments to reduce false alarms coming from noise, modelling errors, subsystem interactions, and disturbances. A false alarm due to the disturbance Ei will not appear in the detection scheme if Lemma 2.4 holds. Lemma 2.4 Given that || e//(w) || < Ae where Ej(n) = (ef^ (« ),..., e jfn ),..., e^m+2(«))r £ R(m +2)"*;. Let Xm ax and Tjj denote a maximum eigenvalue and a threshold value, respectively. If: T-. Ae< [^(H fjV ijV jH ij^m + l ) } ^ then a false alarm due to the disturbance Ei will not appear in the detection scheme. Proof: False alarm will not appear in a detection scheme if the evaluation function of the residual is less than the threshold value in the absence of faults, i.e., r ij( % r o < TU where 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since E f^H jjV ijV ljH ijE in ) < H ^E j(n )E t(n) we have n = k Given that any element of £,(w) € R(m+2K- denoted by e^(«) e R"x < ' is satisfying ||e,y(«) || < Ae, then we have: nj(k)|Fy_o < [ + a )(w + 2)Ag] ^ = [Knax{Hjj VijVjjHij) {m + 2)]’/ 2Ae (2.26) Now o < Tij i f [Xmax{Hjj VijV j'JHij){m + 2)\^2Ee < TtJ T O P A ^ I I [Xmax(H[jVjjV? Hjj)(m + 2)]1 /2 One can increase the upper bound on Ae by selecting a large threshold value Tjj, however this will increase the size of faults that can be detected. 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lemma 2.5 Let Vjj — (vjjfl, v,-^/,..., vy^) where vy G R Suppose that all elements o f the fault vector F y G R(m+1) are equal and denoted by f y G R and let |.| denotes an absolute value. If: \ M > \v ij,0 + vij,l + ■ • • + Vjj,m I then the fault f y is detectable by using the threshold value Ty. Proof: To detect a fault, the evaluation function of the residual should be greater than or equal the threshold value in the presence of faults, i.e., fij{k) > Tjj where i £+a n j = [ - — I {VljHijEM) + VfjFij(n))T(VfjHijEj(n) + V^Fu (n))]^2 ' n=k Since i k+a i k+a > Ittt I (t'ir l FiA"))T(< 'ir ,FiJ m '/2 - [ - n — I (V ^jE ^Y [V hH uE M ]'!2 1 a n=k L ^ u n=k ■v“ ry(*)i£/-o r‘ j(% r o where 1 k+a i k+a n i = [-rrr S(^fiyW )r(^W )]l/2 = [T -r S ( ^ ( » ) ) 2 ]1 /2 1 + a n=k 1 + “ n=/t 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Given that Fij{n) =fj[l, 1,..., l]r and V? = by.o.v/y,/ ,. .. , v/y>] we have: r i j ( k ) |£j=0 - \ f i j 1 1 v /y,0 + v ij,l + • • • + Vjj^m Now nj(k) > Tij i f or or From (2.26), the inequality (2.27) holds if: ~ > TU + { ^ { H j j V i j V l j H ^ m + 2)]1 /2Ae J \Vij,0 + VijJ + ... + Vjj>m\ is satisfied □. 2.2.2.4 Fault Isolation The location of a faulty sensor is the most important information required by the supervi sion system. For this reason, m ,- of residual generators are used in each of the N local units of the DSFDI scheme. In this case, the m, residuals react differently to any number of sensors that fail simultaneously. These different reactions (sensor failure codes) of the residuals are used to find the location of faulty sensor(s). In practice, two or more sensors 28 (nj(k)|£ (= 0 > Tij + rij(k)y \fj\\vU,0 + vij,l + • • • + vij,m\ > Tij + fij(k)\F jj-o Tv+nj(k)\, o \fij\ > vijfi + V m ,/ + ...+ V /j n (2.27) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. rarely fail at the same time. Therefore, the simultaneous isolation of two or more failed sensors may be unnecessary. One can use a different approach by designing these resid ual generators where each one is driven by all local inputs and all local outputs except one. This approach can be used to isolate one faulty sensor at a time, perhaps improving the performance of fault detection and isolation scheme. The reason for not choosing the second approach is that it will involve more on line computational operations than the approach we followed. Computational complexity is one aspect that always needs to be considered for the large order systems with many sensors and actuators. 2.2.3 Supervision Scheme The supervision scheme is designed to integrate the diagnostic information with recon- figurable controllers. At the subsystem level, each local controller is driven by /w , local sensor outputs and therefore m ,- reconfigurable controllers are off-line designed for any case of sensor faults. If one of the sensors fails, one of the reconfigurable controllers will be activated based on the information provided by a local unit of sensor fault detection and isolation scheme. There are N supervision units and each unit is locally responsible for: • Removing a faulty sensor from the closed loop system: the location of the faulty sensor, that is provided by a local sensor fault detection and isolation unit, is sup plied to the corresponding supervision unit to isolate the faulty sensor from the local closed-loop system 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • Selecting a new set o f sensors for a reconfigurable controller: after the faulty sensor is removed from the closed loop system, a new set of sensors is selected by removing a faulty sensor from the set and adding a new healthy one • Switching to a predesigned reconfigurable controller: once the faulty sensor is removed from the feedback path and a new set of sensors is selected, a switch ing logic replaces the current local controller driven by a sensor fault with a pre designed reconfigurable controller designed for that case of the failed sensor. The predesigned reconfigurable controller is designed for a new set of sensors assum ing that one sensor fails at a time. With this approach, one needs only replace the local controller driven by a sensor fault with a new controller without replac ing other local controllers. This approach will save memory space, reduce on-line computational operations, and prevent multiple switching • Restructuring a local sensor fault detection and isolation unit: to continue moni toring the system after accommodating a sensor failure, the input signals (sensor measurement signals) to the local unit of sensor fault detection and isolation are replaced with a new set of sensor output signals. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Segmented Telescope Test Bed 3.1 Introduction The proposed design of the DFTC system is demonstrated on a segmented telescope test bed. The test bed is used to simulate a more complex Next Generation Space Telescope (NGST) that will be used to replace the Hubble Space Telescope (HST). Fig. 3.1 shows an example of deployed NGST in space. The main objective of the NGST is to gather information from the farther reaches of space. The information will be used for under standing the structure of the universe, the birth and formation of stars, and the origins and evolution of planetary systems and galaxies. The essential element to achieve these goals is a space telescope with a large reflector mirror. By increasing the diameter aperture of the reflector mirror to 8 m from the 2.4 m in the HST, the space telescope will be capable of collecting information from regions ten times farther away than the regions covered by the HST. The NGST will be built with a number of segmented reflectors instead of monolithic ones that are cast from a single piece of glass. 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.1: An example of deployed NGST in space. The size and weight limitations as well as the limitations associated with the launch ve hicles are the main reasons for not using a monolithic reflector. Although multiple mirror designs have many advantages, a number of major difficulties are also associated with this technique. Specifically, the ability to provide phasing of the separate beams (keeping the optical paths from the multiple reflectors constantly to the fraction o f the wavelength of light) is especially difficult. Because multiple mirror telescopes are independent de vices, combining their images on a single focal plane is also difficult. This problem requires special consideration in the optical design so that the individual focal planes can be properly aligned. The mirrors can be easily misaligned due to disturbances; therefore designing a controller to provide a single desired shape of the segmented reflectors is 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. essential in order for the images to be reflected at the central panel. Furthermore, de signing a fault-tolerant control system for the NGST is essential because the operating orbit of the NGST will be close to 1.5 million kilometers from Earth, making servicing impractical and uneconomical by space shuttle crews. The test bed located at the Structures Pointing and Control Engineering (SPACE) Labo ratory of California State University, Los Angeles, is used to address the issues and the difficulties related to the NGST system. The test bed was constructed by a team of faculty and students from: California State University, Los Angeles, California State University, Long Beach, University of Southern California, and University of California, Berkeley. The National Aeronautics and Space Administration (NASA) funded the development of the test bed and experiments. 3.2 Structure of the Test Bed The structure of the test bed consists of four main parts (Fig. 3.2): a primary mirror, a secondary mirror, a supporting truss, and an isolation platform. The dimensions of the test bed are shown in Fig. 3.3. The telescope focal length is 2.4 m and the primary mirror diameter is 2.61 m. 3.2.1 Primary Mirror The primary mirror consists of 6 active panels surrounding a fixed center panel where each panel has a hexagonal shape with diameter of 101 cm. The central panel is locked 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.2: The structure of the segmented telescope test bed at SPACE Lab. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.3: Test bed dimensions. to the isolation table. To support the weight of the primary mirror, it is mounted on a lightweight flexible truss structure to which each active panel is attached at three node points through its three actuators. All active panels are attached to 18 actuators (three per panel) to generate power to move the panels. Furthermore, 24 edge sensors are used to measure the relative displacements among the panels. There are 2 edge sensors between every pair of panels. Fig. 3.4 shows the location of actuators and edge sensors under the primary mirror. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 Actuator r— ■ Edge sensor Figure 3.4: The location of actuators and edge sensors under the primary mirror. 3.2.2 Secondary Mirror The secondary mirror, which is a six-sided pyramidal mirror, is used to reflect light from the primary mirror to the focal plane in the center panel. The secondary mirror is located 2.4 m above the primary mirror. The secondary mirror (Fig. 3.5) is actively controlled by 3 actuators attached to the secondary truss at 3 nodes. It is also equipped with 3 position sensors to provide the location of the secondary mirror with respect to its housing. 3.2.3 Supporting Truss The truss structure is designed to support the primary and secondary mirrors. It is at tached to the primary mirror at 18 nodes and to the secondary mirror at 3 nodes. The truss is made of stainless steel to provide the highest strength with lowest mass. 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.5: Secondary mirror. 3.2.4 Isolation Platform The test bed is positioned on an isolation platform (Fig. 3.6) to support the structure’s weight. The isolation platform is isolated from the ground with dampers in order to dampen any external vibrations propagated from the ground. The platform is made of an aluminum honeycomb core with a stainless steel top and bottom skin. It is mounted on three pneumatic cylinders providing a passive isolation system for the structure. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Isolation pillar " " Figure 3.6: Isolation platform. 3.3 Data Acquisition System Fig. 3.7 shows the data flow through the test bed system. The readings from the edge sensors that measure the relative displacements of the panels from their nominal posi tions corresponding to the desired primary mirror shape are sampled using a number of A/D cards. A channel sensor display is used to display in real time the sensor readings for monitoring purposes. The digital signal processor, DSP card-A, receives the sam pled measurements of the all edge sensors and sends the corresponding measurements of the panels no. 1 and 2 to the DSP card-E and the corresponding measurements of the panels no. 3 and 4 to the DSP card-F. The control algorithms implemented using the 3 DSPs are used to generate the control commands that drive the 18 actuators. The 18 con trol commands are passed through a number of D/A Cards to the 3 banks of amplifiers and then to the corresponding actuators. For data collection, an external memory disk (SCSI) is used to save the on line edge sensor outputs and control commands. A host 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. □ Banks of amplifiers To actuators Host computer Disk storage j Data Bus Three digital signal processors Primary mirror DSP card A DSP card E DSP card F system Data Bus Sensor rack Outoutsofedg^ensors Figure 3.7: Data flow through the system. computer supported by MATLAB software is used to plot the results. The characteristics of the sensors, actuators and other important components of the experimental setup are described below. 3.3.1 Edge Sensors There are 24 edge sensors (inductive sensors) use to indicate any deviation in the primary mirror shape from its desired shape by measuring the relative displacements between the panels. The sensors are KDM-Series 8200, model 6U1, provided by Kaman Instrument 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Corp. The inductive sensor is used to measure a target displacement with respect to a sensor reference plane. Fig. 3.8 shows the structure of the inductive sensor KDM-8200. The sensor operates as follows: an electromagnetic field is propagated from a sensor coil Electro-magnetic Voltage output Reference plane Figure 3.8: The structure of the inductive sensor KDM-8200. due to Ac current in the sensor coil. This electromagnetic field will generate an induced current in a conductive target, which will generate another electromagnetic field opposite to the original one. When the target is moving, the intensity of the electromagnetic filed in the coil changes and the result of this changing is a new sensor impedance. A detector senses this new impedance and a converter is used to provide a voltage directly proportional to the target displacement. The inductive sensor KDM-8200 has a low noise level, high resolution of 0.1 jUm, wide range of measurements up to 6 mm, bandwidth of 50 kHz, and output range of +/-15 volts. 3.3.2 Actuators and Amplifiers High performance linear electromagnetic force actuators are used to actively control each active panel and support its weight. For each active panel, three actuators are used to 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. change its position and the overall shape of the primary mirror. A custom designed high performance actuator (model ML2-3002-100LB Fig. 3.9) has been developed for the test bed by Northern Magnetics in order to meet the following requirements: low noise level, generation of a substantial force over a wide mechanical range, strength to support the weight of a panel, sufficient bandwidth to accommodate the spectrum of expected dis turbance, freedom from friction, compact in size, and low thermal energy dissipation. These actuators have a low noise level, a bandwidth of 100 Hz, and can generate forces up to 53.5 Newton. The linear amplifiers are used to amplify the signal received from the Figure 3.9: Actuator model ML2-3002-100LB Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D/A converter before sent to the actuators. For the test bed, GA4555P Linear Amplifiers made by Glentek are designed to work with permanent magnet actuators. These ampli fiers have a high bandwidth and can be operated in velocity or current mode. In the test bed, the amplifiers are operated in current mode. The input range of the amplifier is +/- 15 volts and its gain can be calibrated to three different gains. 3.3.3 A/D and D/A Converters For the purpose of control application, dual cards of A/D and D/A converters (Pentek model 6102) are used to provide a number of channels of A/D and D/A conversion. The A/D converter has the appropriate analog input range consistent with the range of the output voltage of the sensor. The A/D converter is used to convert the output voltage of the sensor into a proportional binary output and the D/A converter is used to convert the controller command into an analog signal for the actuator. These converters have 16-bit resolution where resolution refers to the number of bits needed to properly resolve or quantify an analog input signal. These converters have a sampling frequency range from 320 Hz to 250 kHz. 3.3.4 Digital Signal Processors The digital signal processor (DSP) uses the sampled signals from the A/D converters and performs arithmetic processing on these signals. Three processors TMS320C40’s, labelled as DSP-A, DSP-E, and DSP-F, are used to implement the DFTC algorithms. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Each processor has 2 megabits of RAM. Because all processors are linked together by common ports, all three processors can use the 6 megabits of RAM simultaneously. Fur thermore, a data rate of 20 megabites/sec can be transferred between the processors. A VME bus card (VME stands for Versa Module Eurocard) along with other cards are used to provide the interface between the DSP, the A/D and D/A converters, the personal computer host, other input/output (I/O) peripherals such as the disk storage for DSP. The three processors are connected to a host computer by a mix-card and the connection is supported by a Pentek’s SwiftNet software. For data collection, an external memory disk (SCSI) is used to save the data from the test bed. In the test bed, the DFTC algorithms are written in C++ language and then converted to assembly language using a C-compiler. The assembly codes are required in order to program the DSPs. 3.4 Primary Mirror System Model A mathematical model of the primary mirror system was obtained using frequency do main techniques [37]. The relationship between the edge sensor outputs and actuator inputs is given by: ( \ y l V * / / G n(s) ... G\a(s) \ G6\(s) ... Gte{s) ( \ M l v m6 / (3.1) 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Gjj(s) G C3x3 is the transfer function matrix from the y'th input vector to the ith output vector, and w , G M3 and yi G M3 is the input and output vectors consist of the actuator inputs and edge sensor outputs of the active panel number i. The transfer function matrix (3.1) is stable and non minimum phase. The resonance peaks of the first modes of the system range between 14 to 25 Hz. The interconnections among the adjacent subsystems are strong as shown in Table 3.1. The state space representation of Table 3.1: Maximum singular values Subsystem Gn G22 G3 3 G44 G5 5 G66 Maximum singular value 34.23 36.32 37.61 31.02 35.05 38.01 Adjacent subsystem G\2 Gn G34 G45 ^56 G& \ Maximum singular value 33.15 33.02 35.51 32.13 38.21 36.53 the primary mirror system obtained from (3.1) is given by: x = Ax + B\u\ + . . . + 5 6M 6 yt = CiX i= 1, -.., 6, where jc G M320 is the state vector, and A, 5, and C ,- are known matrices of appropriate dimensions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Implementation of Decentralized Fault-Tolerant Control for the Segmented Telescope Test Bed 4.1 Performance Requirements In the future mission of a real segmented space telescope system, information from far ther reaches of space will be collected by light that hits the telescope’s primary mirror and is then reflected by the secondary mirror to a focal plane in a central panel for in formation collection. Therefore, it is important for the primary mirror to behave as a single surface. The shape of such mirrors can easily deviate from their desired shape in the presence of disturbances. The deviation of the primary mirror shape from the desired shape is characterized by the edge sensor outputs. In the test bed, 6 shape error Set values 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for the 6 panels are used to indicate how far the primary mirror is from its desired shape and they are defined as: where y,- is the z'th local output vector of the subsystem i after filtering measurement noise. The fault-tolerant control design objectives are as follows: • The Set value for each active panel is less than or equal to 1 jUm at steady state, i.e., Set < 1 for i = 1 ,..., 6 • The effects of disturbance on the Set value are reduced by the ratio of 100:1 at steady state. 4.2 Flowchart of the Fault-Tolerant Control Algorithm The proposed design of the DFTC system is implemented in the segmented telescope system. Three digital signal processors are used in parallel to handle the computational operations required for the DFTC algorithm. The algorithm is written in C++ language and its flowchart is shown in Fig 4.1. Each of the three digital signal processors is used to implement a part of the DFTC algorithm. The DSP-A is responsible for panels no. 5 and 6, the DSP-E is responsible for panels no. 1 and 2, and the DSP-F is responsible for panels no. 3 and 4. The local data of controllers, sensor fault detection and isolation, and switching controllers are saved in the corresponding local memory space of the three 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Implemented in DSP-F Implemented in DSP-A Implemented in DSP-E Start', I- Stan 3 ’’ Systejp^. N a tio n Counter = 0? Counter == 0? Counter ==0? Receive sensor data Receive sensor data * 2 local controllers 2 local controllers 2 local controllers Any sensor fails? No Any sensor fails'? No Any sensor fails _ No Send control signals Send control signals Send control signals Action Action Action Receive control signals y | Write to D/As | I On-line storage Figure 4.1: The flowchart of the DFTC algorithm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. digital signal processors. The DSP-A is the master processor, because it is used to send sensor readings to the DSP-E and DSP-F and also to receive command signals from the DSP-E and DSP-F. Furthermore, the DSP-A is used to save the data of sensor readings, command signals, and residuals in the external memory disk (SCSI). 4.3 Decentralized Control The control of large segmented telescopes is challenging due to the complexity and high order of the system. The high order dynamics lead to high order controllers that re quire more memory and faster computations for implementation. While this may not pose a serious problem for a small number of segments, as the number of segments in creases the computational requirements become enormous. In the test bed, the objective is to design and implement different simple decentralized control schemes, compare their performance and computational effort, and come up with candidate controllers that meet the performance requirements with the least computational effort. Three decentralized control designs were selected for implementation. These include a decentralized state feedback proportional plus integral (DSFPI) controller, a decentralized output feedback proportional plus integral (DOPFI) controller, and a decentralized direct adaptive output feedback (DDAOF) controller. To design these controllers, 6 subsystems for the primary mirror system are first defined by dividing its inputs/outputs into 6 local input/output vectors. Each of the local input and output vectors consists of 3 actuators and 3 sensors. 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Then 6 local controllers are designed following the proposed design. To implement the ses controllers, the three decentralized controllers are discretized using a sampling period of 1 ms and then implemented to the primary mirror system. The real time results for these three controllers are shown in Figs. 4.2-4.7 for the 18 edge sensor outputs and 18 control commands. In order to examine the effectiveness of the controllers, initial shape errors are created on the position of the primary mirror panels by adding constant loads on the panels. For the DSFPI control, the number of states for each subsystem is 40. The number of computational operations required to implement the DSFPI control is large and cannot be handled by the available DSPs. For this reason, we reduced the order of the 6 decoupled system models and designed a new controller with 17 states for each lo cal controller. The 18 edge sensor outputs and 18 control commands for the closed-loop system with reduced order DSFPI control are shown in Figs. 4.2-4.7. These plots indicate that the closed-loop system is stable, the effect of the disturbances is reduced, and the control effort for all 18 actuators is within the limits (+/-1 V). For the DOFPI controller, the number of computational operations is dramatically reduced compared to the DSFPI controller, since the number of controller states is now 3 for each local controller. The closed-loop system performance, however, is worse than that of the DSFPI controller as shown in Fig. 4.8. The DDAOF controller reduces the effect of the disturbance on the primary mirror panels by 100:1 faster than the other two designs as shown in Fig. 4.8. The time it takes to compute any 18 control command samples by the 3 DSPs (the sam pling time is 1 ms) is: 0.78 ms for the DSFPI control; 0.2 ms for the DOFPI control; and 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.29 ms for the DDAOF control. Table 4.1 shows the comparison among the three types of controllers. These results indicate that the DOFPI and DDAOF controllers take much less computational power of the 3 DSPs than the DSFPI controller. However, the DOFPI controller performs worse than the other two controllers. Overall, the DDAOF controller performs better and requires less computational effort. Table 4.1: The shape error values for the three types of controllers Panel Initial shape error values in jum Shape error values in jum at 240 s DSFPI DOFPI DDAOF DSFPI DOFPI DDAOF 1 277.34 61.64 107.98 1.43 2.6 0.66 2 318.62 71.13 251.20 0.72 15.7 0.72 3 259.21 151.66 363.24 1.01 1.44 1.05 4 214.91 52.09 233.56 0.27 1.94 1.03 5 272.45 55.38 159.36 4.57 2.69 0.5 6 278.81 102.00 124.33 2.08 7.20 0.3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 600 — sensor no. 1 — sensor no. 2 — sensor no. 3 400 200 < / > 3 & 3 o § -200 0 ) 0 5 -400 -600 Time (s) (a) (b) 10 20 Time (s) (c) 600 400 E 3 200 -2 3 Q . 3 O c -200 ® 0 ) -400 -600 Time (s) 0.4 0.3 0.2 0 .1 0 -0.1 -0.2 -0.3 -0.4 0 1 0 20 30 Time (s) (a) (b) 0.4 — actuator no. 1 — actuator no. 2 — actuator no. 3 0.3 S ' 0.2 0 f 0 .1 1 0 0 r o 3 1 -0.1 -0.2 -0.3 -0.4 Time (s) 0.4 0.3 3 Q. C o -0.1 < 0 3 3 - 0.2 -0.3 -0.4 Time (s) (c) Figure 4.2: Real-time results of the mirror no. 1, for three cases: (a) with DSFPI control; (b) with DOFPI control; (c) with DDAOF control. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 600 0.4 -400 -600 400 E -5 200 ( / ) ! • -200 600 400 ? 200 £ 3 ” 3 0 O L _ n S C -200 a > « -400 -600 10 20 Time (s) (a) 30 sensor no. 4 sensor no. 5 sensor no. 6 10 20 Time (s) (b) 30 600 400 200 (A 3 Q . 3 o § -200 a ) « -400 -600 Time (s) 0.3 (c) -0.3 -0.4 a 0.2 § B 01 I 0 -0.1 -0.2 10 20 Time (s) (a) (b) (c) 30 0.4 — actuator no. 4 — actuator no. 5 — actuator no. 6 0.3 S' 0.2 0 f 0.1 1 0 I -0.1 § -0.2 -0.3 -0.4 Time (s) 0.4 0.3 S' 0 .2 f 0 -1 1 0 I -0.1 | -0.2 -0.3 -0.4 Time (s) Figure 4.3: Real-time results of the mirror no. 2, for three cases: (a) with DSFPI control; (b) with DOFPI control; (c) with DDAOF control. 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 600 0.5 -400 -600 400 200 U J 3 CL -200 1 0 2 0 Time (s) (a) 30 600 — sensor no. 7 — sensor no. 8 — sensor no. 9 400 E •5 200 < 0 3 Q. o 8 c 0 ) CO -200 -400 -600 Time (s) (b) 600 400 E 3 200 w 3 Q . 3 o § -200 CO -400 -600 Time (s) (c) -0.5 1 0 20 Time (s) (a) 0.5 a . actuator no. 7 — actuator no. 8 — actuator no. 9 -0.5 Time (s) (b) 0.5 S' o >, o t o 3 1 3 < -0.5 Time (s) (c) Figure 4.4: Real-time results of the mirror no. 3, for three cases: (a) with DSFPI control; (b) with DOFPI control; (c) with DDAOF control. 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 600 0.5 400 200 0 -200 -400 -600 a 3 o 8 c ® CO 1 0 2 0 Time (s) (a) 30 sensor no. 10 sensor n o .11 sensor no. 12 1 0 2 0 Time (s) (b) 600 400 200 3 a 3 o 8 c -200 w CO -400 -600 Time (s) (c) a. c o -0.5 1 0 20 Time (s) (a) 0.5 V) o > 3 Q. C o actuator no. 10 — actuator no. 11 — actuator no. 12 -0.5 Time (s) (b) (c) 0.5 < 0 § 3 -0.5 Time (s) Figure 4.5: Real-time results of the mirror no. 4, for three cases: (a) with DSFPI control; (b) with DOFPI control; (c) with DDAOF control. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 600 400 E • 2 = 200 W 3 Cl 0 1 -200 ® CO -400 -600 Time (s) (a) sensor n o .13 sensor no. 14 sensor no. 15 — 200 t r r - f r a 1 0 2 0 Time (s) (b) - 200 r 1 0 2 0 Time (s) (C) 0.4 0.3 « 0.2 o > C O 3 Q . C | -0.1 | -0.2 -0.3 -0.4 Time (s) (a) 0.4 actuator no. 13 actuator no. 14 actuator no. 15 0.3 £ 0.2 § • 0 .1 b C O | -0.2 -0.1 -0.3 -0.4 Time (s) (b) 0.4 0.3 0 .1 | -01 | -0.2 -0.3 -0.4 Time (s) (o ) Figure 4.6: Real-time results of the mirror no. 5, for three cases: (a) with DSFPI control; (b) with DOFPI control; (c) with DDAOF control. 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 600 400 200 3 0 O § -200 0 ) CO -400 -600 Time (s) (a) 1 0 20 Time (s) <b) 30 600 400 E 3. 200 B 3 & 3 o § -200 CO -400 -600 Time (s) 0.4 0.3 2 0.2 I B 3 Q . C o -0.2 -0.3 -0.4 Time (s) 600 0.4 400 — sensor n o .16 — sensor n o .17 0.3 — sensor n o .18 B 0.2 § a a1 E 3 200 w 3 3 0 i Ifc i liMii flnfl i c 0 o § -200 © CO ! -0.1 3 -0.2 -400 -600 -0.3 -0.4 (a) actuator no. 16 actuator no. 17 actuator no. 18 Time (s) (b) 0.4 0.3 B 0.2 0 f 0 . 1 1 0 I -0.1 | -0.2 -0.3 -0.4 Time (s) (c) (c) Figure 4.7: Real-time results of the mirror no. 6, for three cases: (a) with DSFPI control; (b) with DOFPI control; (c) with DDAOF control. 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 400 6 350 5 300 E J 250 4 o o i 200 3 © 2 150 .c to 2 100 1 50 0 0 400 350 300 E ^ 250 o a > 200 © JS 150 to 1 0 0 50 0 1 0 2 0 Time (s) (a) 30 mirror no. 1 mirror no. 2 mirror no. 3 mirror no. 4 mirror no. 5 mirror no. 6 10 20 Time (s) (b) 30 400 350 300 . 250 o i 200 < 2 150 1 0 0 50 0 Time (s) 200 240 2 0 15 1 0 5 200 240 210 220 230 Time (s) (b) Time (s) (c ) Time (s) Figure 4.8: Shape errors, for three cases: (a) with DSFPI control; (b) with DOFPI con trol; (c) with DDAOF control. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4 Decentralized Fault Detection and Isolation The common types of sensor faults that could happen for the inductive sensor KDM- 8200 are shown in Table 4.2. For the identification of a faulty sensor location, 3 banks of Table 4.2: Commonly inductive sensor faults Cause Symptom Power supply off Outage Incorrect calibration Drift, degradation, or amplifying actual reading Open sensor coil Saturated output at 10 volts Short circuit Unchanging output voltage at low voltage Intermitted cable Random output signal residual generators are designed for each of the 6 subsystems. The residuals are designed based on the parity equations method. After generating the 3 residual signals for each of the 6 subsystems, the different reactions (sensor failure codes) of these residuals are used to find the location of faulty sensor(s) on each subsystem according to the Table 4.3. To Table 4.3: Sensor failure codes for the subsystem i Residuals No fault S/i S/2 S/3 S/i&Sa S/1&S/3 S/2&S/3 S/1&S/2&S/3 hi 0 1 0 0 1 1 0 1 r,2 0 0 1 0 1 0 1 1 r/3 0 0 0 1 0 1 1 1 S jf. Sensor no. j of the subsystem i 1 : The absolute value of the residual evaluation function exceeds the threshold value 0: The absolute value of the residual evaluation function dose not exceed the threshold value examine the effectiveness of the DSFDI scheme, the following 6 cases are considered. These examples show how the residuals will behave against: 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1. Sensor fault free'. in this case, all sensors operate normally. 2. Sensor fault free with external disturbances: external disturbances are applied on the top of the active panels to see how the residuals will react against these distur bances. 3. One sensor is saturated: saturated fault is created on one sensor by programming a C++ code to include this type of fault on the selected sensor. The fault took place at 5 sec with a stack value of 10 volts to simulate a real situation when a sensor coil is opened. 4. One sensor is randomly failed: the fault took place at 2 sec with different values of sensor output and then stacked at 2 volts. With this fault, we tried to simulate a situation when a sensor cable is connected and disconnected at random. 5. The output o f one sensor is unchanged with a low voltage: the fault represents a short circuit. In this case, a fault took place at 30 sec with unchanged sensor output of 2 volts. 6. With external disturbances, two sensors are failed with a random fault and a low voltage fault', external disturbances are added on the top of the active panels. Then one sensor is randomly caused to fail at 2 sec and another one is caused to fail with unchanged low voltage at 20 sec. Fig. 4.9 shows the residual signals for all 6 cases. In these results, all residuals generated by the non-faulty sensors showed small values compared to other residuals generated 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. by the faulty sensors. Also notice that all faulty residuals stack (not drift) because we imposed magnitude limits on all actuators inputs to prevent system breakdown when we apply these created faults. Furthermore, these residuals are evaluated to find the best threshold values that will reduce the risk of false alarms coming from system noise, external disturbances, subsystem interactions, and modelling errors. Table 4.4 shows the fault occurrence time and isolation time for the last 4 cases with thresholds values selected to be 0.3 for all residuals. Table 4.4: Fault occurrence and isolation time Case no. Fault(s) occurrence time (sec) Fault(s) isolation time (sec) 3 5 5.001 4 2 5.502 5 30 30.001 6 2 and 20 5.502 and 20.001 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Sensor fault free Sensor fault free with disturbances O - 0.6 0.5 0.4 0.3 0.2 0 .1 0 -0.1 -0.2 0.04 0.03 0.02 IN 0.01 o 10 20 30 Time (s) Sensor no. 1 is failed fault occurrence 7 (5 s) 0.5 cn o o -0.5 CO r o O residual of sensor no. 1 -1.5 Time (s) Sensor no. 18 is failed residual of sensor no. 18 X fault occurrence (30 s) X o -0.01 -0.02 -0.03 -0.04 10 20 Time (s) 30 40 Sensor no. 4 is failed 0.6 0.4 residual of sensor no. 4 0 .2 o c 0 ) tf l 8 0 'fault occurrence (2 s) -0.2 -0.4 Time (s) Sensors no. 3 and 8 are failed 10 20 30 Time (s) 40 0.7 residua! of sensor residual of sensor no. 8 no. 3 0.3 CO o c o w 1 0 O -0.3 ifault occurrence '(2 s and 20 s) -0.7 Time (s) Figure 4.9: The Residual signals for all 6 cases. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 Decentralized Fault-Tolerant Control Once the sensor fault is detected and isolated by the DSFDI scheme and the location of the faulty sensor is provided to the supervision system, the local controller effected by the sensor fault is removed from service and a switching controller design for that case of faulty sensor is adapted to service. Four cases are presented here to show the ability of the DFTC system to tolerate sensor failures. These cases are the last four cases considered for testing the DSFDI scheme. In Fig. 4.10, sensor no. 1 is failed at 5 sec by fixing the sensor output at 10 volts. The results show that with the nominal decentralized control only, a number of actuator inputs are saturated and the shape error for subsystem no. 1 is far higher than the desired shape error (<ljum). However, using the DFTC system, the corresponding local unit of the DSFDI scheme quickly detects and isolates this fault and a corresponding switching controller is immediately adapted to service. All 18 actuator inputs are within the limits and the 6 shape errors are less than 1 /im at 120 sec. The same conclusions can be made when sensor no. 18 is failed at 30 sec with unchanging output voltage of 2 volts (Fig. 4.11). In Fig. 4.12, sensor no. 4 is failed with the random fault at 2 sec and the results are two actuators inputs belong to subsystem no. 2 are saturated and all 6 shape errors are larger than ljum (these results are clearly shown in Table 4.5). But with the use of DFTC system, the closed loop system performance is much better with expense of transience at switching time. The same conclusions can be made for the last case (Fig. 4.13) where in this case sensors no. 3 and 8 are failed in the presence of external disturbances at 2 sec and 20 sec with the random fault and unchanging low 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. voltage volt fault, respectively. Table 4.5 shows the steady state shape errors with and without using the DFTC system for all 6 subsystems. Table 4.5: The steady state shape errors (jim ) Case no. 3 4 5 6 Subsystem no. Without FTC With FTC Without FTC With FTC Without FTC With FTC Without FTC With FTC 1 570 0.3 2 0.9 30 0.6 2753 0.5 2 3 0.9 13 0.5 2980 0.8 20 0.4 3 7 0.5 2900 0.1 20 0.1 287 1.2 4 21 0.8 11 0.4 16 0.7 3170 2.9 5 13 0.1 6 0.3 9 0.4 137 0.9 6 105 0.3 4 0.4 17 1.1 121 1.4 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2000 sensor n o .4 1500 | 1000 sensor n o .3 w Q . 5 0 0 3 O n -500 -1000 sensor n o .1 -1500 Time (s) ictuator no. 1 0.4 £ 0.2 CO I 0 0 1 - 0.2 ts < -0.4 actuator no. 3 actuator no. 4 actuator no. 2 30 40 0 10 20 40 80 Time (s) 120 Time (s) 40 80 Time (s) 120 700 600 " E 500 - mirror no. 1 - mirror no. 2 - mirror no. 3 - mirror no. 4 - mirror no. 5 mirror no. 6 £ 400 300 55 200 100 Time (s) 1.4 e o f c til ( 1 ) Q . < 5 x: CO 0.4 0.2 1 2 0 Time (s) Figure 4.10: Sensor no. 1 is failed (case no. 3): without using DFTC (left column); with using DFTC (right column). 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2000 sensor no. 17. E 3 . If) 3 Q . 3 O fc -4000 w c a > CO -6000 -2000 sensor n o .16 sensor n o .18 -8000 Time (s) 1 0 0 0.6 0.4 « £ 0.2 in 1 0 actuator no. 17 o actuator no. 1 § - 0.2 to < -0.4 actuator no. 18 -0.I Time (s) O o -100 -50 40 80 Time (s) 120 40 80 Time (s) 120 4000 3500 3000 2500 mirror no. 1 mirror no. 2 mirror no. 3 mirror no. 4 mirror no. 5 mirror no. 6 2000 S' 1500 1000 500 Time (s) 0.6 m 0.4 0.2 120 Time (s) Figure 4.11: Sensor no. 18 is failed (case no. 5): without using DFTC (left column); with using DFTC (right column). 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8000 6000 | 4000 Q . 3 o 2000 sensor n o .4 sensor n o .6 o (/) c < u < / 5 -2000 Time (s) 600 400 E a. 200 £ 3 Q _ 3 O § -200 a) c o -400 -600 120 Time (s) 0.2 I -0.2 o 15 3 T 3 < actuator no. 4 actuator no. -0.4 actuator no. - 0.6 Time (s) 0.2 - 0.2 < -0.4 - 0.1 120 Time (s) 4000 3500 3000 2500 2000 1000 500 Time (s) 1 5 0 — mirror no. 1 — mirror no. 2 mirror no. 3 ~ — mirror no. 4 mirror no. 5 — mirror no. 6 3. 100 120 Time (s) Figure 4.12: Sensor no. 4 is failed (case no. 4): without using DFTC (left column); with using DFTC (right column). 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8000 4000 O o C O -4000 -8000 sensor n o .3 sensor no. 8-* 10 20 Time (s) 30 40 40 80 Time (s) 10 20 30 40 Time (s) 40 80 Time (s) 120 3500 3000 E- 2500 g 2000 1500 5) 1000 500 Time (s) 200 mirror no. 1 mirror no. 2 mirror no. 3 mirror no. 4 mirror no. 5 mirror no. 6 - 150 t 100 40 80 Time (s) 120 Figure 4.13: Sensor no. 3 and 8 are failed in the presence of external disturbances (case no. 6): without using DFTC. (left column); with using DFTC (right column). 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Fault-Tolerant Control for a Class of Nonlinear Systems 5.1 Introduction In recent years, the field of designing FTC systems has received considerable atten tion ([11], [12], [21], [32], [47], [52] , [53], [61], and [62]). For the case of actuator fault, most of this research had addressed fault accommodation for system subject to parameter variation or frozen output. Other types of actuator fault have been rarely con sidered. In this chapter, a general methodology for designing FTC system for a class of nonlinear systems subject to a reduction in the actuator saturation limit is presented. More specifically, the class of nonlinear systems considered in this study is the one where the dynamics can be described by a linear parameter varying (LPV) model, and the states can be measured. The LPV model is used to simplify the FTC system design since non linear models are in general complicated for design. In the case of using an analytical approach, the main idea behind fault tolerance is the use of fault diagnostic and accom modation schemes. A fault diagnostic scheme driven by plant measurements is used to 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. detect, locate, and estimate a fault; while a fault accommodation scheme driven by fault information from the diagnostic scheme is used to modify the nominal control law in order to reduce the fault effect on the system. Based on the above idea, the total task of the proposed FTC system is divided into three parts: 1. Plant control: attempts to stabilize the closed-loop system and provide the desired tracking properties in the absence of faults. The controller is designed using LPV technique ([2], [3], [26], [4], [10], [36], and [67]), where a number of LMIs are solved for controller parameters. 2. Fault diagnosis: deals with the problem of saturation fault detection, location, and level estimation. To achieve that, a suitable LPV model is derived to describe the faulty system. Then the results in [48] are used to construct the diagnostic scheme. 3. Fault accommodation: attempts to reduce the fault effect on the system by modify ing the nominal control law through the reference reshaping filter and feed-forward gain. The accommodation scheme is designed with the help of the bounded real lemma for LPV system presented in [26]. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2 Problem Statement Consider a class of nonlinear systems of the form: (5.1) y { t ) = xs(t) (5.2) where xs(t) e R" is the state vector, u(t) 6 Rm is the control signal, andy(f) e R" is the measured output signal. / ( . , . ) : R" x Rm — > R” is a known function of the states and inputs describing the system dynamics. Furthermore, it is assumed that the dynamics of the nonlinear system (5.1 )-(5.2) can be described by: where As(<p(t)) = Af0 + l!jL\ <Pi(t)4fp <Pi(t) is the real time bounded function, and Afj and Bs are known constant matrices with appropriate dimensions. The actuator saturation fault considered in this study is given by Definition 5.1. Definition 5.1 (Actuator Saturation Fault) The actuator saturation fault is defined mathematically as: xs(t) = As(<p(t))xs(t)+Bsu(t) (5.3) y ( t ) = x s (t) (5.4) afiSjuj) = < Uj\ < 8jUj 8jUjSign{uj) \uj\ > 8jUj 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where uj is the input to the jth actuator, Oj is the output o f the jth actuator, and 0 < 5j < 1 is the reduced level o f the jth saturation limit Uj. Rem ark 5.1 The value o f Sj represents the reduced level o f the actuator saturation limit where 8j — 0 means a complete failure, 8j = 1 means no failure exists, and 0 < 8j < 1 means the saturation limit has been reduced to the value o f ±8jUj. Rem ark 5.2 Actuator saturation fault can occur in many practical examples. For exam ple, in an electro-magnetic actuator a saturation fault occurs when a short circuit occurs in the coil. In this case, the saturation limit is reduced due to the decrease o f magnetic field. In another example, a saturation fault occurs on aircraft control surfaces when their operating range is reduced due to a failure. Now the main problem is presented. Problem 5.1 Design an FTC system fo r the nonlinear system (5.1)-(5.2) such that: • In the absence o f actuator saturation fault, the nominal control objectives are achieved. • In the presence o f actuator saturation fault, the control objectives are achieved as close as possible to the nominal one. 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.3 Structure of a Fault-Tolerant Control System This section presents a structure of an FTC system. The main schemes of the FTC system are shown in Fig. 5.1. These schemes are controller, fault diagnosis, and fault accom modation. Furthermore, the fault accommodation scheme consists of reconfiguration mechanism, feed-forward gain, and reference reshaping filter. The controller is designed to achieve the desired system performances assuming that the system is under normal operation. The fault diagnostic scheme is designed to detect, locate, and estimate a fault. The fault diagnostic scheme is driven by the available system input and output signals. Fault Accommodation Fault Reshaped Reference Reference Signals ~ Outputs Signals Inputs Fault t Controller R e s h a p i n g F il t e r F e e d - F o r w a r d G a in System R e c o n f i g u r a t i o n M e c h a n i s m ■ Fault Diagnosis Figure 5.1: Structure of fault-tolerant control system. The fault information (no fault, fault, location and magnitude of fault) is supplied to the reconfiguration mechanism to trigger an appropriate reconfiguration of the feed-forward gain and reference reshaping filter. The feed-forward gain and reference reshaping filter are designed to fulfill the new physical constraints imposed by the fault. The proposed 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. design of the FTC system has the advantage of not reconfiguring the baseline controller. This approach is desired in many applications to eliminate the cost of designing and implementing a new controller. 5.3.1 Control Design In this section, the problem of designing a controller is considered for a class of systems described by (5.3)-(5.4). In many cases, the number of parameters < p , (/) in (5.3) is large. To reduce design complexity, a smaller number of parameters pft) is used to bound as many <p,(f) as possible. Then (5.3)-(5.4) is written as: xs{t) = As(p(t))xs(t) +Bsu(t) (5.5) y(t) = xs{t) (5.6) where As(p(t)) = ASu + Xfl i Pi(!)dSi, and ASj and Bs are known constant matrices with appropriate dimensions. Furthermore, p(t) = (pi (t),... ,p/v(t))T is the vector of real time varying parameters ranging inside the hyper-rectangle region defined by p,(r) 6 [p .,pi]. Also, its rate p(t) = (pi (?),-• • ,p/v(0)T is ranging inside another hyper-rectangle region defined by p,(r) £ [v( - , v,]. Rem ark 5.3 The matrix Bs is independent o f the parameter vector p(t). The constant Bs can be obtained by adding low pass input filters to make the input matrix independent o f time varying parameters. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The controller is designed based on the concept of affine quadratic stability [26] defined below. Definition 5.2 (Affine Q uadratic Stability) The LPV system x = Ac(p)x is affinely quadratically stable (AQS) if there exists N+l symmetric matrices Pi such that the fol lowing inequalities: P(p) = P0 + p\P\+... + PiPi + • •. + PnPn > 0 (5.7) F(p,p) = Ac(p)TP(p) + P(p)Ac(p) + < 0, (5.8) where — p\P\ + • • • + pi P i + • • ■ + PnPn, hold fo r all admissible trajectories o f the parameter vector p. In this case, the function V (x,p) = xr P(p)x is a quadratic Lyapunov function fo r the LPV system x — Ac(p)x. The difficulty associated with the control design using Definition 5.2 is that the matrix inequality (5.8) is not linear in tenns of P(p) and Ac(p). However, with results proposed by Bara et al. [7], the product terms P(p)Ac(p) and AT c (p)P(p) can be separated. As a result, conservatism is introduced when using Lemma 5.1 [7] instead of Definition 5.2 to design a controller. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lemma 5.1 The LPV system x(t) = Ac(p)x(t) is AQS if there exist a constant matrix W and a symmetric matrix P(p) such that the following LMI: Ac(p)fV + P(p) -P(p) + ™ 0 < 0 W 0 -P{p) t (5.9) holds fo r all admissible trajectories o f the parameter vector p. Since the parameter vector p ranges over a polytope, the LMI (5.9) involves infinite number of constraints. Theorem 5.1 is used to reduce the infinite number of constraints into a finite one, hence simplifying the controller design. Theorem 5.1 I f there exist matrices W , Ri, and symmetric matrices Pi such that the following LMIs: ajt) = 1 , 'Zfli A'(0 = 1, otj(t) > 0, and j3j(t) > 0. Then, the control law u = — (X^=i 0'i(t)Kj)x(t) where Kj — R,W~l stabilizes the system (5.5)-(5.6). ~(W + W T) (As(Ve,jW — BsRi + P(Vej))r WT As (Ve,j W — BsRj + P( Ve,) -P{Vei) - P 0 +P{Vet) 0 <0(5.10) \ 0 p(ve,) y ^ a ^ a ^ are feasible fo r all (Ve,, Vei) where p — X,= i Oti(t)Vej, and p = X,= i 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Proof: The hyper-rectangle region of the parameter vector p has 2N vertices denoted by Fe, and defined as: Fe, e {(vi,...,v/y) : v ,- e {p.,P<}}- Similarly, the hyper-rectangle region of the parameter rate vector p has 2N vertices denoted by Ve\ and defined as: Fe, e {(vj,..., yv) : V j 6 {y,-, v,}}. With these vertices the parameter vector and its rate can be expressed as p = a,(r)Fe,- and p = X?*, where X?*, a,-(f) = 1, X?=i j3,-(f) = 1, (Xi(t) > 0, and j3,(t) > 0. Using these expressions, the following matrices can be expressed as: 2^ As(p) = Zai(t)As(Ve,) (5.11) i=i 2^ P(p) - X a,(/)P (F e,) (5.12) i=i 2 n ~ p - = - P o + X A ( 0 ^ ( ^ i) (5-13) i=i In LMI (5.9), take Ac(p) = A (p ) -5 s (X L i «,■(/).£,•) an(l substitute (5.11)-(5.13) into (5.9). Also, let /?, = /f,IF. Then we get 2 n X «»-(0 /=i ^ ~{W + WT) {As{Vei)W-BsRi + P{Vei))T WT ^ As{Vei)W — BsRi + P{Vei) -P (F e ,)- P 0 + l t \ 0 y IV 0 - P ( ^ , ) y < 0 which depends affinely on the parameters a, and /3,. Consequently, if this LMI holds at all vertices (Fe;, Lie/) then its holds for all o:,, /3 , > 0 with X?=i a,(t) = 1 and X?=i A (0 = 1 - Furthermore, the feasibility of (5.10) implies that W + WT > 0 which means W is positive 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. definite or W is invertible □. To implement the control law u = — (£/=i a.i{t)Ki)x(t), (Xj(t) must be available on-line. A ^ a,(t) can be computed from the relation p = Xf= 1 ai(l)Vej using the known Vet and p(t). 5.3.2 Fault Diagnosis In this section, the diagnosis of actuator saturation fault is considered for a class of sys tems described by (5.3)-(5.4). The fault diagnostic architecture is illustrated in Fig. 5.2 for the fault in the jth actuator. The fault diagnostic scheme consists of three units: fault Outputs F^ult Model Estimation System Fault Diagnosis Fault Detection jth Saturation Level Figure 5.2: Fault diagnostic architecture. model estimation, fault detection, and fault estimation units. The fault model estimation 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. unit is driven by the system input/output signals and is constructed from the nominal sys tem model and adjustable parameter. The fault detection unit driven by the signals from the fault model estimation unit is used to make a decision regarding whether a fault has occurred or not. If a fault is detected, the fault estimation unit estimates the saturation level using the value of the adjustable parameter and the jth control output. To diagnosis a fault, consider the case where the saturation limit of the jth actuator is reduced due to a fault. Then, equation (5.3) is written as: *s — As((p)xs + B S]u\ + ... + B Sj<Jj(SjUj) + . . . + B Smum (5.14) For fault diagnosis, equation (5.14) is expressed in terms of control outputs by defining: 2 f e ) _ i uj ^ 0 Uj = 0 and then writing (5.14) as: xs - < ^s((p )xs(t) 4*B su T X jB SjUj u j ^ 0 As(<p)xs + B su Uj = 0 Theorem 5.2 is used to estimate the value of A y which will be used to detect the fault and to estimate the saturation level 8j. 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Theorem 5.2 Consider the estim ated model: xs = As((p)xs + Bsu + X j B SjUj + G(xs - x s) where xs G R" is the estimated state vector, G is the constant matrix with negative eigen values, and Xj E R is the estimated parameter o f Xj adjusted as: X X T Xj = n B ^ U j f e - x r - y ^ n B ^ U j f e ; Ay(0) = 0 (5.15) I'M where e — x — x , is the estimated error, T is the positive definite matrix, and X is the indicator function fo r the projection algorithm (to prevent parameter drift) defined as: X 0 (|Ay| < M) or (|Ay| = MandXjr(BSjuj)Te < 0) 1 (|Ay| = Mand XJ T ( B SjU j)Te > 0) Then A y is uniformly bounded, and limt taae(t) = 0. Proof'. The proof is similar to the one in [48] and is presented here for the sake of completeness. Consider the following dynamical error: e = - p e — XjBSjUj (5.16) 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where G is taken as G — —pi, p > 0, without any loss of generality, and Xj = Xj — Xj is the parameter estimation error. Using the Lyapunov function: Z = [ 2 eTe+>2XJ T~ ' i l The time derivative of Z along the solution of (5.15) and (5.16) is: v i i2 t r i y \Usjuj z = —p\e\ - x X jX j Xfr(BSjUj)f e Now we show that: ~r „ Xfr(Bs.Uj)Te XXjXj J \ lJJ >0 l%l2 For ^ = 0, the inequality holds trivially. For % = 1 and given that: \Xj\ = M and Xfr(BSiUj)Te > 0 , then we need to show that: XfXt > 0. The term XJ Xj can be expressed as: XjXj = XjCXj + Xj) = U\Xj\2 + \Xj\2-\Xj\2) > l -(M -\Xj\2) Since by assumption j A y j < M, we get XjXj> 0. Therefore Z < — p |e |2. Hence, Z is uniformly bounded which implies e, Xj, and Xj are also uniformly bounded. Since e is square integrable we conclude that limt = 0 □. 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For fault detection, A y is tested for the likelihood of saturation fault. A decision about the existence of saturation fault is made by using the following logic: Vj < £j =>• No saturation fa u lt Vj > £j = > Saturation fa u lt where Vy = jlll'a(XJ(x))2dT}^2 is the average energy of Ay over the time interval [t,t + a], a is the detection window, and £ y is the threshold. For saturation level estimation, when the saturation fault exists (i.e., Vy > £ y ), A y has the value: A y = 0l^ jU jS > — \ — ~ 8 )u i s,sn^ u ^ _ [ — Xlii. _ p Then the saturation level 5, can be J U j Uj \uj\ j estimated as: 5y = ^ (A y + 1). 5.3.3 Fault Accommodation This section presents a method of accommodating actuator saturation fault. The approach is based on the conceptual tools of LMls and consists of adding a reference reshaping filter and feed-forward gain to the closed-loop system in order to fulfil the input con straints imposed by the saturation fault. The idea of using the reference reshaping filter and feed-forward gain to fulfil the input constraints can be understood by considering the input constraint \uj(t)\ < 8jUj. To enforce this constraint the jth system input Uj(t) is generated from the jth control output uCj{t), which is a function of a modified reference 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. signal ^generated by the reference reshaping filter, and the jth feed-forward signal Uj\ in order to get | ufit) = uCj{t) + uj/(t) \ < 8jUj. In this case, the structure of the original con troller is unchanged. Only its input/output signals are modified in order to accommodate for a saturation fault. Furthermore, the modified reference signal F should be designed to fulfil the input constraint while deviation from the reference signal r is minimized. Based on these suggestions, Problem 5.2 is addressed as follows. Problem 5.2 Given the: • Saturation fault level: Sj Design a reference reshaping filter and feed-forward gain with the following structures: • System: xc{t) = Ac(p)xc(t) + Bc(p)r(t) + Cc(p)xs(t) • Controller: < Uc(t ) = Dc(p)xc(t) +Ec(p)r(t) + Fc(p)xs(t) • Reference reshaping filter: xr(t) = Ar(Sj)xr(t) +Br(8j)r(t) Fit) = Cr(5j)xr(t) +Dr(8j)r(t) Feed-forward: uj{t) = F{8j)r(t) Such that: • The whole system is stable • IUj{t) = UCj(t) +Ufj{t)\ < SjUj 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • IKO — ^ (0 II 2 is minimized. The reference reshaping filter and feed-forward gain are designed based on the concept of affine quadratic //„ performance [26] defined below. Definition 5.3 (Affine Q uadratic //„ Performance) The LPV system: x(t) = A(p)x(t) + B(p)r(t) (5.17) uj(t) = Cj(p)x(t) + Dj(p)r(t) (5.18) has affine quadratic performance Yj if there exist N + 1 symmetric matrices Pi such that: P(p) — Po + P\P\ + • ■ • + PiPi + ■ • • + PnPn > 0 (5-19) \ fnl R(nl A(p)P(p)+P(p)AT(p)+P(p)-P0 P(p)Cj (p) B(p) Cj(p)P(p) - Y jl Dj(p) \ BT{p) DT j{p ) - y j l j < 0 (5.20) holds fo r all admissible parameter vector p. In such a case, the Lyapunov function V(x,p) = xTP(p)x establishes that: 1) the system (5.17) is asymptotically stable, and 2) its L2 gain does not exceed Yj- That is, \uj{t)\ < Yj\\r(i)\\2 f or all Lj-bounded r(t) (provided that x(0) = 0). Rem ark 5.4 Definition 5.3 is stated for zero initial conditions. However, the nonzero initial conditions can be included in Definition 5.3 by considering the following transfer 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. function of(5.17)-(5.18) fo r a frozen p: T(s) = (Cj(p)(sl -A(p)) 1 B(p) +Dj(p))r(s) + Cj(p)(sl-A(p))~'x( 0) = (Cj(p)(sI-A(p))-'{B(p) I] + [Dj(p) 0})(rT(s),xT(0))T. Then, replacing B(p), Dj(p), and r(s) in Definition 5.3 by [ B (p ) /], [Dj(p) 0], and (rT(s),xT(0))T, respectively. The difficulty of using Definition 5.3 to design the reference reshaping filter and feed forward gain resides in the fact that matrix inequality (5.20) is not linear in terms of P(p) and A(p). Therefore, the matrix inequality (5.20) is not convex and thus difficult to solve. To convert the problem into an LMI problem and make it tractable, the following relaxations and selections are proposed: • To get a LMI, Ar(Sj) and Cr(8j) are predesigned and denoted as: A,.(8j) — Arj and Cr(8j) = C,.j, where Arj has negative eigenvalues 8 u • To satisfy \uj{t)\ < 8jUj, y} should satisfy: yy < max[Q t)h) • Minimizing ||F(/) - r(t) |]2 is relaxed and making F(t) — Ajr(t) is considered at the steady state, that is Dr(8j) = Ay + CrA~'Br(8j), where Ay is a constant diagonal matrix with its elements 0 < PiJ < 1 • The structures of the remaining design matrices are selected as: Br(8j) = SjBrj and F(8j) = 8jFj , where Brj and Fj are constant design matrices. 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The state space representation of the plant with controller, reference reshaping filter, and feed-forward gain is given by: x(t) = A(p)x(t) + B(p)Hjr(t) (5.21) Uj{t) = Cj{p)x(t)+Dj{p)Hjr{t) (5.22) where * ( 0 = W ( 0 .*c'(0 >*r (0 )7 -. ( \ As(p)+BsFc(p) BsDc{p) BsEc{p)Crj A(p) = \ Cc(p) Ac(p) Bc{p)Crj 0 0 Ar / B(p) = (((5J£ c(p ))r )5 j ( p ) )0)r i((5y5s£ c(p)Cr/ - , )r ,(5y5c( p ) C ^ - , )7 ',5y/)7 ’: (djBl 0,0)r ), Cy -(p)=jthrow of {(/r c(p),D c(p),£'c(p)C0 )},and D j(p)= jth row of {(^c(p):(5y£c(p)Cr/i4" 1):5y/)}- Corollary 5.1 is used to convert Problem 5.2 into an LMI problem. Corollary 5.1 Consider the LPV system (5.2 1 )-(5.22) with known Sj, Arj, and Crj. I f there exists a solution (Pi, Hj, y f to the following LMI problem: maximize Pij 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. subject to: P{p) ~ Pq + p\P\ + • • ■ + PiPi + ■ • • + PnPn > 0 ^ ( P )P(p)+P{p)AT{p)+ P{p) - P0 P(p)Cj(p) B (p)H j - Y jl D j(p )H j <0(5.24) (.D j( p )H j) T -yjlj (5.23) \ M(p) Cj(p)P(p) (•B (p)H j)T J J Yj - "*«x(||r(/)||2) (5.25) 0 < Pij < 1 (5.26) fo r all admissible parameter vectors p. Then, 1) the system (5.21) is asymptotically stable, 2) \uj(t)\ < 5jUj, and3) r(t) — Ajr(t) as t — > (providedthat r(t) is a bounded constant reference). Proof: Consider the LPV system (5.21 )-(5.22) with known < 5 y , Arj, and Crj. If there exist Pi, Hj, and Yj that satisfy (5.23)-(5.26) then from Definition 5.3 the system (5.21) is asymptotically stable and \uj(t)\ < 8jUj. Furthermore, D r(8j) = Ay + CrjA~JlBr(8j) implies that r(t) = Ajr(t) as t — > for a constant reference signal r(t) □. It is difficult to use Corollary 5.1 to find a solution (Pi, Hj, Yj) because the LMIs (5.23)- (5.24) need to be solved for all admissible parameter vectors p which imply infinite num ber of LMIs. However, the infinite number of LMIs can be reduced to a finite number of LMIs using the following procedures: 8 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1. Write the matrices in (5.21)-(5.22) as: A(p) =A0 + S /li PiAi, B(p) = B0 + SjL, PiBi, Cj(p) = Cj„ + Xf=1 Pic h , and Dj(p) = Djo + l f =1 p tDJ : 2. Use the matrix expressions in step 1 to write the LMI (5.24) as: N N N N M{p) = M0 + £ l plMd0l+ £ p ,M i + X P‘PlM‘l + X P?M“ < 0 (5-27) /= 1 (= 1 i,l=\,i?l i= 1 where / M0 = AoPo + PoAl P0CT B0H CjPo -Yj1 Dj Hj J) j M -dot \ Pi 0 0 0 0 0 y 0 0 0 j 1 AoPi + AiPo + PoAj + P'At 0 PoCj'+PiCl BjHj X Mi = C j . f i + C j P 0 (BiHj)T 0 ( D j f i j ) Dj'Hj A iPi +A ,P,+ PjAJ + P,A J PjCl+PlCj' 0 Mu C j f l + Cj,Pi 0 0 0 0 0 , and 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / AjPj + Pi A J PjCj. 0 Mu 0 0 \ 0 0 0 / 3. To reduce the number of parameters (p„ p;, p,p/, and p f) and hence to reduce the design complexity, define fewer parameters a,, i = 1 , 2 , . . . , A T , to bound the parameters p,, p,, p,p/, and p f. Then, the LMIs (5.23) and (5.27) will be a function of a, . 4. Solve the LMIs (5.23), (5.25), (5.26), and (5.27) for Pj, Hj, and jj at all vertices of the a parameter space. In this case, the existing solutions guarantee the feasibility of the LMIs for all admissible parameter vector cr, see for example [2], Algorithm 5.1 is used to reconfigure the reference reshaping filter and feed-forward de signed using the above procedures. Algorithm 5.1 1. I f there is no saturation fault, i.e. vj(k) < £j, set Fj = 0 and r(t) = r(t), then stop. Otherwise, go to step 2. 2. Given the estimated level 8j £ Sj, where S j = { k j < 8j < K j ' . 0 < K j , K j < 1}, adapt the reference reshaping filter and feed-forward gain designed using the above procedures V Sj £ Sj, then stop. 8 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Algorithm 5.1 is implemented inside the reconfiguration mechanism scheme in order to adapt the right reference reshaping filter and feed-forward gain after estimating the level of saturation fault. 5.4 Illustration Example In this section a second-order LPV model is used to illustrate the proposed design. The LPV model is given by: ( \ xs\ ^ •^ 2 J ( p (0 i o -1 \ ( \ ( \ l + M p ) \V X y = xsi (5.28) (5.29) where -0.5= p <p(t)< p =0.1. 5.4.1 Control System To design a controller using the result of Theorem 5.1, the LPV model (5.28) is written in the polytopic form xs = (aiAs(p) + 0C 2As(p))xs + Bsu, where a\ = (p(t) — p )/(p — p), and (X2 — (~p(t) + p )/(p - p). Then the results of Theorem 5.1 are used to design the controller to stabilize the closed loop system, and to track the desired constant reference signal. Figure 5.3 shows that the controlled system output reaches the desired set point 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w ith ’’almost” perfect tracking. 1 8 1 6 1 4 1 2 1 0 3 3 a e 8 6 4 2 0 • 2 1 2 1 0 6 > > 3 B 3 o 6 - - re fe re n c e 4 2 0 0 2 4 6 8 1 0 T im e (s) 1 2 1 4 1 6 1 8 2 0 Figure 5.3: Simulation results of nominal control system. 5.4.2 Fault Diagnosis The proposed design of saturation fault diagnosis is simulated for the LPV model (5.28). Figure 5.4 shows the results of fault diagnostic system with the saturation level of 8 — 0.5. The fault is occurred at 5.00 sec and detected at 6.12 sec using the threshold value of 0.05. The results of Theorem 5.2 are used to estimate the saturation level. The learning 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. rate T is chosen as r = 10, and the filter pole is set to p = -6 0 . Figure 5.4 shows that after the occurrence of the fault, the level of the saturation fault is converged to the true value within about 3 sec. The result indicates that the estimation scheme provides an accurate estimation of the saturation level within a reasonable time. .0 5 0 .0 5 •0 .1 (5 .0 0 s e c ) .1 5 -0.2 .2 5 0 2 4 6 8 1 0 1 2 14 0 .2 (6 .1 2 s e c ) 0.1 T h r e s h o ld ■ 0 .0 5 0 0 2 4 6 8 1 0 1 2 14 0 .7 . 0 .5 0 .3 0 .1 0 0 2 6 6 1 0 1 2 14 4 T im e (s) Figure 5.4: Simulation results of fault diagnostic scheme. 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.4.3 Fault Accommodation Once the saturation level is estimated and sent to the reconfiguration mechanism scheme. An appropriate reconfiguration of the feed-forward gain and reference reshaping filter is triggered to accommodate the fault. Figure 5.5 shows the simulation results of the control system with and without fault accommodation for the case of 0.2 reduction in the saturation limit. Without fault accommodation, the input signal reaches its limit and the output diverges from the desired set point. With fault accommodation, on the other hand, the input signal is reduced within its new limit and the output tracks the desired set point. i - 8 T im e <s) Figure 5.5: Simulation results with and without fault accommodation. 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 Application of Fault-Tolerant Control for an F-16 Aircraft Longitudinal Motion 6.1 Nonlinear Equations The nonlinear equations of an F-16 aircraft longitudinal motion derived from [63] are: . , cosa qScosa _ qSsince „ V = g s in ( a - 0 ) + T + l C*,v+ - C2,v (6.1) m m m a = | c o s ( a - e ) - ^ T + (l + ^ ^ C Xia)q + - ^ C 2 } a (6.2) 4 = - ~ y C Xtqq + j ^ C z^ (6.3) ' *yy e = q (6.4) h = F sin (a —0) (6.5) 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Nomenclature V, a, 6 q, h T , 8 e m,g q,c S, lyy X -C g l Xc gr Cx,CZ ) CX q, CZ q Cm i Cm q Cx,v — Cx(ot, 8e) + 2 y C Xq(cc)q Cz,v = Cz(oc, 8e) + ^j^CZq((x)q Qc,a = Czq(a)cosa— Cxq(a)s\na Cz,a = Cz(a ,5 e) c o s a - Q ( a ,5 e)sin a Cx q — c C m q ( ( X ) A c(X C gr X C g )Czq(O C ) Cz^ q — Cm(a , 8e) + (XC gr XC g)Cz((X) q = ^(2.377 x 10“3(1 - 0.703 x 1(T5/j)414)F 2 Velocity (ft/sec), Angle of attack (deg), Pitch angle (deg) Pitch rate (rad/sec), Altitude (ft), Engine thrust (lb), Elevator deflection (deg) Airplane mass (slugs), Gravitational constant (ft/sec2) Dynamic pressure (psi), Mean aerodynamic chord (ft) Wing platfonn area (ft2), Pitch moment of inertia, slug (ft2) Center of gravity location (ft), Reference eg location (ft) Aerodynamic force coefficients Aerodynamic moment coefficients Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The control surfaces and the three body axes of rotation of an F-16 aircraft are shown in Fig. 6.1. The values of the aerodynamic coefficients are determined from a look- Vertical axis Yaw Lateral axis. Rudder Elevators Pitch C \ ') t'N Longitudinal axis '"Roll Ailerons Figure 6.1: Control surfaces and three body axes of rotation of an F-l 6 aircraft. up table in the wind tunnel database derived from NASA-Langley wind-tunnel experi ments [43]. The data in the look-up table are valid for: 100<V< 900 ft/sec, -10<a< 45 deg, and 5000<h<40000 fit. The aerodynamic coefficients are also expressed analyti cally in [40], The analytical expression of the aerodynamic coefficients Cx, Cz, and Cm are decomposed into two terms: the first term is a polynomial of a; and the second term is a nonlinear function of a and 8e. These coefficients are written as: Cx(a,Se) — Cxl(a) + Cx2(a,8e)8e, Cz(a,8e) = Cz](a)+Cz2{a,8e)8e, and Cm(a,8e) = Cm\{a) + 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Cm2 ( a ,8 e)8e. After basic manipulations, the nonlinear equations (6.1)-(6.5) are written as: V = auV + a\2q + gsin(fe) + b\\T + b n8e (6.6) q = ci2\V + ai2q + b228e (6.7) fe = a2\V + a-$2q+ ^ c o s ( /e ) + * 3 1T + * 32Se (6.8) h = Fsin {fe) (6.9) where fe = a - 6, = |y ( G i ( a ) c o s a + Czi(a)sino:), = § ( c *?(°Ocosa + G ? (a )sin a ), l _ co s a b" ~ ~m~’ *12 = ^(C * 2( a , 5e)c o sa + Cz 2(a ,5 e)sin a), a 21 — (C/nl ( ° 0 + Q l (°0 (^cgr _ ^cg))i °22 = 2/^K 6 z^(tt)(A^gr — Acg)), *22 = ^ ( C m2(oc,8e)), «3i = ^ j ( C zi( a ) c o s a - C ,i( a ) s in a ) , a32 = ^ ( C z, ( a ) c° s a - C t, ( a ) sina ), 6 31 = ^ a , a n d *32 ” (6^2 (t^ ; *e) COS O J Cx2{o>^ *e) s in t)j). 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.2 Linear Parameter Varying Model The objective of this section is to derive an LPV model from the nonlinear equations (6.6)-(6.9) and then use the LPV model to design a FTC system. At the beginning, we approximate cos(fe) and sin(/e) using Taylor series as: cos(/e) rs 1 - ^ ^ ^ and sin(/e) ~ fe — "yr + ^ — %;■ Then, substitute these approximations into (6 .6 )- (6.9). After simple manipulations, the nonlinear equations (6.6)-(6.9) are written as: / • \ V < 7 fe h / t \ a ii a\2 a n 0 a2\ a22 0 0 031 a 32 033 0 Cl 4 1 0 <343 0 \ / \ V b\\ b1 2 ( \ q 0 & 2 2 T + fe ^31 ^32 \ 5e ) h ) { 0 0 } (6 .1 0) where a n — g(l — + r)> 031 — 031 + ^ j, 0:33 — y{ — 27 + ^ — '^r)> a4i — fe, and 043 = V( — 'yi—|- ^ — 4^-). The matrices of the LPV model (6.10) are function of the states: V, a, fe, h, and control input 8e. The LPV model (6.10) can be simplified by approximating some matrix entries as constant values. This is done for: -25<<5e<25 deg, 100<V< 900 ft/sec, -10< a< 45 deg, and 5000<h<40000 ft. Based on these bounds, the following matrix entries are approximated as: a n « 32.13, b\ \ « ~ -0.025, 033 ~ -0.01, 631 ss 0, and 043 rs — 0.035. Furthermore, the actuator dynamics are in cluded in the model (6 .10). The aircraft thrust engine is modelled as and the elevator actuator is modelled as - 7^0 , see [63]. Then the LPV model (6.10) is written as: s + 20 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( . \ V < 1 fe h x, V s* / a 11 (321 (331 (3 4 1 0 0 (312 (3 2 2 -0 .0 2 0 0 0 32.13 0 0.85 0 0 - 4 0 0 0 \ b\i ( \ V ( 0 \ 0 b22 < 7 0 0 bn fe + 0 0 0 h 0 0 0 xt 4 0 —2 0 j { xSeJ 1 ° 2 0 j f \ T \ 5 e j (6.11) The validity of the LPV model (6.11) is examined using simulations to compare the output response of the LPV model with that of the nonlinear model for different input excitation signals. The constant coefficient values of the F-16 aircraft longitudinal equa tions used in the simulation are given in Table 6.1. Table 6.1: The constant coefficient values of the F-16 aircraft longitudinal equations Symbol Value m 637.1604 slugs S 300 f t 1 c 11.32 ft XC S r 0.35cft g 32.174 ft/s e c 1 Fig. 6.2 shows the input signals and output responses of the nonlinear and LPV models at the starting speed 400 ft/sec and altitude 5000 ft. The results indicate that the outputs of the two models are almost identical except for the altitude where there exists a small 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. approximation errors. For the starting speed 600 ft/sec and altitude 40000 ft, Fig. 6.3 shows the input signals and output responses of both models. The results indicate that the miss-match increases compared to the previous results. To test the LPV model at different frequencies, rectangular input signals are used to excite both models. Fig. 6.4 shows the input signals and output responses of the nonlinear and LPV models at the starting speed 400 ft/sec and altitude 5000 ft. The results show that the outputs of both models are almost identical with small approximation error. For the starting speed 600 ft/sec and altitude 40000 ft, Fig. 6.5 shows the input signals and output responses of both models. The results indicate a good match of the responses with small approximation errors for some of the output responses. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T (lb) 1001 1000.5 1000 999.5 999 1 < D S, 0.5 5 € O 0 50 100 150 50 100 150 400 _ 300 " o ' < 1 ) | 200 > 100 Nonlinear - LPV 0 50 100 150 at ( D ■ o -20 100 150 2 1 2 0 50 100 150 6000 5000 S- 4000 J Z 3000 2000 100 150 T (sec) 250 „ 240 o a > ( / ) i - 230 > 220 144 146 148 150 5.4 S ’ 5.3 144 146 148 150 0.2 o a ) a t id T 3 cr - 0.1 150 144 146 148 3100 3000 2900 2800 148 150 144 146 T (sec) Figure 6.2: Input signals (constant) and output responses of nonlinear and LPV models at starting speed 400 ft/sec and altitude 5000 ft. 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T (lb) 1001 1 1000.5 1000 999.5 999 0 50 x 10 100 150 800 ■ o ' 600 > 400 Nonlinear - LPV 200 100 150 0) -10 100 150 2 1 0 2 0 50 100 150 4.5 3.5 100 150 T (sec) a > S 0.5 50 100 150 235 g 230 225 147 148 149 150 ■ 6 .5 ■ 7 — 147 148 149 150 0.05 o 0) jo O ) 0) x> -0.05 cr - 0.1 149 150 147 148 x 10 3.44 E- 3.42 ■ C 3.4 149 150 147 148 T (sec) Figure 6.3: Input signals (constant) and output responses of nonlinear and LPV models at starting speed 600 ft/sec and altitude 40000 ft. 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 600 500 1 3 =- 400 300 200 0 50 100 150 0.2 3 o.i a ) S- o 0 ) < o - 0 1 - 0.2 50 100 150 700 — N onlinear - LPV 600 g 500 400 300 0 50 100 150 630 620 g 610 600 590 147 148 149 150 15 10 o i 5 0 0 50 100 150 u > C D ■ o 14 13 12 11 > — 147 148 150 149 2 0 1 2 0 50 100 150 x 10 0.5 100 150 T (sec) -0 .5 147 148 149 150 x 10 1.38 it? 1.36 1.34 148 149 150 147 T (sec) Figure 6.4: Input signals (rectangular) and output responses of nonlinear and LPV mod els at starting speed 400 ft/sec and altitude 5000 ft. 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 600 500 400 I- 300 200 0 50 100 150 0.2 CD 1 0 > S 0 t u - 0 ,1 L J - 0.2 0 50 100 150 900 N onlinear - LPV „ 800 o' < u t 7 0 0 > 600 500 100 150 810 o' 800 > 790 780 147 148 149 150 1 0 5 0 0 50 100 150 S ’ 8 .2 ■ o 147 148 149 150 2 1 S ? 0 1 ■ 2 150 0 50 100 0.2 o © © T 3 - 0.1 - 0.2 147 148 149 150 x104 x104 4.5 50 100 150 4.79 s? 4.78 • “ = 4.77 4.76 4.75 150 148 149 147 T (sec) T (sec) Figure 6.5: Input signals (rectangular) and output responses of nonlinear and LPV mod els at starting speed 600 ft/sec and altitude 40000 ft. 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.3 Control System The difficulty associated with the control design using the LPV model (6.11) is the large number of parameters (a,y and bjj) which increases the complexity of the FTC system design. One way to reduce the design complexity is to reduce the number of parame ters by using similar or identical bounds for as many parameters as possible. To do so, the extreme values of the parameters a,y and bjj are obtained based on the given control input limits and flight envelope. Then, the two parameters defined as: -15= p { <pi< Pi — - 0 . 17 and -20= p 2 < P 2< p2 =2 are used to bound the values of a,j and bjj as shown in Table 6.2. Table 6.2: Parameter bounds Parameter Lower bound Upper bound Pi bound on -0.4 0.05 P2/40 a\2 -38 -0.1 2p2 b\2 -140 20 10p2 a2\ -0.15 0.05 P2 / 4 0 C 1 2 2 -13 -0.2 pi b 2 2 -40 0 2 p2 < 231 -0.001 -0.02 Pi/10 bn -0.15 -0.03 P l/10 < 241 -0.87 0.87 P2 / 2 0 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N ow the LPV model (6.11) is written as: / • \ V ( ei 40 2 p 2 32.13 0 0.85 m \ 10p2 ( \ V / 0 \ 0 < 7 P 2 40 Pi 0 0 0 2p 2 q 0 0 f e a 1 0 - 0 .0 2 5 - 0 .0 1 0 0 a 1 0 f e + 0 0 ( ~ \ T ( h a 20 0 - 0 .0 3 5 0 0 0 h 0 0 [ S e ) xt 0 0 0 0 - 4 0 xt 4 0 u 1 ° 0 0 0 0 - 2 0 j V 5 J 1 ° 2 0 ) X s MP\,Pi) Bs (6.12) Rem ark 6.1 The LPV model (6.12) has the same structure o f(5.5). The matrix As(p i, P2) depends affinely on the parameters p\ and p2, and the matrix Bs is a constant matrix due to including the actuator dynamics. To design a controller using the result of Theorem 5.1, the LPV model (6.12) is writ ten in the polytopic form as xs — (X?=i ajAs(Vej))xs + Bsu where As(Ve\) = As(pi,p2), As(Ve2) = As(p],p2),As(Ve3) =As{puP2), and As(Ve4) = As(p{ ,p 2), with a , = (1 - x)(l - y ) , a 2 = x y , a 3 = (1 - x ) y , and 0C 4 = x ( l - y ), w h erex = (pt - p i ) / ( p i — p , ), andy = (p2 — p2)/(p2 - p 2). Then, the LMIs (5.10) are used to design the F-16 aircraft longitudinal controller. The engine thrust and elevator deflection are used to control the longitudinal motion. The control objectives are: • Track the velocity and altitude command signals with almost zero steady state er rors 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • The control system should operate for a velocity range o f 100<V< 900 ft/sec, an angle o f attack range of-10< a< 45 deg, and an altitude range of5000<h<40000 ft- The structure of the F-16 aircraft longitudinal control system is shown in Fig. 6.6. The controller parameters Ks, Kj, and K j are designed using the LMIs (5.10) to stabilize A lphaj E ngine T hrust R eference S ignals K ,( i> Q - S tales _> Elevator D eflection In teg rato r D erivative Actuators Outputs to be controlled F-16 Nonlinear Longitudinal Model Figure 6.6: F-16 aircraft longitudinal control system structure. the closed loop system, cancel the steady state errors, and improve the output response. The controller parameters are function of velocity, angle of attack, and altitude which are assumed to be available on-line for adapting the controller. The system of nonlinear equations of F-16 aircraft longitudinal motion with F-16 aircraft longitudinal controller is simulated for different flight maneuvers at different operating points. The following three scenarios are considered here: 1. The aircraft at an altitude of 5000 ft and speed of 400 ft/sec and the desired next maneuver is to reach 9000 ft with speed of 450 ft/sec within 30 sec (Fig. 6.7). 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2. The aircraft at an altitude of 5000 ft and speed of 400 ft/sec and the desired next maneuver is to reach 9000 ft with speed increasing linearly from 400 to 720 ft/sec within 80 sec (Fig. 6.8). 3. The aircraft at an altitude of 30000 ft and speed of 600 ft/sec and the desired next maneuver is to reach 10000 ft with speed of 650 ft/sec within 100 sec (Fig. 6.9). The simulation results demonstrate that for the case where there is no saturation fault the controlled system outputs reach their desired altitude and speed with almost perfect tracking and within the desired flight region. 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 0 ■ 5 30 40 50 0 1 0 20 14000 12000 10000 8000 K 6000 4000 2000 0 1 0 20 30 40 50 4 3 2 1 output reference 0 1 30 40 50 0 1 0 20 460 450 440 8 430 < / > £ > 420 output reference 410 400 390 0.5 -0.5 O ) 2 . -1.5 8 -2.5 -3 -3.5 Time (s) 10000 9000 8000 £- 7000 sz 6000 5000 4000 Time (s) Figure 6.7: Simulation results of nominal control system (scenario no. 1). 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. h (ft) 5 0 •5 80 100 0 20 40 60 7000 6000 5000 4000 ■ Q K 3000 2000 1000 100 0 20 40 60 80 4 3 output reference 2 1 0 1 80 100 20 40 60 0 750 700 650 ~ 600 o Q ) § 550 > 500 output reference 450 400 350 100 0.5 0 -0.5 1 -1.5 ■ 2 -2.5 3 100 60 80 20 40 0 10000 9000 8000 7000 6000 5000 4000 100 Time (s) Figure 6.8: Simulation results of nominal control system (scenario no. 2). 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T (lb) 5 0 ■ 5 20 40 60 80 100 0 14000 12000 10000 8000 6000 4000 2000 0 20 40 60 80 100 0.4 0.2 ra -0.2 0 ) 2 £ -0.4 -0.6 output reference -0.8 100 20 40 60 80 0 700 680 660 £ 640 output reference 620 600 580 100 0.5 -0.5 u > a ) 2. a -2.5 100 Time (s) 3.5 2.5 100 Time (s) Figure 6.9: Simulation results of nominal control system (scenario no. 3). 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4 Fault Diagnosis The proposed design of saturation fault diagnosis is simulated for the nonlinear equations of an F-16 aircraft longitudinal motion. In this case, saturation faults on the aircraft en gine and elevator control surface are considered where a single fault is assumed to occur at a given time. To isolate a saturation fault, two banks of fault diagnosis are designed for each of the two actuators. To examine the effectiveness of the fault diagnostic scheme, the following simulation experiments are performed: 1. Fault free: in this case, the aircraft engine and elevator control surface are normally operated without an occurrence of a saturation fault (Fig. 6.10). 2. Aircraft engine fails with the saturation level o f 8 = 0.4: the fault occurs at 5.75 sec and is detected at 5.87 sec using the threshold value of 0.05 (Fig. 6.11). 3. Aircraft engine fails with the saturation level o f 8 — 0.8: the fault occurs at 5.81 sec and is detected at 6.13 sec using the threshold value of 0.05 (Fig. 6.12). 4. Elevator control surface fails with the saturation level o f 8 = 0.8: the fault occurs at 5 sec and is detected at 6.09 sec using the threshold value of 0.01 (Fig. 6.13). 5. Elevator control surface fails with the saturation level o f 8 — 0.4: the fault occurs at 2.52 sec and is detected at 2.89 sec using the threshold value of 0.01 (Fig. 6.14). Table 6.3 shows the fault occurrence and detection time for the last 4 cases. The threshold value of 0.05 is selected for cases no. 2 and 3 and the threshold value of 0.01 is selected 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for cases no. 4 and 5. To estimate the saturation level, the results of Theorem 5.2 are used Table 6.3: Fault occurrence and detection time Case no. Fault occurrence time (sec) Fault detection time (sec) 2 5.75 5.87 3 5.81 6.13 4 5.00 6.09 5 2.52 2.89 for that. The learning rate T is chosen as T = 0.2, and the filter pole is set to p = —25 for the aircraft engine fault estimation and p = — 3 for the elevator control surface fault estimation. Figs. 6.11-6.14 show the estimations of the saturation level for the last four cases. The figures show the levels of the saturation fault for cases no. 2, 3, 4, and 5 are converged to the true values within about 5, 2.5, 2.7, and 4.1 sec, respectively, after the occurrence of the fault. These results indicate that the estimation scheme provides an accurate estimation of the saturation level within a reasonable time. 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Aircraft Engine E levator Control Surface 8 T 3 L L -0.5 O > . S > a ) c 1 1 1 a > a > 0 ! -0.5 10 20 30 40 Time (s) 10 20 30 Time (s) Figure 6.10: Simulation results of fault diagnostic scheme (case no. 1). 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Average Energy o f X Fault Indication (X ) Aircraft Engine 0.2 Fault occurrence (5.75 sec) - 0.2 -0.4 - 0.6 -0 .1 0.6 0.4 Fault detection (5.87 sec) .Threshold = 0.05 0.2 -0.2 < B > 0 ) W o 0.6 c o r o E 0.4 0.2 Time (s) Figure 6.11: Simulation results of fault diagnostic scheme (case no. 2). 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Average Energy o f X Fault Indication (X) Aircraft Engine 0.05 Fault occurrence (5.81 sec) -0.05 -0.1 -0.15 -0.2 -0.25 0.2 0.15 0.05 Fault detection ■ (6.13 sec) Threshold = 0.05 -0.05 0 ) > < D 0.8 c /j o 0.6 c 0 1 0.4 0 ) U J 0.2 Time (s) Figure 6.12: Simulation results of fault diagnostic scheme (case no. 3). 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Elevator Control Surface Fault occurrence (5 sec) ^ -0.05 o s -0.1 D r o u. -0.15 -0.2 -0.25 0.06 0.05 ° 0.04 a > iS 0.03 Fault detection (6.09 sec) ® u > E 5 0.02 > < 0.01 Threshold = 0.01' 1 0.8 0.6 0.4 0.2 0 18 14 16 1 0 1 2 6 8 0 2 4 Time (s) Figure 6.13: Simulation results of fault diagnostic scheme (case no. 4). 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Average Energy o f X Elevator Control Surface 0.2 Fault occurrence (2.52 sec) ■ 2 - 0.2 -0.4 -0.6 -0.8 0 2 4 6 8 1 0 1 2 14 16 18 0.2 0.15 Fault detection (2.89 sec) 0.05 Threshold = 0.01 -0.05 0.5 0.4 0.3 0.2 0 .1 0 18 14 16 6 8 1 0 1 2 0 2 4 Time (s) Figure 6.14: Simulation results of fault diagnostic scheme (case no. 5). 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.5 Fault Accommodation Once the level of the saturation fault is estimated, the fault information is sent to a recon figuration mechanism scheme. Then, an appropriate reconfiguration of the feed-forward gain and reference reshaping filter is triggered to accommodate the fault. To evaluate the performance of the proposed FTC system, the following saturation faults are simulated using the nonlinear equations of an F-16 aircraft longitudinal motion: 1. A 0.5 reduction of the saturation limit in the aircraft engine is simulated at time 1.0 sec. For this case, the aircraft is at an altitude of 30000 ft and speed of 600 ft/sec and the desired next maneuver is to reach 36000 ft with speed of 650 ft/esc and within 25 sec (Fig. 6.15). 2. A 0.8 reduction of the saturation limit in the aircraft engine is simulated at time 5.0 sec. For this case, the aircraft is at an altitude of 30000 ft and speed of 400 ft/sec and the desired next maneuver is to increase the altitude and speed linearly to 33000 ft and 500 ft/esc, respectively, within 30 sec (Fig. 6.16). 3. The saturation limit of the elevator surface is reduced to ±1 deg. For this case, the aircraft is at an altitude of 30000 ft and speed of 400 ft/sec and the desired next maneuver is to increase the altitude and speed linearly to 33000 ft and 500 ft/esc, respectively, within 30 sec (Fig. 6.17). 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.15 shows that, without fault accommodation, the engine thrust signal T reaches its limit and the aircraft speed and altitude diverge from the desired values after the oc currence of the fault. However, with fault accommodation the engine thrust signal is reduced within its new limit (0.5 of 19000 lb) at the same time the aircraft speed and altitude track the desired values. The same conclusions can be made from Fig. 6.16 for the case of 0.8 reduction of the engine thrust limit. In Fig. 6.17, without fault accom modation, the elevator deflection reaches its new limit imposed by the fault. The results are high overshoot in the aircraft speed and the flight path angle f e diverges from the de sired value. However, with fault accommodation the elevator deflection signal is reduced within its new limit and the aircraft speed and altitude track the new modified reference signals. 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V (ft/sec) T (lb) x 104 -0.5 0 10 20 30 o > < D ■ a -1 0 0 10 20 30 700 650 600 550 500 10 20 30 0 4 3 2 O ) a) T J 1 0 1 30 0 10 20 0.5 - Without fault accom. Reference With fault accom. -0.5 O ) CD ;o 8 -2.5 -3 -3 .5 Time (s) 3.6 3.4 3.2 Time (s) Figure 6.15: Simulation results with and without fault accommodation (case no. 1). 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. h (ft) V (ft/sec) T < lb) x 10 -0.5 -1.5 0 10 20 30 at a ■ o 600 550 500 450 400 350 300 30 0 10 20 4 3 2 O ) 1 0 1 20 30 0 10 x 10 3.35 3.3 3.25 3.2 3.15 3.05 2.95 Time (s) 0.5 — - Without fault accom. - - Reference With fault accom. -0.5 at a 'a -1.5 -2.5 Time (s) Figure 6.16: Simulation results with and without fault accommodation (case no. 2). 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V (ft/sec) T < lb) x 10' 1 0.5 0 -0.5 1 0 5 10 15 20 25 30 1.5 1 0.5 -0.5 -1 -1.5 0 5 10 15 20 25 30 500 400 300 200 100 0 5 10 15 20 25 30 6 4 2 0 2 0 5 10 15 20 25 30 1 0 1 2 3 ■ 4 30 20 25 5 10 15 0 4.5 3.5 .c 2.5 Time (s) Figure 6.17: Simulation results with and without fault accommodation (case no. 3). 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 7 Conclusions and Future Research Directions This chapter presents the main results of what was achieved during the course of this work as well as the future research directions. 7.1 Results • An algorithm of designing fault-tolerant control (FTC) system is developed and analyzed for a class of large scale systems. The FTC system consists of three decentralized schemes: controller; fault detection and isolation; and reconfigurable controller. • The proposed design of an FTC system is implemented in real time on a large scale system; segmented telescope test bed. Three commercial processors work ing in parallel are used to implement the proposed algorithm. Real time results demonstrate the capability of the proposed decentralized FTC system to tolerate sensor failure in large scale systems. 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • Fault-tolerant control method with consideration of new physical constraints caused by the fault is developed for a class of nonlinear systems. The idea is to modify the controller and system mission to fulfill the new physical constraints imposed by the fault. • A new method for tolerating saturation fault is developed for a class of nonlinear systems. • A nonlinear model of an F-16 aircraft longitudinal motion is used to demonstrate the effectiveness of the proposed FTC system. Simulation results demonstrate the effectiveness of the proposed FTC system when a saturation fault is occurred on the aircraft control surfaces and engine. 7.2 Future Research Directions • Decentralized control design of polytopic linear parameter varying systems. • Fault-tolerant control of polytopic linear parameter varying systems subject to ac tuator and sensor faults. • Fault-tolerant control of uncertain polytopic linear parameter varying system. • Develop a toolbox for different methods of fault detection, isolation, and estima tion. 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1] B. Anderson, and J. Moore. Optimal Control. Prentice-Hall, New Jersey, 1990. [2] P. Apkarian, P. Gahinet, and G. Becker, “Self-scheduled //< * > control of linear parameter-varying systems: a design example,” Automatica, vol. 31, pp. 1251- 1261,1995. [3] P. Apkarian and R. J. Adams, “Advanced gain-scheduling techniques for uncertain systems,” IEEE Transaction on Control Systems Technology, vol. 6, pp. 21-32, Jan. 1998. [4] P. Apkarian,Hoang Duong Tuan, and J. Bemussou, “Continuous-time analysis, eigenstructure assignment, and H2 synthesis with enhanced linear matrix inequal ities (LMIs) characterizations,” IEEE Transaction on Automatic Control, vol. 46, pp. 1941-1946, Dec. 2001. [5] G. Bajpai, B.C. Chang, and A. Lau, “Control laws with hierarchical switch logic to accommodate eme-induced sensor failures,” in Proc. 18th Digital Avionics Systems Conference, 1999, pp. 10.C.3-1 - 10.C.3-4. [6] G. J. Balas, I. Fialho, A. Packard, J. Renfrow, and C. Mullaney. On the design of LPV controllers for the F-14 aircraft lateral-directional axis during powered ap proach. In .Proc. American Control Conference, (1): 123-127, 1997. [7] G. 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Abdullah, Ali A.
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Fault -tolerant control in complex systems with real-time applications
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