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Integrated control of traffic flow

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Integrated control of traffic flow

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Content Integrated Control of Trac Flow by Yihang Zhang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree Doctor of Philosophy (ELECTRICAL ENGINEERING) December, 2018 Copyright 2018 Yihang Zhang Acknowledgements I would like to thank Prof. Petros Ioannou, my advisor, for his guidance in regard to research as well as life. I greatly appreciate the life-long-lasting training, support, encouragement, and unlimited patience he has given me during these six and a half years I have been working with him. I also thank Prof. Maged Dessouky, Prof. Ketan Savla, Prof. Mihilo Jovanovic, and Prof. Paul Bogdan for being on my qualication committee and thesis committee. It has truly been a rewarding and amazing journey to study and research at USC with Prof. Ioannou for the past six and a half years. My deepest thanks go to my wife Liyang Hao, for her support, encouragement and tolerance to me, and my parents Dawei Zhang and Xiaojing Zhang, for their unconditional and innite love. ii Table of Contents Acknowledgements ii List of Tables vi List of Figures vii Abstract x 1 Introduction 1 1.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Existing Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Outline of the Report . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Macroscopic Highway Trac Flow Models 14 2.1 First Order Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.1 The Lighthill, Whitham and Richards (LWR) Model . . . . 14 2.1.2 The Cell Transmission Model . . . . . . . . . . . . . . . . . 16 2.2 Second Order Models . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 The Payne Model . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.2 METANET and its Variations . . . . . . . . . . . . . . . . . 19 2.3 A Switching Second Order Model with Variable Speed Limit . . . . 20 3 Combined Variable Speed Limit and Lane Change Control 24 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.1 Model of highway bottleneck . . . . . . . . . . . . . . . . . . 26 3.2.2 VSL conguration and cell transmission model . . . . . . . . 27 3.2.3 Eects of Lane Change Control . . . . . . . . . . . . . . . . 29 3.3 Design of the Lane Change Controller . . . . . . . . . . . . . . . . . 31 3.3.1 Lane Change Recommendation Messages . . . . . . . . . . . 32 3.3.2 Length of LC Control Segment . . . . . . . . . . . . . . . . 32 3.4 Heuristic Variable Speed Limit Controller . . . . . . . . . . . . . . . 34 iii 3.4.1 Virtual Ramp Metering Strategy . . . . . . . . . . . . . . . 34 3.4.2 Constraints on VSL commands . . . . . . . . . . . . . . . . 35 3.4.3 Combination of VSL Control and LC Control . . . . . . . . 36 3.5 Feedback Linearization Variable Speed Limit Controller . . . . . . . 37 3.5.1 Desired Equilibrium Point . . . . . . . . . . . . . . . . . . . 37 3.5.2 Feedback Linearization VSL Controller . . . . . . . . . . . . 39 3.5.3 Robustness with respect to varying demands . . . . . . . . . 42 3.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.6.1 Simulation Network . . . . . . . . . . . . . . . . . . . . . . . 44 3.6.2 Evaluation of the Heuristic VSL Controller . . . . . . . . . . 46 3.6.2.1 Monte Carlo Simulation and Scenarios . . . . . . . 46 3.6.2.2 Performance Measurements . . . . . . . . . . . . . 46 3.6.2.3 Controller Parameters . . . . . . . . . . . . . . . . 47 3.6.2.4 Simulation Results . . . . . . . . . . . . . . . . . . 48 3.6.3 Evaluation of the Feedback Linearization VSL Controller . . 50 3.6.3.1 Simulation Network and Scenarios . . . . . . . . . 51 3.6.3.2 Macroscopic Simulation . . . . . . . . . . . . . . . 51 3.6.3.3 Microscopic Simulation . . . . . . . . . . . . . . . . 55 3.6.3.4 Consistency between microscopic and macroscopic models . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.6.3.5 Performance Measurement and Criteria . . . . . . 58 3.6.3.6 Evaluation Results . . . . . . . . . . . . . . . . . . 59 4 Coordinated Variable Speed Limit, Ramp Metering and Lane Change Controller 63 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2.1 Eect of VSL on the Fundamental Diagram . . . . . . . . . 64 4.2.2 Cell Transmission Model with Ramp Flows . . . . . . . . . . 65 4.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3.1 Design of VSL . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3.2 Design of the RM Controller . . . . . . . . . . . . . . . . . . 68 4.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.4.1 Scenario Setup . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 71 5 Comparison of Feedback Linearization and Model Predictive Strate- gies in Variable Speed Limit Control 75 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2 Nonlinear Model Predictive Control . . . . . . . . . . . . . . . . . . 76 5.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.3.1 Scenario setup . . . . . . . . . . . . . . . . . . . . . . . . . . 78 iv 5.3.2 Performance and Robustness Analysis with Macroscopic Sim- ulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.3.3 Performance and Robustness Analysis with Microscopic Sim- ulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6 Stability Analysis of Cell Transmission Model under All Operating Conditions 86 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.2 Stability of Trac Flow in a Single-Section Road Segment . . . . . 87 6.3 Stability of Trac Flow in a Multi-Section Road Segment . . . . . . 95 7 VSL Control of the Cell Transmission Model under All Operating Conditions 100 7.1 Control of Trac Flow: Single Section . . . . . . . . . . . . . . . . 100 7.2 N-section Road Segment with VSL Control . . . . . . . . . . . . . 107 7.3 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 115 8 Robust VSL Control of Cell Transmission Model with Disturbance121 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.2 Robust Control of Trac Flow in a Single-Section Road Segment . 122 8.3 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 134 9 Conclusion 135 Reference List 137 Appendix A Parts of Proof of Theorem 6.2.1 . . . . . . . . . . . . . . . . . . . . . . . 145 A.1 Case a), i.e. I2 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 A.2 Case b), i.e. I2 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 A.3 Case c), i.e. I2 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 A.4 Case d), i.e. I2 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 A.5 Case e), i.e. I2 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Appendix B Proof of Theorem 6.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Appendix C Proof of Theorem 7.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Appendix D Proof of Theorem 7.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 v List of Tables 3.1 Simulation Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Performance Measurements of Scenario 1 . . . . . . . . . . . . . . . 47 3.3 Performance Measurements of Scenario 2 . . . . . . . . . . . . . . . 48 3.4 Performance Measurements of Scenario 3 . . . . . . . . . . . . . . . 48 3.5 Evaluation Results of Scenario 1 . . . . . . . . . . . . . . . . . . . . 60 3.6 Evaluation Results of Scenario 2 . . . . . . . . . . . . . . . . . . . . 60 3.7 Evaluation Results of Scenario 3 . . . . . . . . . . . . . . . . . . . . 60 4.1 Evaluation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.1 Evaluation Results with Original Parameters . . . . . . . . . . . . . 83 5.2 Evaluation Results under Dierent w 1 . . . . . . . . . . . . . . . . . 84 5.3 Evaluation Results under Dierent cb . . . . . . . . . . . . . . . . 84 5.4 Evaluation Results under Dierent w b . . . . . . . . . . . . . . . . . 84 vi List of Figures 1.1 Trac Flow Control Signs on Highway . . . . . . . . . . . . . . . . 2 2.1 Fundamental Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Triangular Fundamental Diagram . . . . . . . . . . . . . . . . . . . 16 2.3 Discretization of Highway Segment . . . . . . . . . . . . . . . . . . 17 2.4 Rendered ow-density curves (1) . . . . . . . . . . . . . . . . . . . . 20 2.5 Rendered ow-density curves (2) . . . . . . . . . . . . . . . . . . . . 20 2.6 A switch model of VSL . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1 Highway Bottleneck . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Fundamental Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 Conguration of VSL Control System . . . . . . . . . . . . . . . . . 28 3.4 Fundamental Diagram with and without LC Control . . . . . . . . 29 3.5 under dierent trac demands . . . . . . . . . . . . . . . . . . . . 33 3.6 System Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.7 Desired Equilibrium Point . . . . . . . . . . . . . . . . . . . . . . . 37 3.8 Steady State q b under Dierent Demands . . . . . . . . . . . . . . . 45 3.9 Simulation Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.10 Trac Condition in Discharging Section . . . . . . . . . . . . . . . 48 3.11 Controller Performance without Constraints . . . . . . . . . . . . . 52 3.12 Controller Performance with Constraints . . . . . . . . . . . . . . . 53 3.13 System Behavior without Control . . . . . . . . . . . . . . . . . . . 54 3.14 Growth and Discharge of the Queue . . . . . . . . . . . . . . . . . . 55 3.15 Comparison of Macroscopic and Microscopic Models . . . . . . . . . 56 vii 3.16 Fundamental Diagram with Combined Controller . . . . . . . . . . 57 4.1 Eects of LC and VSL on Fundamental Diagrams . . . . . . . . . . 64 4.2 Conguration of the Highway Segment . . . . . . . . . . . . . . . . 65 4.3 Geometry of Simulation Network . . . . . . . . . . . . . . . . . . . 71 4.4 Bottleneck Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5 Vehicle Densities w/ and w/o Control . . . . . . . . . . . . . . . . . 71 4.6 Density Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.7 Queue Length w/ and w/o Control . . . . . . . . . . . . . . . . . . 72 5.1 Simulation System . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 Simulation System . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3 7 with FL and MPC . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.4 Performance sensitivity of no control (black), FL (blue), and NMPC (red) to perturbations on demand d. . . . . . . . . . . . . . . . . . 80 5.5 Performance sensitivity of no control (black), FL (blue), and NMPC (red) to perturbations on C b . . . . . . . . . . . . . . . . . . . . . . . 81 5.6 Performance sensitivity of no control (black), FL (blue), and NMPC (red) to perturbations on d,c . . . . . . . . . . . . . . . . . . . . . . 81 5.7 Performance sensitivity of FL (blue) and NMPC (red) to increasing levels of standard deviation in measurement noise. . . . . . . . . . . 82 6.1 Single Road Section . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.2 All Possible Operating Scenarios . . . . . . . . . . . . . . . . . . . . 90 6.3 Fundamental Diagram for I2 1 . . . . . . . . . . . . . . . . . . . 91 6.4 Fundamental Diagram for I2 2 . . . . . . . . . . . . . . . . . . . 92 6.5 Fundamental Diagram for I2 3 . . . . . . . . . . . . . . . . . . . 93 6.6 Fundamental Diagram for I2 4 . . . . . . . . . . . . . . . . . . . 94 6.7 Fundamental Diagram for I2 5 . . . . . . . . . . . . . . . . . . . 95 6.8 Multiple Section Road Network . . . . . . . . . . . . . . . . . . . . 96 7.1 Road Section with VSL Control . . . . . . . . . . . . . . . . . . . . 102 7.2 Fundamental Diagram of the VSL Zone . . . . . . . . . . . . . . . . 103 7.3 Switching Logic of VSL Controller . . . . . . . . . . . . . . . . . . . 106 viii 7.4 VSL Controlled Road Segment . . . . . . . . . . . . . . . . . . . . . 108 7.5 Fundamental Diagram of Section i . . . . . . . . . . . . . . . . . . . 109 7.6 Phase portrait when I2 1 . . . . . . . . . . . . . . . . . . . . . . 116 7.7 Phase portrait when I2 2 . . . . . . . . . . . . . . . . . . . . . . 116 7.8 Phase portrait when I2 3 . . . . . . . . . . . . . . . . . . . . . . 117 7.9 Phase portrait when I2 4 . . . . . . . . . . . . . . . . . . . . . . 117 7.10 Phase portrait when I2 5 . . . . . . . . . . . . . . . . . . . . . . 118 7.11 Flow rate when I2 4 . . . . . . . . . . . . . . . . . . . . . . . . . 118 7.12 Flow rate with Perturbed v f when I2 4 . . . . . . . . . . . . . . 118 8.1 Design Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.2 State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 8.3 System Behavior of the Perturbed Closed-loop System . . . . . . . 134 ix Abstract Highway congestion is detrimental to trac mobility, safety and the environment. Numbers of studies have been conducted to avoid or relieve highway congestion with dierent trac ow control strategies such as Variable Speed Limit (VSL), Ramp Metering (RM) and Lane Change (LC) recommendation and their combinations. While consistent improvement on trac safety is reported under existing trac ow control strategies in macroscopic and microscopic simulations, the results are rather controversial when it goes to the improvement on trac mobility and the environmental impact, especially in microscopic simulations. Some researchers at- tribute the inconsistencies to the complexity of underlying reasons of the congestion and highly disordered and stochastic behavior at the bottleneck. Therefore, it is necessary to investigate the dynamical behavior of the open-loop trac ow sys- tems under all possible demand levels as well as initial densities in order to nd out the reasons of the chaotic behavior at the bottleneck, and based on which nd an integrated trac ow controller which is able to provide consistent improve- ment in trac mobility, safety and the environmental impact under dierent trac scenarios. In this dissertation, we discover that one of the major reasons of the disordered behavior is the forced lane changes at vicinity of the bottleneck. A lane change controller is proposed which provides lane change recommendations to upstream vehicles in order to avoid the capacity drop. Two types of variable speed limit con- trollers are designed to improve the ow rate at highway bottleneck together with the lane change controller. The combined lane change and feedback linearization x variable speed limit controller, which is build on the rst order cell transmission model, can analytically guarantee the global exponential convergence to the de- sired equilibrium point at which maximum possible ow rate is achieved. Then the combined LC and VSL controller is extended to coordinate with ramp metering controllers. The coordinated VSL, RM and LC controller is able to improve system performance, maintain the queue length on ramps and keep the fairness between mainline and on-ramp ows. Microscopic simulations show consistent improvement under dierent trac demand and scenarios. The proposed controller is compared to the widely used MPC control strategy. Both macroscopic and microscopic simu- lations show that the performance and robustness with respect to model parameter errors and measurement noise of our controller is better than that of the MPC controller. Furthermore, we modify the cell transmission model to include the ef- fect of capacity drop and the decreasing discharging ow of the road section and rigorously investigate its stability properties under all possible trac ow scenar- ios. The analysis is used to motivate the design of variable speed limit control to overcome capacity drop without lane change control and achieve the maximum possible ow under all feasible trac situations. We also consider the case where the system disturbance is included and extend the VSL controller by adding the integral action in order to reject the disturbance while avoiding the capacity drop. xi Chapter 1 Introduction Due to the rapidly increasing demand for transportation and mobility, congestion has become a signicant problem all around the world. Congestion has negative impact on trac mobility, safety and the environment. In the United States during 2014, the yearly delay time per auto commuter due to congestion was 42 hours, which is increased by 13.5% compared to 37 hours in 2000. The fuel wasted in congestion is 19 gallons per commuter per year in 2014, which increased by 26.7% when compared to 15 gallons per commuter per year in 2000 [1]. Unstable trac ow conditions on highway segments are known to increase the possibility of crash [2]. In highway trac, bottlenecks often arise due to incidents, construction, merge or diverge points and other road conditions. When trac demand is higher than the capacity of the bottleneck, congestion occurs. One possible way to solve highway congestion problem is to expand highway networks, which is usually constrained by long building period and limited capital investment. Hence, increasing the utility of existing road infrastructure with advanced trac control strategies is a more attractive solution to highway congestion. To prevent or relieve highway conges- tion, dierent Intelligent Transportation Systems (ITS) techniques, e.g. dynamic routing, driver information systems, variable speed limit (VSL), ramp metering 1 (a) Variable Speed Limit (b) Ramp Metering (c) Lane Change Control Figure 1.1: Trac Flow Control Signs on Highway (RM) etc., are widely studied and applied to improve the eciency of existing road networks [3, 4, 5, 6, 7]. 1.1 Problem Description There are various factors that may lead to highway congestion. For example, capac- ity drop at highway bottlenecks which deteriorates the maximum possible through- put of a highway, overloaded mainline trac which creates shockwave propagating upstream and excessive on-ramp ow which disturbs the mainline trac. Due to the variety and complexity of underlying reasons of highway congestion, it has never been an easy task to nd a control strategy which is able to eciently regulate the trac ow, improve trac mobility, safety and the environment impact when congestion occurs on highway. Numbers of previous studies have been conducted to relieve or postpone highway congestion with dierent control strategies. Variable speed limit (VSL), ramp me- tering (RM) and lane change (LC) control are among the most intensively studied and applied highway trac ow control strategies. Variable speed limit dynami- cally changes the speed limits along a highway segment thus regulate the trac ow and improve trac condition at the bottleneck. Ramp metering limits the number of vehicles entering the highway in unit time from on-ramps in order to maintain 2 an appropriate demand on highway and attenuate the disturbance of ramp ows to the mainline. Lane change control provide lane change instructions to vehicle drivers therefore help the trac ow avoid closed lanes and eciently distributed to all open lanes. Existing works on the development and evaluation of VSL, RM and LC control have reported consistent improvements in trac safety in theories, macroscopic simulations, microscopic simulations and eld tests [8, 9, 10], while the impact on trac mobility and environment is rather controversial. Although most of previous studies are able to show improvements of trac mobility in macroscopic simula- tions with dierent trac ow control strategies, when it comes to microscopic simulations and eld tests, these improvements are not consistent under dierent trac conditions or incident scenarios. In some cases, the travel time is improved and in others deteriorated due to the deployment of trac ow controllers which raises questions as to the ability of VSL, RM and LC to improve trac mobility [11, 12, 13, 14, 15, 16]. Most researchers attribute the inconsistencies in travel time improvement to the highly disordered and stochastic trac conditions at congested bottlenecks, which are dicult to predicts and regulate [16, 17, 13, 8]. While these arguments have an element of truth, some questions we need to ask here is as follows: 1. What is the behavior of the trac ow in a road network? Under what condition the road network will get congested and what are the reasons of the disordered behavior of the trac ow at the bottleneck? 2. Is it possible to reduce the disorder at the bottleneck, therefore the consistency between macroscopic and microscopic simulations can be achieved? 3. Is it possible to nd ecient VSL, RM and LC control strategies which are able to improve the trac mobility at highway bottlenecks and robust to dierent incident scenarios? 3 4. Given the complexity of underlying reasons of highway congestion, is it pos- sible to apply multiple trac ow control strategies simultaneously in a inte- grated and systematic manner, such that dierent control strategies can work along with each other coordinately without deteriorate the benet introduced by other control strategies. 5. Is it possible to nd a trac control strategy that can improve the trac mobility under all possible trac scenarios and capacity constraints as well as initial conditions? In this study, to answer the above questions, the problem of analysis of trac ow systems and the design, analysis and evaluation of integrated VSL, RM and LC controller for highway trac is addressed. The goal of the integrated controller is to stabilize and homogenize the trac ow upstream a highway bottleneck, there- fore improve the trac mobility, safety and the environmental impact. We also evaluate the robustness of the integrated controller with respect to dierent levels of trac demand, model parameters and measurement noise in both macroscopic and microscopic simulations. Furthermore, the open-loop stability properties of the modied cell transmission trac ow model (CTM) which takes the capacity drop phenomenon into consideration under all possible trac ow scenarios are investigated, which motivates the design of a VSL controller which is able to avoid the capacity drop, stabilize the system and maximize the ow rate at the bottle- neck. The VSL controller is extended with integral action in order to reject system disturbance. 1.2 Existing Work In the past several decades, numerous studies have been conducted to explore the eect of VSL, RM and LC control on trac mobility, safety and the environmental impact. 4 VSL control has been one of the widely studied highway trac control tech- nologies since the 1990s [18]. Papageorgiou et al. studied the eect of VSL on the fundamental diagram in [19]. It is shown that VSL control decreases the slope of the fundamental diagram when the vehicle density is lower than the critical value and increases the critical density. The ow at the same density would be higher with VSL in over critical conditions. Muralidharan et al. proposed a MPC VSL controller based on the LN-CTM model that is able to recover the bottleneck from capacity drop and obtain an op- timal trajectory in the absence of capacity drop [20]. In 2014, Frejo et al. proposed a hybrid MPC controller which combines VSL with ramp metering. The proposed method reduced the computation load of the receding horizon optimization by using genetic and exhaustive algorithms while achieving a good performance in simulation [21]. In [22], Khondaker and Kattan designed a MPC VSL controller based on a microscopic car following model with the assumption of a connected vehicle en- vironment. The proposed method predicts trac conditions on the microscopic level and optimizes a cost function which is the weighted sum of TTT and time to collision (TTC), therefore improves both trac mobility and safety. The method was evaluated using a microscopic simulation model based on the commercial soft- ware, VISSIM. Signicant improvement on travel time is demonstrated. However, the authors assumed that all vehicle information is available in real time and the vehicle states can be accurately predicted, which is very dicult, if at all possible. In 2013, Carlson et al. [23] proposed two local feedback VSL controllers. The lo- cal feedback controllers were compared to a nonlinear optimal controller via macro- scopic simulations. Results showed that the simple feedback controllers can provide similar improvement with respect to the total time spent (TTS) as the optimal controller by using much lower computational eort. The method is extended to 5 multiple bottlenecks in [24] and evaluated to be also eective in microscopic sim- ulations in [25]. In [26], Jin and Jin proposed a proportional-integral (PI) VSL controller to maximize the bottleneck throughput with only one VSL sign by lo- cally stabilizing the vehicle density at a critical value. Since the analysis is local there is no guarantee that a trac disturbance would not lead to a capacity drop and unstable situation. In addition it is not clear how the design for one section can be extended to multiple sections upstream the bottleneck. In [27], Hegyi et al. proposed the SPECIALIST VSL controller strategy based on shockwave theory. The SPECIALIST method detects the shockwave upstream the bottleneck and uses VSL to make the shockwave accumulate slower and dis- sipate faster thus dampen the shockwave and improve trac mobility. In [28], a local feedback VSL control strategy integrated with ramp metering is proposed based on the fundamental diagram. An extended version of this control strategy is evaluated in [29] with microscopic simulations. The method is shown to be able to improve freeway eciency as well as be robust with respect to modeling error and measurement noise. The eects of VSL on trac safety and the environment is also assessed. In [30], Abdel-Aty et al. showed that well-congured VSL strategies can decease the crash likelihood but large gaps of speed limit in time and space may increase it. No improvement in travel time is observed in this study. In [31], a genetic algorithm was used to choose the control parameters in order to minimize the rear-end collision risks near freeway recurrent bottlenecks. With the proposed control strategy, the VSL control reduced the rear-end crash potential by 69.84% for the high demand scenario and by 81.81% for the moderate demand scenario. [32] evaluated the environmental impact of the VSL and LC control method proposed in [33] with a microscopic emission model CMEM and a macroscopic one MOVES. It is shown that the environmental benets are evaluated to be qualitatively similar with both models while the microscopic CMEM is more sensitive to transient process. 6 In[34], a MPC VSL strategy was proposed using a car-following model to reduce both total time spent (TTS) and total emissions. It is shown that a reduction of TTS alone may not reduce the total emissions. [22] showed that in case of 100% penetration rates of connected vehicles, optimizing for safety alone is enough to achieve simultaneous and optimum improvements in all measures. However, in case of lower penetration rate, a higher collision risk was observed when optimizing for only mobility or fuel consumption. The aim of RM is to adjust the on-ramp ow into the mainline in order to improve the overall trac condition. RM has been widely used in United States and the Europe [35, 36]. ALINEA, one of the most popular RM strategies, is a heuristic local feedback control method with integral action [6]. In [37], ALINEA is expanded to MALINEA, which includes the mainline oc- cupancy upstream the on-ramp in the feedback loop. MALINEA addresses two main disadvantages to ALINEA. The rst is that although ALINEA optimizes the occupancy downstream of the entrance ramp, congestion can still occur upstream of the ramp. The second is that the optimal detector location can be dicult to determine. [38] proposed FL-ALINEA which includes feedback downstream ow rate in- stead of occupancy and ALINEA/Q algorithm which combines queue control with ALINEA. MALINEA addresses two main disadvantages to ALINEA. The rst is that although ALINEA optimizes the occupancy downstream of the entrance ramp, congestion can still occur upstream of the ramp. The second is that the optimal detector location can be dicult to determine. Its formula is identical to the for- mula used for traditional occupancy-based ALINEA, except that it measures ow, and tries to reach a set point ow rather than a set point occupancy. However, when the occupancy is over the critical occupancy, the metering rate is set to the minimum rate, since the freeway is already over capacity. ALINEA/Q algorithm calculates two metering rates. The rst rate is calculated exactly the same as in the 7 traditional ALINEA algorithm. The second rate that is calculated is the minimum rate needed to keep the ramp queue at or below the maximum allowable queue length. The nal calculated rate is the greater of either the ALINEA rate or the queue control rate. Some model-based RM algorithms are also developed. Coordinated ramp meter- ing is based on a second order trac ow model and an optimal control approach that decides the metering rates of multiple ramps in a coordinated manner [39]. Coordinated ramp metering is basically a vectorization of the ALINEA equation, which uses vectors of occupancy, and 2 control gain matrices to return a vector of metering rates. SWARM is a data-based ramp metering strategy which uses linear regression of measured data to predict the density [40]. Despite the intensive application of RM, it is recognized that ramp metering can only control the vehicle density immediately downstream the on-ramp therefore barely improves the overall trac condition in practice, especially when the mainstream demand is high [41, 42]. The above limitations of RM motivates the investigation of combining ramp metering with mainline trac control strategies such as VSL. Previous eorts to study the eect of lane changes at bottlenecks and develop trac ow control strategies with consideration of lane management include the following: In 1986, Rathi et al. [43] developed a microscopic simulation model to evaluate the eect of LC control in a freeway work zone at dierent driver compliance rate. In 1988, Mahmassani et al. [44] applied a macroscopic simulation model to evaluate lane closure strategy for planned work zone. The work in [43, 44] is focused on long-term lane closure strategies rather than temporary lane closures. In 1998, Schaefer et al. [45] assessed the eectiveness of overhead lane control signals. The signals are placed at 1/2 mile intervals ahead of the highway incident area and indicate lane closure with red \x" symbols. A microscopic simulation 8 using SLAM was used to evaluate the performance of the lane change signal on time delay. In 1999, Jha et al. [46] evaluated three dierent lane control signal settings for the tunnel of I-93 South. Yellow and red overhead signals were applied ahead of incident location and evaluated with microscopic simulator MITSIM. The study showed that under incident condition, TTT is sensitive to upstream road geometry and driver compliance rate. Carelessly congured LC signal settings may result in increase of TTT. In [47], Jin stated that systematic lane changes can seriously deteriorate trac safety and eciency during lane drop, merge, and other types of bottleneck. The author introduced an entropy condition for the multi-commodity LWR model and solve the Riemann problem inside a homogeneous lane-changing area. In [48], Laval and Daganzo also conrmed that lane changes at the bottleneck reduce the ow rate and result in capacity drop at the bottleneck. In recently years, researchers start to examine the combination and integration of dierent trac ow control schemes. In [49], Baskar et al. proposed a MPC approach to nd optimal speed limits and lane allocations for platoons. The method is simulated on a 2-lane highway segment and reported to improve travel time by 5% - 10%. It is assumed that all vehicles are controlled by road-side controllers. In 2014, Roncoli et al. [50] proposed a MPC-based trac control strategy for multi- lane motorways, which integrates VSL, ramp metering and lane allocation. The authors adopted the rst order ow model and treated each lane as dierent cells. MPC is designed based on a cost function which penalizes TTS, queue length on the ramps and amplitude of oscillations. Simulation results show that VSL performs much better when combined with ramp metering and lane allocation. The coordination of RM and VSL involves consideration of network mobility, on-ramp queues and fairness between the mainline and the ramps. The objective is to keep a balanced delay time between vehicles on the mainline and the ramps and avoid long queues on the ramps from spilling back to the urban road network. 9 Past eorts to integrate ramp metering with variable speed limit control include the following: [51, 52] chose the optimal VSL and RM commands based on a second order model in an open-loop manner. [53] developed a combined VSL and RM controller by using model predictive control (MPC) based on the METANET model. [54] combined VSL and coordinated RM using an optimal control approach. [41] used a MPC approach to generate the VSL commands which coordinate with pre- existing RM controllers. [7] designed a MPC-based RM controller with a linearized rst-order model which is equipped with a heuristic VSL controller. The design of the coordinated VSL, RM and LC controller is based on the rst-order cell transmission trac ow model, which during the recent years was used to develop variable speed limit (VSL) control strategies. In [55], Hadiuzzaman et al. proposed a model predictive control (MPC)-based VSL control strategy to relieve congestion caused by active bottleneck which introduces capacity drop. No signicant improvement was shown in bottleneck throughput. The reason given by the authors of [55] for the lack of improvements by the proposed VSL was that the model and data used were not accurate enough. In [20], an MPC-based coordinated VSL and ramp metering (RM) controller is proposed based on the link-node CTM. The VSL and RM control commands are computed by relaxing the receding-horizon optimization problem into linear programming. In [56], the CTM model is expressed in a piecewise ane switching-mode form, based on which an MPC-based VSL controller is developed to attenuate shockwave. In [57], Gomes et al. performed a thorough analysis of the equilibrium points and their stability properties of the CTM model. However, the authors did not take the capacity drop phenomenon into consideration. In addition, the convergence rate at which the system states converge to the equilibrium points is not specied. Ref- erence [58] developed sucient conditions for the stability of the equilibrium points of CTM in terms of connectivity of a graph associated with the trac network. The results of [57] and [58] are established based on the monotonicity of CTM. However, 10 if the CTM is modied to account for capacity drop and the fact that the discharg- ing ow rate of a congested road section decreases with density [5, 59, 26, 60], then the CTM is no longer monotone. A nite horizon optimal routing and ow control strategy is proposed in [58]. The stability and convergence of the closed-loop sys- tem to a desired equilibrium however has not been established. In [61], the authors analyzed the equilibrium points and their stability properties under feasible and infeasible demand, however the capacity drop phenomenon and trac ow control is not considered. In [62], sucient conditions for global asymptotic stability and global exponential stability of the equilibrium points of discrete-time CTM model are developed using vector Lyapunov functions. In [63], the authors proposed a feedback control law that guarantees the global exponential stability of the desired equilibrium point of the CTM model. The control input in this case is the ow rate itself. It is not clear, however, how to implement the ow controller with VSL control. 1.3 Contribution Motivated by the questions in Section 1.1, this report dedicates to the design, analysis and evaluation of integrated trac ow control strategies which is able to provide consistent improvement in trac mobility, safety and environmental impact. Main contributions of this report is summarized as follows: Discover that forced lane changes at highway bottleneck is one of the main reasons of capacity drop phenomenon. Therefore capacity drop can be re- lieved and avoided by providing appropriate lane change recommendations to vehicle drivers. A routine of designing the lane change controller for highway bottleneck is proposed. Combined with the lane change controller, an heuristic integral variable speed limit controller and a feedback linearization variable speed limit controller 11 are designed. The combined LC and VSL controller guarantees the global exponential stability of the optimal equilibrium point of the rst order cell transmission model at which maximum bottleneck ow is achieved. Extend the combined VSL and LC controller to coordinate with ramp meter- ing controller while the global exponential convergence of the optimal equilib- rium point is still achieved. With the coordinated VSL, RM and LC controller, mobility of the bottleneck, queue length on the on-ramps and the fairness be- tween on-ramp and mainline trac ows are considered simultaneously. Evaluate the integrated trac control strategy with microscopic simulator VISSIM [64]. Simulation results show that the integrated controller is able to provide signicant and consistent improvement in trac mobility, safety and environmental impact. Compare the proposed integrated trac controller with model predictive con- troller in both macroscopic and microscopic simulations. Comparison results show that the integrated controller proposed in this report has similar or bet- ter performance under dierent levels of trac demand, model perturbations and measurement noise. Modify the CTM to include the eect of capacity drop and the decreasing discharging ow of the road section. We then consider all possible trac ow scenarios, identify all equilibrium points and analyze their stability properties for both single-section and multi-section road segment. Design a VSL controller which can deal with all possible demand levels and ca- pacity constraints as well as under all initial density conditions. The proposed VSL controller guarantees that the density in each section converges exponen- tially fast to a single equilibrium point which corresponds to the maximum 12 possible ow and speed that is dictated by the demand level and capacity constraints. Design a VSL controller with integral action which is able to reject disturbance to the system and guarantee global stability of the desired equilibrium point. 1.4 Outline of the Report The rest of this report is organized as follows: Chapter 2 presents the models of highway trac ow. The design and analysis of two types of combined variable speed limit and lane change controller are demonstrated in Chapter 3. The con- troller is extended to a combined variable speed limit, ramp metering and lane change controller in Chapter 4. Chapter 5 devotes to comparison of feedback lin- earization and model predictive VSL controller. Chapter 6 presents the stability analysis of the open-loop CTM. Chapter 7 presents the design of the VSL con- troller which avoids capacity drop and guarantees the global stability of the desired equilibrium point, which is extended in Chapter 8 to deal with system disturbance. 13 Chapter 2 Macroscopic Highway Trac Flow Models Macroscopic trac ow models describe the dynamics and evolvement of aggregated trac states, e.g. vehicle density , ow rate q and ow speed v. Based on the order of system dynamics, macroscopic trac ow models can be classied into rst order models and second order models. Macroscopic models are more suited for model-based controller design techniques. We will talk about several popular ones in this chapter. Only general theory and formulation of the trac models will be discussed here. Boundary conditions and detailed analysis of the models will be discussed when used in following chapters. 2.1 First Order Models 2.1.1 The Lighthill, Whitham and Richards (LWR) Model The modeling of macroscopic trac ows starts from the analog between trac ows and compressible uid ows. The Lighthill, Whitham and Richards (LWR) Model is the one of the earlies continuum trac ow models, which is proposed in 14 Figure 2.1: Fundamental Diagram 1955 however still widely used nowadays [65, 66]. The LWR model is described by the following three equations: @(x;t) @t + @q(x;t) @x = 0 (2.1) q = v (2.2) q = Q() (2.3) in the equations above, the vehicle density and ow speed q are both continuous functions of the location x and the time t. Equation (2.1) is the rst order partial dierential equation which describes the mass conservation law, which holds at all times. Equation (2.2) describes the relationship between ow speed, vehicle density and ow rate. By denition, ow rateq is the product of vehicle density and ow speed v. [65] and [66] both observed that the equilibrium ow rate q is a function of the density , as shown in (2.3), which is usually referred to as the fundamental diagram. Equation (2.3) is usually achieved by empirical data. Fig. 2.1 shows the general appearance of the fundamental diagram of a highway segment. From Fig. 2.1, we can see that the ow rate increases with vehicle density when is less than a critical 15 Figure 2.2: Triangular Fundamental Diagram value c . When> c , the ow rate will decrease as drivers slow down the vehicles to ensure safety. The ow rate becomes zero again when reaches the jam density j , as the speed decreases to zero. Other than the parabolic fundamental diagram, the triangular fundamental diagram shown in Fig. 2.2 is also widely used. The fundamental diagram describes the relationship between the equilibrium ow rate and density. Therefore, one basic assumption for all rst order trac ow models is that the ow speed always equals the the equilibrium speed which is a static function of the density , given by v =Q()=. 2.1.2 The Cell Transmission Model The LWR model is a continuum model in both time and space. An closed form solution of the equations (2.1)-(2.3) is dicult or even impossible nd. In 1994, Daganzo proposed the cell transmission model (CTM) in [67] by discretizing the LWR model in space and adopting the triangular fundamental diagram. In the cell transmission model, a highway segment is divided into sections with typical length of 500m - 1000m. The vehicle density and ow speed are assumed to be homogeneous within a single section. Fig. 2.3 shows 3 successive sections of a 16 1 i   section i 1 i q  i q Figure 2.3: Discretization of Highway Segment highway segment (i1;i andi+1). The partial dierential equation (2.1) becomes the following ordinary dierential equation, d i (t) dt = [q i (t)q i+1 (t)]=L i (2.4) where i ,q i andL i are the density, in- ow rate and length of sectioni. By adopting the triangular fundamental diagram, we have q i (t) = minfv i1 i1 ;C i ;w i ( j;i i )g (2.5) In equation (2.5), the term v i1 i1 is the attempting ow from section i 1 to sectioni, or the demand part of the fundamental diagram. C i denotes the capacity of section i, that is, the maximum possible ow rate of section i. Therefore, the term minfC i ;w i ( j;i i )g denes the maximum ow that sectioni can receive, or the supply part of the fundamental diagram. In this report, the integrated trac ow controller is designed based on the rst order CTM model. 17 2.2 Second Order Models 2.2.1 The Payne Model In the rst order models, the ow speed is assumed to be equal to the equilibrium speed at all times, which neglects the transient behavior of speed following. In eort to catch the transient eects, Payne modied the LWR model to a second order one by adding the acceleration equation (2.6) [68]. @v @t = v @v @x | {z } convection + 1 [V e ()v | {z } relaxation @ @x |{z} anticipation ] (2.6) The acceleration equation consists of three terms: 1. Convection This term describes the eect of downstream ow speed on the acceleration. If the dierence between downstream speed and v(x;t) is large ( @v @x large), vehicles need larger acceleration/deceleration to catch up with downstream. 2. Relaxation This term is proportional to the dierence between v(x;t) and the desired equilibrium speed V e () which is given by the fundamental dia- gram. It catches the eect of tracking the equilibrium speed. 3. Anticipation The anticipation term describes the eect of downstream den- sity on the acceleration by assuming the drivers look ahead. If the downstream density is lower than (x;t), the driver will accelerate and vice versa. This term is also inversely proportional to (x;t) itself for safety consideration. Equations (2.1)-(2.3) and (2.6) yields a complete second order motorway traf- c model. The major dierence with the LWR model is the ow speed becomes dynamic instead of static. 18 2.2.2 METANET and its Variations Since METANET, the modied spatial and temporal discrete version of Payne's second order model, is quite popular for the design and simulation of trac control strategies in the literature, it is presented here as follows, v i (k + 1) = v i (k) + T L i [v i (k)v i1 (k)v i (k) 2 ] + T [V e ( i (k))v i (k)] T L i i+1 (k) i (k) i (k) + T L i i r i (k)v i (k) i (k) + (2.7) i (k + 1) = i (k) + T L i i [q i1 (k)q i (k) +r i (k)s i (k)] (2.8) q i (k) = i (k)v i (k) i (2.9) V e ( i (k)) = v f exp[ 1 m ( i (k) c ) m ] (2.10) where m > 0 is a parameter of the parabolic fundamental diagram. The existing research work modied theV e () term in the speed dynamic equa- tion (2.6) to incorporate eects of VSL. One way to modify the equilibrium speed curve V e () is to scale the parameters of the curve, such as the free ow speed and the critical density, by the ratio of imposed VSL to the original speed limit ([19, 23]). Let b i (k) = V R;i (k) v f , the three parameters of V e ( i (k)) were rendered as follows, v f [b i (k)] = v f b i (k) c [b i (k)] = c f1 +A m [1b i (k)]g (2.11) m [b i (k)] = m [E m (E m 1)b i (k)] 19 where v f , c , m denote the specic non-VSL values of the parameters; A m and E m are constant parameters to be estimated from data. An illustration of the rendered fundamental diagram using (2.11)is shown in Figure 2.4. Another modication of the curve is to replaceV e () by the minimum of imposed VSL and the original V e (), which was proposed in [53] as follows: V e ( i (k)) = min n v f exp[ 1 m ( i (k) c ) m ]; (1 +b)V R;i (k) o (2.12) where b is the non-compliance factor. An illustration of the rendered fundamental diagram using (2.12)is shown in Figure 2.5. A third modication ([69]) is to replace V e () by V R . Figure 2.4: Rendered ow-density curves (1) Figure 2.5: Rendered ow-density curves (2) 2.3 A Switching Second Order Model with Variable Speed Limit The second order Payne model and METANET model are both originally estab- lished without consideration of VSL. In [19], Papageorgiou et al. introduced the eect of VSL by assuming that VSL changes the parameters in the parabolic fun- damental diagram. However, in fact, when we consider the drivers, who are the 20 \actuators" of the VSL control, their behaviors switch between the car following mode and the speed limit tracking mode subject to safety constraints. When external speed commands are imposed (assume they are regulatory in- stead of advisory, otherwise, a noncompliance factor could be added), drivers will follow the commands and ignore the downstream trac condition if it is safe to do so. Otherwise, drivers will follow the vehicle in front of them and ignore the speed commands. For example, if a vehicle is traveling at 70 km/h and the leading vehicle is not decelerating; at the same time, the displaying speed limit is 50 km/h, the driver will start to decelerate towards 50 km/h. If on the other hand, the leading vehicle is decelerating, the driver of the following vehicle will ignore the 50 km/h speed limit and start to decelerate with the leading vehicle towards a new safe headway. Once a new safe headway is reached, the driver will consider to follow the displaying speed limit if his/her speed is still above the speed limit. Considering the fact above, Wang and Ioannou proposed a switching second order model for VSL in [17]. In this model the acceleration equation is given as @v @t = 8 > > > > > > > > < > > > > > > > > : u(t;x); if V R <v f and v>V R and u()<f(); f(t;x); otherwise. (2.13) where u(t;x) = a(V R v) describes the dynamics of vehicle following the speed limit. f(t;x) is the original acceleration equation (2.6) in Payne model. 21 For a freeway segment divided into N sections, discretize both in time and space the second order macroscopic model with VSL control (2.13) and (2.1)-(2.3), the new model is, v i (k + 1) =v i (k) + 8 > > > > > > > > < > > > > > > > > : u i (k); if V R;i (k)<v f and v i (k)>V R;i (k) and u i (k)<f i (k;d(k)); f i (k;d(k)); otherwise. (2.14) where v i (k) is the speed of section i at time kT and T is the simulation time step length; V R;i (k) is the imposed variable speed limit in section i at time kT ; and u i (k) = K P [V R;i (k)v i (k)] =K P e i (k) (2.15) e i (k) = V R;i (k)v i (k) (2.16) where K P > 0 is a model parameter; and the term f i (k;d(k)) derived from the car following model is f i (k;d(k)) = T L i i1 (k) i (k + 1) + v i1 (k)[ p v i (k)v i1 (k)v i (k)] + T [V e ( i (k))v i (k)] (k)T L i i+1 (kd(k)) i (k) i (k) + T L i i r i (k)v i (k) i (k) + (2.17) where i (k) and r i (k) are the density and on ramp ow of section i at time kT , respectively; L i is the length of section i; i is the number of lanes of section i; 22 i u p K 1 1 z , R i V i e i v ( ) f Figure 2.6: A switch model of VSL ;;; are model parameters; and (k) is a time varying model parameter and d(k) is a time varying delay, (k) = 8 > < > : high ; if i+1 (k) i (k); low ; otherwise. (2.18) d(k) = 8 > < > : d high ; if i+1 (k) i (k); d low ; otherwise. (2.19) This switching VSL model is illustrated in Figure 2.6. The switching from car following mode to speed limit tracking mode only take place if the VSL command is lower than the default speed limit, lower than the current speed and if the change in speed is less than that of the predicted car following eects. In other words when the posted VSL is lower than the current speed the vehicles will respond without been in uenced by the density and speed values of the vehicles ahead as vehicle following will switch to speed tracking without violating any safety considerations. 23 Chapter 3 Combined Variable Speed Limit and Lane Change Control 3.1 Introduction As introduced in Chapter 1, inconsistent performance of variable speed limit and ramp metering controllers have been reported in existing studies. Some researchers attribute the inconsistencies to the highly disordered and stochastic behavior at highway bottlenecks. One of the main factors of the disordered behavior at highway bottlenecks is the capacity drop phenomenon, where the maximum achievable trac ow rate decreases when queues form [70, 71]. Under certain speed limit, when the density at the vicinity of the bottleneck increases to be higher than some critical value, a queue forms upstream of the bottleneck which decreases the capacity of the bottleneck. Capacity drop makes the dynamics of the trac ow at bottleneck highly unstable, which is dicult for VSL control to maintain a high ow rate. [23] claims that one of the main factors that introduce capacity drop is the inecient acceleration of vehicles at the bottleneck, thus by providing an acceleration section with reasonable length and regulating the density with VSL, capacity drop can be avoided. Such an approach however has the following drawbacks. First it is dicult to establish in cases of incidents and second enforcing an acceleration section may require reducing the ow upstream considerably. The method in [20] is developed 24 under the assumption that the bottleneck never returns to capacity drop mode from free ow mode, i.e., once the VSL controller recovers the bottleneck from capacity drop, the capacity drop never occurs again. While there is no reason to doubt the reported results, our studies and observations of trac show clearly that forced lane changes in close proximity to the incident or bottleneck is the major cause of capacity drop and once it takes place VSL control will have limited or no eect in improving travel time. Most likely in the reported results which show signicant benets the scenarios did not involve signicant forced lane changes or as in the case of [23] it was prevented by creating an acceleration area before the bottleneck. It should be intuitively clear that once the forced lane changes bring down the speed of vehicles in neighboring lanes there is no way for an VSL control technique to eliminate the capacity drop. In this chapter, we rst proposed a lane change (LC) controller which can avoid or relieve the capacity drop at the bottleneck. Two types of VSL controller are designed to combine with the LC controller. The rst one is an heuristic local feedback controller with integral action. The second one is a feedback lineariza- tion controller which is designed based on the rst order cell transmission model. Together with a lane change controller, the feedback linearization VSL controller guarantees stability of the trac ow and convergence of trac densities to an equilibrium density with an exponential rate of convergence. In contrast to previ- ous studies which relied on linearized models, our approach is based on feedback linearization and the results obtained are global. Therefore from the macroscopic point of view the proposed VSL and lane change control guarantees no capacity drop and maximum ow at the bottleneck. The lane change controller is based on a space model as in this case the control variable is the location of the lane change control commands. This location is found to depend on demand and num- ber of lanes closed. The proposed combined lane change and VSL control design is evaluated using microscopic Monte Carlo simulations under dierent scenarios. 25 Figure 3.1: Highway Bottleneck The microscopic results generated are very consistent with the macroscopic ones and demonstrate consistent improvements to trac mobility and impact on the environment for all the simulated scenarios. 3.2 System Modeling 3.2.1 Model of highway bottleneck Consider a highway segment without on-ramps and o-ramps. A bottleneck is the point with lowest ow capacity. Due to the bottleneck a queue of vehicles forms as trac demand increases. The ow rate of the bottleneck determines the throughput of the entire highway segment. Therefore, the modeling of the bottleneck trac ow is crucial to the design of an ecient trac control strategy. A bottleneck can be introduced by lane drop, incident lane blockage, merge point or other road conditions. Fig. 3.1 shows a highway segment with 5 lanes. A bottleneck is introduced by an incident which blocks one lane. The length of the bottleneck is denoted by L b . We assume that the capacity of the highway segment before the incident is C. Then the ideal capacity of the bottleneck after the incident should be C b = 4 5 C. As we can see in Fig. 3.1, if L b is small, the eect of the density within L b is negligible and will not aect the bottleneck ow. The ow rateq b at the bottleneck is determined by d , the vehicle density of the immediate upstream section of the bottleneck, which is referred to as the discharging section in Fig. 3.1. We adopt the 26 Figure 3.2: Fundamental Diagram assumption of triangular fundamental diagram, that is, when the value of d is low, q b = v f d , where v f is the free ow speed. However, when d is higher than some critical value d,c , i.e. the demand of the bottleneck is higher than its capacity C b , a queue forms at the discharging section which propagates upstream. Forced lane changes performed by the vehicles in the queue reduce the speed of ow in the open lanes. Therefore, the capacity would drop to C 0 b = (1)C b once the queue forms [26, 20, 60]. The relationship between d and q b is shown as solid line in Fig. 3.2 and is described by the equation q b = 8 < : v f d ; d d,c (1)C b ; d > d,c (3.1) where C b =v f d,c ; 2 (0; 1). 3.2.2 VSL conguration and cell transmission model As shown in Fig. 3.3, the upstream highway segment of bottleneck is divided into N sections. The lengths of dierent sections are expected to be similar but not necessarily identical. VSL signs are installed at the beginning of section 1 through section N 1. The speed limit in section N, which functions as the discharging 27 Figure 3.3: Conguration of VSL Control System section in Fig. 3.1, is constant and equals v f , the maximum possible speed given by the fundamental diagram, which would let vehicles in open lanes get through the bottleneck as fast as possible, under the assumption of triangular fundamental diagram. For i = 1; 2;:::;N, we denote the length, vehicle density and the in ow rate of section i with L i , i and q i respectively. For i = 1;:::;N 1, we denote the variable speed limit in section i with v i . The variables i ; q i ; v i are all functions of time t. By conservation law, the dynamics of densities i are described by the dierential equations _ i = (q i q i+1 )=L i ; i = 1; 2;:::;N 1 _ N = (q N q b )=L N (3.2) Under the assumption of triangular fundamental diagram, the ow rate q i can be found as follows: q 1 = minfd;C 1 ;w 1 ( j;1 1 )g q i = minfv i1 i1 ;C i ;w i ( j;i i )g; i = 2; 3;:::;N (3.3) whered is the demand ow of this highway segment assumed to be constant relative to the other variables. j;i is the jam density of section i, at which q i would be 0. w i is the backward propagating wave speed in section i, C i the capacity, i.e. the 28 Figure 3.4: Fundamental Diagram with and without LC Control maximum possible ow rate in section i, given by C i = v i w i j;i =(v i +w i ). We should note that for i = N, C N and N;c are not the same as C b and d,c . When N reaches d,c , q b decreases but section N still has enough space for vehicles in section N 1 to ow in. Therefore, N;c > d,c , C N > C b . The goal of the VSL controller is to stabilize the system described in (3.1) - (3.3) and maximize the ow rate q b . According to (3.1), maximum q b is obtained at N = b , which is a discontinuity point of the fundamental diagram. From the macroscopic point of view, it is possible to nd a VSL controller to maintain that N = d,c [26]. However, microscopic simulations in [33] demonstrate that when congestion occurs at the bottleneck, the queue accumulates so fast that VSL control can hardly reduce the density back to d,c , therefore it fails to maintain maximum ow. The reason is explained in the following subsection. 3.2.3 Eects of Lane Change Control In order to study the eect of lane change control, we build a hypothetical highway segment as shown in Fig. 3.1, which is straight, 8 km long and with 5 lanes, 29 with the microscopic trac ow simulated using the commercial software VISSIM [64]. The VISSIM model is calibrated with typical freeway road geometry and driving behavior. The bottleneck is formed by an incident which blocks the middle lane. We investigate the relationship between the ow of the bottleneck q b and the density d in the 500 m long discharging section immediately upstream the bottleneck under dierent levels of trac demand. Fig. 3.4 shows the relationship between q b and d without any VSL control. The small blue circles describe the fundamental diagram in the case of lane change control. The red asterisks show the corresponding fundamental diagram in the absence of lane change control. The design procedure of LC controller is described in Section 3.3. Observing Fig. 3.4, we can see that when LC control is applied, the capacity of the bottleneck is around 7600 veh/h, which is achieved at d 135 veh/mi. However, when there is no LC control, q b stops increasing even before d reaches 135 veh/mi (around d = 100 veh/mi). The highest ow rate is around 6300 veh/h. The reason why the ow rate in the no control case fails to reach higher level is demonstrated in Fig. 3.1. When vehicles approach the incident spot without being aware that their lane is blocked they are forced to slow down considerably and change lanes. These forced lane changes at low speed cause the trac to slow down in the open lanes before and after the incident leading to lower volume, while the average density of the discharging section, d , is still low. Other parts of the fundamental diagram in the no control case t equation (3.1) very well. Compared to the fundamental diagram with LC control, we can calibrate the parameters as, d,c = 135 veh/mi, C b = 7600 veh/h and = 0:16. The above stated behavior of the bottleneck makes it dicult for VSL control to increase q b at the bottleneck, as VSL is only able to regulate the average density d in the discharging section, but cannot eliminate the forced lane changes at the vicinity of the bottleneck. On the other hand, with the LC control, we can see that 1. no obvious capacity drop is observed at d = d,c ; 30 2. q b at d > d,c is approximately linear with a negative slope w b , which repre- sents the wave propagation rate; 3. most data points scatter close to d = d,c . The points of high density are rare. These observations show that the LC controller is able to reduce the number of vehicle stops in the queue at bottleneck and decrease the vehicle density, which makes the system continuous and easier for the VSL controller to stabilize. As a consequence of the LC control action, in the cell transmission model the relationship between N and q b can be modeled as: q b = 8 < : v f N ; N d,c w b ( j,d N ); N > d,c (3.4) where j,d =v f d,c =w b + d,c . Although the lane change control is able to recover the triangular shape of the fundamental diagram, when the demand is higher than the capacity C b , a conges- tion will still occur at the bottleneck. Now the goal is to design a VSL controller to stabilize system (3.2) - (3.4) by homogenizing the densities in all sections and have them converge to an equilibrium which corresponds to the maximum possible ow as shown in the following section. 3.3 Design of the Lane Change Controller The design of LC controller includes the pattern of the LC recommendation mes- sages and the length of LC controlled segment. As we will explain below the control variable for LC control is the location of the LC recommendation which depends on a nonlinear spatial model that we developed. 31 3.3.1 Lane Change Recommendation Messages Suppose a general highway segment has m lanes, with Lane 1 (Lane m) being the right (left) most lane in the direction of ow. We select the LC recommendation message R i for lane i;i = 1; 2;:::;m using the following rules: 1. For 1im, if lane i is open, R i = \Straight Ahead"; 2. For i = 1(i =m), if lane i is closed, R i = \Change to Left (Right)"; 3. For 1 < i < m, if lane i is closed, lane i 1 and lane i + 1 are both open, R i = \Change to Either Side"; 4. For 1<i<m, if lanei is closed, lanei 1 (lanei + 1) is closed but lanei + 1 (lane i 1) is open, R i = \Change to Left (Right)"; 5. For 1 < i < m, if lane i is closed, lane i 1 and lane i + 1 are both closed, then we check R i1 and R i+1 . If R i1 =R i+1 , then R i =R i1 =R i+1 , else if R i1 6=R i+1 , R i = \Change to Either Side". Rules (1)-(5) determine the LC recommendation messages depending on the inci- dent location. The 5 rules covers all incident cases and are also mutually disjoint. Therefore they are well-dened and self-consistent. 3.3.2 Length of LC Control Segment The control variables in the LC control case are the length of the LC control seg- ment and the location of the LC recommendation. Within that segment, a LC recommendation is given at each section within the segment. The length of the LC controlled segment need to be long enough in order to provide adequate space and time for upstream vehicles to change lanes. Intuitively, if more lanes are closed at the bottleneck, a longer LC control distance is required. In addition, the capacity of the bottleneck and demand will also aect the LC control distance. On the other 32 Figure 3.5: under dierent trac demands hand if the length of LC control segment is too long it may cause other problems as the blocked lane will appear empty to drivers inviting more lane changes in and out of the blocked lane which is going to deteriorate performance in terms of un- necessary maneuvers. We used extensive microscopic simulation studies to develop the following empirical model that allows us to generate the control variable d LC which is the length of the LC controlled section given by the following equation: d LC =n; (3.5) where n is the number of lanes closed at the bottleneck, a design parameter related to the capacity of bottleneck and the trac demand which in our case is found to have the relationship shown in Figure 3.5. For a specic highway segment, the minimum value of required under dierent trac demands can be found by simulation. Since LC signs are only deployed at the beginning of sections, we choose the number of LC controlled sections M, as M = argmin P N i=NM+1 l i d LC ; where l i represents the length of section i. More details can be found in [33]. Here we assume that the LC controlled segment has no on-ramp or o-ramps. The model (3.5) is empirical and more spacial than temporal despite the dependence of on demand which may be time varying. The purpose of the LC control is to ask drivers to start changing lanes before the incident. It is an o and on controller i.e change 33 lanes or not required to change lanes. It is dierent than the VSL controller which is purely dynamic. 3.4 Heuristic Variable Speed Limit Controller 3.4.1 Virtual Ramp Metering Strategy Given the LC controller is applied, VSL controller is expected to work together with LC in order to regulate the vehicle density at the critical value. Our rst attempt to design a VSL controller is a non-model based heuristic one, which adopts the idea of ramp metering algorithm, ALINEA. ALINEA adjusts the on-ramp ow rate to keep downstream density at a desired level [72]. We generalize it to VSL control by regarding each highway section as the on-ramp of its downstream sections and regulating downstream density with VSLs. Unlike ramp metering, VSL cannot di- rectly control the ow rate by stopping vehicles, therefore a multi-section structure as shown in Fig. 3.3 is applied to ensure control eect. The VSL controller in each section is expected to regulate the vehicle density of its downstream sections. The VSL control law is described as follows. Let i (k) = P N j=i j (k)l j = P N j=i l i denote the average vehicle density of section i through section N at time step k, For each 1iN 1, the VSL command of Section i at time step k can be expressed as: V i (k) =V i (k 1) +K I [ d;c i (k)] (3.6) where V i (k) is the speed limit command of section i in control period k. K I is the feedback gain, d;c denote the critical density of the discharging section. In equation (3.6), VSL commands respond to the dierence to a xed reference density, in order to suppress the shockwave and keep the density in discharging section. 34 3.4.2 Constraints on VSL commands To ensure safety, we apply the following constraints to VSL commands in (3.6). 1. Finite Command Space. VSL commands would be hard to comply if take value from a continuous space. Hence, we round VSL commands V i (k) in (3.6) to multiples of 5 mi/h and apply lower/upper bounds to it. This makes the commands clear for drivers and adds dead-zone characteristics to the controller therefore avoid control chattering. 2. Saturation of Speed Limit Variations. It is dangerous to decrease the speed limit too fast in both time and space. The decrease should be within some threshold C v > 0 between successive control periods and highway sections. We don't bound the speed limit variation if the speed limit increases. In this study, C v = 10 mi/h(16km/h). The above described constraints can be presented as follows V i (k)V i (k + 1)C v ; 1iN 1 (3.7) V i (k)V i+1 (k)C v ; 1i<N 1 (3.8) V min V i (j)V max ; 1iN 1 (3.9) Hence, the virtual mainline ramp metering VSL controller can be formulated as follows: V i (k) =V i (k 1) + [K I ( d;c i (k))] 5 (3.10) ~ V i (k) = maxf V i (k);V i (k 1)C v ;V i1 (k 1)C v g (3.11) V i (k) = 8 > > > < > > > : V max ; if ~ V i (k)>V max V min ; if ~ V i (k)<V min ~ V i (k); otherwise (3.12) 35 Figure 3.6: System Block Diagram In (3.10), [] 5 is the operator which rounds a real number to its closest whole 5 number. In (3.12),V max andV min are the upper and lower bounds of VSL commands respectively. 3.4.3 Combination of VSL Control and LC Control As described in Section 3.3 and Section 3.4, the LC controller is designed based on bottleneck layout and trac demand. The VSL controller takes LC controlled segment as the discharging section and deploys VSL signs at upstream of it to keep desired density and smooth the trac ow. The eect of LC controller helps the VSL controller to be more eective in generating the desired benets. The block diagram of combined VSL & LC control system is shown in Fig. 3.6. 36 Figure 3.7: Desired Equilibrium Point 3.5 Feedback Linearization Variable Speed Limit Controller In this section, we designed a feedback linearization VSL controller based on the cell transmission model (3.2)-(3.4). 3.5.1 Desired Equilibrium Point The fundamental diagram under LC control is shown in Fig. 3.7. We consider the demandd>C b , which may introduce congestion at the bottleneck. From the non- linear system (3.2) - (3.4), we calculate the equilibrium point by setting the deriva- tives in (3.2)-(3.4) to be zero. Let e = [ e 1 ; e 2 ;:::; e N ] T andv e = [v e 1 ;v e 2 ;:::;v e N1 ] T denote the vector of equilibrium density and the corresponding equilibrium speed limits in each section respectively. The desired equilibrium point should be the one at which maximum possible ow rate C b is achieved and the upstream trac ow is homogenized. According to the triangular fundamental diagram (3.4), since the 37 speed limit is constant and equals v f in section N, therefore the optimum equilib- rium density for maximum ow is e N = C b =v f . For section 2 through N 1, we set e 2 = = e N =C b =v f ; v e 2 = =v e N1 =v f : (3.13) hence at the desired equilibrium point, the densities and speed limits in section 2 through N would be the same and the upstream trac ow of the bottleneck is homogenized. Sinced>C b , we need to lower the speed limit in section 1 in order to suppress the trac ow entering the controlled segment. According to (3.3), the equilibrium point satises: v e 1 e 1 =w 1 ( j;1 e 1 ) =C b : which gives e 1 = j;1 C b =w 1 ; v e 1 =C b w 1 =( j;1 w 1 C b ) (3.14) The equilibrium point described in (3.13) - (3.14) is the desired equilibrium point which maximizes the ow at the bottleneck and homogenizes the upstream trac. In addition, it minimizes the average travel time according to the fundamental diagram. Without loss of generality, we assume the length of all sections are the same and equal to unit length. The system (3.2) - (3.4) can be expressed as follows: _ 1 =w 1 ( j;1 1 )v 1 1 _ i =v i1 i1 v i i ; for i = 2;:::;N 1 _ N = 8 < : v N1 N1 v f N ; N d,c v N1 N1 w b ( j,b N ); N > d,c (3.15) In (3.15), the only switching point is N = d,c . This is consistent with real- world, since the capacities of upstream sections are much larger thanC b . As long as 38 system (3.15) converges to the desired equilibrium point, the steady-state bottleneck ow is maximized and upstream trac is homogenized. 3.5.2 Feedback Linearization VSL Controller For the design and analysis of the VSL controller we dene the deviations of the state of (3.15) from the desired equilibrium (3.13) - (3.14) by dening the error system as: e i = i e i for i = 1; 2;;:::;N and u i =v i v e i for i = 1; 2;:::;N 1. Substitute into (3.15), we have _ e 1 =w 1 e 1 v e 1 e 1 u 1 1 _ e i =v e i1 e i1 +u i1 i1 v e i e i u i i for i = 2;:::;N 1 _ e N = 8 < : v e N1 e N1 +u N1 N1 v f e N ; e N 0 v e N1 e N1 +u N1 N1 +w b e N ; e N > 0 (3.16) The transformation of (3.15) to (3.16) shifts the non zero equilibrium state of (3.15) to the zero equilibrium point of (3.16). The nonlinear terms in (3.16) are u i i for i = 1; 2;:::;N1. Now the problem is to selectu 1 throughu N1 in order to stabilize system (3.16) and force all the errors or deviations from the equilibrium state to converge to zero. We introduce the following feedback controller which `kills' all nonlinearities and forces the closed loop system to be linear, an approach known as feedback linearization [73]. We choose u i = (v e i e i i e i+1 )= i ; for i = 1;:::;N 2 u N1 = 8 > > < > > : N1 e N v e N1 e N1 +v f e N N1 ;e N 0 N1 e N v e N1 e N1 w b e N N1 ;e N > 0 (3.17) 39 where i > 0 for i = 1;:::;N 1 are design parameters. This is a switching controller, whose switching logic is based on the value of e N . Since we avoid the capacity drop by applying the LC control, the controller is continuous at the switch- ing point. With the feedback linearization controller (3.17), the closed loop system becomes: _ e 1 =w 1 e 1 + 1 e 2 _ e i = i1 e i + i e i+1 ; for i = 2:::;N 2 _ e N1 = 8 < : N2 e N1 N1 e N +v f e N ; e N 0 N2 e N1 N1 e N w b e N ; e N > 0 _ e N = N1 e N (3.18) The stability properties of the closed loop system (3.18) are described by the following Theorem. Theorem 3.5.1. The equilibrium pointe i = 0, i = 1; 2;:::;N of the system (3.18) is isolated and exponentially stable. The rate of exponential convergence depends on the control design parameters i , i = 1; 2;:::;N 1. Proof For i = 1; 2;:::;N, setting _ e i = 0 in (3.18), the only equilibrium point is e i = 0. From (3.18), we can see that the state e N is decoupled from other states, i.e. _ e N = N1 e N :, whose solution is e N (t) =e N (0) exp( N1 t);8t> 0: (3.19) Since exp( N1 t)> 0 for all t, e N (t) and e N (0) have the same sign for all t> 0, i.e. ife N (0) 0, thene N (t) 0, ife N (0)> 0, thene N (t)> 0 for allt> 0. In other wordse N is either non increasing or non decreasing which means that the state e N 40 will not switch betweene N 0 ande N > 0. Therefore, the dynamics of statee N1 can be written as _ e N1 = 8 < : N2 e N1 N1 e N +v f e N ; e N (0) 0 N2 e N1 N1 e N w b e N ; e N (0)> 0 Let us dene e = [e 1 ;e 2 ;:::;e N ] T , then the system (3.18) can be written in the compact form _ e = 8 < : A 1 e; e N (0) 0 A 2 e; e N (0)> 0 (3.20) where A i = 2 6 6 6 6 6 6 6 6 6 4 w 1 1 1 2 . . . . . . N2 N1 + i N1 3 7 7 7 7 7 7 7 7 7 5 ; i = 1; 2 and 1 =w b ; 2 = v f . A 1 and A 2 are both upper triangular matrices with all diagonal entries being negative real numbers, i.e. A 1 ;A 2 are both Hurwitz. Hence, system (3.20) is exponentially stable. Therefore (3.18) is also exponentially stable. In addition, for a given sign of e N (0) there is no switching taking place in (3.20). The rate of convergence to the equilibrium depends on the design parameters i ;i = 1; 2;::::N 1 which can be tuned to achieve a desirable convergence rate. It would also depend on the sign of the initial condition e N (0) as the dynamics that drive the error system depend on whether the initial condition e N (0) is negative or positive. Q.E.D. The feedback linearization controller (3.17) is continuous in time. To apply it on real highway, we discretize the controller and apply the constraints described in Section 3.4.2. 41 Let u i (k) denotes u i computed by equation (3.17) at t =kT c . We have, v i (k) = [v e i +u i (k)] 5 (3.21) ~ v i (k) = maxf v i (k);v i (k 1)C v ;v i1 (k)C v g (3.22) v i (k) = 8 > > > < > > > : v max ; if ~ v i (k)>v max v min ; if ~ v i (k)<v min ~ v i (k); otherwise (3.23) for i = 1; 2;:::;N 1;k = 0; 1; 2;:::. The above modications will in uence the ideal performance of the VSL con- troller described by Theorem 1. Such modications are necessary in every control application [17, 74, 75] and the way to deal with possible deterioration from the ideal performance is to use the design parameters 1 ; 2 ;:::; N1 to tune the sys- tem using intuition and practical considerations. The selection of the feedback gains 1 ; 2 ;:::; N1 has to consider the trade o between stability and robustness with respect to modeling errors. 3.5.3 Robustness with respect to varying demands In the analysis above, we assume that the demand d is a constant and d > C b . As explained below, the proposed VSL controller is robust with respect to dierent demands. If d<C b , vehicles in the controlled segment would discharge and the densities in each section would be lower than the desired density. The VSLs in each section would increase, but saturated atv f . This situation is easy as due to the low demand congestion can be avoided or managed very well. When d>C b and keeps increasing, according to Theorem 3.5.1, the controller lowers the speed limit in section 1 and limits the number of vehicles that enter the downstream network. Therefore, a queue would be created whose size will 42 be increasing upstream the ow. It appears, at rst glance, as if we are moving congestion from the sections under VSL and LC control to upstream sections. The important question we need to answer is how many vehicles there are in this queue and how fast it grows with and without VSL and LC control in the sections under consideration. In order to analyze the queue size upstream of section 1, we modify the system (3.2) - (3.4) by introducing a new stateQ, which represents the number of vehicles in the queue upstream section 1. We assume thatQ = 0 at steady state ow before the incident. Using the ow conservation equation, we have _ Q =dq 1 (3.24) where d is the trac demand. The in ow rate of section 1, q 1 then becomes q 1 = 8 < : minfd;C 1 ;w 1 ( j;1 1 )g; Q 0 minfC 1 ;w 1 ( j;1 1 )g; Q> 0 (3.25) Equation (3.25) assumes that as long as the queue upstream section 1 is not fully discharged, the in ow rate of section 1 will be as high as the maximum ow rate that section 1 can receive under current 1 . Note that the introduction of Q does not make any dierence to system (3.2) - (3.4) before and during the incident. It only tracks the growth and discharge of the queue upstream section 1. Therefore the stability of the closed-loop system (3.18) is not aected. Hence, with the combined VSL and LC controller, the queue size is measured with Q. In the no control case, a queue forms at section N, whose size is denoted by ^ Q. The following Lemma holds. Lemma 3.5.1. If the demand d > C b , ^ Q grows faster than Q at steady state. In particular, _ Q _ ^ Q =C b < 0 (3.26) 43 Proof Similar to Equation (3.24), we can estimate ^ Q with the following equation _ ^ Q = d ^ q b , where _ ^ Q is the growth rate of ^ Q, ^ q b is the out ow rate of section N without control. Since d > C b , q 1 converges according to Theorem 3.5.1 to the desired ow rate C b exponentially with the combined VSL and LC controller. ^ q b would decrease to ^ q b = (1)C b due to capacity drop. Substituting the steady state values of q 1 and ^ q b in the above equations we obtain (3.26). i.e. at steady state, the growth rate of Q is less than that of ^ Q. Q.E.D. From the analysis above, it is clear that if the demand d increases from below the bottleneck capacity C b to greater than C b and keeps increasing, the combined VSL and LC controller is able to protect the bottleneck from getting congested by suppressing the speed limit in section 1 therefore N can be stabilized at the desired value. On the other hand, in the no control case, the bottleneck is directly exposed to the excessive demand, therefore N increases and leads to capacity drop. Fig. 3.8 plots the steady state bottleneck ow q b with respect to demandd. When d<C b , the bottleneck would not be congested. When d>C b , the bottleneck ow would be stabilized at the maximum value C b by the combined controller in the controlled case. In the no control case, the ow rate would decrease to (1)C b due to capacity drop. Therefore, the combined VSL and LC controller is robust with respect to dif- ferent levels of trac demand. The queue of vehicles grows slower in the controlled case than in the case with no control. 3.6 Numerical Results 3.6.1 Simulation Network We evaluate the combined VSL & LC control method using a microscopic and macroscopic model of the trac ow on a 10 mile (16 km)-long southbound segment of I-710 freeway in California, United States (between I-105 junction and Long 44 Figure 3.8: Steady State q b under Dierent Demands | With Control, - - -Without Control Figure 3.9: Simulation Network Beach Port), which has a static speed limit of 65 mi/h (105 km/h). We build this freeway network in VISSIM and calibrate the microscopic model using historical data provided by [76]. The car following and lane change behavior of the VISSIM model is calibrated and validated using real measurements under static speed limit of 65 mi/h. The studied highway segment has 3-5 lanes at dierent locations. As shown in Fig. 3.9, we assume the bottleneck is introduced by an incident which blocked one lane. The upstream segment of the bottleneck is divided to 10 500m-600m sections. The bars across the highway in Fig. 3.9 are where VSL signs and LC signs deployed. In VISSIM, incidents are simulated by placing stopped bus in certain lane. 45 Table 3.1: Simulation Scenarios Scenario No. Total No. of Lanes Bottleneck Pattern 1 3 Lane 2 Closed 2 3 Lane 3 Closed 3 4 Lane 3 Closed 3.6.2 Evaluation of the Heuristic VSL Controller This section demonstrates the evaluation results of the heuristic VSL controller. 3.6.2.1 Monte Carlo Simulation and Scenarios In this case, we load the network with a demand of 6500 veh/h, 30% of which are trucks. This proportion is much higher than that in reality. We use this setup to test the performance on extremely high truck volume. To verify that the proposed control method generates consistent results under dierent trac conditions, we set up 3 dierent scenarios on the highway network to perform a general evaluation of the proposed method and take 10 sets of Monte Carlo simulation for each scenario. The nal performance measurements are averages of the Monte Carlo simulation results. In the simulation, all lanes are open at the beginning of simulation. 20 min after simulation begins, certain lane is closed near the incident spot in Fig. 3.9 and the controller is activated. The simulation terminates when 2000 vehicles pass through the bottleneck. We x the total number of vehicles that passed through the bottleneck in each simulation, so that the measurements are comparable. Other conguration of scenarios are listed in TABLE 3.1. 3.6.2.2 Performance Measurements We introduce the following measurements to evaluate the performance of the pro- posed control method. To be precise, all measures start from the time instant of lane closing and terminate with the simulation. 46 Table 3.2: Performance Measurements of Scenario 1 Performance Measurement Cars Trucks No Control LC Percentage Changed VSL Percentage Changed VSL+LC Percentage Changed No Control LC Percentage Changed VSL Percentage Changed VSL+LC Percentage Changed Travel Time (min) 29561 20486 -31% 29780 1% 20574 -30% 9539 6925 -27% 9447 -1% 7047 -26% No. of Stops 27503 3007 -89% 25721 -6% 3099 -89% 6757 719 -89% 6344 -6% 783 -88% No. of LC 12344 12089 -2% 11134 -10% 10630 -14% 1245 1314 6% 1094 -12% 1142 -8% Fuel (g/mi/veh) 141.46 120.76 -15% 130.78 -8% 109.64 -22% 599.24 582.77 -3% 520.60 -13% 505.71 -16% CO2 (g/mi/veh) 422.40 354.76 -16% 394.44 -7% 325.56 -23% 1917.86 1864.23 -3% 1665.80 -13% 1617.36 -16% NOx (g/mi/veh) 0.49 0.47 -4% 0.42 -15% 0.39 -20% 22.10 20.38 -8% 20.03 -9% 18.65 -16% Control eects on trac mobility are evaluated by total travel time (TTT) of all vehicles that passed through the highway network (in hours). Let t i;in and t i;out denote the time instant vehiclei enters and exits the network respectively. TTT is given by TTT = P 2000 i=1 (t i;out t i;in ). Control eects on trac safety are evaluated by total number of stops s tot = P 2000 i=1 s i and total number of lane changes c tot = P 2000 i=1 c i , where s i , c i are number of stops and lane changes performed by vehicle i re- spectively. For environmental impact, we measure fuel consumption ratefr = P 2000 i=1 f i =(2000 P 2000 i=1 d i )wheref i ,d i are the fuel consumption and distance traveled in the network by vehicle i respectively. The denition of CO 2 emission rate E CO 2 and NO x emis- sion rate E NOx are similar to fr. 3.6.2.3 Controller Parameters In our simulation, the default speed limit when VSL controller is not active is V f = 65mi/h. VSL decrease threshold C v = 10mi/h (16km/h). Bounds of VSL V min = 30mi/h (48km/h), V max = 65mi/h. Feedback gain K I = 2. 47 (a) Flow Rate (b) Density Figure 3.10: Trac Condition in Discharging Section Table 3.3: Performance Measurements of Scenario 2 Performance Measurement Cars Trucks No Control LC Percentage Changed VSL Percentage Changed VSL+LC Percentage Changed No Control LC Percentage Changed VSL Percentage Changed VSL+LC Percentage Changed Travel Time (min) 29076 19914 -32% 28403 -2% 19854 -32% 9273 6862 -26% 9280 0% 6842 -26% No. of Stops 23889 2541 -89% 22464 -6% 2321 -90% 7206 573 -92% 6665 -7% 535 -93% No. of LC 12404 12944 4% 11254 -9% 11585 -7% 1354 1543 14% 1233 -9% 1373 1% Fuel (g/mi/veh) 141.60 120.67 -15% 128.50 -9% 109.67 -23% 599.35 582.86 -3% 516.70 -14% 502.49 -16% CO2 (g/mi/veh) 421.32 353.24 -16% 386.03 -8% 323.71 -23% 1918.83 1864.56 -3% 1653.81 -14% 1607.09 -16% NOx (g/mi/veh) 0.50 0.48 -5% 0.42 -17% 0.41 -19% 22.22 20.37 -8% 19.94 -10% 18.56 -16% Table 3.4: Performance Measurements of Scenario 3 Performance Measurement Cars Trucks No Control LC Percentage Changed VSL Percentage Changed VSL+LC Percentage Changed No Control LC Percentage Changed VSL Percentage Changed VSL+LC Percentage Changed Travel Time (min) 30033 20378 -32% 30033 0% 20426 -32% 9524 6938 -27% 9650 1% 6914 -27% No. of Stops 27544 2797 -90% 25763 -6% 2681 -90% 6729 695 -90% 6568 -2% 650 -90% No. of LC 12475 12380 -1% 11295 -9% 11084 -11% 1276 1331 4% 1152 -10% 1162 -9% Fuel (g/mi/veh) 143.37 120.71 -16% 132.38 -8% 110.05 -23% 601.58 583.50 -3% 523.66 -13% 506.20 -16% CO2 (g/mi/veh) 427.21 354.32 -17% 398.29 -7% 326.01 -24% 1925.31 1866.57 -3% 1675.56 -13% 1618.96 -16% NOx (g/mi/veh) 0.51 0.47 -6% 0.43 -15% 0.40 -21% 22.16 20.40 -8% 20.12 -9% 18.67 -16% 3.6.2.4 Simulation Results In scenario 1-3, we compare the simulation results under the following control modes: 1) No control; 2) LC control only; 3) VSL control only; 4) Combined VSL & LC control. Fig. 3.10a and Fig. 3.10b show the density of discharging section and the bot- tleneck ow rate during the simulation in scenario 1. After the incident happens at 1200s, the density of discharging section increase dramatically to 250 veh/km and the bottleneck ow rate drops by 50% if LC control is not applied. When VSL control is applied alone, the density of discharging section increases slower however 48 cannot be kept at a lower level. When LC control is applied, the bottleneck ow rate only deceased by about 30%. Since we lose 1 lane out of 3, the ow rate per lane has no drop. LC control ensures a high discharging rate of the bottleneck therefore avoids the congestion. Comparing the ow rate and density curve with and without VSL control, system oscillation is damped by VSL, therefore trac safety improved. Fuel consumption and emissions are also tend to be reduced, which is shown below. The eects of dierent control mode on performance measurements dened in Section 3.6.2.2 are shown in Table 3.2 - Table 3.4. We can observe that the combined control method provides signicant improvement on each measurement, which is also consistent with respect to dierent scenarios. The combined VSL & LC con- trol strategy reducesTTT by 26%-32%,s tot by about 90%,c tot by 3%-14%,fr and E CO 2 by 16%-24%, E NOx by 16%-21%. To study the roles of VSL control and LC control in the combined control strat- egy respectively, we also analyze the case VSL control and LC control are applied to the trac system alone. LC control considerably decreases travel time and number of stops, but cannot reduce number of lane changes, since it only spreads forced lane changes along the LC controlled sections, instead of avoiding them. On the other hand, VSL control homogenizes the density and speed in each section. Drivers are not tend to change lane if densities and speeds are similar in all lanes, therefore VSL control reduces number of lane changes in VSL controlled sections. This is very important for trac safety in truck-dominant highways. Trucks not only take long time and large space to change lane, their large size also blocks the eye sight of drivers, which makes lane change much more dangerous than usual. The environmental evaluation is interesting. VSL and LC control has dier- ent performance on dierent measurements and vehicle types. For trucks, fr and E CO 2 are highly sensitive to accelerations. Large portion of fuel consumption and CO 2 emission are produced by speeding up and down in shock waves. Therefore, 49 although LC control reduced the travel time of trucks by 26%-27%, fr and E CO 2 of trucks are only reduced by 3%. On the other hand, VSL control suppresses the shockwave and smooth the speed of all vehicles, which reducefr andE CO 2 of trucks by 13%-14%. For cars, fr and E CO 2 are not as sensitive to accelerations as those of trucks. Engine eciency, which increases with speed, is also a major factor. LC Control signicantly increases the average speed and engine eciency of cars, therefore decreasefr and E CO 2 of cars by 15%-17%. In the meantime, VSL control also re- duce fr and E CO 2 of cars by 7%-9%. NO x is major toxic road trac emission. Since we assume cars are all gasoline- based, the NO x emission of cars is very small comparing to that of trucks. Both VSL control and LC control have contributions on reduction of NO x . From the simulation results and analysis above, combined VSL & LC control method can improve the bottleneck ow rate, smooth and homogenize the trac ow simultaneously, hence is able to provide signicant and consistent improvement on trac mobility, safety and environmental impact in truck-dominant highway networks. 3.6.3 Evaluation of the Feedback Linearization VSL Controller In this section, we design and evaluate a combined VSL and LC controller for the simulation of a real world highway segment. We use both macroscopic and microscopic trac ow models and carry out Monte Carlo simulations for dierent incident scenarios in order to evaluate consistency with respect to performance improvements. 50 3.6.3.1 Simulation Network and Scenarios We use the same network in Fig. 3.9 to evaluate the performance of the feedback linearization VSL controller. To demonstrate the performance, robustness and con- sistency of the proposed controller under dierent incident conditions, we consider 3 dierent scenarios with dierent incident durations. We simulate each scenario under dierent demand ows. In each scenario, the incident occurs 5 minutes after simulation begins and lasts for 30 min in scenario 1, which simulates the case of an incident of moderate duration which may be due to an accident; for 10 min in scenario 2 which simulates the case of a short incident due to a vehicle breakdown or minor accident. The incident is not removed after occurrence in scenario 3, which simulates a long time lane closure or a construction site or a physical bottle- neck. We evaluate the combined VSL and LC control performance for each scenario with constant demand ows of 6000 veh/h and 6500 veh/h which is higher than the capacity of the bottleneck. 5% of the demand are trucks. 3.6.3.2 Macroscopic Simulation In this section, we use a macroscopic model to evaluate the performance of the pro- posed VSL controller. Since the macroscopic model used does not take into account lane changes and their eect close to the incident, we apply the LC controller to the corresponding microscopic model and use the microscopic model data to validate the macroscopic cell transmission model. The desired equilibrium point of the I-710 highway segment is calculated to be e 1 = 174:6 veh/mi; e 2 = e 3 = = e 10 = 90 veh/mi v e 1 = 33:5 mi/h; v e 2 =v e 3 = =v e 9 = 65 mi/h The LC recommendation sign is deployed at the beginning of section 9 and section 10 in Fig. 3.9, and recommends vehicles to change lanes by moving to the 51 (a) Vehicle Density (b) VSL Command Figure 3.11: Controller Performance without Constraints open lanes on either side. For the VSL controller, the following parameters are used: C v = 10 mi/h, v max = 65 mi/h, v min = 10 mi/h, T c = 30 s. We choose 1 = 2 = = 9 = 20. We should note that as mentioned in Section 3.6.1, the capacity of the bottleneck with incident is 4500 veh/h. However, in the macroscopic model, we are assuming a strict triangular fundamental diagram and the capacity C b is calibrated to be v f e 10 = 5850 veh/h. The reasons for this dierence are explained in the following section. Since the logic of our VSL controller is to stabilize the density at the critical value, the accurate value of equilibrium density is more important than the value of ow rate. The densities and variable speed limits for the case of scenario 1 with demand d = 6500 veh/h are plotted in Fig. 3.11. For clarity of presentation, we only plot the densities in section 1, 9 and 10 and VSL commands in section 1 and 9. Fig. 3.11 demonstrates what is predicted by theory. That is the density in section 1 converges to the desired density of 174.6 veh/mi and the densities in sections 9, 10 to the desired density of 90 veh/mi till the incident is removed at t = 35 min, in which case the densities converge to 105 veh/h, which is higher than the pre-incident value. This is because the queue formed at section 1 during the incident needs to discharge, therefore the temporary demand of the bottleneck after the incident is higher than the demand of the overall network. 52 (a) Vehicle Densities (b) VSL Commands Figure 3.12: Controller Performance with Constraints We then apply the constraints (3.21) - (3.23) to the VSL controller. The densi- ties and VSL commands with constraints are shown in Fig. 3.12. Fig. 3.12a demon- strates that the density in the discharging section converges to 10 = 85 veh/mi, which is lower than e 10 = 90 veh/mi. According to the fundamental diagram in Fig. 3.7, the steady state ow would be a bit lower than the desired ow rate. However, the dierence is negligible. The VSL command in section 1 converges to v 1 = 30 mi/h and the VSL command in section 9 converges to v 9 = 55 mi/h, which are not exactly the same as the desired values due to the application of the constraints. In Fig. 3.11, 9 and 10 converge to the corresponding equilibrium point in less than 10 min while 1 converges to e 1 much slower (in about 20 min). The reason of this phenomenon is the dierent values of e 1 and e 9 . As discussed in [75], a low value of speed limit would suppress the capacity of the section. After the incident occurs, v 1 decreases to a low value and 1 increases rapidly, since because of the out ow of section 1, q 2 is suppressed by v 1 . Then the process of adjusting 1 from the overshoot to e 1 takes long time due to the low level of q 2 . On the other hand, from Fig. 3.12, we can see that with the constrained VSL, 1 converges fast and no overshoot is observed. This is becausev 1 is constrained by (3.21) - (3.23) thus fails to adjust 1 back to e 1 after overshooting, however, as stated 53 (a) Vehicle Densities without Control (b) Bottleneck Flow with and without Control Figure 3.13: System Behavior without Control before, the dierence is negligible. Similarly, in Fig. 3.12b, the VSL command v 1 converges to 30 mi/h in less than 10 min and stays at that value. Since the VSL commands only take whole 5 mi/h values due to (3.21), small variation ofv 1 in the continuous case are all rounded up. Therefore, in the constrained case, there are no variations of v 1 around 30 mi/h. Fig. 3.13a demonstrates how vehicle densities evolve in scenario 1 without any control. The density increases dramatically in the discharging section to 370 veh/h and propagates upstream. Even after the incident is removed at t = 35 min, the shockwave continues propagating backwards and takes longer time to discharge. Fig. 3.13b shows the ow rate at the bottleneck with and without control. During the incident, the ow rate decreases to less than 3000 veh/h due to capacity drop in the case of no control, while the bottleneck ow converges to 5600 veh/h with the combined VSL and LC controller. Again, the ow rate under control is higher than the real capacity of the bottleneck due to the assumption of triangular fundamental diagram. We use scenario 1 to examine the growth of the queue at the entrance to the controlled network. The numbers of vehicles in the queues are plotted in Fig. 3.14 with respect to the time t. When the demand d = 6500 veh/h, the maximum number of vehicles in the queue is 1700 in the case of no control, while the number 54 (a) (b) Figure 3.14: Growth and Discharge of the Queue is less than 500 in the control case, which demonstrates that the combined VSL and LC controller reduces the queue size signicantly. The queues grow slower and discharge faster with lower demand, as less vehicles arrive at the tail of the queue. 3.6.3.3 Microscopic Simulation In this section, we use a microscopic trac model that is closer to the real environ- ment in order to conrm the improvements predicted by theory and demonstrated by the macroscopic model. In addition, the microscopic model allows us to evaluate additional performance criteria such as number of stops and lane changes that aect safety as well as the environmental impact of VSL and LC controllers. We simulate the I-710 trac ow network shown in Fig. 3.9 for the above mentioned 3 trac scenarios. The simulated demand consists of 85% light duty passenger vehicles and 15% trucks. This ratio represents the highest truck ratio at peak hours on I-710, therefore shows the worst trac condition [76]. To show consistency of the results, we conducted 10 sets of Monte-Carlo simulations with dierent random seeds for each scenario. The curves in Fig. 3.15 are generated from a single simulation. The evaluation results in Table 3.5 - 3.7 are the average of 10 simulations. 55 (a) (b) (c) (d) (e) (f) Figure 3.15: Comparison of Macroscopic and Microscopic Models | Microscopic without Control, - - - Macroscopic without Control | Microscopic with Control, - - - Macroscopic with Control 3.6.3.4 Consistency between microscopic and macroscopic models Fig. 3.15 shows the density and ow rate of the discharging section in both micro- scopic and macroscopic simulations. We can see that the density curve in macro- scopic and microscopic simulations match each other. The microscopic ow rates 56 Figure 3.16: Fundamental Diagram with Combined Controller in the no control cases are very similar and consistent with those in macroscopic simulations. However, when the combined VSL and LC controller is applied, the ow rates in microscopic simulations are lower than those in macroscopic simula- tions, which means that the ow speed in the discharging section in microscopic simulations is lower than what we get from the macroscopic model. The deviation in speed is due to the following factors: 1. Modeling error. In the macroscopic model, we use a simplied triangular fundamental diagram to model the discharging section, which implies that the ow speed at the desired density is v f . However, the actual speed would be lower than v f . Especially when the LC controller is applied, drivers are usually conservative when merging to the open lanes. 2. Speed limit following delay. In the macroscopic model, we assume that the ow speed follows the speed limit exactly with no delay. However, in the microscopic model, the trac ow needs time and space to accelerate to the desired speed limit. When vehicles change lanes, they do not adjust to new speeds instantaneously. 3. Friction eect. The friction eect re ects the empirically observed drivers' fear of moving fast in the open lanes when an incident or slowly moving 57 vehicles exist in neighboring lanes [77]. In microscopic simulation, this phe- nomenon is captured and has an eect when compared with the macroscopic simulations. Fig. 3.16 demonstrates the relationship between 10 and q b at the equilibrium state under the combined VSL and LC controller in microscopic simulations. In Fig. 3.16, the negative slope part, i.e. the congested part of the fundamental diagram is not observed even when the demandd is higher than the capacity, since the controller protects the bottleneck from getting congested. For dierent levels of demand, the data points concentrate in dierent clusters which shows that the controller homogenizes the trac ow. Furthermore, when d 3000 veh/h, the data points stay close to the line with the slope v f = 65 mi/h. When d keeps increasing, the data points move to the right side of the line due to the factors we explained above. 3.6.3.5 Performance Measurement and Criteria We use the following measurements to evaluate the performance of the proposed controller. To be precise, in scenario 1 and 2, the measurements start at the time instant that the incident begins (t = 5 min) and terminate at the time instant 10 minutes after the incident ends (t = 45 min in scenario 1 and t = 25 min in scenario 2), so that the trac states can achieve steady state. In scenario 3, where the incident is not removed, the measurements start at the time instant that the incident begins (t = 5 min) and terminate at t = 45 min. In each scenario, we collect the data of all vehicles that pass through the bottleneck during the above dened measuring periods and calculate the following values: (a)Average travel time T t . (b)Average number of stops s. (c)Average number of lane changes c. (d)Average fuel consumption rate. (e)Average CO2 emission rate. (f)Average NOx emission rate. (g)Average PM25 emission rate. Control eects on trac mobility are evaluated using the average travel time. Let N v denote the number of vehicles 58 pass through the bottleneck during the measuring period. Average travel time T t is dened as T t = Nv X i=1 (t i;out t i;in )=N v where t i;in and t i;out denote the time instant vehicle i enters and exits the network respectively. Note that our simulation network has enough space upstream of the controlled segment, therefore the time waiting in the queue is also counted. Control eects on trac safety are evaluated by the average number of stops and average number of lane changes. Less stops and lane changes indicate smoother trac ow and lower probability of crash [16]. s and c are dened as s = Nv X i=1 s i =N v ; c = Nv X i=1 c i =N v wheres i ,c i are number of stops and lane changes performed by vehiclei respectively. For environmental impact, we measure the average fuel consumption rate and the average emission rates of CO2, NOx, and PM25. These rates are uniformly dened as: R = Nv X i=1 E i = Nv X i=1 d i where E i denotes the fuel consumed or a certain type of emission generated by vehicle i in the highway network, d i represents the distance traveled by vehicle i in the network, and R denotes the fuel consumption rate or the tailpipe emission rate of CO2, NOx, or PM25. The fuel consumption rate and emission rates are calculated using the MOVES model of the Environment Protection Agency (EPA) based on the speed and acceleration prole of each vehicle [78]. 3.6.3.6 Evaluation Results Table 3.5, 3.6 and 3.7 demonstrate the results of microscopic evaluation of all 3 scenarios under dierent trac demands. From the results, we can see that 59 Table 3.5: Evaluation Results of Scenario 1 Demand 6000 veh/h 6500 veh/h Control No Control LC Only VSL Only Control Improvement No Control LC Only VSL Only Control Improvement T t 18.85 17.12 18.95 16.85 -10.59% 20.72 17.67 21.21 16.83 -18.76% s 11.16 2.45 3.61 1.90 -83.00% 12.10 2.55 3.78 1.91 -84.21% c 4.00 4.75 4.74 3.78 -5.60% 4.67 5.54 5.88 4.31 -7.71% NOx 1.56 1.49 1.61 1.49 -4.43% 1.64 1.58 1.60 1.53 -6.71% CO2 558.56 543.22 577.59 536.01 -4.04% 589.46 556.47 605.59 537.21 -8.86% Energy 178.65 173.67 184.76 171.40 -4.06% 186.78 177.93 193.73 170.31 -8.82% PM25 0.049 0.048 0.047 0.050 0.66% 0.054 0.054 0.053 0.050 -7.73% Table 3.6: Evaluation Results of Scenario 2 Demand 6000 veh/h 6500 veh/h Control No Control LC Only VSL Only Control Improvement No Control LC Only VSL Only Control Improvement T t 12.41 11.87 13.46 11.63 -6.25% 13.58 12.62 15.02 12.42 -8.54% s 5.16 0.75 2.16 0.65 -87.37% 5.72 1.58 2.33 0.91 -84.09% c 3.68 3.80 3.90 3.52 -4.31% 4.27 4.81 5.01 3.91 -8.33% NOx 1.42 1.41 1.44 1.39 -2.48% 1.48 1.49 1.51 1.42 -4.05% CO2 483.37 479.17 497.81 470.16 -2.73% 508.13 504.16 524.36 487.18 -4.12% Energy 154.53 151.65 159.18 150.36 -2.70% 161.04 161.15 167.66 154.18 -4.26% PM25 0.041 0.041 0.041 0.041 -0.77% 0.046 0.047 0.047 0.045 -2.17% Table 3.7: Evaluation Results of Scenario 3 Demand 6000 veh/h 6500 veh/h Control No Control LC Only VSL Only Control Improvement No Control LC Only VSL Only Control Improvement T t 19.84 17.25 18.16 16.69 -15.89% 21.25 16.75 20.45 16.55 -22.13% s 15.46 2.13 4.00 1.74 -88.75% 16.12 2.54 3.72 1.83 -88.65% c 4.61 4.55 5.11 4.21 -8.60% 4.58 5.36 6.36 4.10 -10.48% NOx 1.58 1.51 1.58 1.50 -4.95% 1.58 1.55 1.66 1.50 -4.95% CO2 570.72 538.41 564.54 529.76 -7.18% 568.96 550.32 597.94 523.25 -8.04% Energy 182.55 172.17 180.58 169.39 -7.21% 182.85 175.99 191.26 168.11 -8.06% PM25 0.052 0.047 0.047 0.050 -3.74% 0.052 0.053 0.053 0.050 -3.74% 60 the combined VSL & LC controller is able to provide signicant improvements in trac mobility, safety and environment. For trac mobility, the proposed controller reduces the average travel time of each vehicle by 6.25% - 22.13%. For trac safety, the combined VSL and LC controller dramatically decreases the average number of stops by 83% - 88.75% in dierent scenarios, therefore dras- tically reduces the instances of the stop-and-go trac, smooths the trac ow and damps the shockwave. Average number of lane changes is also decreased by 5.6% - 10.48%. The combined VSL and LC controller homogenizes the density and speed in each section. Drivers tend to not change lane if densities and speeds are similar in all lanes, therefore the VSL control reduces the number of lane changes in the network under consideration. This is highly important for trac safety in highway segments with high truck ratio. Trucks not only take long time and large space to change lane, their large size also blocks the eye sight of drivers, which makes lane changes of trucks much more dangerous than other vehicles. The proposed controller reduces the fuel consumption rate and tailpipe emission rate from two perspectives. First, it reduces the travel time of vehicles, therefore decreases the emission levels of vehicles waiting in the queue. Second, it smooths the trac ow and suppresses the acceleration and deceleration, therefore decreases the emission in these transient states. In the simulation, fuel consumption rate is decreased by 4.26% - 8.82%. The improvement in CO 2 emission rate is approx- imately proportional to the improvement of fuel consumption rate, since CO 2 is the main product of fuel burnt. The proposed controller reduces NO x emission rate by about 3.54% - 6.71%. The emission rate of PM25 is also decreased by 3.74% - 7.73%. Therefore, the combined VSL and LC controller is able to bring environmental benets. The question how much of these improvements is due to VSL and LC controller alone is also answered using these simulation studies. From Table 3.5 - Table 3.7, 61 we can see that when the LC controller is applied alone, all evaluation criteria im- prove except for the average number of lane changes. The improvements onT t and s are signicant, while other criteria are only improved slightly. As discussed in Section 3.2.3, the LC controller is able to recommend upstream vehicles to make lane changes before stopping at the queue and avoid the capacity drop therefore reduce the average travel time and average number of stops. Improvements on environmental criteria are results of improvements of trac mobility. However, for the average number of lane changes, the LC controller only makes the lane changes take place in advance, instead of avoiding them, thus fails to reduce c. Further- more, when the VSL controller is applied alone, only the average number of stops is reduced. Other criteria are not improved and in some cases are even deteriorated by the VSL controller. This is because the VSL controller (3.17) is designed based on the assumption that the capacity drop has already been removed by the LC controller. When the LC controller is absent, VSL is not able to improve the bot- tleneck ow and reduce the vehicle density. But when the VSL controller is applied together with the LC controller, all criteria are further improved since the VSL stabilizes the vehicle densities at the desired equilibrium point and homogenizes the trac ow. When the trac ow is homogenized in each section and lane, the drivers do not tend to change lanes frequently, hence the average numbers of lane changes are also reduced. Comparing the three scenarios, the improvement on each measurement criteria in scenario 2 appears to be less signicant than the other 2 scenarios. The reason is that the incident duration in scenario 2 is very short. 62 Chapter 4 Coordinated Variable Speed Limit, Ramp Metering and Lane Change Controller 4.1 Introduction The coordination of RM and VSL considers network mobility, on-ramp queues and fairness between the mainline and the ramps. The objective is to keep a balanced delay time between vehicles on the mainline and the ramps and avoid queues on the ramps from spilling back to the urban roads. In this chapter, we use an analytical method to design a coordinated VSL and RM controller based on a cell transmission macroscopic model with triangular fundamental diagram which together with a lane change controller guarantees stability of the trac ow and convergence of trac density to the desired equilibrium point exponentially fast. Considering the fact that RM controllers have been widely deployed in the United States, we assume that the RM control command is determined before the VSL and design the VSL controller to coordinate with the RM and stabilize the trac ow. The coordinated VSL and RM controller with lane change is evaluated using Monte Carlo microscopic simulations and shows signicant improvement in trac mobility, safety and the environment impact. 63 (a) w/ and w/o LC (b) w/ and w/o VSL Figure 4.1: Eects of LC and VSL on Fundamental Diagrams 4.2 System Modeling 4.2.1 Eect of VSL on the Fundamental Diagram Consider the highway bottleneck shown in Fig. 3.9. A bottleneck is introduced by an incident that blocks one lane. The speed limit upstream the bottleneck is the free ow speed v f = 65 mi/h. As discussed in Section 3.2.3, the lane change controller can avoid the capacity drop. However, as shown in Fig.4.1a, in the fundamental diagram with lane change control, the low d part is very close to its triangular approximation, which means that the ow speed is close tov f , while the ow speed decreases as d approaches d;c . In last chapter we attribute the reduction of speed to modeling error, delay of speed limit following and driver's caution when passing the incident site. This deviation of speed will not harm the benet of VSL with respect to trac mobility when designing the VSL controller based on the triangular fundamental diagram as long as d is stabilized at d;c . However, if the speed limit upstream the bottleneck is v f , vehicles need to decelerate when approaching the bottleneck, which leads to shock waves that propagate upstream. If we decrease the speed limit upstream the bottleneck tov d , such that 0<v d < v f , according to [19], the critical density in the fundamental diagram will be shifted to higher value and the slope of the under-critical part of the fundamental diagram 64 Figure 4.2: Conguration of the Highway Segment will be decreased and made closer to a straight line. Our microscopic simulations conrm this statement. The black solid line in Fig. 4.1b shows the fundamental diagram under a speed limit of 40 mi/h. Compared to the one under 65 mi/h, which is shown as the blue solid line in Fig. 4.1b, the capacity of the bottleneck is not decreased despite under a lower speed limit as the critical density is increased from ~ d;c to d;c . As we can see in the gure, this fundamental diagram is very close to its triangular approximation, that is, the speed deviation at d;c is very small. If we design the coordinated VSL and RM controller based on this fundamental diagram and let the VSL command converge to v d at the equilibrium state, the shockwave upstream the bottleneck will be attenuated. We demonstrate this with microscopic simulations in Section 4.4. To conclude, under speed limit of v d , the highway bottleneck can be modeled with high accuracy as equation 3.4 4.2.2 Cell Transmission Model with Ramp Flows The highway segment to be controlled by the coordinated VSL and RM controller is shown in Fig. 4.2. The bottleneck is introduced by a lane closure. The highway segment upstream the bottleneck is divided into N + 1 sections, which are indexed as section 0 through section N. For i = 0; 1;:::;N, i ;q i ;r i ;s i represent the vehicle density, mainline in- ow rate, on-ramp ow rate and o-ramp ow rate in section i respectively, where i ;s i are measurable,r i are determined by the RM controller, therefore also measurable. Fori = 0; 1;:::;N 1,v i denote the variable speed limit 65 in section i. In section N, the speed limit is a constant denoted by v d . q b denotes the ow rate through the bottleneck. Let R i =r i s i be the net ramp ow andL i the length of section i, for i = 0; 1;:::;N. According to the ow conservation law, we have _ i = 1 L i (q i q i+1 +R i ); for i = 0; 1;:::;N 1 _ N = 1 L N (q N q b +R N ) (4.1) The ow rate and bottleneck model is the same as (3.3) and (3.4). For the sake of completeness, we write the equations here. q 0 = minfd;C 0 ;w 0 ( j;0 0 )g q i = minfv i1 i1 ;C i ;w i ( j;i i )g; i = 1;:::;N (4.2) q b = 8 < : v d N ; N d,c w b ( j,d N ); N > d,c (4.3) 4.3 Controller Design In this section, the coordinated VSL and RM controller is designed. We rst design the VSL controller by assuming that the RM control command is given. Then we choose the ramp metering strategy, ALINEA/Q, to manage the ramp ows and the queue lengths on ramps. 4.3.1 Design of VSL The goals of designing the VSL controller include: (1) Given any type of RM controller, the VSL controller should be able to coordinate with it and stabilize the density N in the discharging section at the critical value d;c , in order to keep q b at the highest level. (2) Homogenize the trac ow upstream the bottleneck in 66 order to improve the trac safety and bring environmental benets. Consider the subsystem which includes section 1 through section N. Dene the error states e i = i d;c ; for i = 1; 2;:::;N We have _ e i = 1 L i (v i1 i1 v i i +R i ); for i = 1; 2;:::;N 1 _ e N = 8 < : v N1 N1 v d N +R N L N ; N 0 v N1 N1 w b ( j;b N )+R N L N ; N > 0 (4.4) Let v i = i L i+1 e i+1 +v d d;c P N j=i+1 R j i ; for i = 0; 1;:::;N 2 v N1 = 8 < : N1 L N e N +v d N R N N 1 ; N d;c N1 L N e N +w b ( j;b N )R N N 1 ; N d;c (4.5) Substitute the controller (4.5) into the open-loop system (4.4), we have the following closed-loop system: _ e i = i1 e i + L i+1 L i i e i+1 ; for i = 1; 2;:::;N 2 _ e N1 = 8 < : N2 e N1 + L N L N1 ( N1 v d )e N ; N 0 N2 e N1 + L N L N1 ( N1 +w b )e N ; N > 0 _ e N = N1 e N (4.6) Theorem 4.3.1. e i = 0, for i = 1; 2;:::;N is the unique and isolated equilibrium point of the closed-loop system (4.6) and is guaranteed to be globally exponentially stable. The rate of exponential convergence depends on the control design parame- ters i , i = 0; 1;:::;N 1. The proof of Theorem 4.3.1 is similar to the proof of Theorem 3.5.1. According to Theorem 4.3.1, the steady state value of i is i;ss = d;c , i = 1;:::;N. The 67 steady state value ofv i isv i;ss =v d P N j=i+1 R i = d;c ,i = 1;:::;N 1. Therefore, by applying the coordinated VSL and RM controller, 1 through N are stabilized and homogenized. The eect of a ramp ow is compensated by its upstream VSL and does not aect downstream trac. If R i = 0, then v i;ss = v d , for i = 1;:::;N 1. That is the upstream speed limit converges to v d . By adjusting the value of v d , we can guarantee that the shockwave resulted by speed deviation between actual trac ow and the triangular fundamental diagram is eliminated. Now let us consider the dynamics of 0 and v 0 . Since q 1 converges to v d d;c , if the demand d>v d d;c , 0 will increase. Once 0 > j;0 d=w 0 , we have _ 0 = 1 L 0 (w 0 ( j;0 0 )v 0 0 +R 0 ) (4.7) Substitute (4.5) into (4.7), we have _ 0 = 1 L 0 (w 0 ( j;0 0 )v d d;c + N X j=0 R j ) Assume that P N j=0 R j is constant, then 0 = j;0 + P N j=0 R j v d d;c w 0 is a stable equilibrium point. As long as P N j=0 R j < v d d;c , 0 will not exceed the jam density j;0 and v 0 will not go negative, thus the VSL controller is feasible. For driver's acceptance and safety, we as well apply the constraints (3.21) - (3.23) to the VSL controller (4.5). 4.3.2 Design of the RM Controller According to Theorem 4.3.1, the VSL controller (4.5) can stabilize the system and improve the mobility as long as the net ramp ow is lower than the bottleneck 68 capacity. It seems that RM control is unnecessary. However, if no RM is applied and large ramp ows ush into the mainline, the merging of ramp ows will severely disturb the mainline ow. Furthermore, when the net ramp ow is high, the VSL controller (4.5) will suppress the mainline ow in order to spare the capacity for the ramp ows. That is, without RM control, the ramp ow will always have priority which may harm the fairness between the ramp ows and the mainline ow, or even make the VSL controller infeasible. Furthermore, the RM controller should be able to manage the queue on the ramps so that the queues do not spill backwards to the urban road network. We adopt the ALINEA/Q, which modies the classic ALINEA ramp metering strategy with queue adjustment. The original ALINEA/Q method proposed in [38] includes the downstream occupancy and the queue length in the feedback loop. In this paper, to be consistent with the VSL controller, we use the downstream density instead of occupancy. For an on-rampi, two RM rates,r d i (k) andr q i (k), are decided respectively based on the downstream density and the queue length on the ramp at each time step t =kT c . The nal RM rate r i (k) is the maximum of the two. i.e. r d i (k) =r(k 1) + d [( d;c i (k))] r q i (k) = q (w r i w i (k)) +d i (k 1) r i (k) = maxfr d i (k);r q i (k)g (4.8) where i (k) is the density in the highway section that connects to ramp i, w i (k) is the queue length on ramp i at time step k, d i (k 1) is the demand from ramp i within time step k 1, w r i is the reference queue length of ramp i. r d i (k) is an integral feedback controller that regulates i (k) to be close to d;c , which helps maintain the vehicle density on mainline at the desired equilibrium value. r q i (k) adjusts the RM rate in order to prevent the queue length from being too large, i.e. if w i (k) is larger than w r i , the RM rate will increase to discharge excessive vehicles in the queue and newly arrived vehicles. Since the nal RM rate is the maximum 69 of the two, the ramp ow will get the priority to pass the bottleneck if the ramp queue is large, while the mainline ow will get the priority if the vehicle density on the mainline is high. In this way, the ALINEA/Q strategy maintains the fairness between the ramp ows and the mainline ow and avoids the ramp queues from piling up towards the urban road. 4.4 Numerical Simulations In this section , we use the microscopic simulator VISSIM to carry out Monte Carlo simulations to evaluate the performance of the coordinated VSL, RM and lane change control on trac mobility, safety and the environment. 4.4.1 Scenario Setup We evaluate the proposed controller on the highway segment in Fig. 3.9. To coordinate with the ramps, we divide the highway segment in to 8 sections, the VSL signs are deployed at the beginning of section 0 through 6. An incident blocks the middle lane at the end of section 7 and creates a bottleneck. 4 on-ramps, which are equipped with RM, and 5 o-ramps are connected to the highway segment. The lane change control is deployed at the beginning of section 7. The incident occurs at 5 minutes after simulation starts, and lasts for 30 min. The capacity of the highway segment is 6800 veh/h without incident. During the incident, the ideal bottleneck capacity is about 4500 veh/h. We load the network with the real demand at 5pm on Monday, which is a peak hour. The mainline demand is 4500 veh/h, the on-ramp demand from upstream to downstream are 400 veh/h, 500 veh/h, 300 veh/h, 300 veh/h respectively. 70 Figure 4.3: Geometry of Simulation Network Figure 4.4: Bottleneck Flow |with control, |no control (a) Density in section 7 (b) Density in section 0 Figure 4.5: Vehicle Densities w/ and w/o Control |with control, |no control 4.4.2 Simulation Results Fig. 4.4 shows the bottleneck ow with and without the coordinated VSL, RM and lane change control. When there is no control, the ow rate decreases immediately to around 3000 veh/h due to the lane blockage and capacity drop, and increases right away after the incident is removed as the queue in the bottleneck area ushes downstream. When the controller is applied, the ow rate decreases to around 4200 veh/h, which is higher than that in the no control case since the capacity drop is 71 (a) v d = 40 mi/h (b) v d = 65 mi/h Figure 4.6: Density Contours (a) Queue length on r 11 (b) Queue length on r 4 Figure 4.7: Queue Length w/ and w/o Control |VSL + RM, |RM only avoided by the lane change control and VSL stabilizes the vehicle densities. The bottleneck ow starts increasing about 10 min after the incident is removed as the high density area is held in section 0 by the VSL controller. The high density wave moves forward from section 0 and the ow rate q b starts increasing once the wave front reach the bottleneck. Fig. 4.5 shows the curve of 7 and 0 , which are the vehicle density of the discharging section and the rst VSL controlled section, respectively. When there is no control, 7 starts increasing immediately as the incident occurs at t = 5 min. In addition the shockwave propagates upstream, which makes 0 starts increasing 72 at t = 25 min and reaches 500 veh/mi. The high density in section 0 does not dis- charge until 15 min after the incident is removed. When the coordinated controller is applied, 7 increases slightly and is stabilized at 110 veh/mi. 0 increases im- mediately after the incident since v 0 decreases to reduce the ow into downstream sections and is stabilized at around 400 veh/h which is lower than that without control. Fig. 4.6 demonstrates the contour plot of vehicle densities with respect to time and space with dierent values of v d . When v d = 40 mi/h, high density is held in section 0 during the incident, while downstream sections are highly homogenized. 2 is higher than d;c at the beginning of the incident as the ramp ows r 11 and r 12 ush in but then discharged under control. The density in section 6 is slightly higher than d;c as vehicles receive the lane change recommendations and make lane changes thus slightly disturbs upstream ow. When v d = 65 mi/h, as explained in Section 4.2.1, a shockwave propagates upstream. After the incident is removed, the vehicles in section 0 ush downstream and meet with the shockwave, which leads to a high density area in section 2. However in this case, the discharging section is still well protected. As the shockwave propagates upstream, vehicle densities converge to d;c gradually from downstream section to upstream section. This is because we use the cascade structure of VSL controller in Fig. 3.3, which attenuates the shockwave section by section. Thus the controller is robust to parameter selection. Fig. 4.7 shows the queue length on rampr 11 andr 3 , with RM control alone and with the coordinated controller. With RM control alone, the queues pile up fast as the densities in mainline increase. Due to the queue adjustment mechanism of ALINEA/Q, the queue lengths are maintained around the reference value. With the coordinated controller, the queue lengths increase in the transient process when the incident begins and the mainline density is being adjusted to the desired level and then discharge fast. After the incident is removed, large ow ushes downstream, 73 Table 4.1: Evaluation Results Control Type No Control RM + VSL Improvement T t (min) 15 11 27% s 23 4 82% c 5.1 4.6 10% CO2 (g/veh/mi) 585 538 8% Fuel (g/veh/mi) 187 172 8% the RM controller decrease the rate to give priority to the mainline, therefore the queue lengths increase. We use the following metrics to evaluate the performance of the coordinated controller. To evaluate trac mobility, we use: (a) average travel time T t ; for trac safety, use (b) average number of stops s and (c) average number of lane changes c; for the environment, we use (d) average emission of CO2 and (e) average fuel consumption. The detailed denition of the above metric can be found in [5]. Table 4.1 shows the evaluation results. The improvement in trac mobility, safety and the environment is signicant. The average travel time is reduced by about 27% as the bottleneck throughput is increased. For trac safety, the number of stops dramatically decreased by 81% as the lane change control prevented vehicles from stopping at the bottleneck and waiting for lane changes. The 10% reduction in number of lane changes is contributed by both homogenization of mainline ow and the regulated merging behavior of ramp ows. For the environment metrics, the reductions of CO2 emission and energy consumption are usually proportional to each other, which are both around 8% in this case. 74 Chapter 5 Comparison of Feedback Linearization and Model Predictive Strategies in Variable Speed Limit Control 5.1 Introduction Given the fact that LC control is able to relieve or eliminate the capacity drop, one important question arising at this point is that if other VSL control strategies are combined with the LC control, will the system performance exceed the performance under the FL controller? Intuitively, since MPC control follows an optimization based routine, it should provide the `optimal' performance to some extent. However, FL controller guarantees exponential stability of the equilibrium point with highest bottleneck ow rate. Therefore, by tuning the feedback gain, the FL controller should be able to force the system to converge as fast as possible, only limited by the saturation of control input. In this chapter, we propose FL and MPC schemes for VSL-actuated highway trac, where we assume that an LC controller is active just upstream of the bot- tleneck. Both controllers are designed with a CTM-based model representing the ideal system. TTS performance and robustness with respect to perturbations on model parameters and measurement noise of the proposed controllers are evaluated 75 via simulation studies. Results show both VSL controller is able to improve the total time spent under dierent levels of perturbation and measurement noise. Fur- thermore, feedback linearization VSL can provide better performance than model predictive VSL with much less computational eort. 5.2 Nonlinear Model Predictive Control Model predictive control strategy generates the control command at each control step by solving a nite horizon optimal control problem in a receding horizon man- ner. In this chapter, we formulate the cost function of the MPC problem as the quadratic error of the states of system (3.16). To take into consideration the vehi- cles that are blocked upstream the VSL controlled segment, we augment the system by add a new state Q, that is _ Q =dq 0 ; (5.1) with Q = 0 at t = 0. Therefore, if the number of vehicles upstream of section 0 is greater than the number at time 0, Q > 0, otherwise Q 0. We should note here that the introduction ofQ is only for the purpose of evaluating the TTS. Both the FL and MPC controllers are implemented based on system (3.16). The performance metric TTS is dened as follows: TTS = Z T 0 Q(t) + N X i=0 i (t)L i dt (5.2) The open-loop highway system (3.16) can be implicitly expressed as _ e =f(e;u) (5.3) 76 Here we formulate the problem of nding the VSL commands u() that try to maintain system (5.3) at the equilibrium point as the following nite-horizon constrained optimal control problem (OCP): minimize u() Z kTc+Tp kTc e() T ~ Qe() +u() T ~ Ru()d subject to e(kT c ) = ^ e(kT c ) _ e =f(e;u);82 [t;t +T p ] v min v e u()v max v e ; (5.4) wheret is the current control sampling instant in time, ^ e(t) is the measurement on error states taken at that instant, ~ Q and ~ R are weighting matrices on error and control input, respectively, whereas T p is the prediction horizon. The optimization problem is solved at the beginning of each control step kT c , with ^ e(kT c ) as the initial condition. Constraint (3.23) has already been included in the constraints of the optimization problem. (3.21) and (3.22) are also applied to the MPC VSL commands before applied to the system. Due to the continuous-time dynamics, the OCP (5.4) is an innite dimensional optimization problem. We resort to approximating it as a nite dimensional nonlin- ear program (NLP) via the direct multiple shooting method [79]. Details on direct methods from numerical optimal control literature can be found in [80]. 5.3 Numerical Simulation In this section, macroscopic simulation is used to evaluate the performance and robustness of the FL and NMPC schemes combined with LC. 77 5.3.1 Scenario setup The FL and MPC controllers have evaluated on the network shown in Fig. 3.9 In our simulation, the incident happens 5 minutes after the simulation starts, and it lasts for 30 min. The nominal demand is 6000 veh/h. The desired equilibrium point of this network is calibrated to be: e 0 = 278 veh/mi e 1 = e 2 = = e 7 = 110 veh/mi v e 0 = 15:8 mi/h v e 1 =v e 2 = =v e 7 = 40 mi/h For the FL controller, we choose i = 50 for i = 0; 1;:::; 6. The NMPC controller is implemented using the direct multiple shooting method via the CasADi toolbox [81] in MATLAB 8.5.0 (R2015a), on a 64-bit Windows PC with 3.4-GHz Intel Core i7 processor and 8-GB RAM, where IPOPT [82] is used for solving the NLPs. In our simulation, we choose the prediction horizon T p = 10 min, which is much greater than the control time step T c = 30 s. Weight matrices are chosen as ~ Q = I and ~ R = 0:1I, with I denoting the identity matrix of appropriate dimensions. The computation time of NMPC is around 0.35 seconds, whereas it is negligible for FL. The NMPC scheme is still computationally tractable, as its computation time of 0.35 s per step is negligible with respect to the control time step of 30 s. 5.3.2 Performance and Robustness Analysis with Macroscopic Simulations To compare the performance and robustness of the FL and MPC VSL controllers, we evaluate the following criteria for the two controllers: 1) Total time spent (TTS) as dened in (5.2), and sensitivity of TTS with respect to 2) perturbation on trac 78 Figure 5.1: Simulation System Figure 5.2: Simulation System demand, 3) perturbation on model parameters and 4) measurement noise. In the simulation, the FL and MPC controllers are synthesized with the ideal model (5.3), but the control command are applied on a perturbed model. The structure of the simulation system is shown in Fig. 5.1. For the trac demand, we add up to20% perturbation on the nominal demand 6000 veh/h. For the model parameters, as shown in Fig. 5.2, we respectively add up to20% perturbation on the nominal value of d;c and C b , which directly alter the shape of the fundamental diagram of the bottleneck section. For the measurement noise, we use Gaussian white noise with dierent levels of standard deviation up to = 0:1 cb to match the scale of the density measurements. 79 Figure 5.3: 7 with FL and MPC Figure 5.4: Performance sensitivity of no control (black), FL (blue), and NMPC (red) to perturbations on demand d. 80 Figure 5.5: Performance sensitivity of no control (black), FL (blue), and NMPC (red) to perturbations on C b . Figure 5.6: Performance sensitivity of no control (black), FL (blue), and NMPC (red) to perturbations on d,c . 81 Figure 5.7: Performance sensitivity of FL (blue) and NMPC (red) to increasing levels of standard deviation in measurement noise. Fig. 5.3 shows the behavior of the vehicle density in the discharging section under FL and MPC controller. Both controllers are able to maintain the density around the desired value e 7 = 110 veh/h after the incident occurs att = 5 min. The oscillation is introduced by the roundup-to-5 constraint. However, the MPC con- troller introduces higher frequency chattering and a sharp decrease at the beginning of the incident. A series of simulation experiments are conducted with dierent levels of per- turbation and measurement noise. Figure 5.4 shows how TTS varies with varying demand levels. The gure showcases that both controllers are able to function properly under various levels of demand, the TTS increases and decreases approx- imately linear with the demand. This demonstrates that both MPC and FL VSL controllers are robust with respect to the variation of demand, which is due to the selection of the desired equilibrium point (3.13) - (3.14). At the equilibrium point, the speed limit in section 0 is decreased to block excessive trac demand at upstream of the entire control segment, therefore the bottleneck ow is not af- fected. Furthermore, under dierent levels of perturbation, the performance of FL and MPC controller are similar. But the TTS of FL is always slightly lower than 82 Table 5.1: Evaluation Results with Original Parameters TTT (hr) Stops LC CO (g/veh/mi) Nox (g/veh/mi) CO2 (g/veh/mi) Energy (g/veh/mi) No Control mean std 1270 42 23.2 1.3 6.6 0.2 3.4 0.1 1.8 0.1 605 20 194 6 Improvement - - - - - - - LC Only mean std 1075 40 10.5 0.9 5.9 0.3 3.4 0.1 1.7 0.1 552 16 176 5 Improvement 15% 55% 11% 0% 6% 9% 9% FL mean std 1036 36 9.9 1.3 5.5 0.2 3.0 0.1 1.6 0.1 529 13 169 4 Improvement 18% 57% 17% 12% 11% 13% 13% MPC mean std 1018 41 8.7 1.2 5.5 0.2 3.0 0.1 1.6 0.1 525 15 168 5 Improvement 20% 63% 17% 12% 11% 13% 13% that of MPC, which shows that MPC fails to beat FL in TTS although the control commands are generated by solving the optimization problem in receding horizon fashion. In gures 5.5 and 5.6, the change in TTS is plotted with respect to dierent values of perturbation on C b and d,c , respectively. These results show that both controllers achieve signicant improvements over the no control case and are able to operate properly even under situations with high amount of uncertainty in these model parameters. With perturbation on C b , the TTS under FL and MPC are increased by 45% and 43% in the worst case, respectively. Considering the fact that in this case the bottleneck capacity is decreased by 20% as a baseline, the TTS does not increase too much due to the modeling error and is still much lower than that in the no control case. The worst case for the perturbation on d,c is 27% worse than the non-perturbed value for FL, and 16% for NMPC. The sensitivity of TTS performance in the case of varying levels of standard deviation in measurement noise is given in gure 5.7, which shows that the TTS under both controllers increases with the standard deviation of measurement noise. However, the system does not diverge as the no control case. The performance of FL is always better than that of NMPC in this case. 83 Table 5.2: Evaluation Results under Dierent w 1 TTT (hr) Stops LC CO (g/veh/mi) Nox (g/veh/mi) CO2 (g/veh/mi) Energy (g/veh/mi) FL w 1 =9 mean std 1036 36 9.9 1.3 5.5 0.2 3.0 0.1 1.6 0.1 529 13 169 4 Improvement 18% 57% 17% 12% 11% 13% 13% w 1 =14 mean std 1036 36 9.9 1.3 5.5 0.2 3.0 0.1 1.6 0.1 529 13 169 4 Improvement 18% 57% 17% 12% 11% 13% 13% w 1 =6 mean std 1036 36 9.9 1.3 5.5 0.2 3.0 0.1 1.6 0.1 529 13 169 4 Improvement 18% 57% 17% 12% 11% 13% 13% MPC w 1 =9 mean std 1096 55 12.3 2.4 5.5 0.2 3.1 0.1 1.6 0.1 533 16 170 5 Improvement 14% 47% 17% 9% 11% 12% 12% w 1 =14 mean std 1018 41 8.7 1.2 5.5 0.2 3.0 0.1 1.6 0.1 525 15 168 5 Improvement 20% 63% 17% 12% 11% 13% 13% w 1 =6 mean std 1226 61 12.1 1.9 5.6 0.3 3.1 0.1 1.6 0.1 546 20 174 6 Improvement 3% 48% 15% 9% 11% 10% 10% Table 5.3: Evaluation Results under Dierent cb TTT (hr) Stops LC CO (g/veh/mi) NOx (g/veh/mi) CO2 (g/veh/mi) Energy (g/veh/mi) FL cb = 100 mean std 1024 44 8.8 2 5.5 0.2 3.0 0.1 1.6 0.1 528 14 169 5 Improvement 19% 62% 17% 12% 11% 13% 13% cb = 110 mean std 1036 36 9.9 1.3 5.5 0.2 3.0 0.1 1.6 0.1 529 13 169 4 Improvement 18% 57% 17% 12% 11% 13% 13% cb = 120 mean std 1031 43 9.4 2.2 5.5 0.2 3.0 0.1 1.6 0.1 526 15 168 4 Improvement 19% 59% 17% 12% 11% 13% 13% MPC cb = 100 mean std 1236 41 11.4 0.3 5.5 0.2 3.1 0.1 1.6 0.1 544 16 174 5 Improvement 3% 51% 17% 9% 11% 10% 10% cb = 110 mean std 1018 41 8.7 1.2 5.5 0.2 3.0 0.1 1.6 0.1 525 15 168 5 Improvement 20% 63% 17% 12% 11% 13% 13% cb = 120 mean std 1242 35 11.6 1.0 5.5 0.2 3.1 0.1 1.6 0.1 542 17 173 6 Improvement 2% 50% 17% 9% 11% 10% 11% Table 5.4: Evaluation Results under Dierent w b TTT (hr) Stops LC CO (g/veh/mi) Nox (g/veh/mi) CO2 (g/veh/mi) Energy (g/veh/mi) FL w b =20 mean std 1025 36 9.6 1.0 5.5 0.2 3.0 0.1 1.6 0.1 527 13 169 4 Improvement 19% 59% 17% 12% 11% 13% 13% w b =40 mean std 1036 36 9.9 1.3 5.5 0.2 3.0 0.1 1.6 0.1 529 13 169 4 Improvement 18% 57% 17% 12% 11% 13% 13% w b =60 mean std 1042 34 10.2 1.8 5.5 0.2 3.0 0.1 1.6 0.1 526 15 168 4 Improvement 18% 56% 17% 12% 11% 13% 13% MPC w b =20 mean std 1098 58 12.4 2.4 5.5 0.2 3.1 0.1 1.6 0.1 533 16 170 5 Improvement 14% 47% 17% 9% 11% 12% 12% w b =40 mean std 1018 41 8.7 1.2 5.5 0.2 3.0 0.1 1.6 0.1 525 15 168 5 Improvement 20% 63% 17% 12% 11% 13% 13% w b =60 mean std 1092 53 12.3 2.2 5.5 0.2 3.1 0.1 1.6 0.1 529 15 169 5 Improvement 14% 47% 17% 9% 11% 13% 13% 84 5.3.3 Performance and Robustness Analysis with Microscopic Simulations Table 5.1 shows the microscopic simulation results with calibrated model parameter set: w 1 = 14 mi/h; w b = 40 mi/h; cb = 110 veh/mi The performance of the MPC controller is similar to that of the FL controller. Table 5.2 - Table 5.4 demonstrate the simulation results of MPC and FL con- troller under dierent values of model parameters. From the result, we can see that the FL controller is robust with respect to the perturbations on w 1 , w b and cb . As to MPC, the mobility performance is signicantly adversed by the pertur- bations on w 1 and cb , which both change the value of the equilibrium point. But MPC is robust with respect to the perturbations on w b which does not change the equilibrium point and can be compensated by the control input. 85 Chapter 6 Stability Analysis of Cell Transmission Model under All Operating Conditions 6.1 Introduction In Chapter 3 and Chapter 4, we designed a coordinated variable speed limit, ramp metering and lane change control based on the rst-order cell transmission model. However, the analysis of dynamical behavior and stability properties of the open- loop cell transmission model which takes capacity drop into consideration is missing from the previous work, which makes it dicult for us to perform an analytical comparison of the open-loop and closed-loop performance of the VSL controlled cell transmission model. In addition, the analysis of the closed-loop behavior in Chapter 3 and Chapter 4 is performed with a simplied CTM, i.e. consider only the region in the state space near the desired equilibrium point (3.13) and under the assumption that the demand is higher than the bottleneck capacity. It remains unclear whether the global stability of the desired equilibrium point is still valid with the complete CTM and in other operating scenarios. In [57], Gomes et al. performed a thorough analysis of the equilibrium points and their stability properties of the CTM model. However, the authors did not take the capacity drop phenomenon into consideration. Reference [58] developed sucient conditions for the stability of the equilibrium points of CTM in terms of 86 connectivity of a graph associated with the trac network. The results of [57] and [58] are established based on the monotonicity of CTM. However, if the CTM is modied to account for capacity drop and the fact that the discharging ow rate of a congested road section decreases with density [5, 59, 26, 60], then the CTM is no longer monotone. Therefore, in this chapter, We use the CTM which take into consideration the eect of capacity drop which is due to microscopic phenomena such as forced lane changes at a bottleneck [5] and the decreasing discharging ow of the road section, then consider all possible trac ow scenarios, identify all equilibrium points and analyze their stability properties for a single road section, then extend the results to arbitrary number of sections under dierent trac demand levels and capacity constraints as well as under all initial density conditions, based on which the design of the VSL controller which guarantees global stability of the closed-loop system with complete CTM and under all possible operating scenarios is perform in the next chapter. 6.2 Stability of Trac Flow in a Single-Section Road Segment Consider a single road section of unit length with an in ow q 1 and out ow q 2 , expected to meet a demand of ow d as shown in Fig. 6.1. We assume that the  d 2 q 1 q Figure 6.1: Single Road Section vehicle density is uniform along the section, i.e. it is independent of distance 87 from the entrance to the exit of the section and does not vary across the lanes in the vertical direction. Under these assumptions, the evolution of with respect to time is given by the following dierential equation: _ =q 1 q 2 ; 0(0) j ; (6.1) where q 1 = minfd;C;w( j )g; q 2 = 8 < : minfv f ; ~ w(~ j ); (1())C d g if C d <C minfv f ; ~ w(~ j );C d g otherwise ; v f c =w( j c ) = ~ w(~ j c ) =C; 0< c < j ; 0< ~ w<w;v f > 0; () = 8 > < > : 0 if 0 C d v f 0 otherwise ; (6.2) and the constants in equation (6.1),(6.2) are dened as follows: C: the capacity of the road section. w: the back propagation speed. j : jam density, the highest density possible, at which q 1 = 0. v f : free ow speed of the road section. ~ w: the rate that the out ow q 2 decreases with , when c . ~ j : the jam density associated with out ow q 2 . c : the critical density of the road section, at which v f c = w( j c ) = ~ w(~ j c ) =C. C d : the downstream capacity. 88 In equation (6.2), the in ow q 1 is dictated by the upstream demand d as well as the potential ability of the section to absorb trac ow, which is the value minfC;w( j )g. If c , the section can absorb as much ow as the capacity C, however if > c , the section's ability to absorb upstream ow decreases with at a rate w. When = j , q 1 = 0 as the section is completely congested. The out owq 2 is dictated by the ability of the section to send trac ow to downstream and the downstream capacity. When c , the section's ability to send trac ow increases with , but when > c , this ability decreases with at a rate ~ w[83, 84, 59, 85]. Sincew> ~ w, we have ~ w(~ j )>w( j ) for all> c , which captures the phenomenon that if the downstream segment has enough capacity, the density in a congested road section upstream will eventually decrease to a value less than or equal to c . The capacity of the downstream segment isC d . IfC d <C and C d v f , then the out ow q 2 = v f can increase up to C d . However, when > C d v f , the section generates more ow than C d , a queue will form at the outlet, which may cause forced lane changes which in turn reduce the ow speed leading to the reduction of ow to lower than the capacity C d i.e. to (1 0 )C d [5, 26]. This phenomenon is known as capacity drop. The original CTM is modied to include the capacity drop eect as shown in equation (6.2). The model (6.1) - (6.2) with 0 = 0 is the CTM of [67]. The 0 > 0 denotes the level of capacity drop, in which case, despite the availability of ow, q 2 is restricted from reaching the capacity C d . Note that capacity drop can only occur when the downstream capacity C d is lower than the capacity of the section C. In system (6.1)-(6.2), we model the capacity drop using a reduction in the downstream capacity C d which has been veried by microscopic simulations using VISSIM in [5]. The modeling of capacity drop has been discussed in [60] more extensively where dierent models are considered. These models do not change the methodology and results of this chapter, which can be easily extended to dierent capacity drop models. 89 The purpose of this section is to analyze the stability properties of the model (6.1)-(6.2). Since these properties will depend on the characteristics of the road section dened by the constants C;C d , the demand d which could vary and the magnitude of capacity drop 0 which may depend on microscopic eects [5, 60], the following ve possible operating scenarios are identied and represented by the sets i , i = 1; 2;:::; 5. The union of these sets 5 S i=1 i , as shown in Fig. 6.2, covers all possible situations. Let I = (C d ;C;d; 0 ) be the state of the road section. We d C d CC  0 (1 ) d dC   0 (1 ) d dC   0 (1 ) dd C d C     d dC  1  2  3  4  5  d CC  Figure 6.2: All Possible Operating Scenarios analyze the stability properties of the dynamical model (6.1)-(6.2) when I2 i , i = 1; 2;:::; 5. Theorem 6.2.1 presents the results of the analysis. Theorem 6.2.1. For constant but otherwise arbitrary demand d, we have the fol- lowing results: a) Let I2 1 . Then8(0)2 [0; j ], (t) converges exponentially fast to d v f . b) Let I2 2 . Then 8(0)2 [0; C d v f ], (t) converges exponentially fast to d v f = (1 0 )C d v f . 8(0)2 ( C d v f ; j d w ], (t) =(0);8t 0. 8(0)2 ( j d w ; j ], (t) converges exponentially fast to j d w = j (1 0 )C d w . c) Let I2 3 . Then 8(0)2 [0; C d v f ], (t) converges exponentially fast to d v f . 90 8(0)2 ( C d v f ; j ], (t) converges exponentially fast to j (1 0 )C d w . d) Let I 2 4 . Then8(0) 2 [0; j ], (t) converges exponentially fast to j (1 0 )C d w . e) Let I2 5 . Then8(0)2 [0; j ], (t) converges exponentially fast to minfd;Cg v f . Proof. a) When I2 1 , we plot the relationship of q 1 ;q 2 given by equation (6.2) in Fig. 6.3. From the density equation (6.1), the equilibrium points of the system C q d C 0 (1 ) d C   d f v w w 1 q 2 q  c  j  j  f d v d f C v 0 (1 ) j d C w     j d w   Figure 6.3: Fundamental Diagram for I2 1 are the values of for which _ = 0, which happens when q 1 = q 2 . It is clear from Fig. 6.3 that the only intersection of q 1 and q 2 is the point e = d v f , which implies that this is the only equilibrium of in the region [0; j ] of feasible values of. We dene the Lyapunov function V () = (d=v f ) 2 2 ; whose time derivative _ V () = ( d v f ) _ =( d v f )(q 2 q 1 ): We show in Appendix A.1 that _ V( d v f ) 2 ; 91 where = minfv f ; (1 0 )C d d j d=v f ; ( ~ ww)[c( j d w )] j d=v f g > 0. Hence converges exponen- tially fast to d v f with a rate greater than or equal to for all possible initial con- ditions in [0; j ][86]. The rate of convergence is guaranteed to be greater than or equal to as it is clear from the value of V and _ V . b) When I 2 2 , the plot of q 1 ;q 2 generated from equation (6.2) is given in Fig. 6.4. In this case, q 1 and q 2 intersect at one point = d v f and q 1 = q 2 for all C q d C f v w w 1 q 2 q  c  j  j  j d w   f d v d f C v 0 (1 ) d Cd  j d w   Figure 6.4: Fundamental Diagram for I2 2 2 ( C d v f ; j d w ]. Therefore, we have one isolated equilibrium point e 1 = d v f and an equilibrium manifold which is the interval ( C d v f ; j d w ]. From Fig. 6.4, we know that82 [0; C d v f ]; q 1 = (1 0 )C d = d and q 2 = v f which gives _ =v f +d;8(0)2 [0; C d v f ]; whose solution is (t) = d v f + ((0) d v f )e v f t C d v f : Hence8(0)2 [0; C d v f ] we have (t)2 [0; C d v f ]; 8t 0 and according to the solution above, (t) converges exponentially fast to d v f = (1 0 )C d v f . For (0)2 ( C d v f ; j d w ], we have q 1 = q 2 , therefore _ = 0, which implies that (t) =(0);8t 0, for all (0)2 ( C d v f ; j d w ]. If (0) 2 ( j d w ; j ], it is clear from Fig. 6.4 that q 2 > q 1 which implies that _ < 0 until (t) = j d w at which time _ = 0. This implies that for all 92 (0)2 ( j d w ; j ], (t) converges at least asymptotically with time to j d w . In Appendix A.2 we show that this rate of convergence is exponential, i.e. j(t) ( j d w )jc 0 e t ;8(0)2 ( j d w ; j ]; where c 0 > 0 and = minfw;w ~ wg> 0. c) When I 2 3 , q 1 and q 2 described by equation (6.2) are plotted in Fig. 6.5. From Fig. 6.5, it is clear that the only values of for which q 1 = q 2 are C q d C f v w w 1 q 2 q  c  j  j  f d v d f C v 0 (1 ) d C   d 0 (1 ) j d C w     j d w   0 (1 ) j d C w     Figure 6.5: Fundamental Diagram for I2 3 d v f and j (1 0 )C d w , which implies that the system has two isolated equilibrium points e 1 = d v f and e 2 = j (1 0 )C d w when I2 3 . We show below that e 1 = d v f is exponentially stable with a region of attraction [0; C d v f ] and e 2 = j (1 0 )C d w is exponentially stable with a region of attraction ( C d v f ; j ]. For (0)2 [0; C d v f ], we have q 1 = d;q 2 = v f , therefore _ =v f +d;8(0)2 [0; C d v f ], whose solution is =e v f t ((0) d v f ) + d v f ; which implies that (t)2 [0; C d v f ];8t 0 and (t) converges exponentially fast to e 1 = d v f . 93 Consider the equilibrium point e 2 and choose the Lyapunov function V () = ( e 2 ) 2 2 ; then _ V =( e 2 )(q 2 q 1 ). We show in Appendix A.3 that _ V( e 2 ) 2 ; where = minf d(1 0 )C d e 2 C d =v f ;w; (w ~ w)g > 0; 8(0)2 ( C d v f ; j ] which implies expo- nential convergence to the equilibrium point e 2 = j (1 0 )C d w ,8(0)2 ( C d v f ; j ]. d) When I2 4 , q 1 and q 2 described by equation (6.2) are plotted in Fig. 6.6. From Fig. 6.6, it is clear thatq 1 =q 2 when = e = j (1 0 )C d w , which is a unique C q d C f v w w 1 q 2 q  c  j  j  d f C v 0 (1 ) d C   d 0 (1 ) j d C w     j d w   0 (1 ) j d C w     Figure 6.6: Fundamental Diagram for I2 4 equilibrium when I2 4 . Choose the Lyapunov function V () = ( e ) 2 2 ; then _ V =( e )(q 2 q 1 ). We show in Appendix A.4 that _ V =( e ) 2 ; where = minf dC d e ; d(1 0 )C d e C d =v f ;w; (w ~ w)g> 0;82 [0; j ], which implies expo- nential convergence to the equilibrium point e = j (1 0 )C d w ,8(0)2 [0; j ]. 94 e) WhenI2 5 ,q 1 andq 2 described by equation (6.2) are plotted in Fig. 6.7. In C q f v w w 1 q 2 q  c  j  j  d j d w   f d v (a) d<C C q f v w w 1 q 2 q  c  j  j  (b) dC Figure 6.7: Fundamental Diagram for I2 5 this case it is clear that there is only one equilibrium point e = minfd;Cg v f , depending whether the demand d<C or dC. We choose the Lyapunov function V () = ( e ) 2 2 and show in Appendix A.5 that _ V =( e ) 2 ; where = minfv f ; ( ~ ww)[c( j d w )] j d=v f g> 0 if d<C and = minfv f ; (w ~ w)g> 0 if dC,82 [0; j ], which implies exponential convergence to the equilibrium point e = minfd;Cg v f ,8(0)2 [0; j ]. 6.3 Stability of Trac Flow in a Multi-Section Road Segment The equilibrium points and their stability analysis of the single section CTM can be extended to the general N section case. Consider a road segment which is divided 95 into N (N 2) sections as in Fig. 6.8. Without loss of generality, we assume that the geometry of all sections is identical and each section has unit length. In the single section case, we assume the density to be the same along the section. We extend this to the case of multiple sections 1 to N where each section has its own density. The capacity of all sections remains the same constantC and the capacity at the outlet is C d whereas the demand d appears at the entrance of section 1 as shown in Fig. 6.8. It is well-known that the CTM in the multiple section case may include discontinuities in the values of densities when transitioning from one section to another. The control objective to be achieved via VSL, will require all section densities to converge to the same value in order to have smooth ow. N i 1 2 N-1 1 N q  1  2  i  1 N   N  N q 1 N q  i q 1 q 2 q d d C Figure 6.8: Multiple Section Road Network Let = [ 1 ; 2 ;:::; N ] T be the state vector of the trac ow system, where i represents the density in sectioni. Sectioni can absorb the ow minfC;w( j i )g from upstream and can generate the ow minfv f i ; ~ w(~ j i )g into the down- stream section. Therefore, the dynamics of the vehicle densities in each section are formulated as: _ i =q i q i+1 ; 0 i (0) j ; for i = 1; 2;:::;N; q 1 = minfd;C;w( j 1 )g; q i = minfv f i1 ; ~ w(~ j i1 );C;w( j i )g;i = 2;:::;N; q N+1 = 8 < : minfv f N ; ~ w(~ j N ); (1( N ))C d g if C d <C minfv f N ; ~ w(~ j N );C d g otherwise ; (6.3) 96 where ( N ) = 8 > < > : 0 if 0 N C d v f 0 otherwise and 0< 0 < 1 denotes the level of capacity drop at the outlet of the Nth section. Since we assume that the capacities of all sections 1 to N have the same value C, the capacity drop can only happen at the outlet of sectionN, whenC d <C, which aects the value of q N+1 . We know that8t 0, the density vector (t) belongs to the feasible set S =fj0 i j ; for i = 1; 2;:::;Ng: Let e = [ e 1 ; e 2 ;:::; e N ] T be the equilibrium vector of system (6.3), obtained by setting _ i = 0, for i = 1; 2; ;N. Let q e i denote the value of q i when = e , then the equilibrium condition of system (6.3) is given by q e 1 =q e 2 =::: =q e N+1 ; (6.4) due to _ i =q i q i+1 = 0, for i = 1; 2; ;N. Dene the vector of initial condition (0) = [ 1 (0); 2 (0);:::; N (0)] T and the parameter vectorI = (C d ;C;d; 0 ), whose partition sets are the same as in the case of a single section and are shown in Fig. 6.2. Then the equilibrium states of (6.3) for all possible I in the sets 1 to 5 and corresponding stability properties are given by the following theorem. Theorem 6.3.1. Let 1 = [1; 1;:::; 1] T be a vector with N elements each equal to 1. For constant but otherwise arbitrary demand d, we have the following results: a) Let I 2 1 . The equilibrium state of (6.3) is equal to e = d v f 1 and it is exponentially stable, i.e for all (0)2 S, (t) converges exponentially fast to e = d v f 1. 97 b) Let I2 2 . System (6.3) has an isolated equilibrium state e = d v f 1, which is locally exponentially stable, and an innite number of equilibrium states dened by the set S e =f( j d w ) 1g[fj i = d v f ;i = 1; 2;:::;N 1; C d v f < N < j d w g [ [ N1 [ i=1 fj d v f i < j d w ; k = d v f ; 1k<i; r = j d w ;i<rNg]: All equilibrium states e 2S e are stable in the sense that for any > 0;9> 0, such that8(0) that satisfyk(0) e k < , (t) converges to a e 2 S e that satisesk e e k<. Furthermore,8(0)2fj0 i C d =v f ;i = 1; 2;:::;Ng, (t) converges to e = d v f 1 exponentially fast, and8(0) = 2 fj0 i C d =v f ;i = 1; 2;:::;Ng, 9 e 2 f d v f 1g[ S e , such that (t) converges to e asymptotically with time. c) Let I2 3 . System (6.3) has two isolated equilibrium states e1 = d v f 1 and e2 = ( j (1 0 )C d w )1, which are both locally exponentially stable. Furthermore, 8(0)2fj0 i C d =v f ;i = 1; 2;:::;Ng, (t) converges to e1 exponentially fast and8(0) = 2fj0 i C d =v f ;i = 1; 2;:::;Ng, (t) converges to either e1 or e2 exponentially fast. d) Let I2 4 . The equilibrium state of (6.3) is equal to e = ( j (1 0 )C d w ) 1 and is exponentially stable, i.e for all(0)2S, (t) converges exponentially fast to e = ( j (1 0 )C d w ) 1. e) Let I 2 5 . The equilibrium state of (6.3) is equal to e = minfd;Cg v f 1 and is exponentially stable, i.e for all (0)2 S, (t) converges exponentially fast to e = minfd;Cg v f 1. The proof of Theorem 6.3.1 is given in Appendix B. The above stability properties show that depending on the situation classied by the operating scenarios 1 to 5 and initial density value in the section, the density 98 will reach an equilibrium that is not always the one that corresponds to maximum ow rate. In fact when I2 2 there are an innite number of equilibrium points and when I2 3 , there are two equilibrium points. One in the free ow region and one in the congested region depending on the initial density condition. The objective of feedback is to close the loop so that the system converges to a single equilibrium point for the density which also corresponds to the maximum possible ow rate and speed. The feedback control variable is variable speed limit that provides speed commands to the upstream section in order to control the in ow to the section in a way that guarantees the maximum possible out ow from the downstream section. Such a design is presented in the next section. 99 Chapter 7 VSL Control of the Cell Transmission Model under All Operating Conditions 7.1 Control of Trac Flow: Single Section The stability analysis of the ow in Section 6.2 shows that if C d C, i.e. the downstream capacity is higher than the capacity of the section, i.e. I2 5 then the density(t) converges exponentially fast to a unique equilibrium point minfd;Cg v f , which corresponds to the maximum possible ow. The steady state speed of ow in the section isv f and the steady state section ow will be at the maximum possible value q =q 1 =q 2 = minfd;Cg according to the model (6.1)-(6.2). In this case no control action is needed. WhenC d <C andd< (1 0 )C d , i.e. I2 1 , the demand is lower than the dropped capacity of the downstream segment and therefore the density converges exponentially fast to d v f and the steady state ow speed and ow rate in the section will be v f and d respectively. In this case, no control action is needed as the section operates at the maximum possible ow rate level dictated by the demand d. The problem arises when C d < C and d (1 0 )C d . where we have the following control problem cases: (i) (1 0 )C d =d<C d <C, i.e. I2 2 . (ii) (1 0 )C d <dC d <C, i.e. I2 3 . 100 (iii) C d <d;C d <C, i.e. I2 4 . In case (i) we showed in previous section that a maximum ow of d = (1 0 )C d can be maintained at an innite number of density equilibrium points specied by an isolated point and an equilibrium manifold, which include low and high density values with steady state speeds v ss v f . In this case, the control objective is to maintain the maximum ow ofd = (1 0 )C d with a lowest possible density which in this case is d v f = (1 0 )C d v f with free ow speed v f . In case (ii), we showed that we have two stable equilibrium points for density. One at low density which is equal to d v f and one at high density equal to j (1 0 )C d v f . In this case, maximum ow in the section corresponds to the density equilibrium point = d v f therefore the control objective is to choose the VSL in a way that the density converges to d v f for all possible initial density conditions. In case (iii), there is only one equilibrium point for density which is in the high density region and corresponds to the steady state ow of (1 0 )C d . In this case, the maximum possible ow is C d and corresponds to the density of C d v f . However, the convergence of to C d v f does not guarantee thatq 1 andq 2 converge toC d due to the capacity drop. From equation (6.2) and Fig. 6.6, we know that q 2 is a function of . For 2 [0; C d v f ];q 2 = v f , and for 2 ( C d v f ; ~ j (1 0 )C d ~ w ];q 2 = (1 0 )C d . Therefore we have lim !( C d v f ) q 2 () = lim ! C d v f v f =C d and lim !( C d v f ) + q 2 () = lim ! C d v f (1 0 )C d = (1 0 )C d ; i.e., if converges to C d v f from the left side, thenq 2 converges to the maximum value C d . However, if converges to C d v f from the right side, q 2 converges to (1 0 )C d . Therefore, the control objective in this case is to choose the VSL so that (t) satises the following conditions: 9t 0 > 0, such that8t t 0 ; (t) C d v f and lim t!1 (t) = C d v f . 101 Therefore for all cases (i), (ii) and (iii), the control objective is to choose the VSL control so that (t) converges to the desired equilibrium point minfd;C d g v f , and the ow rate q 1 and q 2 converge to the maximum possible level which is equal to minfd;C d g. A reasonable control action is to use VSL control to restrict the incoming owq 1 to the level that is within the capacity constraints of the section at the bottleneck so that the density and ow rate converge to the desired possible values. As shown  VSL Zone f v v 1 q 2 q d Figure 7.1: Road Section with VSL Control in Fig. 7.1, we apply the VSL command v in the upstream segment of the section under consideration, which is referred to as the VSL zone. All vehicles are asked to follow the speed limit v in the VSL zone and follow the free ow speed limit v f inside the section. Decreasing the speed limit leads to lower ow q 1 from the VSL zone to the section as shown in Fig. 7.1. Fig. 7.2 shows how the changing of the speed limit v can control the ow rate q 1 entering the section through a nonlinear relationship. Suppose the VSL zone has similar characteristics as the road section under consideration. If the VSL command is set tov<v f , the fundamental diagram of the VSL zone is distorted such that the parameters j ;w; ~ w remain unchanged, while the maximum possible ow is decreased to vw j v+w , as shown in Fig. 7.2, obtained by simple geometric considerations [26, 55, 56]. In Fig. 7.2, the red line s v denotes the ow rate that the VSL zone can absorb from upstream under dierent densities in the VSL zone and the blue lined v denotes the ow rate that the VSL zone sends to the section under consideration. However, since the single section model does not include the density in the VSL zone, the ow into the road section from the 102 C q f v w w  c  j  j  j w vw   j vw vw   v v s v d Figure 7.2: Fundamental Diagram of the VSL Zone VSL zone is assumed to be minfd; vw j v+w g, where vw j v+w is the maximum possible ow in the VSL zone under speed limit v. Then the density in the section is given by the following equation: _ =q 1 q 2 ; 0(0) j ; q 1 = minfd; vw j v +w ;C;w( j )g; q 2 = minfv f ; ~ w(~ j ); (1())C d g: (7.1) We design a VSL controller to overcome capacity drop and achieve the control objectives in all cases, by rst considering the most complicated case I2 4 , in which d>C d . Since in equation (7.1), vw j v+w is the only term in q 1 that depends on v, we derive the VSL controller using feedback linearization under the assumption that q 1 = vw j v+w . Then we show in Theorem 7.1.1 that, for the general equation where q 1 = minfd; vw j v+w ;C;w( j )g, the derived controller can still guarantee that converges to C d v f andq 1 ;q 2 converge to the maximum valueC d . Furthermore, we also show in Theorem 7.1.1 below that, when I2 3 S i=1 i , i.e. dC d , the same controller guarantees the convergence of to the desired equilibrium point d v f and the convergence of q 1 ;q 2 to the maximum level d. As discussed above, when I 2 4 , the desired equilibrium point is e = C d v f . Dene the error state x = C d v f and recall that the control objective is to force 103 to converge to C d v f , i.e. x converge to 0 from the left side ( C d v f ). If x(0) 0, that is (0) C d v f , we choose v so that q 1 =q 2 x; (7.2) where > 0 is a design constant to be selected. Thus we have _ x = _ =q 1 q 2 =x; which implies that8x(0) 0 andt 0,x(t) 0 andx converges to 0 exponentially fast. Since we assume that q 1 = vw j v+w , solving equation (7.2) for v, we have, v = w(q 2 x) w j (q 2 x) ; (7.3) whose denominator is guaranteed to be greater than 0 as we show in detail in the proof of Theorem 7.1.1. If x(0)> 0, i.e. (0)> C d v f we choose v such that q 1 =q 2 (x + 1 ); (7.4) where 1 > 0 is a design constant. Then we have8x(0)> 0 _ x = _ =q 1 q 2 =(x + 1 ): Thus x will decrease exponentially toward the value 1 < 0. At some nite time t = t 0 > 0, x(t 0 ) = 2 , i.e. (t 0 ) = C d v f 2 , where 0 < 2 < minf 1 ; C d v f g, thus (t 0 ) is in the region of (7.2),(7.3). At the time instant t = t 0 , we have x(t) 0 and controller (7.3) is switched on which guarantees as shown above that x(t) will 104 converge to zero exponentially fast. Assuming that q 1 = vw j v+w and solving (7.4) for v, we have v = w(q 2 (x + 1 )) w j (q 2 (x + 1 )) : (7.5) The use of the design constant 1 is to reduce the incoming ow via VSL so that the density of the section reduces to be within the set [0; C d v f ], which guarantees convergence to the equilibrium point which corresponds to maximum ow and speed. The choice of 1 will depend on how aggressively we want the density to move to the \good" free speed region. Using the above VSL controller derivation and assuming that the speed is not allowed to go below zero or exceed the speed limitv f , the following equations summarize the VSL controller for the section under the assumption that q 1 = vw j v+w , which we will relax subsequently. When I2 4 S i=1 i , the VSL control is generated as follows: v 1 = w[q 2 (x + 1 )] w j [q 2 (x + 1 )] ; v 2 = w(q 2 x) w j (q 2 x) ; v i = medf0; v i ;v f g;i = 1; 2; v = 8 > > > > > > > < > > > > > > > : v 1 if (0)> C d v f and (t)> C d v f 2 v 2 if (0)> C d v f and (t) = C d v f 2 v 2 if (0) C d v f and (t) C d v f ; (7.6) where x = C d v f , and 1 > 0; 0< 2 < minf 1 ; C d v f g; 0<< v f w j C d are design constants and medfg denotes the median of the numbers, which indicates that the VSL command saturates at the upper bound v f and the lower bound 0. 105 The upper bound v f w j C d of guarantees that the denominator of v is not 0, which we will show in the proof of Theorem 7.1.1. The shape of the functionv as it varies with is shown in Fig. 7.3. v 1 v 2 v  j  d f C v 2 d f C v   0 Figure 7.3: Switching Logic of VSL Controller For I2 5 , the VSL control is v =v f : (7.7) In Theorem 7.1.1 below, we show that the above controller also works for any value of q 1 = minfd; vw j v+w ;C;w( j )g, and guarantees the exponential convergence of the density to the desired equilibrium point and the exponential convergence of the ow rate to the maximum possible value of q 1 =q 2 =C d . Furthermore, when I2 3 S i=1 i , i.e. dC d , controller (7.6) guarantees the exponential convergence of to the desired equilibrium point d v f and the convergence of q 1 ;q 2 to the maximum level d. Theorem 7.1.1. For q 1 = minfd; vw j v+w ;C;w( j )g, we have the following: a) Let I2 4 S i=1 i , i.e. C d <C, and consider the VSL controller (7.6). The closed- loop system (7.1), (7.6) has a unique equilibrium point e = minfd;C d g v f . In addi- tion,8(0)2 [0; C d v f ],(t) converges to e exponentially fast and8(0)2 ( C d v f ; j ], (t) decreases to C d v f 2 exponentially fast which brings it to the region where 106 (t) converges to e exponentially fast. The ow rate and speed converge to the desired values of minfd;C d g and v f respectively with the same rate. b) Let I 2 5 , i.e. C d C, and consider the VSL controller (7.7). System (7.1),(7.7) has a unique equilibrium point e = minfd;Cg v f . In addition,8(0)2 [0; j ], (t) converges to e exponentially fast. The ow rate and speed converge exponentially fast to the desired values of minfd;Cg and v f respectively. The proof of Theorem 7.1.1 is given in Appendix C. Theorem 7.1.1 shows that the VSL controller (7.6), (7.7) guarantees that for all cases I 2 5 S i=1 i , the density, ow rate and ow speed converge exponentially fast to unique values that correspond to maximum possible ow through the section for all initial density conditions within the set [0; j ]. Theorem 7.1.1 shows in an analytically rigorous manner that VSL control can stabilize the ow in the section and force it to converge to the maximum possible ow under any situation. This maximum ow depends on the characteristics and relationships between demand d and capacities C;C d as well as capacity drop level 0 . It is also clear from the analysis of the open-loop system that without the VSL control the ow can reach steady states that do not correspond to maximum possible ow. From equation (7.6), we can see that the logic of the VSL controller is to deacti- vate the capacity drop withv 1 by suppressing the in ow suciently and then force the system state to converge to the desired equilibrium point with v 2 . This logic and the feedback linearization technique can always be used to design a VSL con- troller if dierent capacity drop models such as those presented in [60] are included in the CTM. 7.2 N-section Road Segment with VSL Control The analysis in Section 6.3 shows that the stability properties of the open-loop N-section system are similar to those of the single-section system. For the cases 107 I 2 1 and I 2 5 , (t) converges exponentially fast to the unique equilibrium state e = d v f 1 and e = minfd;Cg v f 1 respectively, which corresponds to the maximum possible ow rate. In these two cases no control action is needed. When I2 2 [ 3 , the control objective is to stabilize the system at the equi- librium state e = d v f 1, at which the maximum possible ow rate d is achieved and the densities in each section are stabilized at the lowest possible value whereas the speed of ow converges to the free ow speed v f . When I2 4 , the maximum possible ow rate is C d , which corresponds to the equilibrium state e = C d v f 1. From equation (6.3), we know that due to capacity drop lim N !( C d v f ) q N+1 ( N ) = lim N ! C d v f v f N =C d and lim N !( C d v f ) + q N+1 ( N ) = lim N ! C d v f (1 0 )C d = (1 0 )C d : Therefore, in this case, in order to achieve the maximum possible ow rate C d , we want to choose the VSL control so that there exits t 0 0 such that8t t 0 , N (t) C d v f and i (t) converges to C d v f , for i = 1; 2;:::;N. Furthermore we want to achieve a steady state ow speed v f in all sections. Similar to the single section case, the VSL controller is applied to theN-section road segment as shown in Fig. 7.4. All vehicles in the upstream segment of section 0 v d C 1 v 2 v i v 1 N v  f v 1  2  i  1 N   N  1 q 2 q i q 1 N q  N q 1 N q  d Figure 7.4: VSL Controlled Road Segment 1 are asked to follow the VSL command v 0 and all vehicles in section i follow the 108 VSL command v i , for i = 1; 2;:::;N 1. The speed limit in section N is set to the constant free ow speed v f . If the speed limit of section i is set to be v i v f , i = 1; 2;:::;N 1, then the fundamental diagram of section i is distorted as shown in Fig. 7.5. In Fig. 7.5, C q f v w w c  j  j  j v  j i i vw vw   () ii s  () ii d  j i w vw   i  () j vi w  i v () j i w  Figure 7.5: Fundamental Diagram of Section i s i ( i ) denotes the ability of section i to absorb trac ow from section i 1. We haves i ( i ) = minf v i w j v i +w ;w( j i )g;i = 1; 2;:::;N1. d i ( i ) denotes the trac ow generated by sectioni to go into sectioni+1. We haved i ( i ) = minfv i i ; v i w j v i +w g;i = 1; 2;:::N 1. Therefore, in Fig. 7.4, we have q i = minfd i1 ( i1 );s i ( i )g = minfv i1 i1 ; v i1 w j v i1 +w ; v i w j v i +w ;w( j i )g;i = 2;:::;N1: For the road segment upstream section 1, i.e. the segment with speed limit v 0 , whose density is not included in system (6.3), we assume the ow rate generated by this segment to be d 0 = minfd; v 0 w j v 0 +w g, which is independent of the density in the section with speed limit v 0 , therefore q 1 = minfd 0 ;s 1 ( 1 )g = minfd; v 0 w j v 0 +w ; v 1 w j v 1 +w ;w( j 1 )g: 109 The speed limit in section N is constant v f , therefore section N can absorb a ow of s N ( N ) = minfC;w( j N )g, therefore q N = minfd N1 ( N1 );s N ( N )g = minfv N1 N1 ; v N1 w j v N1 +w ;C;w( j N )g: For the sake of simplicity, we omit the term ~ w(~ j v i ) from d i ( i ), where ~ j v is ~ j distorted by the VSL. As shown in Fig. 7.5, for i = 1;:::;N 1, if the out ow q i+1 = ~ w(~ j v i ), then the in ow q i s i ( i ) = w( j i ) < q i+1 will force i to decrease until q i+1 6= ~ w(~ j v i ). Therefore, this simplication does not aect the results. The system model with VSL control inputs can be formulated as follows: _ i =q i q i+1 ; 0 i (0) j ; for i = 1; 2;:::;N; q 1 = minfd; v 0 w j v 0 +w ; v 1 w j v 1 +w ;w( j 1 )g; q i = minfv i1 i1 ; v i1 w j v i1 +w ; v i w j v i +w ;w( j i )g;i = 2; 3;:::;N 1; q N = minfv N1 N1 ; v N1 w j v N1 +w ;C;w( j N )g; q N+1 = minfv f N ; (1( N ))C d ; ~ w(~ j N )g: (7.8) Similar to the single section system, the objective is to design a VSL controller that can overcome the capacity drop and achieve the control objectives in all cases. We derive the VSL controller using feedback linearization for the case of I2 4 then show in Theorem 7.2.1 that the controller also works for all other scenarios. When I2 4 , i.e. d > C d , we need to decrease v 0 to suppress q 1 so that the ow from upstream can be handled by the downstream capacity C d . We start by assuming thatq 1 = v 0 w j v 0 +w , which is the only term in the equation ofq 1 that depends onv 0 and then show that the VSL controller works for all values ofq 1 . Furthermore, in this case, the desired equilibrium density is e i = C d v f for i = 1; 2;:::;N and the equilibrium ow speed and ow rate are v e i =v f andq e i+1 =v e i e i =C d respectively 110 for i = 1; 2;:::;N 1. Therefore, we initially assume that q i = v i1 i1 for i = 2; 3;:::;N, which we relax in Theorem 7.2.1 below. Let x = [x 1 ;x 2 ;:::;x N ] T ; where x i = i C d v f ;i = 1; 2;:::;N. If x N (0) 0, i.e. N (0) C d v f , we choose v = [v 0 ;v 1 ;:::;v N1 ] T , such that q i =q i+1 i1 x i ;i = 1; 2;::;N; (7.9) where i > 0;i = 0; 1;:::;N 1 are design constants. Thus we have _ x i = _ i =q i q i+1 = i1 x i ;i = 1; 2;:::;N; which implies that x i (t) converges to 0 exponentially fast and x N (0) 0;8t 0. Since we assume that q 1 = v 0 w j v 0 +w and q i =v i1 i1 for i = 2;:::;N, solving (7.9) for v gives v 0 = (q 2 0 x 1 )w w j q 2 + 0 x 1 ; v i = q i+2 i x i+1 i ;i = 1; 2;:::;N 1: (7.10) If x N (0)> 0, i.e. N (0)> C d v f , we choose v such that q i =q i+1 i1 x i ;i = 1; 2;::;N 1; q N =q N+1 N1 (x N + 1 ); (7.11) where 1 > 0 is a design constant. Then we have _ x i = _ i =q i q i+1 = i1 x i ;i = 1; 2;:::;N 1; _ x N = _ N =q N q N+1 = N1 (x N + 1 ); which implies that8x N (0)> 0, x N (t) will decrease exponentially toward 1 < 0. Therefore there exists t 0 > 0, such that x N (t 0 ) = 2 , i.e. N (t 0 ) = C d v f 2 , where 0 < 2 < minf 1 ; C d v f g, which is in the region of (7.9), (7.10). At t = t 0 , we have 111 x N (0)< 0 and the controller (7.10) is switched on, in which case x(t) converges to 0 exponentially fast as shown above. Solving (7.11) for v, we have v 0 = (q 2 0 x 1 )w w j q 2 + 0 x 1 ; v i = q i+2 i x i+1 i ;i = 1; 2;:::;N 2; v N1 = q N+1 N1 (x N + 1 ) N1 : (7.12) Using the above VSL controller and assuming that the speed is not allowed to go below zero or exceed the speed limitv f , the following equations summarize the VSL controller for theN-section road system under the assumption that q 1 = v 0 w j v 0 +w and q i = v i1 i1 ;i = 2; 3;:::;N, which we will relax subsequently. For all I2 4 S i=1 i , the VSL commands are generated as follows: v 0 = (q 2 0 x 1 )w w j q 2 + 0 x 1 ; v i = 8 > < > : q i+2 i x i+1 i i > 0 v f i = 0 ;i = 1; 2;:::;N 2; v N1;1 = q N+1 N1 (x N + 1 ) N1 ; v N1;2 = 8 > < > : q N+1 N1 x N N1 N1 > 0 v f N1 = 0 ; v N1 = 8 > > > > > > > < > > > > > > > : v N1;1 if N (0)> C d v f and N (t)> C d v f 2 v N1;2 if N (0)> C d v f and N (t) = C d v f 2 v N1;2 if N (0) C d v f and N (t) C d v f ; v i = medf0; v i ;v f g;i = 0; 1;:::;N 1; (7.13) 112 where 1 > 0; 0< 2 < minf 1 ; C d v f g, i >v f , fori = 1; 2:::;N1, 0< 0 < v f w j C d . In controller (7.13), 0 < v f w j C d guarantees that the denominator ofv 0 is always greater than 0. i > v f , for i = 1; 2:::;N 1 guarantees the exponential convergence of the density states, which we will show in the proof of Theorem 7.2.1 below. The switching logic ofv N1 is similar to that ofv shown in Fig.7.3 for the single section case. For I2 5 , the VSL command is v i =v f ;i = 0; 1;:::;N 1: (7.14) Similar to the single section case, we can show thatv 0 is well-dened as its denom- inator is always greater than 0. For i = 1; 2;:::;N 1, v i is also well-dened by setting v i =v f when its denominator is equal to 0. In Theorem 7.2.1, we show that the controller (7.13) also works for the general ow equations (7.8), and guarantees that i (t) converges to C d v f exponentially fast, fori = 1; 2;:::;N and the ow ratesq i converge exponentially fast to the maximum possible ow rate which in this case is equal to C d , for i = 1; 2;:::;N + 1. We also show in Theorem 7.2.1 that when I2 3 S i=1 i , controller (7.13) guarantees that i converges to d v f exponentially fast, for i = 1; 2;:::;N and q i converges to d, which is the maximum possible ow rate, for i = 1; 2;:::;N + 1. Theorem 7.2.1. We consider the trac ow model described by (7.8) with the VSL controller (7.13), (7.14): a) Let I2 4 S i=1 i , i.e. C d < C. The closed-loop system (7.8), (7.13) has a unique equilibrium state e = minfd;C d g v f 1. In addition,8(0)2fj0 N C d v f g, the density vector(t) converges to e exponentially fast and8(0)2fj C d v f g< N j , N (t) decreases to C d v f 2 exponentially fast, which brings it to the region where the density vector (t) converges to e exponentially fast. Furthermore, the ow rates q i ;i = 1; 2;:::;N + 1 and ow speeds v i ;i = 0; 1;:::;N 1 converge 113 to minfd;C d g and v f respectively which is the state which corresponds to the maximum possible ow. b) Let I2 5 , i.e. C d C. The closed-loop system (7.8), (7.14) has a unique equilibrium state e = minfd;Cg v f 1. In addition,8(0)2 S, the density vec- tor converges exponentially fast to e . Furthermore, the ow rates and ow speeds converge exponentially fast to minfd;Cg and v f respectively, achieving the maximum possible ow at steady state. The proof of Theorem 7.2.1 is presented in Appendix D. Theorem 7.2.1 shows that the VSL controller (7.13)-(7.14) guarantees that for all cases I2 5 S i=1 i , the steady state densities, ow rates and speeds of ow are stabilized at the desired values which correspond to the maximum ow rate through the road segment while achieving homogeneous density distribution. We should note that in Theorem 7.2.1, the design of the VSL controller (7.13) and the stability analysis of the closed-loop system (7.8) are performed under the assumption that we have perfect knowledge of system parameters of the open-loop system (6.3) and accurate measurement of the density vector . However when I 2 4 , since the desired equilibrium point of the closed-loop system (7.8), i.e. e = C d v f 1 lies exactly on the discontinuity plane of the fundamental diagram, which isfj N = C d v f g, when(t) = C d v f 1 at steady state, any disturbance in model parameters or measurement noise may push the density in section N to N > C d v f , which may lead to temporary capacity drop which the controller tries to correct leading to a possible oscillation around the desired equilibrium point. Even though such oscillations may not have any signicant impact in an actual trac situation, the proposed controller can be easily modied to avoid such oscillatory response. This is achieved by setting the desired equilibrium point to be e = ( C d v f ) 1, where > 0, in order to provide a margin between e and the discontinuity at N = C d v f . Thus in (7.13), x i = i ( C d v f ). With suciently large feedback gains 0 ;:::; N , controller (7.13) is able to stabilize the density state at a point 114 that is arbitrarily close to e = ( C d v f )1, therefore avoid the capacity drop. We will demonstrate this with numerical simulations in Section 7.3. Thus, although the controller (7.13) is designed for accurate system model, it can be robust with respect to system disturbance with simple modication. How to modify the controller of this paper to be robust with respect to a wide range of uncertainties is currently under investigation and it is outside the scope of this paper. However the ideal properties of the controller of this paper form the basis for comparison of any other controller under less ideal situations and for this reason it has its own merit. 7.3 Numerical Experiments In this section, we use numerical simulations to demonstrate the analytical results of the previous sections, for both open-loop and closed-loop systems. The simu- lations are performed on a two-section road network, whose parameters are: C = 6500 veh/h; w = 20 mi/h; j = 425 veh/mi; v f = 65 mi/h; ~ w = 10 mi/h; ~ j = 750 veh/mi; c = 100 veh/mi. WhenI belongs to 1 to 4 , we set the downstream capacity C d = 5200 veh/h, which is less than C, and 0 = 0:15. When I belongs to 5 , we set C d = 7000 veh/h, which is greater than C. The upstream demand d is set to be 4000 veh/h, 4420 veh/h, 5000 veh/h, 6000 veh/h and 6000 veh/h for the cases of I in 1 to 5 respectively. We apply controller (7.13),(7.14) to the two-section system with the following design constants: 1 = 2 = 70 mi/h; 1 = 20 veh/mi; 2 = 5 veh/mi. Among the abbreviated units we used above, \veh" stands for number of vehicles, \mi" stands for miles and \h" stands for hours. Fig. 7.6 - Fig. 7.10 show the phase portraits of the two-section open-loop and closed-loop systems when I belongs to 1 to 5 . When I 2 1 , all the density state trajectories of the open-loop system, shown in Fig. 7.6a, converge to the unique equilibrium state e = ( d v f ; d v f ) = (61:5; 61:5), indicated by the red dot. In Fig.7.6b, all density state trajectories of the closed-loop system converge to the same 115 0 50 100 150 200 250 300 350 400 ρ 1 0 50 100 150 200 250 300 350 400 ρ 2 (a) Open-loop 0 100 200 300 400 1 0 50 100 150 200 250 300 350 400 2 (b) Closed-loop Figure 7.6: Phase portrait when I2 1 (C d < C;d< (1 0 )C d ). Both the open- loop and closed-loop densities converge to the same low density equilibrium state. Single low density equilibrium state. 0 50 100 150 200 250 300 350 400 ρ 1 0 50 100 150 200 250 300 350 400 ρ 2 (a) Open-loop 0 100 200 300 400 1 0 50 100 150 200 250 300 350 400 2 (b) Closed-loop Figure 7.7: Phase portrait when I2 2 (C d < C;d = (1 0 )C d ). The open-loop system has an innite number of equilibrium density states which do not correspond to the maximum possible ow speed. Closed-loop system has a single low density equilibrium state. Equilibrium state; Equilibrium manifold. equilibrium state as in the open-loop case as expected from the analysis. When I 2 2 , all density state trajectories of the open-loop system shown in Fig.7.7a converge to the isolated equilibrium state e = ( d v f ; d v f ) = (68; 68), indicated by the red dot in Fig. 7.7a, or to the equilibrium set S e =fj 1 = d v f ; C d v f < 2 j d w g[fj d v f 1 j d w ; 2 = j d w g =fj 1 = 68; 80< 2 204g[fj68 1 204; 2 = 204g; 116 0 100 200 300 400 1 0 50 100 150 200 250 300 350 400 2 (a) Open-loop 0 100 200 300 400 1 0 50 100 150 200 250 300 350 400 2 (b) Closed-loop Figure 7.8: Phase portrait when I 2 3 (C d < C; (1 0 )C d < d C d ). The open-loop system has two equilibrium density states one in the low density and the other in the high density region. The closed-loop system has a unique equilibrium state at low density. Low density equilibrium state; High density equilibrium state. 0 100 200 300 400 1 0 50 100 150 200 250 300 350 400 2 (a) Open-loop 0 100 200 300 400 1 0 50 100 150 200 250 300 350 400 2 (b) Closed-loop Figure 7.9: Phase portrait when I2 4 (C d < C;d > C d ). The open-loop system has a unique equilibrium state in the high density region. The closed-loop system has a unique equilibrium state at low density. Low density equilibrium state; High density equilibrium state. indicated by the red line in Fig. 7.7a. When the VSL control is applied, all the density state trajectories of the closed-loop system converge to the unique equilibrium state e = ( d v f ; d v f ) = (68; 68), as shown in Fig. 7.7b. When I2 3 , each density state trajectory of the open-loop system shown in Fig.7.8a converges to one of the two isolated equilibrium states, e 1 = (d=v f ;d=v f ) = (77; 77) and e 2 = ( j (1 0 )C d w ; j (1 0 )C d w ) = (204; 204), indicated by the red dot and red 117 0 50 100 150 200 250 300 350 400 ρ 1 0 50 100 150 200 250 300 350 400 ρ 2 Figure 7.10: Phase portrait whenI2 5 (C d C). Same open-loop and closed-loop response. Single low density equilibrium state. 0 5 10 15 20 25 30 t (min) 4000 4500 5000 5500 6000 6500 Flow Rate (veh/hr) q 1 q 2 q 3 (a) Open-loop 0 2 4 6 8 10 t (min) 0 1000 2000 3000 4000 5000 6000 Flow Rate (veh/hr) q 1 q 2 q 3 (b) Closed-loop Figure 7.11: Flow rate when I2 4 0 2 4 6 8 10 t (min) 0 1000 2000 3000 4000 5000 Flow Rate (veh/hr) q 1 q 2 q 3 (a) Original Controller 0 2 4 6 8 10 t (min) 0 1000 2000 3000 4000 5000 Flow Rate (veh/hr) q 1 q 2 q 3 (b) Modied Controller Figure 7.12: Flow rate with Perturbed v f when I2 4 star respectively. All closed-loop state trajectories shown in Fig.7.8b converge to the unique equilibrium state e 1 = (d=v f ;d=v f ) = (77; 77). Fig. 7.9a shows that when I2 4 , all the density state trajectories converge to the unique equilibrium 118 state e = ( j (1 0 )C d w ; j (1 0 )C d w ) = (204; 204), indicated by the red dot. The phase portrait of the corresponding closed-loop system is plotted in Fig. 7.9b. As shown in Theorem 7.2.1, all density state trajectories converge to the desired equilibrium state e = (C d =v f ;C d =v f ) = (80; 80), indicated by the red dot in Fig. 7.9b. Furthermore, 2 converges to 2 = 80 when the initial condition satises 2 (0) 80. If 2 (0) > 80, 2 (t) decreases to 2 = 75 rst, then increases and converges to 80, which guarantees the steady-state ow rate C d = 5200 veh/h. WhenI2 5 , capacity drop will not occur since the downstream capacity is higher than the capacity of the road sections. All state trajectories in Fig. 7.10 converge to the unique equilibrium state e = (minfd;Cg=v f ; minfd;Cg=v f ) = (92:3; 92:3). The open-loop and closed-loop behavior when I2 5 are the same as expected. Fig.7.11 shows the ow rate time responses of the open-loop and closed-loop systems when I2 4 with initial condition = (110; 110). From Fig. 7.11a, we can see that at t = 0, q 1 (0) = d = 6000 veh/h;q 2 (0) = v f 1 (0) = 6500 veh/h < q 1 (0) and decrease to the steady state value of 4420 veh/h. On the other hand, q 3 = (1 0 )C d = 4420 veh/h remains constant during the entire simulation time (30 min). In Fig. 7.11b, q 3 = (1 0 )C d = 4420 veh/h at the beginning of the simulation, then jumps to 5200 veh/h, then oscillates a little and converges to C d = 5200 veh/h. The jump in the value of q 3 is due to the fact that 2 decreases and crosses the value C d v f , at whichq 3 jumps from (1 0 )C d toC d . The values ofq 1 andq 2 also have a jump betweent = 1 min andt = 2 min. This jump is caused by the switching of the VSL control (7.13) which at this time does not aect q 3 since q 3 is only a function of 2 , and does not jump when the VSL switches. Fig. 7.12 shows the performance of the closed-loop system in the same scenario as in Fig. 7.11, however with perturbed v f . In this case, the actual free ow speed v f = 0:9v fn , where v fn is the nominal value of v f , based on which the controller (7.13) is designed. That is, the VSL controller is over-estimating the ow rate at the bottleneck, therefore sends more ow to section 2 than it can handle, 119 which leads to temporary capacity drop, which the controller corrects creating an oscillation around an average that corresponds to the desired ow as shown in Fig. 7.12a. Our controller however can be easily modied to take care of the uncertainty without changing the fundamentals of the design and analysis. As shown in Fig. 7.12b, we modied the controller (7.13) as stated in Section 7.2 by setting x i = i ( C d v fn 5) = i 75;i = 1; 2 and increasing the feedback gains to be i = 100 mi/h, i = 0; 1. The modied controller tries to stabilize the density vector at e = (75; 75), which gives a margin between e and the boundary of capacity drop. The increased feedback gains are able to suppress the steady state error to make sure that the steady state value of is close to e thus capacity drop does not occur. With the modied VSL controller, the steady state density is = (74:2; 78:44), and the steady state trac ow is q 1 =q 2 =q 3 = 4590 veh/h, as shown in Fig. 7.12b. This a simple case how an uncertainty can be dealt with by the proposed controller. The robustness of the proposed controller with respect to a wide range of uncertainties is currently under investigation and it is outside the scope of this paper that focuses on the control design and analysis under ideal conditions. The results form the basis for comparison as uncertainties are included in the model. 120 Chapter 8 Robust VSL Control of Cell Transmission Model with Disturbance 8.1 Introduction Chapter 7 shows the possibility of achieving the maximum possible ow rate at a bottleneck and avoiding capacity drop, under the assumption that we have perfect knowledge of model parameters of the CTM and accurate measurement of the vehicle densities. We have shown as well in Chapter 7 that due to the discontinuous nature of the desired equilibrium point, any disturbance or measurement noise may lead to a oscillatory behavior of the closed-loop system. We also demonstrate with numerical simulations that with simple modication, the VSL controller can help the system avoid the oscillation and stabilize the density at an equilibrium point close to the desired one. In this chapter, we modify the VSL controller by adding the integral action in order to reject the constant disturbance which may be introduced by the ramp ows or biased measurement etc. 121 8.2 Robust Control of Trac Flow in a Single- Section Road Segment Consider a single road section of in Fig. 7.1, with a constant disturbance , which may be introduced by the ramp ows or biased measurement of the ow rate, the evolution of with respect to time is given by the following dierential equation: _ =q 1 q 2 +; 0(0) j ; (8.1) where q 1 = minfd; vw j v +w ;C;w( j )g; q 2 = minfv f ; ~ w(~ j ); (1())C d g:; v f c =w( j c ) = ~ w(~ j c ) =C; 0< c < j ; 0< ~ w<w;v f > 0; () = 8 > < > : 0 if 0 C d v f 0 otherwise ; (8.2) We assume the constant disturbance satisfy thatjj m C d , that is, com- paring to the bottleneck capacity, the magnitude of the disturbance is very small, which also guarantees that 0(t) j ;8t 0. We dene constants m < L ? < U < C d v f as shown in Fig. 8.1 to help the design of the controller. In the equation ofq 1 , the only term that can be controlled by the VSL controller is vw j v+w . Let q 1v = vw j v+w and assume d < C without loss of generality. We haveq 1 = minfd;q 1v ;w( j )g. Letx 1 = ? , system (8.1)-(8.2) can be rewritten as: _ x 1 =q 1 q 2 +; ? x 1 (0) j ? (8.3) 122 q d C 0 (1 ) d C   f v w 2 q  j  d f C v 0 (1 ) j d C w       U  L  Figure 8.1: Design Constants where q 1 = minfd;q 1v ;w( j )g q 2 = minfv f ; ~ w(~ j ); (1())C d g q 1v = medianf0; q 1v ;Cg: (8.4) In equation (8.4), the constraint q 1v = medianf0; q 1v ;Cg is applied to guarantee that 0 v v f , where q 1v is the unconstrained control input to be designed. When the road section is congested, i.e., x 1 is high, we try to decrease q 1 to bring x 1 back to the low region by letting q 1v =q s (8.5) where q s < minfv f L ; (1 0 )C d ; ~ w(~ j j )g is a small constant ow which guar- antees that8 L , i.e.8x 1 L ? , q s <q 2 , which implies that _ x 1 < 0. Thus, there exists a nite time instant t 0 > 0, at which x 1 (t 0 ) = L ? and we set q 1v =q 2 1 x 1 2 Z t t 0 x 1 d +c (8.6) where 2 > 0 and 1 > maxf2 p 2 ;v f + 2 v f g> 0. c is a constant we use to guarantee that converges to ? asymptotically, which we will show later. Controller (8.6) is a PI controller which tries to reject the disturbance and stabilize x 1 at x 1 = 0. Furthermore, once x 1 decreases to the uncongested region x 1 C d v f ? , we do not 123 want it to go back to the capacity drop region again. Therefore, ifx 1 increases and reaches U ? , we switch back to controller (8.5). To summarize, the controller with hysteresis characteristics can be formulated as below: q 1v =k(t) k(0) = 8 < : k 1 (0) if x 1 (0) L ? k 2 (0) otherwise k(t) = 8 < : k 2 (t) if k(t ) =k 1 and x 1 (t) = L ? k(t) otherwise ;8t> 0 (8.7) where k 1 (t) =q s k 2 (t) =q 2 1 x 1 2 Z t t 0 x 1 d 1 x 1 (t 0 ) m 2 (8.8) Mapping the ow rate control input into the VSL command, we have v = w j q 1v wq 1v where q 1v = medianf0; q 1v ;Cg. In the equation of k 2 (t), if k(0) = k 2 (0) at t = 0, thent 0 = 0 , and ifk(t) switches fromk 1 (t) tok 2 (t) att =t 0 , thent 0 is the switching time instant. When k(t) =k 2 (t), we have that x 1 (t) U ? , i.e. U < C d v f , thus q 2 =v f =v f ? +v f x 1 and q 1 = medianf0;d; q 1v g, due to d<C. Let x 2 (t) = Z t t 0 x 1 (t)d 1 x 1 (t 0 ) m 2 2 ;tt 0 : (8.9) Then we have _ x 2 =x 1 . Since x 2 (t) =x 2 (t 0 ) + Z t t 0 _ x 2 d; tt 0 ; 124 and substitute it into (8.9), we have x 2 (t 0 ) = 1 x 1 (t 0 ) m + 2 Then with k(t) =k 2 (t), system (8.3) can be written as _ x 1 = medianf0;d; q 1v g (v f ? +v f x 1 ) + _ x 2 =x 1 ;8tt 0 x 1 (t 0 ) L ? ;x 2 (t 0 ) = 1 x 1 (t 0 ) m + 2 (8.10) where q 1v =k 2 (t) = (v f ? +v f x 1 ) 1 x 1 2 (x 2 + 2 ) (8.11) and 2 > 0; 1 > maxf2 p 2 ;v f + 2 v f g > 0. Let x = [x 1 ;x 2 ] T be the state vector of system (8.10). We rst ignore the capacity drop, i.e. assume that (8.10) holds for allx 1 2<, and investigate the stability property of system (8.10) in the follow- ing lemma, which we will use to analyze the stability the system which takes the capacity drop into consideration. Lemma 8.2.1. Consider system (8.10), if d + v f ? , we have the following results: 1) System (8.10) has a unique equilibrium point x e = [0; 0] T . 2)8x(t 0 )2< 2 , x(t) converges to x e asymptotically. 3)8x(t 0 )2S =fxjv f ? d ( 1 v f )x 1 + 2 x 2 v f ? g\fxj v f ? v f < x 1 < v f ? d v f g, x 1 (t)2S;8tt 0 . Proof. Consider the value of q 1 , we have the following 3 cases: 125 1. When 0 q 1v d, i.e. x2S 1 =fxjv f ? d ( 1 v f )x 1 + 2 x 2 v f ? g, we have q 1 = q 1v , the dynamics of system (8.10) become: _ x 1 = 1 x 1 2 x 2 _ x 2 =x 1 (8.12) 2. When q 1v < 0, i.e. x2S 2 =fxj( 1 v f )x 1 + 2 x 2 >v f ? g, we have q 1 = 0, the dynamics of system (8.10) become: _ x 1 =v f x 1 v f ? + _ x 2 =x 1 (8.13) 3. When q 1v > d, i.e. x2 S 3 =fxj( 1 v f )x 1 + 2 x 2 < v f ? dg, we have q 1 =d, the dynamics of system (8.10) become: _ x 1 =v f x 1 v f ? + +d _ x 2 =x 1 (8.14) Therefore the state space is divided as shown in Figure 8.2. 1 x 2 x Saturation Boundaries 1 S 2 S 3 S (0,0) 1 f f v x v     1 f f vd x v     S Figure 8.2: State Space It is easy to show that system (8.13) and (8.14) have no equilibrium point. System (8.12) has a unique equilibrium point x e = [0; 0] T 2 S 1 . Therefore the 126 system (8.10) has a unique equilibrium point x e = [0; 0] T . Consider the Lyapunov function V (x) =x T Px; (8.15) where P = 2 4 2 1 1 2 1 + 2 2 3 5 It is easy to check that matrixP is symmetric and positive denite. ThereforeV (x) is positive denite. For all x2S 1 , we have _ x =Ax where A = 2 4 1 2 1 0 3 5 Therefore the derivative of the Lyapunov function is _ V (x) =x T (A T P +PA)x =x T Qx; where Q = 2 4 2 1 2 1 2 3 5 : Thus _ V (x)< 0 for all x2S 1 nf0g. Now we are going to show that if9t 0 0, such that x(t 0 )2S 3 , then9t 1 >t 0 , such that x(t 1 )2 S 1 . If9t 0 0, such that x(t 0 )2 S 3 , let = v f ? d, according to (8.14), we have that dx 2 dx 1 = x 1 v f x 1 = 1 v f + v 2 f x 1 +v f dx 2 = 1 v f dx 1 + v 2 f x 1 +v f dx 1 127 Take the integral of both sides, we have that x 2 (t)x 2 (t 0 ) = 1 v f (x 1 (t)x 1 (t 0 )) + v 2 f ln(v 2 f x 1 (t) +v f ) ln(v 2 f x 1 (t 0 ) +v f ) x 2 (t) = 1 v f x 1 (t) + v 2 f ln(v 2 f x 1 (t) +v f ) +x 2 (t 0 ) + 1 v f x 1 (t 0 ) v 2 f ln(v 2 f x 1 (t 0 ) +v f ) (8.16) From (8.14), we know thatx 1 (t) approaches v f whenx2S 3 . According to (8.16), x 2 approaches innite as x 1 (t) approaches v f . Thus at some nite time instant t 1 , x 2 (t 1 ) is large enough and ( 1 v f )x 1 + 2 x 2 =v f ? d, i.e. x2S 1 . Then we show that if9t 0 0, at which x(t 0 ) lies on the boundary between S 1 and S 3 , and x(t) moves into S 3 , then9t 1 >t 0 , at which x(t 0 ) lies on the boundary between S 1 and S 3 , and x(t) gets into S 1 . Furthermore, V (x(t 0 ))>V (x(t 1 )). The normal vector of the boundary line which points toS 3 isn = [v f 1 ; 2 ] T . If9t 0 0, at which x(t 0 ) lies on the boundary between S 1 and S 3 , and x(t) moves into S 3 , we have that x(t 0 ) = [x 1 (t 0 ); 1 v f 2 x 1 (t 0 ) + 2 ] T and _ x(t 0 ) = [v f x 1 (t 0 );x 1 (t 0 )] T Since _ x(t 0 ) points to S 3 , therefore n T _ x(t 0 )> 0, that is [v f ( 1 v f ) 2 ]x 1 (t 0 ) (v f 1 )> 0 x 1 (t 0 )> (v f 1 ) v f ( 1 v f ) 2 > v f due to 1 > v f + 2 v f . According to (8.14), if x(t 0 )2 S 3 and x 1 (t 0 ) > v f , then x 1 (t) > v f and _ x 1 (t) < 0;8t > t 0 , as long as x(t) stays in S 3 . Together with (8.16), we know that9t 1 >t 0 , at whichx(t) crosses the boundary and gets intoS 1 , 128 and x 1 (t 1 )> v f and x 1 (t 1 )<x 1 (t 0 ). For all points on the boundary between S 1 and S 3 , the Lyapunov function is evaluated as V (x) = 2x 2 1 + 2 1 x 1 x 2 + ( 2 1 + 2 2 )x 2 2 ; whose partial derivative with respect to x 1 along the boundary line is @V @x 1 = 4x 1 + 2 1 x 2 + 2 1 x 1 @x 2 @x 1 + 2( 2 1 + 2 2 ) @x 2 @x 1 (8.17) Since x(t 0 ) = [x 1 (t 0 ); 1 v f 2 x 1 (t 0 ) + 2 ] T on the boundary line, we have that @x 2 @x 1 = 1 v f 2 : Substituting into (8.17), we have @V @x 1 =ax 1 +b where a = 4 4 1 1 v f 2 + 2( 2 1 + 2 2 )( 1 v f 2 ) 2 and b = 2[ 1 ( 2 1 + 2 2 )] 2 : Note that a = 2[1; 1 v f 2 ] T P [1; 1 v f 2 ]> 0 due to P is positive denite. And a( 2 ) +b =[2 2 1 ( 1 v f ) + 2 1 2 ]> 0 due to< 0 and 1 >v f + 2 v f . Therefore,8x 1 > 2 , @V @x 1 =ax 1 +b> 0. Therefore, 2 <x 1 (t 1 )<x 1 (t 0 ) indicates that V (x(t 0 ))>V (x(t 1 )). 129 Similarly, we can show that if9t 0 0, such thatx(t 0 )2S 2 , then9t 1 >t 0 , such that x(t 1 )2 S 1 . And if9t 0 0, at which x(t 0 ) lies on the boundary between S 1 and S 2 , and x(t) gets into S 2 , then9t 1 > t 0 , at which x(t 0 ) lies on the boundary between S 1 and S 2 , and x(t) gets into S 1 . Furthermore, V (x(t 0 ))>V (x(t 1 )). Summarizing the behavior of the Lyapunov functionV (x), we can conclude that 8x(t 0 )2< 2 , x(t) converges to x e = [0; 0] T asymptotically. Since in (8.10), v f x 1 (v f ) _ x 1 v f x 1 (v f d); thus for all v f ? v f <x 1 (t 0 )< v f ? d v f , v f ? v f <x 1 (t)< v f ? d v f ,8tt 0 . Also we have shown that if x(t)2 S 1 \fxj v f ? v f < x 1 < v f ? d v f g, x(t) will not leave S 1 . Therefore,8x(t 0 )2 S =fxjv f ? d ( 1 v f )x 1 + 2 (x 2 ) v f ? g\fxj v f ? v f <x 1 < v f ? d v f g, x 1 (t)2S;8tt 0 . Lemma 8.2.1 shows that, if (8.10) holds for8x 1 2<, then the unique equilibrium point x e = [0; 0] T is globally asymptotically stable. However, the proof of Lemma 8.2.1 also shows that if x(t 0 )2S 2 , it is possible that x 1 (t) approaches 2 , which is already in the capacity drop region where (8.10) does not hold. Fortunately, in system (8.10), the initial conditions are set to be in a certain region. We only need to show that for some specic x(t 0 ), (8.10) holds for all tt 0 and x 1 (t) converges to 0. Theorem 8.2.1. Consider the system (8.3)(8.4) with controller (8.7), if d + v f ? , then8x 1 (0)2 [ ? ; j ? ], x 1 (t) converges to 0 asymptotically. Proof.8x 1 (0) < L ? , k(0) = k 2 (0). If8t 0; x 1 (0) < U ? , then (8.10) holds for all t 0, thus x 1 (t) converges to 0 asymptotically according to Lemma 8.2.1. If9t> 0, such that x 1 (t) = U ? , then k(t) switches to k 1 (t), then for all x 1 (t) L ? , _ x 1 (t) =q s q 2 < 0, thus9t 0 > 0, such that x 1 (t 0 ) = L ? and k(t 0 ) = k 2 (t 0 ). Similarly,8x 1 (0) > L ? , k(0) = k 1 (0), then9t 0 > 0, such that 130 x 1 (t 0 ) = L ? and k(t 0 ) = k 2 (t 0 ). Therefore, we only need to consider the case that x 1 (t 0 ) = L ? and k(t 0 ) = k 2 (t 0 ). As shown in (8.9) - (8.11), in this case, as long as k(t) =k 2 (t), the system (8.3)(8.4) can be written as _ x 1 = medianf0;d; q 1v g (v f ? +v f x 1 ) + _ x 2 =x 1 ;8tt 0 x 1 (t 0 ) = L ? ;x 2 (t 0 ) = 1 x 1 (t 0 ) m + 2 Since x 2 (t 0 ) = 1 x 1 (t 0 )m+ 2 , thus 1 x 1 (t 0 ) + 2 x 2 (t 0 ) = m > 0. And since m v f ? , we also have 1 x 1 (t 0 ) + 2 x 2 (t 0 ) < v f ? . Furthermore, x 1 (t 0 ) = L ? > v f ? m v f > v f ? v f . Similarly,x 1 (t 0 )< v f ? d v f . Thereforex(t 0 )2S. According to Lemma 8.2.1, as long as x 1 (t) < U ? , then x(t)2 S, and (8.12) holds, i.e. _ x =Ax x 1 (t 0 ) = L ? ;x 2 (t 0 ) = 1 x 1 (t 0 ) 2 where A = 2 4 1 2 1 0 3 5 : Therefore, x(t) =e A(tt 0 ) x(t 0 ); where e At =L 1 [(sIA) 1 ] andL 1 [] is the inverse Laplace transform operator. We can show that (sIA) 1 = 2 4 s s 2 + 1 s+ 2 2 s 2 + 1 s+ 2 1 s 2 + 1 s+ 2 s+ 1 s 2 + 1 s+ 2 3 5 131 Since 1 > 2 p 2 , therefore 2 1 4 2 > 0, the equations 2 + 1 s + 2 = 0 has 2 real negative roots, i.e. p 1 = 1 + p 2 1 4 2 2 ; p 1 = 1 p 2 1 4 2 2 It is easy to check that 0>p 1 >p 2 . We can calculate that e At =L 1 [(sIA) 1 ] = 2 4 a 11 (t) a 12 (t) a 21 (t) a 22 (t) 3 5 where a 11 (t) = 1 2 (e p 1 t +e p 2 t ) 1 p 1 p 2 (e p 1 t 2e p 2 t ) and a 12 (t) = 2 p 1 p 2 (e p 1 t e p 2 t ): Therefore x(t) =x 1 (t 0 )a 11 (tt 0 ) +x 2 (t 0 )a 12 (tt 0 ) = 1 2 x 1 (t 0 )(e p 1 (tt 0 ) +e p 2 (tt 0 ) ) 1 x 1 (t 0 ) + 2 x 2 (t 0 ) p 1 p 2 (e p 1 (tt 0 ) e p 2 (tt 0 ) ) Sincep 1 >p 2 ;x 1 (t 0 ) 0 and 1 x 1 (t 0 )+ 2 x 2 (t 0 )> 0, we have thatx(t)< 0;8tt 0 . Consequently, (8.14) holds for allt>t 0 . According to Lemma 8.2.1,x 1 (t) converges to 0 asymptotically. Theorem 8.2.1 shows that if d + v f ? , then controller (8.7) forces x 1 to converge to 0 asymptotically. In the case that d +<v f ? , the dynamics and the stability properties of the closed-loop system is shown in the following theorem. Theorem 8.2.2. Consider the system (8.3)(8.4) with controller (8.7), if d + < v f ? , then8x 1 (0)2 [ ? ; j ? ], x 1 (t) converges to d+ v f ? asymptotically. 132 Proof. Sinced +<v f ? , then9> 0, such thatd +v f ? . Similar to the case in Theorem 8.2.1, we only need to consider the case where x 1 (t 0 ) L ? and k(t 0 ) =k 2 (t 0 ). According to (8.3)(8.4), we have that _ x 1 dq 2 + =d + (v f ? +v f x 1 ) thus for all x 1 (t 0 ) L ? , x 1 (t) d+ v f ? ;8tt 0 . Therefore x 2 (t) =x 2 (t 0 ) + Z t t 0 x 1 (t)d 2 x 2 (t 0 ) 2 (tt 0 );tt 0 which decreases to negative innity as t increases. Therefore q 1 saturates at d. _ x 1 =d + (v f ? +v f x 1 ) Thus x 1 converges to d+ v f ? . Therefore, with the controller (8.7), if the sum of the upstream demand d and the disturbance is greater than or equal to the predetermined equilibrium ow, the density in the section will converge to the equilibrium point ? . If the sum of the upstream demandd and the disturbance is less than the predetermined equilibrium ow, the density converges to d+ v f , at which the steady state ow is d +, which is the maximum possible value. Note that the selection of L aects the distance from the switching point to the desired equilibrium point. According to the proof of Theorem 8.2.1, we can select L = ? which minimizes the distance while still guarantees the convergence. In addition, since(t) always converges to ? from the left side, ? can be arbitrarily close to C d v f . 133 8.3 Numerical Experiments In this section, we use numerical simulations to demonstrate the analytical re- sults of the previous sections. The simulations are performed on a single-section road network, whose parameters are: C = 6500 veh/h; w = 20 mi/h; j = 425 veh/mi; v f = 65 mi/h; ~ w = 10 mi/h; ~ j = 750 veh/mi; c = 100 veh/mi;d = 6000 veh/h; = 300 veh/h. We apply controller (8.7) to the perturbed single- section system with the following design constants: 1 = 200; 2 = 900; ? = L = 75 veh/mi; U = 79 veh/mi; m = 350 veh/h;q s = 3200 veh/h. The initial condi- tion (0) = 120 veh/mi. 0 5 10 15 20 25 30 t (min) 70 80 90 100 110 120 density (veh/mi) (a) Density 0 5 10 15 20 25 30 t (min) 3000 3500 4000 4500 5000 5500 Flow Rate (veh/hr) q1 q2 (b) Flow Rate 0 5 10 15 20 25 30 t (min) 12 14 16 18 20 22 24 VSL (mi/hr) (c) VSL Figure 8.3: System Behavior of the Perturbed Closed-loop System Figure 8.3 shows the behavior of the density, ow rate and the VSL commands respectively. We can see that the density converges asymptotically to the predeter- mined value ? = 75 veh/mi. There is a dierence of 100 veh/h between the steady state values ofq 1 andq 2 , since in steady stateq 1 q 2 + = 0. The out owq 2 starts at the value of 3500 veh/h, which is the dropped capacity of the bottleneck. Then suddenly jump to 4250 veh/h since the density decreases and becomes lower than the critical density C d =v f = 80 veh/h, thus the capacity drop is removed. Then q 2 decreases as keeps decreasing until = L . Then the PI controller takes over, and q 2 increases and converges to the steady state value eventually. 134 Chapter 9 Conclusion In this report, we reviewed the rst order and second order models of the highway trac ow network. Based on the rst order cell transmission model, we conducted design, analyze and evaluate the performance of several integrated highway trac ow control strategies in both macroscopic and microscopic simulations. We dis- covered that forced lane change at vicinity of the bottleneck is a major reason of the capacity drop phenomenon. We proposed a lane change controller which provides lane change recommendations to upstream vehicles in order to avoid the capacity drop. Two types of variable speed limit controllers are designed to improve the mobility, safety and environmental impact at highway bottleneck together with the lane change controller. The combined LC and feedback linearization VSL controller can theoretically guarantee the global exponential convergence to the desired equi- librium point at which maximum possible ow rate is achieved. Furthermore, the combined LC and VSL controller is extended to coordinate with ramp metering controllers. The coordinated VSL, RM and LC controller is able to improve system performance, maintain the queue length on ramps and keep the fairness between mainline and on-ramp ows. Microscopic simulations show consistent improvement under dierent trac demand and scenarios. The proposed controller is compared to the widely used MPC control strategy. Both macroscopic and microscopic simu- lations show that the performance and robustness with respect to model parameter 135 errors and measurement noise of our controller is better than that of the MPC con- troller. 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Thus q 2 q 1 =v f ( d v f ) (A.1) therefore _ V =v f ( d v f ) 2 ;82 [0; C d v f ]. When C d v f < ~ j (1 0 )C d ~ w , d v f > 0, q 1 =d and q 2 = (1 0 )C d . Thus q 2 q 1 = (1 0 )C d d (1 0 )C d d j d=v f (d=v f ) (A.2) 145 due to (1 0 )C d d > 0 and d=v f < j ;82 ( C d v f ; j ], which implies 0 < d=v f j d=v f 1. Therefore _ V (1 0 )C d d j d=v f ( d v f ) 2 ;82 ( C d v f ; ~ j (1 0 )C d ~ w ]. When ~ j (1 0 )C d ~ w < j d w , d v f > 0, q 1 =d and q 2 = ~ w(~ j ). Thus q 2 q 1 = ~ w(~ j )d ~ w[~ j ( j d w )]d = ~ w[~ j ( j d w )]w[ j ( j d w )] = ~ w(~ j c )w( j c ) + ( ~ ww)[ c ( j d w )] =CC + ( ~ ww)[ c ( j d w )] ( ~ ww)[ c ( j d w )] j d=v f (d=v f ); (A.3) due to 0 < d=v f j d=v f 1, ~ ww < 0 and c ( j d w ) < 0. Therefore, _ V ( ~ ww)[c( j d w )] j d=v f (d=v f ) 2 ,82 (~ j (1 0 )C d ~ w ; j d w ]. When j d w < j , d v f > 0, q 1 =w( j ) and q 2 = ~ w(~ j ). Thus q 2 q 1 = ~ w(~ j )w( j ) = ~ w(~ j c )w( j c ) + ( ~ ww)( c ) ( ~ ww)[ c ( j d w )] ( ~ ww)[ c ( j d w )] j d=v f (d=v f ); (A.4) since 0 < d=v f j d=v f 1, ~ w w < 0 and c ( j d w ) < 0. Therefore _ V ( ~ ww)[c( j d w )] j d=v f (d=v f ) 2 ,82 ( j d w ; j ]. From (A.1) to (A.4), we can conclude that _ V( d v f ) 2 ;8(0)2 [0; j ] (A.5) where = minfv f ; (1 0 )C d d j d=v f ; ( ~ ww)[c( j d w )] j d=v f g > 0; 8(0)2 [0; j ], which implies exponential stability of the equilibrium point e = d v f and exponential convergence of (t) to d v f ,8(0)2 [0; j ]. The rate of convergence of to the equilibrium d v f is greater or equal to and can be shown by substituting for ( d v f ) 2 = 2V in (A.5) and integrating both sides of the inequality. 146 A.2 Case b), i.e. I2 2 From Fig. 6.4, the situation where (0)2 ( j d w ; j ] is divided into two cases: Case I: (0)2 ( j d w ; ~ j d ~ w ]. Case II: (0)2 (~ j d ~ w ; j ]. In case I, q 1 =w( j ) and q 2 =d as long as 2 ( j d w ; ~ j d ~ w ];8t 0, which we need to show. We have _ =q 1 q 2 =w +w j d; whose solution is (t) = ( j d w ) + [(0) ( j d w )]e wt : Since 0< j d w <(0) ~ j d ~ w andw> 0, it follows that2 ( j d w ; ~ j d ~ w ];8t 0 and (t) converges exponentially fast to j d w according to the above equation. In case II, when(0)2 (~ j d ~ w ; j ], it follows from Fig. 6.4 thatq 1 =w( j ), q 2 = ~ w(~ j ) and _ =(w ~ w) + (w ~ w) c ; as long as (t)2 (~ j d ~ w ; j ], whose solution is (t) = c + ((0) c )e w 0 t ( j d w ) + ((0) c )e w 0 t ; wherew 0 =w ~ w> 0 according to model (6.2). Since c < ~ j d ~ w , it follows that (t) will decrease exponentially to the value of ~ j d ~ w at which instant _ switches to case I which guarantees exponential convergence to j d w . The above equation implies that j(t) ( j d w )jc 0 e t ; where c 0 =(0) ( j d w ) and = minfw;w ~ wg. A.3 Case c), i.e. I2 3 Consider the Lyapunov function V () = ( e 2 ) 2 2 ; 147 where e 2 = j (1 0 )C d w . Then _ V =( e 2 )(q 2 q 1 ). As shown before, when C d v f < j d w , we have e 2 < 0 and q 2 q 1 = (1 0 )C d d (1 0 )C d d C d =v f e 2 ( e 2 ); (A.6) due to 0 < ( e 2 ) C d =v f e 2 < 1;82 ( C d v f ; j d w ] and (1 0 )C d d < 0. Therefore _ V (1 0 )C d d C d =v f e 2 ( e 2 ) 2 . When j d w < ~ j (1 0 )C d ~ w , q 2 q 1 = (1 0 )C d w( j ) =w[ ( j (1 0 )C d w )] =w( e 2 ) : (A.7) Therefore _ V =w( e 2 ) 2 . When ~ j (1 0 )C d ~ w < j , we have e 2 > 0 and q 2 q 1 = ~ w(~ j )w( j ) = (w ~ w)( c ) (w ~ w)( e 2 ); (A.8) due to w ~ w> 0 and e 2 > c . Therefore, _ V (w ~ w)( e 2 ) 2 . From (A.6) to (A.8), we conclude that82 ( C d v f ; j ], _ V( e 2 ) 2 ; where = minf d(1 0 )C d e 2 C d =v f ;w; (w ~ w)g> 0;8(0)2 ( C d v f ; j ] which implies exponen- tial stability of the equilibrium point e 2 = j (1 0 )C d w and exponential convergence of (t) to e 2 ,8(0)2 ( C d v f ; j ]. A.4 Case d), i.e. I2 4 Consider the Lyapunov function V () = ( e ) 2 2 ; where e = j (1 0 )C d w . Thus _ V =( e )(q 2 q 1 ): From Fig. 6.6, it is clear that82 [0; C d v f ], e < 0 and q 2 q 1 C d d dC d e ( e ); 148 therefore, _ V dC d e ( e ) 2 ;82 [0; C d v f ]: Similar to the case I2 3 , we have82 ( C d v f ; j ], _ V minf d (1 0 )C d e C d =v f ;w; (w ~ w)g( e ) 2 : Therefore,82 [0; j ], the time derivative of the Lyapunov function satises _ V( e ) 2 ; where = minf dC d e ; d(1 0 )C d e C d =v f ;w; (w ~ w)g > 0, which implies exponential con- vergence to the equilibrium point e ;8(0)2 [0; j ]. A.5 Case e), i.e. I2 5 Consider the Lyapunov function V () = ( minfd;Cg=v f ) 2 2 : Then if d < C, _ V =(d=v f )(q 2 q 1 ). According to equation (6.2) and Fig. 6.7a, when 0 c , we have that q 1 =d, and q 2 =v f . Thus q 2 q 1 =v f (d=v f ): (A.9) Therefore _ V =v f (d=v f ) 2 . When c < j d w , we have d=v f > 0, q 1 =d and q 2 = ~ w(~ j ). Using equation (A.3), we have q 2 q 1 = ~ w(~ j )d ~ w[~ j ( j d w )]d ( ~ ww)[ c ( j d w )] j d=v f (d=v f ): (A.10) Therefore _ V ( ~ ww)[c( j d w )] j d=v f (d=v f ) 2 . When j d w < j , we have d=v f > 0, q 1 = w( j ) and q 2 = ~ w(~ j ), which together with equation (A.4) gives q 2 q 1 = ~ w(~ j )w( j ) ( ~ ww)[ c ( j d w )] j d=v f (d=v f ); (A.11) 149 Therefore _ V ( ~ ww)[c( j d w )] j d=v f (d=v f ) 2 . From (A.9) - (A.11), we conclude that 82 [0; j ], _ V(d=v f ) 2 ; where = minfv f ; ( ~ ww)[c( j d w )] j d=v f g > 0, which guarantees exponential stability of the equilibrium point e = d=v f and exponential convergence of (t) to e , 8(0)2 [0; j ]. If d C,82 [0; c ];q 1 = C;q 2 = v f , and82 ( c ; j ];q 1 = w( j );q 2 = ~ w(~ j ). Therefore _ V = ( v f ( c ) 2 ; if 2 [0; c ] (w ~ w)( c ) 2 ; if 2 ( c ; j ] ; which implies that _ V minfv f ; (w ~ w)g( c ) 2 . The properties of V and _ V imply exponential stability of the equilibrium point e = c = C v f and exponential convergence of (t) to e ,8(0)2 [0; j ], due to w ~ w> 0. 150 Appendix B Proof of Theorem 6.3.1 For the proof of Theorem 6.3.1, we use the following two lemmas: Lemma B.0.1 gives the region of e within the setS. For a setA< N and a pointx 0 2< N , the distance between x 0 and A is dened as: d(x 0 ;A) = inf x2A kxx 0 k: Then we have the following lemma. Lemma B.0.1. Let e be an equilibrium state of system (6.3), then we have the following results: a) If C d < C, i.e. I 2 4 S i=1 i , then e 2 S I , where S I = fj minfd;Cg v f i j (1 0 )C d w ;i = 1; 2;:::;NgS. Furthermore,8(0)2S, d((t);S I ) converges to 0 exponentially fast. b) If C d C, i.e. I2 5 , then e 2 S I , where S I =fj minfd;Cg v f i c ;i = 1; 2;:::;Ng S. Furthermore,8(0)2 S, d((t); S I ) converges to 0 exponen- tially fast. Proof of Lemma B.0.1: a) For I2 4 S i=1 i , we rst show that e i minfd;Cg v f , for i = 1; 2;:::;N. Assume that 0 e 1 < minfd;Cg v f , then w( j e 1 ) C due to e 1 < minfd;Cg v f c . Therefore the corresponding equilibrium ow rate q e 1 = minfd;C;w( j e 1 )g = minfd;Cg; q e 2 = minfv f e 1 ; ~ w(~ j e 1 );C;w( j e 2 )gv f e 1 ; which implies that _ 1 =q e 1 q e 2 minfd;Cgv f e 1 > 0; as e 1 < minfd;Cg v f ; 151 which violates the equilibrium condition (6.4), hence e 1 minfd;Cg v f . For any i = 1; 2;::;N 1, assume e i minfd;Cg v f and check the property of e i+1 . If 0 e i+1 < minfd;Cg v f , we have v f e i+1 < minfd;Cg<C <w( j e i+1 ). Thus q e i+1 = minfv f e i ; ~ w(~ j e i );C;w( j e i+1 )g = minfv f e i ; ~ w(~ j e i )g; q e i+2 = minfv f e i+1 ; ~ w(~ j e i+1 );C;w( j e i+2 )gv f e i+1 < minfd;Cg: Ifq e i+1 = ~ w(~ j e i ), then ~ w(~ j e i )v f e i , which implies e i c . Since e i is the equilibrium density in section i, we have q e i =q e i+1 , and w( j e i )q e i =q e i+1 = ~ w(~ j e i ); which implies e i c . Thus e i = c andq e i =q e i+1 = ~ w(~ j c ) =C minfd;Cg> q e i+2 . If q e i+1 =v f e i , then q e i+1 minfd;Cg>q e i+2 due to e i minfd;Cg v f . Therefore, for all possible q i+1 = minfv f e i ; ~ w(~ j e i )g, we have q e i+1 > q e i+2 , which violates the equilibrium condition (6.4). Therefore, the assumption 0 e i+1 < minfd;Cg v f is invalid, which implies that e i+1 minfd;Cg v f . By mathematical induction, we know that e i minfd;Cg v f ;i = 1; 2;:::;N: (B.1) Then we show that e i j (1 0 )C d w , for i = 1; 2;:::;N. Assume that j (1 0 )C d w < e N j , then q e N+1 = minf(1 0 )C d ; ~ w(~ j e N )g; q e N = minfv f e N1 ; ~ w(~ j e N1 );C;w( j e N )gw( j e N ): Since e N > j (1 0 )C d w > c , we have w( j e N ) < (1 0 )C d and w( j e N )< ~ w(~ j e N ). Therefore q e N w( j e N )<q e N+1 , which contradicts the the equilibrium condition (6.4). Thus e N j (1 0 )C d w . Assume e i j (1 0 )C d w , for anyi = 2; 3;:::;N, we check the property of e i1 . If j (1 0 )C d w < e i1 j , then ~ w(~ j e i1 )<C <v f e i1 as e i1 > c . Therefore q e i = minfv f e i1 ; ~ w(~ j e i1 );C;w( j e i )g = minf ~ w(~ j e i1 );w( j e i )g; q e i1 = minfv f e i2 ; ~ w(~ j e i2 );C;w( j e i1 )gw( j e i1 ): Since e i j (1 0 )C d w < e i1 , we have w( j e i1 ) < ~ w(~ j e i1 ) and w( j e i1 ) < (1 0 )C d w( j e i ). Thus q e i1 < q e i , which violates the equilibrium 152 condition (6.4). Therefore e i1 j (1 0 )C d w . By mathematical induction, we have e i j (1 0 )C d w ;i = 1; 2;:::;N: (B.2) Combining the two inequalities (B.1) and (B.2), we can conclude minfd;Cg v f e i j (1 0 )C d w ;i = 1; 2;:::;N: To show that d((t);S I ) converges to 0 exponentially fast8(0) 2 S, it is equivalent to show that8> 0;9T > 0, such that8t>T minfd;Cg v f < i (t)< j (1 0 )C d w +; i = 1; 2;:::;N (B.3) and d((t);S I ) is bounded from above by a decaying exponential function. First we show the left half of inequality (B.3). Sinceq 1 = minfd;C;w( j 1 )g and q 2 v f 1 , we have _ 1 =q 1 q 2 minfd;C;w( j 1 )gv f 1 : (B.4) If9t 0 0, such that 1 (t 0 ) minfd;Cg v f , then for all tt 0 we have the following result: since 1 (t) is uniformly continuous, if 1 (t) keeps decreasing and 1 (t 1 ) = minfd;Cg v f for some t 1 t 0 , then from (B.4) we have _ 1 (t 1 ) 0, which implies that 1 (t) will no longer decrease and 1 (t) minfd;Cg v f ;8t t 0 . Therefore8 1 > 0 and 8tt 0 , 1 (t) minfd;Cg v f 1 . If8t 0; 1 (t)< minfd;Cg v f , then in the region 1 (t)< minfd;Cg v f , we have _ 1 (t) minfd;Cgv f 1 =v f ( 1 minfd;Cg v f ): (B.5) By Lemma 3.2.4 in [86], we have 1 (t)e v f t [ 1 (0) minfd;Cg v f ] + minfd;Cg v f ;8t 0: (B.6) The right side of (B.6) converges to minfd;Cg v f exponentially fast, therefore8 1 > 0, 9T 1 > 0, such that8t>T 1 , 1 (t) minfd;Cg v f 1 . 153 For i = 1; 2;:::;N 1, we assume i minfd;Cg v f i ;8t> 0, where i > 0, then we examine the dynamics of i+1 . We have _ i+1 =q i+1 q i+2 minfv f i ; ~ w(~ j i );C;w( j i+1 )gv f i+1 : Since i minfd;Cg v f i , we have v f i minfd;Cgv f i , therefore _ i+1 minfminfd;Cgv f i ; ~ w(~ j i );w( j i+1 )gv f i+1 : (B.7) Similar to (B.4), we can show that if9t 0 0, such that i+1 (t) minfd;Cg v f i , then i+1 (t) minfd;Cg v f i ;8tt 0 , that is,8 i+1 > i and8tt 0 , i (t)> minfd;Cg v f i+1 . If i+1 (t) < minfd;Cg v f i ;8t 0, then in the region i+1 (t) < minfd;Cg v f i , we have _ i+1 (t) minfd;Cgv f i v f i+1 =v f ( i+1 minfd;Cg v f + i ): By Lemma 3.2.4 in [86], we have i+1 (t)e v f t [ i+1 (0) minfd;Cg v f + i ] + minfd;Cg v f i ;8t 0: (B.8) Similar to (B.6), the right hand side of equation (B.8) converges exponentially fast to minfd;Cg v f i . Therefore, 8 i+1 > i , 9T i+1 > 0, such that8t > T i+1 , i (t) > minfd;Cg v f i+1 . By mathematical induction, we can conclude that for i = 1; 2;:::;N,8 i > 0,9T i > 0, such that8t> P i j=1 T j , i (t) minfd;Cg v f i . If we take N <, T 1 +T 2 +::: +T N <T , then the left side of inequality (B.3) holds. Next we prove the right half of the inequality (B.3). Sinceq N+1 = minfv f N ; ~ w(~ j N ); (1( N ))C d g and q N < minfC;w( j N )g, we have _ N =q N q N+1 minfC;w( j N )g minfv f N ; ~ w(~ j N ); (1( N ))C d g: Similar to (B.4), we can show that if9t 0 0, such that N (t 0 ) j (1 0 )C d w , then N (t 0 ) j (1 0 )C d w ;8t t 0 , that is,80 < N < , and8t 0, N (t) j (1 0 )C d w + N . If N (t)> j (1 0 )C d w ;8t 0, then in the region N > j (1 0 )C d w , we have _ N ( ~ ww)( N ( j (1 0 )C d w )): 154 By Lemma 3.2.4 in [86], we have N (t)e ( ~ ww)t [ N (0) ( j (1 0 )C d w )] + ( j (1 0 )C d w );8t 0: (B.9) Since ~ ww< 0, the right side of (B.9) converges to j (1 0 )C d w exponentially fast. Therefore,80< N <,9T N > 0, such that8t>T N , N (t) j (1 0 )C d w + N . Fori = 1;:::;N1, we assume i+1 j (1 0 )C d w + i+1 ;8t> 0, where i+1 > 0, then we check the dynamics of i . We have _ i =q i q i+1 minfC;w( j i )g minfv f i ; ~ w(~ j i );C;w( j i+1 )g: Since i+1 j (1 0 )C d w + i+1 , we have w( j i+1 ) (1 0 )C d w i+1 . Thus _ i minfC;w( j i )g minfv f i ; ~ w(~ j i ); (1 0 )C d w i+1 g: Similar to (B.4), we can show that if9t 0 0, such that i (t 0 ) j (1 0 )C d w + i+1 , then i (t 0 ) j (1 0 )C d w + i+1 ;8t t 0 , that is,8 i > i+1 , and8t 0, i (t) j (1 0 )C d w + i . If i (t 0 )> j (1 0 )C d w + i+1 ;8t 0, then in the region i > j (1 0 )C d w + i+1 , we have _ i ( ~ ww)( N ( j (1 0 )C d w ) i+1 ): By Lemma 3.2.4 in [86], we have i (t)e ( ~ ww)t [ i (0) ( j (1 0 )C d w + i+1 )] + ( j (1 0 )C d w + i+1 );8t 0: (B.10) Similar to (B.9), the right hand side of equation (B.10) converges exponentially fast to j (1 0 )C d w + i+1 . Therefore,8 i > i+1 ,9T i > 0, such that8t > T i , i (t) j (1 0 )C d w + i . By mathematical induction, we can conclude that for i = 1; 2;:::;N,8 i > 0, 9T i > 0, such that8t > P i j=1 T j , i (t) j (1 0 )C d w + i . If we take 1 < , T 1 +T 2 +::: +T N < T , then the right side of inequality (B.3) holds. Therefore, d((t);S I ) converges to 0 exponentially fast for all (0) in the feasible set S. b) Part b) of Lemma B.0.1 can be proved in a similar manner. Specically, when I 2 2 , the equilibrium points of system (6.3) satisfy the properties given by the following lemma. Lemma B.0.2. Let I 2 2 . If e is an equilibrium state of system (6.3), then the corresponding equilibrium ow rate is q e 1 = q e 2 = ::: = q e N+1 = d = (1 0 )C d . Furthermore, e has the following properties: a) For i = 1; 2;:::;N 1, if d v f < e i j d w , then e k = j d w , for all i<kN. 155 b) For i = 2; 3;:::;N, if d v f e i < j d w , then e k = d v f , for all 1k<i. Proof of Lemma B.0.2: Assume e is an equilibrium state of system (6.3), then using Lemma B.0.1, we have d v f e i j (1 0 )C d w = j d w ;i = 1; 2;:::;N, therefore q e 1 = minfd;C;w( j e 1 )gd; q e N+1 = minfv f e N ; ~ w(~ j e N ); (1( e N ))C d gd: Using the equilibrium condition (6.4), we have that the equilibrium ow q e 1 = q e N+1 =d. Therefore q e i =d, for i = 1; 2;:::;N + 1. For anyi = 1; 2;:::;N1, if d v f < e i j d w , we havev f e i >d and ~ w(~ j e i ) ~ w[~ j ( j d=w)]>w[ j ( j d=w)] =d, therefore d =q e i+1 = minfv f e i ; ~ w(~ j e i );C;w( j e i+1 )g =w( j e i+1 ); which gives that e i+1 = j d=w. By mathematical induction, e k = j d w , for all i<kN. For any i = 2; 3;:::;N, if d v f e i < j d w , we have that w( j e i ) > d, therefore d =q e i = minfv f e i1 ; ~ w(~ j e i1 );C;w( j e i )g = minfv f e i1 ; ~ w(~ j e i1 )g: If q e i = ~ w(~ j e i1 ) = d, then q e i1 w( j e i1 ) < ~ w(~ j e i1 ) < d, which contradicts the fact that q e i1 = d, therefore q e i = v f e i1 = d; e i1 = d v f . By mathematical induction, k = d v f , for all 1k<i. Using the above two lemmas the proof of Theorem 6.3.1 is completed as follows: Proof of Theorem 6.3.1: From the part a) of Lemma B.0.1, we know that for I 2 4 S i=1 i if e is an equilibrium state of system (6.3), then e 2 S I and d((t);S I ) converges to 0 exponentially fast, 8(0) 2 S. Therefore, we only need to nd all equilibrium states of system (6.3) in S I and analyze the dynamics of (t) for all (0)2 S I , where S I =fj minfd;Cg v f i j (1 0 )C d w +;i = 1; 2;:::;Ng and > 0 can be arbitrarily small. 156 From the part b) of Lemma B.0.1, we know that when I2 5 , we only need to nd all equilibrium states of system (6.3) in the set S I and analyze the dynamics of (t) for all (0) in the set S I , where S I =fj minfd;Cg v f i c +;i = 1; 2;:::;Ng: Now we prove the statements of Theorem 6.3.1 from a) to e) respectively. a) When I2 1 ;d < (1 0 )C d . By Lemma B.0.1, we have that minfd;Cg v f e i j (1 0 )C d w , therefore w( j e i ) (1 0 )C d > d, i = 1; 2;:::;N. Thus q e 1 = minfd;C;w( j e 1 )g =d. Using the equilibrium condition (6.4), we have q e i =d; for i = 1; 2;:::;N + 1 Now we show that e i = d=v f , for i = 1; 2;:::;N. For i = 1; 2;:::;N 1, q e i+1 = minfv f e i ; ~ w(~ j e i );C;w( j e i+1 )g. If c e i j (1 0 )C d w , we have w( j e i+1 ) (1 0 )C d >d; v f e i C ~ w(~ j e i )w( j e i )> (1 0 )C d >d; which implies that q e i+1 > d, therefore the assumption c e i j (1 0 )C d w is invalid. Hence d=v f e i < c , which gives that ~ w(~ j e i )>C >v f e i , thus q e i+1 = minfv f e i ;w( j e i+1 )g: By Lemma B.0.1, we have e i+1 j (1 0 )C d w , thusw( j e i+1 ) (1 0 )C d >d. Solving the equation q e i+1 = d gives the unique equilibrium density e i = d=v f . Therefore, we have e i = d=v f ;i = 1;:::;N 1. For i = N, we have q e N+1 = minfv f e N ; ~ w(~ j e N ); (1( e N ))C d g. If C d =v f < e N j (1 0 )C d w , we have q e N+1 = (1 0 )C d > d, therefore the assumption C d =v f < e N j (1 0 )C d w is invalid, which together with Lemma B.0.1 implies that d=v f e N C d =v f . Therefore q e N+1 = v f e N . Solving the equation q e N+1 = d gives a unique solution e N = d=v f . Therefore, the point d v f 1 is the unique equilibrium state of system (6.3) when I2 1 . Using Lemma B.0.1, we have that for all(0)2S I ,d=v f < i (t)< j (1 0 )C d =w +;8t 0;> 0. Therefore w( j i )>w[ j ( j (1 0 )C d =w +)] = (1 0 )C d w 157 and ~ w(~ j i )> ~ w[~ j ( j (1 0 )C d =w +)]>w[ j ( j (1 0 )C d =w +)] = (1 0 )C d w; for i = 1; 2;:::;N. Since (1 0 )C d > d, taking to be suciently small, we have (1 0 )C d w>d. Therefore q 1 = minfd;C;w( j 1 )g =d and _ 1 =q 1 q 2 =d minfv f 1 ; ~ w(~ j 1 );C;w( j 2 )g: (B.11) Combine (6.3) and (B.11), we can show that _ 1 8 > > > > > > > < > > > > > > > : =v f ( 1 d v f ) if d v f < 1 < d v f = 0 if 1 = d v f ( 1 d v f ) if d v f < 1 < j (1 0 )C d =w + ; where = minfv f ; (1 0 )C d wd j d=v f g > 0, which implies that for all (0)2 S I , 1 (t) converges to d=v f exponentially fast. Based on the convergence of 1 , we can show that 2 also converges to d=v f exponentially fast, followed by 3 through N . Therefore,8(0)2S,(t) converges to d v f 1 exponentially fast. b)When I2 2 , d = (1 0 )C d . Using Lemma B.0.2, we have that the equi- librium ow rate q e i = d = (1 0 )C d , for i = 1; 2;:::;N + 1. If 0 e N C d v f , then q e N+1 = d gives e N = d v f . By part b) of Lemma B.0.2, we have e i = d v f , i = 1; 2;:::;N 1. Therefore e = d=v f 1 is a potential equilibrium point of system (6.3). Substituting e = d=v f 1 into equation (6.3), we have q e i = d, for i = 1; 2;:::;N + 1. Therefore, e = d=v f 1 is the only equilibrium state in the region 0 e N C d v f . If e N 2 ( C d v f ; j d w ), according to Lemma B.0.2, we have that e 1 =::: = e N1 = d v f . Substituting any e 2fj 1 =::: = N1 = d v f ; C d v f < N < j d w g into equation (6.3), we haveq e i =d, fori = 1; 2;:::;N +1. Therefore all e 2fj 1 =::: = N1 = d v f ; C d v f < N < j d w g are equilibrium states of system (6.3). If e N = j d w , we nd all the equilibrium states of the system (6.3) by consid- ering the following two cases: Case I: for all i = 1; 2;::;N 1, e i = j d w ; Case II: there existsi2f1; 2;:::;N 1g,d=v f e i < j d w and e i+1 = j d w . 158 Case I contains only one point, that is, e = ( j d w ) 1. Substituting this density state into equation (6.3), we have q e i =d, for i = 1; 2;:::;N + 1. Therefore ( j d w ) 1 is an equilibrium state of system (6.3). For case II, it is clear from Lemma B.0.2 that e 1 = ::: e i1 = d=v f ; e i+1 = ::: = e N = j d w . Taking i = 1; 2;:::;N 1, we have that all potential equilibrium points of system (6.3) in case II are in the set N1 S i=1 fj d v f i < j d w ; k = d v f ; 1 k < i; r = j d w ;i < r Ng. Substituting any point in this set into equation (6.3), we have q e i = d, for i = 1; 2;:::;N + 1. Therefore all e 2 N1 S i=1 fj d v f i < j d w ; k = d v f ; 1k<i; r = j d w ;i<rNg are equilibrium states of system (6.3). To summarize, when I 2 2 , system (6.3) has an isolated equilibrium state d v f 1 and an equilibrium manifold S e =f( j d w ) 1g[fj i = d v f ;i = 1; 2;:::;N 1; C d v f < N < j d w g [ [ N1 [ i=1 fj d v f i < j d w ; k = d v f ; 1k<i; r = j d w ;i<rNg]: We now prove the rest of part b) as follows: rst we show that for all (0) in the feasible space S, (t) converges to one equilibrium state in S e , where S e = S e [f d v f 1g. Then we show that d v f 1 is locally exponentially stable, and that every e 2 S e is stable in the sense of Lyapunov, i.e. 8 > 0;9 > 0, such that 8(0) that satisfyk(0) e k<, we havek(t) e k;8t> 0. Furthermore, (t) converges to some e 2S e that satisesk e e k<. For all (0)2 S I , by letting = 0 in the proof of Lemma B.0.1, we can show (t) 2 S I ;8t 0. From equation (6.3), we know that8 2 S I , q 1 = d and q i d;i = 2; 3;:::;N. Therefore, k X i=1 _ i =q 1 q k 0;k = 1; 2;:::;N: Thus P k i=1 i is monotonically decreasing but bounded from below which implies that it converges to a limit. Therefore, we have = [ 1 ; 2 ;:::; N ] T converges to a constant vector e . From equation (6.3) we know that _ is a piecewise uniformly continuous function of, therefore, as converges to a constant e , _ also converges to a constant, which has to be 0 (otherwisekk goes to innity). Therefore e is a equilibrium point of system (6.3) by denition. Thus e 2 S e . From part a) of Lemma B.0.1, for all (0)2 S, d((t);S I ) converges to 0 exponentially fast. Therefore8(0)2S, (t) converges to a equilibrium point e 2 S e . 159 Next we show that the equilibrium state e = d v f 1 is exponentially stable, and that every e 2S e is stable in the sense of Lyapunov. (1) When e = d v f 1, then for all (0)2fj0 i C d =v f ;i = 1; 2;:::;Ng, q i C d and q i+1 =v f i , for i = 1; 2;:::;N. Thus _ i =q i q i+1 C d v f i , which implies that i (t)C d =v f ,8t 0. Therefore, _ 1 =dv f 1 ; _ i =v f i1 v f i ;i = 2; 3;:::;N; which can be written in the compact form as _ =A( d v f 1) where A = 2 6 6 6 4 v f v f v f . . . . . . v f v f 3 7 7 7 5 : Since v f > 0, we have that A is Hurwitz. Therefore (t) converges to d v f 1 exponentially fast. e is in the interior of the setfj0 i C d =v f ;i = 1; 2;:::;Ng, thus we can always nd a > 0, such thatfjk e k<g fj0 i C d =v f ;i = 1; 2;:::;Ng. Therefore,8(0)2fjk e k<g, (t) converges to e exponentially fast, which implies that e = d v f 1 is exponentially stable. (2) When e = ( j d w ) 1,8(0) that satisfyk(0) e k < , with > 0 suciently small, equation (6.3) gives the ow rates as follows: q 1 = minfd;w( j 1 )g, q i =w( j i ); for i = 2;:::;N, and q N+1 = (1 0 )C d =d. Therefore, _ 1 = minfd;w( j 1 )gw( j 2 ); _ i =w( j i )w( j i+1 );i = 2;:::;N 1; _ N =w( j N )d: Let e i = i e i and e = [e 2 ;e 3 ;:::;e N ] T , then we have e = [e 1 ; e T ] T , where _ e 1 = ( we 2 ; e 1 0 we 1 +we 2 ; e 1 > 0 (B.12) 160 and _ e = 2 6 6 6 4 w w w w . . . . . . w 3 7 7 7 5 e: Since w > 0, it follows that e converges to 0 exponentially fast, i.e. there exists constants ; > 0, such that je i (t)jje i (0)j exp(t);i = 2; 3;:::;N: (B.13) From (B.12), we have that _ e 1 we 1 +we 2 , which together with the continuity of _ e 1 implies that e 1 (t)je 1 (0)j exp(wt) +je 2 (0)j w w 0 [exp( 0 t) exp(wt)]; (B.14) where 0< 0 < minfw;g. Therefore,e 1 is bounded from above by a function that decays exponentially fast to 0 with time. If8t 0;e 1 (t)> 0, then e 1 (t) converges exponentially fast to 0. Otherwise, if9t 0 0, such that e 1 (t 0 ) 0, then we have the following cases: Case I: If e 1 (0) 0, then as long as e 1 (t) 0, we have e 1 (t) =e 1 (0) + Z t 0 we 2 ()dje 1 (0)j Z t 0 wje 2 ()jdje 1 (0)j w je 2 (0)j exp(t); which together with (B.14) implies that for any given > 0, there exists a nite time T such that je 1 (0)je 1 (t);8tT; which implies that the equilibrium e 1 = 0 is stable in the sense of Lyapunov. Case II: If e 1 (0) > 0, note that e 1 (t 0 ) 0, then due to the uniform continuity of e 1 , we have that9t 1 2 (0;t 0 ]; such that e 1 (t 1 ) = 0. Then8t t 1 , as long as e 1 (t) 0, we have e 1 (t) = Z t t 1 we 2 ()d Z t t 1 wje 2 ()jd w je 2 (0)j exp(t 1 )[1 exp((tt 1 ))] w je 2 (0)j exp(t 1 ); which together with (B.14) implies that for any given > 0, there exists a nite time Tt 1 such that w je 2 (0)j exp(t 1 )e 1 (t);8tT; 161 which implies that the equilibrium e 1 = 0 is stable in the sense of Lyapunov. To summarize the above analysis, we have that the equilibrium state e = ( j d w ) 1 is stable in the sense of Lyapunov. (3) When e 2fj d v f i < j d w ; k = d v f ; 1k<i; r = j d w ;i<rNg, i = 1; 2;:::;N1,8(0) that satisfyk(0) e k<, if is suciently small, we can get the ow rates from equation (6.3) as follows: q 1 = d, q k = v f k1 ;k = 2;:::;i, q r =w( j r );r =i + 1;:::;N, and q N+1 = (1 0 )C d =d. Therefore, we have _ 1 =dv f 1 ; _ k =v f k1 v f k ;k = 2;:::;i 1; _ i =v f i1 w( j i+1 ); _ r =w( j r )w( j r+1 );r =i + 1;:::;N 1; _ N =w( j N )d: Let e i = i e i and e = [e 1 ;e 2 ;:::;e N ] T , the dynamics of e can be presented in the compact form as follows: _ e =Ae; where A = 2 6 6 6 6 6 6 6 6 6 4 v f 0 v f v f 0 . . . . . . . . . v f 0 w 0 w w . . . . . . . . . 0 w 3 7 7 7 7 7 7 7 7 7 5 ; Let e = [e 1 ;e 2 ;:::;e i1 ] T and e = [e i+1 ;:::;e N ] T , then _ e = 2 6 6 6 4 v f v f v f . . . . . . v f v f 3 7 7 7 5 e and _ e = 2 6 6 6 4 w w . . . . . . w w w 3 7 7 7 5 e: 162 The above two subsystems are both linear and exponentially stable. Thus e and e both converge to 0 exponentially fast. _ e i =v f e i1 +we i+1 , therefore e i (t) =e i (0) +v f Z t 0 e i1 ()d +w Z t 0 e i+1 ()d: Since ke i1 kk ekk e(0)k 1 exp( 1 t); ke i+1 kkekke(0)k 2 exp( 2 t); where 1 ; 2 ; 1 ; 2 > 0, thus e i =e i (0) + Z t 0 (v f e i1 +we i+1 )d; ke i kke i (0)k + Z t 0 (v f ke i1 k +wke i+1 k)d ke i (0)k +v f Z t 0 k e(0)k 1 exp( 1 t)d +w Z t 0 ke(0)k 2 exp( 2 t)d =ke i (0)k +k e(0)k v f 1 1 (1 exp( 1 t)) +ke(0)k w 2 2 (1 exp( 2 t)): Since we have shown before that8(0)2 S;(t) converges to a constant, i.e. the limit lim t!1 (t) exists, which implies that the limit lim t!1 ke i k also exists. There- fore, lim t!1 ke i kke i (0)k +k e(0)k v f 1 1 +ke(0)k w 2 2 : For all> 0, by selecting(0) suciently close to e , we havek(t) e k;8t 0, i.e. all equilibrium points e 2fj d v f i < j d w ; k = d v f ; 1 k < i; r = j d w ;i<rNg, i = 1; 2;:::;N 1 are stable in the sense of Lyapunov. (4) When e 2fj i =d=v f ;i = 1; 2;:::;N 1; C d v f < N < j d w g ,8(0) that satisfyk(0) e k<, if is suciently small, we get the ow rates from equation (6.3) as: q 1 =d, q i =v f i1 ;i = 2; 3;:::;N, and q N+1 = (1 0 )C d =d. Therefore we have that _ 1 =dv f 1 ; _ i =v f i1 v f i ;i = 2;:::;N 1; _ N =v f N1 d: Let e i = i e i and e = [e 1 ;e 2 ;:::;e N ] T , the dynamics of e can be expressed in the compact form as follows: _ e =Ae; 163 where A = 2 6 6 6 6 6 4 v f v f v f . . . . . . v f v f v f 0 3 7 7 7 7 7 5 : The stability of e can be shown by following a similar analysis as in previous case. Therefore, all e 2S e are stable in the sense of Lyapunov. Recall that8(0)2 S;(t) converges to an equilibrium state in S e , thus8 > 0;9 > 0, such that 8(0) that satisfyk(0) e k < , (t) converges to some e 2 S e that satises k e e k<. c) For the case I 2 3 , from part a) of Lemma B.0.1, we know d v f e i j (1 0 )C d w ;i = 1; 2;:::;N. If d v f e 1 j d w , then w( j e 1 ) d, thus q e 1 = minfd;C;w( j e 1 )g =d andq e i =q e 1 =d;i = 2; 3;:::;N + 1, according to the equilibrium condition (6.4). Solving the equation q e N+1 =d gives only one solution e N = d=v f . For i = 1; 2;:::;N 1, given e i+1 = d=v f ;q e i = q e i+1 = d, we have d =q e i w( j e i ), thus e i j d=w. Since q e i+1 = minfv f e i ; ~ w(~ j e i );C;w( j e i+1 )g and w( j e i+1 )>C >d; as e i+1 =d=v f < c ; ~ w(~ j e i ) ~ w[~ j ( j d=w)]>w[ j ( j d=w)] =d: The equation q e i+1 =d gives only one solution, that is v f e i =d; e i =d=v f : By mathematical induction, we have that e i = d=v f ;i = 1; 2;:::;N. Therefore in the region d v f e 1 j d w , system (6.3) has only one equilibrium state d v f 1. If j d w < e 1 j (1 0 )C d w , we have w( j e 1 )<d<C, thus q e 1 = minfd;C;w( j e 1 )g =w( j e 1 )<d: For i = 2; 3;:::;N, given j d w < e i1 j (1 0 )C d w and q e i1 = q e i = q e 1 = w( j e 1 ), then we have q e i = minfv f e i1 ; ~ w(~ j e i1 );C;w( j e i )g =w( j e 1 ): Sincev f e i1 >C >d as e i1 > c ,v f e i1 6=w( j e 1 ). If ~ w(~ j e i1 ) =w( j e 1 ), then q e i1 w( j e i1 )< ~ w(~ j e i1 ) =w( j e 1 ); 164 which contradicts the fact thatq e i1 =w( j e 1 ), therefore ~ w(~ j e i1 )6=w( j e 1 ). Thus we have q e i =w( j e i ) =w( j e 1 ) and e i = e 1 ; i = 2; 3;:::;N: Therefore j d w < e N j (1 0 )C d w , equation (6.3) gives thatq e N+1 = (1 0 )C d . Using the equilibrium condition (6.4), we have q e i = w( j e i ) = (1 0 )C d for i = 1; 2;:::;N, which gives only one solution, that is, ( j (1 0 )C d w ) 1 is the only equilibrium state of system (6.3) in the region j d w < e 1 j (1 0 )C d w . To summarize, e1 = d v f 1 and e2 = ( j (1 0 )C d w ) 1 are 2 isolated equilibrium states of system (6.3) when I2 3 . Now we are going to show that for all 0(0) j ,(t) converges to either e1 or e2 . According to part a) of Lemma B.0.1, we have that8 > 0;9T > 0, such that d=v f < i < j (1 0 )C d +;i = 1; 2;:::;N. Without loss of generality, let T = 0. If8t 0, N (t) C d v f , then w( j N )>C, due to C d v f < c . We have q N = minfv f N1 ; ~ w(~ j N1 )g; q N1 = minfv f N2 ; ~ w(~ j N2 );C;w( j N1 )gw( j N1 ): Therefore,8 N1 > c , _ N1 =q N1 q N w( j N1 ) ~ w(~ j N1 )< 0, thus lim sup t!1 N1 c . Consequently, we have that lim sup t!1 i c ;i 1; 2;:::;N 1. When e i c ;i = 1; 2;:::;N 1, N C d =v f , equation (6.3) gives thatq 1 =d;q i = v f i1 ;i = 2; 3;:::;N + 1, therefore, _ 1 =dv f 1 ; _ i =v f i1 v f i ;i = 2; 3;:::;N: which can be written in the compact form as _ =A( d v f 1) where A = 2 6 6 6 4 v f v f v f . . . . . . v f v f 3 7 7 7 5 : Since v f > 0, we have that A is Hurwitz. Therefore (t) converges to e1 = d v f 1 exponentially fast. 165 If there exists t 0 0, N (t 0 ) > C d v f , then q N+1 (t 0 ) = (1 0 )C d , due to C d v f < N (t 0 )< j (1 0 )C d +. Recall that q N = minfv f N1 ; ~ w(~ j N1 );C;w( j N )g: and d=v f < N1 < j (1 0 )C d +, we have that v f N1 >dv f ; ~ w(~ j N1 )> ~ w(~ j ( j (1 0 )C d +)) = (w ~ w)( j (1 0 )C d c ) + (1 0 )C d ~ w: therefore _ N =q N q N+1 8 > > > > > < > > > > > : > 1 [ N ( j (1 0 )C d w )]> 0; if N < j (1 0 )C d w = 0; if N = j (1 0 )C d w < 2 [ N ( j (1 0 )C d w )]< 0; if N > j (1 0 )C d w ; (B.15) where 1 = minf dv f (1 0 )C d j (1 0 )C d =w ; (w ~ w)( j (1 0 )C d c ) ~ w j (1 0 )C d =w ;wg and 2 = minf dv f (1 0 )C d (1 0 )C d =w ; (w ~ w)( j (1 0 )C d c ) ~ w (1 0 )C d =w ;wg: When is suciently small, 1 and 2 are both positive. Therefore N (t) > C d v f ;8t t 0 and N (t) converges to j (1 0 ) w exponentially fast. Consequently, i (t) also converges exponentially fast to j (1 0 ) w for i = 1; 2;:::;N 1, that is, (t) converges exponentially fast to e2 . Therefore, for all initial condition(0)2S, (t) converges to one of the two equilibrium states exponentially fast. From the analysis above, we have that8(0)2fj d v f i j (1 0 )C d w + ;i = 1; 2;:::;N 1; C d v f < N j (1 0 )C d w +g,(t) converges to the equilibrium state e2 exponentially fast. Therefore this equilibrium state is locally exponentially stable. 166 Similar to the case I 2 2 , we can show that for all (0) 2 fj0 i C d =v f ;i = 1; 2;:::;Ng,(t) converges to the point e1 exponentially fast. Therefore this equilibrium state is exponentially stable. d) When I2 4 , d > C b . If e is an equilibrium state of system (6.3), then we have minfd;Cg v f e i j (1 0 )C d w ;i = 1; 2;:::;N by using Lemma B.0.1, in this region q e N+1 = minfv f e N ; ~ w(~ j e N ); (1( e N ))C d g = (1 0 )C d ; From the equilibrium condition (6.4), we have that q e i = (1 0 )C d ;i = 1; 2;:::;N. Recall that q e 1 = minfd;C;w( j e 1 )g: Since d > (1 0 )C d and C > (1 0 )C d , q e 1 = (1 0 )C d gives only one solution e 1 = j (1 0 )C d w . Fori = 2; 3;:::;N, given e i1 = j (1 0 )C d w , we check the value of e i . Recall that q e i = minfv f e i1 ; ~ w(~ j e i1 );C;w( j e i )g: Since v f e i1 > C > (1 0 )C d and ~ w(~ j e i1 ) > w( j e i1 ) = (1 0 )C d as e i1 > c , thusq e i = (1 0 )C d givesw( j e i ) = (1 0 )C d , i.e., e i = j (1 0 )C d w . Therefore the point ( j (1 0 )C d w ) 1 is the unique equilibrium state of system (6.3) when I2 4 . For all(0)2S I , we havev f N >dv f and ~ w(~ j N )> ~ w(~ j j + (1 0 )C d w ) by using Lemma B.0.1. Take to be suciently small, we have v f N >C d and ~ w(~ j N )> (1 0 )C d . Thus q N+1 = minfv f N ; ~ w(~ j N ); (1( N ))C d g = (1 0 )C d ; then _ N =q N q N+1 = minfv f N1 ; ~ w(~ j N1 );C;w( j N )g (1 0 )C d : (B.16) Similar to equation (B.15), we have _ N 8 > > > > > < > > > > > : > 1 [ N ( j (1 0 )C d w )]> 0; if N < j (1 0 )C d w = 0; if N = j (1 0 )C d w < 2 [ N ( j (1 0 )C d w )]< 0; if N > j (1 0 )C d w ; 167 where 1 = minf dv f (1 0 )C d j (1 0 )C d =w ; (w ~ w)( j (1 0 )C d c ) ~ w j (1 0 )C d =w ;wg and 2 = minf dv f (1 0 )C d (1 0 )C d =w ; (w ~ w)( j (1 0 )C d c ) ~ w (1 0 )C d =w ;wg: When is suciently small, 1 and 2 are both positive. Therefore N converges to j (1 0 )C d w exponentially fast. Based on the converges of N , we can show that N1 also convergences to j (1 0 )C d w , followed by N2 through 1 . Therefore,8(0)2S,(t) converges to ( j (1 0 )C d w ) 1 exponentially fast. e) For the caseI2 5 The proof of this part can be demonstrated by following the same routine of the case of I2 1 based on part b) of Lemma B.0.1. For the sake of briefness, we omit the detailed proof here. 168 Appendix C Proof of Theorem 7.1.1 a) IfI2 4 S i=1 i , the VSL controller (7.6) is applied. First we show that the controller v is well-dened82 [0; j ]. According to (7.6), v 1 is dened in the region C d v f 2 j , in which q 2 C d and x + 1 > 0. Therefore the denominator of v 1 w j q 2 +(x + 1 )w j C d >w( j c )C d =CC d > 0: Hencev 1 = medf0; v 1 ;v f g is well-dened in the region C d v f 2 j . v 2 is dened in the region 0 C d v f , in which q 2 =v f =C d +v f x. Since 0<< v f w j C d and C d v f x 0, we have that q 2 x =C d +v f xx>C d +v f x 0 and w j (q 2 x)>w j C d (v f v f w j C d )xw j C d (v f v f w j C d )( C d v f ) = 0 due to v f v f w j C d < 0. Therefore, the denominator of v 2 is greater than 0, v 2 = medf0; v 2 ;v f g is well-dened, and v 2 = w(q 2 x) w j (q 2 x) > 0. Now we nd the equilibrium point of system (7.1),(7.6) and analyze its stability properties. We have that8(0)2 (C d =v f ; j ];v =v 1 . Ifv 1 = 0, i.e. v 1 0, we have q 1 = v 1 w j v 1 +w = 0. In the region C d v f 2 j , we have v f C d v f 2 > 0; as 2 < C d v f ; ~ w(~ j ) ~ w(~ j j )> 0; as ~ j > j ; (1())C d (1 0 )C d > 0; as 0 < 1: 169 Therefore, q 2 = minfv f ; ~ w(~ j ); (1())C d g minfC d v f 2 ; ~ w(~ j j ); (1 0 )C d g minfC d v f 2 ; ~ w(~ j j ); (1 0 )C d g j C d =v f + 1 ( C d v f + 1 ) due to C d v f 2 j , which implies 0< C d =v f + 1 j C d =v f + 1 1, since 2 < 1 . Thus we have _ =q 1 q 2 minfC d v f 2 ; ~ w(~ j j ); (1 0 )C d g j C d =v f + 1 ( C d v f + 1 ) (C.1) If v 1 > 0, i.e., v 1 > 0, v 1 = minf v 1 ;v f g v 1 and v 1 w j v 1 +w v 1 w j v 1 +w =w j (v 1 v 1 )w (v 1 +w)( v 1 +w) 0; which implies v 1 w j v 1 +w v 1 w j v 1 +w . Hence, q 1 = minfd; v 1 w j v 1 +w ;C;w( j )g v 1 w j v 1 +w v 1 w j v 1 +w =q 2 (x + 1 ) and _ =q 1 q 2 ( C d v f + 1 )< 0: (C.2) According to equation (C.1) and (C.2), _ ( C d v f + 1 ); where = minf; minfC d v f 2 ; ~ w(~ j j );(1 0 )C d g j C d =v f + 1 g> 0. Using Lemma 3.2.4 in [86], we have (t) C d v f 1 + [(0) C d v f + 1 ]e t : Since C d =v f 1 <C d =v f 2 <C d =v f <(0), (t) will decrease exponentially to the value (t 0 ) = C d =v f 2 at some nite time t 0 , at which v switches to v 2 , in which case the dynamics of (t) are analyzed below. Either the initial condition 0 (0) C d v f or v switches to v 2 from v 1 , there exists a t 0 0, at which 0 (t 0 ) C d v f and v = v 2 . Since v 2 > 0, we have v 2 = minf v 2 ;v f gv f and v 2 w j v 2 +w v f w j v f +w =C <w( j ) as C d =v f < c : 170 Therefore, q 1 = minfd; v 2 w j v 2 +w ;C;w( j )g = minfd; v 2 w j v 2 +w g = minfd; v f w j v f +w ; v 2 w j v 2 +w g = minfd; v 2 w j v 2 +w g = minfd;q 2 xg and q 2 =C d +v f x: Consequently, _ =q 1 q 2 = minfdv f ;( C d v f )g: (C.3) In the case d > C d , we have _ minfv f ;g( C d v f ),82 [0;C d =v f ] and _ = 0 at =C d =v f , which implies that (t) converges exponentially fast to = C d v f , and 8tt 0 ; C d v f , therefore the ow at the exit of the section q 2 =v f converges to C d . In the case dC d , _ = minfdv f ;( C d v f )g 8 > > > > > > > < > > > > > > > : minfv f ;g( d v f ) if 2 [0; d v f ) = 0 if = d v f v f ( d v f ) if 2 ( d v f ; C d v f ] : (C.4) Therefore, (t) converges to d v f exponentially fast, and q 2 = v f converges to d with the same rate. In summary, the closed-loop system (7.1) - (7.6) has a unique equilibrium point e = minfd;C d g v f . In addition,8(0)2 [0; C d v f ], (t) converges to e exponentially fast and8(0)2 ( C d v f ; j ],(t) decreases to C d v f 2 exponentially fast and then converges to e exponentially fast. The ow rate at the exit of the section converges to the maximum possible value minfd;C d g exponentially fast while the speed of ow converges with the same rate to v f . b) Part b) of Theorem 7.1.1 can be derived directly from part e) of Theorem 6.2.1. 171 Appendix D Proof of Theorem 7.2.1 a) In controller (7.13), v 1 through v N1 is well-dened by letting v i = v f when i = 0, for i = 1; 2;:::;N 1. Since 0 < 0 v f w j C d , we can show that v 0 is also well-dened in a similar manner to the single section case in Theorem 7.1.1. For all N (0)2 ( C d v f ; j ]; v N1 = v N1;1 . If v N1 = 0, i.e. v N1;1 0, we have q N = 0, thus in the region C d v f 2 N (t) j , we have _ N =q N+1 = minfv f N ; (1( N ))C d ; ~ w(~ j N )g minfC d v f 2 ; ~ w(~ j j ); (1 0 )C d g j C d =v f + 1 ( N C d v f + 1 ): If v N1 > 0, i.e. v N1 > 0, then q N v N1 N1 v N1 N1 , _ N v N1 N1 q N+1 = N1 ( N C d v f + 1 ): Therefore,8 N (0)2 ( C d v f ; j ], we have _ N ( N C d v f + 1 ); where = minf N1 ; minfC d v f 2 ; ~ w(~ j j );(1 0 )C d g j C d =v f + 1 g> 0. Since > 0 and C d v f 1 < C d v f 2 < C d v f < N (0), and C d v f 2 > 0, N (t) will decrease exponentially fast to the value C d v f 2 at some nite timet 0 , at which v N1 switches to v N1;2 , in which case (t) evolves as analyzed below. Either the initial condition N (0) 2 [0; C d v f ] or v N1 switches to v N1;2 from v N1;1 at N = C d v f 2 , there exists t 0 0, at which time instant N (t 0 ) C d v f and 172 v N1 = v N1;2 . Since N C d v f ;q N+1 0; and N1 0, we have v N1;2 0 from equation (7.13), thus v N1 = minfv f ; v N1;2 g. Therefore _ N v N1;2 N1 q 2 = N1 ( N C d v f ): Without loss of generality, let t 0 = 0, then we have N (t) C d v f + ( N (0) C d v f )e N1 t ; which implies that8t 0; N (t) C d v f , v N1 = v N1;2 . Then we examine the dynamics of N1 . If v N2 = 0, i.e. v N2 0, q N1 = 0, we have _ N1 =q N = minfv N1 N1 ; v N1 w j v N1 +w ;C;w( j N )g: Since N C d v f < c ; w( j N )>C and v N1 = minfv f ; v N1;2 g, we have _ N1 = minf v N1;2 N1 ;v f N1 ; v N1;2 w j v N1;2 +w ; v f w j v f +w ;Cg: Since 0 N C d v f ; N1 >v f and N1 j , we have v N1;2 N1 =C d + (v f N1 )( N C d v f )C d C d j C d v f ( N1 C d v f ); v f N1 v f ( N1 C d v f ); v N1;2 w j v N1;2 +w =w j C d + (v f N1 )x N C d + (v f N1 )x N +w N1 w j v f C d v f w j + N1 C d w j v f C d (v f w j + N1 C d )( j C d v f ) ( N1 C d v f ); v f w j v f +w =C C j C d v f ( N1 C d v f ): Thus _ N1 minf C d j C d v f ;v f ; w j v f C d (v f w j + N1 C d )( j C d v f ) ; C j C d v f g( N1 C d v f ). If v N2 > 0, i.e. v N2 > 0, _ N1 v N2 N2 q N = N2 ( N1 C d v f ): 173 To conclude, _ N1 ( N1 C d v f ); where = minf C d j C d v f ;v f ; w j v f C d (v f w j + N1 C d )( j C d v f ) ; C j C d v f ; N2 g> 0: Therefore lim sup t!1 N1 (t)C d =v f and lim inf t!1 v N1;2 v f due to v N1;2 = C d +(v f N1 )x N C d =v f +x N1 C d N1 , which implies lim t!1 v N1 =v f . Similarly, we can show that lim sup t!1 i (t) C d =v f and lim t!1 v i = v f for i = 1; 2;:::;N 1. Then the dynamics of (t) become _ 1 = minfd; v 0 w j v 0 +w ; v f w j v f +w gv f 1 = minfdv f 1 ;Cv f 1 ; 0 ( 1 C d =v f )g; _ i =v f i1 v f i ;i = 2;:::;N: Note that the rst dierential equation is the same as equation (C.3) in the single- section case. Therefore we can directly take the analysis result of equation (C.3), which shows that 1 converges to 1 = minfd;C d g v f exponentially fast. Consequently, i converges exponentially fast to i = minfd;C d g v f , for i = 1; 2;:::;N. Recall that N (t) C d v f ,8t t 0 , thus q N+1 converges to C d exponentially fast. Consequently, q i converge to C d exponentially fast for i = 1; 2;:::;N. b) This part can be shown directly with part e) of Theorem 6.3.1. 174

Abstract (if available)

Abstract Highway congestion is detrimental to traffic mobility, safety and the environment. Numbers of studies have been conducted to avoid or relieve highway congestion with different traffic flow control strategies such as Variable Speed Limit (VSL), Ramp Metering (RM) and Lane Change (LC) recommendation and their combinations. While consistent improvement on traffic safety is reported under existing traffic flow control strategies in macroscopic and microscopic simulations, the results are rather controversial when it goes to the improvement on traffic mobility and the environmental impact, especially in microscopic simulations. Some researchers attribute the inconsistencies to the complexity of underlying reasons of the congestion and highly disordered and stochastic behavior at the bottleneck. Therefore, it is necessary to investigate the dynamical behavior of the open-loop traffic flow systems under all possible demand levels as well as initial densities in order to find out the reasons of the chaotic behavior at the bottleneck, and based on which find an integrated traffic flow controller which is able to provide consistent improvement in traffic mobility, safety and the environmental impact under different traffic scenarios. ❧ In this dissertation, we discover that one of the major reasons of the disordered behavior is the forced lane changes at vicinity of the bottleneck. A lane change controller is proposed which provides lane change recommendations to upstream vehicles in order to avoid the capacity drop. Two types of variable speed limit controllers are designed to improve the flow rate at highway bottleneck together with the lane change controller. The combined lane change and feedback linearization variable speed limit controller, which is build on the first order cell transmission model, can analytically guarantee the global exponential convergence to the desired equilibrium point at which maximum possible flow rate is achieved. Then the combined LC and VSL controller is extended to coordinate with ramp metering controllers. The coordinated VSL, RM and LC controller is able to improve system performance, maintain the queue length on ramps and keep the fairness between mainline and on-ramp flows. Microscopic simulations show consistent improvement under different traffic demand and scenarios. The proposed controller is compared to the widely used MPC control strategy. Both macroscopic and microscopic simulations show that the performance and robustness with respect to model parameter errors and measurement noise of our controller is better than that of the MPC controller. Furthermore, we modify the cell transmission model to include the effect of capacity drop and the decreasing discharging flow of the road section and rigorously investigate its stability properties under all possible traffic flow scenarios. The analysis is used to motivate the design of variable speed limit control to overcome capacity drop without lane change control and achieve the maximum possible flow under all feasible traffic situations. We also consider the case where the system disturbance is included and extend the VSL controller by adding the integral action in order to reject the disturbance while avoiding the capacity drop.

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Creator Zhang, Yihang (author)

Core Title Integrated control of traffic flow

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 11/19/2018

Defense Date 10/24/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag control systems,lane change,nonlinear systems,OAI-PMH Harvest,ramp metering,Traffic flow,Transportation,variable speed limit

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ioannou, Petros (committee chair), Dessouky, Maged (committee member), Savla, Ketan (committee member)

Creator Email yihangzh@usc.edu,zhang.yihang.china@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-108448

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