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July, 1976 Report ESL-R-671 SENSITIVITY ANALYSIS OF July, 1976 Report ESL-R-671 SENSITIVITY ANALYSIS OF OPTIMAL STATIC TRAFFIC ASSIGNMENTS IN A LARGE FREEWAY CORRIDOR, USING MODERN CONTROL THEORY by Pierre Dersin Stanley B. Gershwin Michael Athans Electronic Systems Laboratory Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology Cambridge, Massachusetts 02139 _-'~~"_I, Room 14-0551 77 Massachusetts Avenue Cambridge, MA 02139 MITLibraries Ph: 617.253.5668 Fax: 617.253.1690 Document Services Email: [email protected] http://libraries. mit. edu/docs DISCLAIMER OF QUALITY Due to the condition of the original material, there are unavoidable flaws in this reproduction. We have made every effort possible to provide you with the best copy available. If you are dissatisfied with this product and find it unusable, please contact Document Services as soon as possible. Thank you. Due to the poor quality of the original document, there is some spotting or background shading in this document. SENSITIVITY A&NALYSIS OF OPTIMAL STATIC TA.FFIC ASSIGN'MiENTS IN A LARGE FREEWAY CORRIDOR, USING MODERN CONTROL THEORY by Pierre Dersin Licencie en Sciences Mathematiques, Universite Libre de Bruxelles, Brussels (1975) SUBMITTED IN PARTIAL FULFILL¥MENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY July, 1976 Signature of Author .................. ...... ............. Department of Matheumatics , July 9, 1976 Certified by ............................... Thesis Co-Supervisor Thesis Co-Supervisor Accepted by .. .. ........................................ Chairman, Departmental Committee on Graduate Students ABSTRACT The system-optimized static traffic assignment problem in a freeway corridor network is the problem of choosing a distribution of vehicles in the network to minimize average travel time. It is of interest to know how sensitive the optimal steady state traffic distribution is to external changes including accidents and variations in incoming traffic. Such a sensitivity analysis is performed via dynamic programming. The propagation of external perturbations is studied by numerical implementation of the dynamic programming equations. When the network displays a certain regularity and satisfies certain conditions, we prove, using modern control theory and graph theory, that the effects of imposed perturbations which contribute no change in total flow decrease exponentially as distance from the incident site increases. We also characterize the impact of perturbations with nonzero total flow. The results confirm numerical experience and provide bounds for the effects as functions of distance. This study gives rise to theoretical results, used in performing our analysis but also of a broader interest. Flow conservation in a class of networks can be described by linear discrete dynamical systems in standard form. The controllability of these systems is intimately related to the structure of the network, and is connected with graph and Markov chain theory. When the cost function is quadratic (which is the case in our traffic context), asymptotic properties of the optimal cost and propagation matrix are derived. In addition to proved results, we formulate some conjectures, verified in the numerical experiments. -2- ACKNOWLEDGEMENT This research was conducted at the Transportation group of the M.I.T. Electronic Systems Laboratory and sponsored by the Department of Transportation under contract DOT/TSC/849 and a graduate fellowship of the Belgian American Educational Foundation. We are grateful to Dr. Diarmuid O'Mathuna and Ms. Ann Muzyka of TSC, and Dr. Robert Ravera of DOT for their encouragement, support, and criticism. -3- - - TABLE OF CONTENTS Page ABSTRACT 2 ACKNOWLEDGMENTS 3 LIST OF.FIGURES 8 CHAPTER 1 INTRODUCTION 11 1.1 Physical Motivation 11 1.2 Literature Survey 18 1.3 Contribution of This Report 22 1.3.1 Contribution to Traffic Engineering 22 1.3.2 Contribution to Optimal Control Theory 26 1.4 Outline of the Report 29 CHAPTER 2 DOWNSTREAM PERTURBATIONS IN A FREEWAY CORRIDOR NETWORK WITH QUADRATIC COST 32 2.1 Introduction 32 2.2 Problem Statement and Solution by Dynamic Programming 33 2.2.1 Summary of the Static Optimization Problem 33 2.2.2 Structure of the Network 36 2.2.3 Linear Constraints 38 2.2.4 Quadratic Expansion of Cost Function 42 2.2.5 Solution by Dynamic Programming 45 2.3 Asymptotic Analysis in the Stationary Case 55 2.3.1 Introduction 55 2.3.2 Numerical Experiments 56 2.3.3 System Transformation. Corresponding 60 Expression for Cost Function 2.3.4 Controllability of Reduced Systems 74 2.3.4_.1 The Meaning of Ccntrollability 75 in this Context 2.3.4.2 Graph Formulation of Control- 78 lability 2.3.4.3 Algebraic Formulation of Control- 90 lability -5- Page 2.3.5 Extension and Application of Linear- 97 Quadratic Optimal Control Theory 2.3.5.1 Extended Linear-Quadratic Problem 97 2.3.5.2 Application to the Original Cost 103 Function 2.3.5.3 Example 110 2.3.6 The Propagation Matrix and Its.Asymptotic 113 Behavior 2.3.7 Sensitivity Analysis 122 2.3.7.1 Physical Motivations 122 2.3.7.2 Mathematical Formulation 125 2.3.8 Illustration 158 2.3.9 Quasi-Stationary Networks 169 2.4 Concluding Remarks 178 CHAPTER 3 UPSTREAM PERTURBATIONS IN A FREEWAY CORRIDOR NETWORK 180 3.1 Physical Description of the Problem 180 3.2 Mathematical Treatment 181 3.2.1 Solution by Dynamic Programming in the General 183 Case 3.2.2 Asymptotic Sensivity Analysis in the Stationary 193 Case 3.3 Concluding Remarks 199 CHAPTER 4 CHARACTERIZATION OF NETWORK STEADY-STATE CONSTANTS 201 BY A VARIATIONAL PROPERTY 4.1 Introduction 201 4.2 Notation and Preliminaries 202 4.3 Approximation Lemma 205 4.4 Main Result 208 4.5 Explicit Expression for c and p in a Special Case 215 CHAPTER 5 AN EXAMPLE OF NONSTATIONARY NETWORK 221 5.1 Introduction 221 5.2 Splitting of Network and Cost Function 222 -6- Page 5.2.1 Notation 222 5.2.2 Decomposition of the Network. 224 5.2.3 Constraints. Green Split Variables 225 5.2.4 Decomposition of the Cost Function 227 5.2.5 Expansion to the Second Order 228 5.2.6 Analytical Expressions for the Derivatives 229 5.2.7 Example 232 CHAPTER 6 CONJECTURES 246 6.1 Introduction 246 6.2 Is D (k) a Stochastic Matrix? 246 6.3 Random Walk-Like Expression for D 250 CHAPTER 7 NUMERICAL EXPERIMENTATION 262 7.1 Introduction 262 7.2 Various Stationary Networks 262 CHAPTER 8 CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH 289 8.1 Summary of this Work 289 8.2 Further Research in Traffic Engineering 291 8.3 Further Research in Optimal Control Theory 292 APPENDIX A 296 APPENDIX B 312 APPENDIX C 331 APPENDIX D 337 APPENDIX E APPENDIX F 342 REFERENCES 353 -7- LIST OF FIGURES Fig. 1.1.1 Example of Freeway Corridor Network Fig. 1.1.2 Upstream and Downstream Perturbations Fig. 1.2.1 Fundamental Diagram Fig. 1.3.1 Example of "Quasi-Stationary Network" Fig. 2.2.2 Splitting of a Network into Subnetworks Fig. 2.2.3 Example of Subnetwork Fig. 2.3.3 Example of Subnetwork of Class 3 Fig. 2.3.4-1 Negative Flow Perturbation Fig. 2.3.4-2 Rapid Control, by Negative Flow Perturbations Fig. 2.3.4-3 Accessibility Graph of the Subnetwork of Fig. 2.3.4-2 Fig. 2.3.4-4 Subnetwork Whose Accessibility Graph has Transient Nodes Fig. 2.3.4-5 Accessibility Graph of the Subnetwork of Fig. 2.3.4-4 Fig. 2.3.4-6 Uncontrollable Subnetwork of Class 1 Fig. 2.3.4-7 Accessibility Graph of the Subnetwork of -Fig. 2.3.4-6 Fig. 2.3.4-8 Controllable Subnetwork of Class 2 Fig. 2.3.4-9 Accessibility Graph of the Subnetwork of Fig. 2.3.4-8 Fig. 2.3.4-10 Uncontrollable Subnetwork of Class 2 Fig. 2.3.4-11 Accessibility Graph of the Subnetwork of Fig. 2.3.4-10 Fig. 2.3.4-12 Subnetwork of Class 1, Controllable but not in One Step Fig. 2.3.4-13 Accessibility Graph of the Subnetwork of Fig. 2.3.4-12 Fig. 2.3.8-1 Eigenvalue of D as a Function of Link Length a Fig. 2.3.8-2 General Controllable Case and Corresponding Accessibility Graph. O<a<-; -1<X<l Fig. 2.3.8-3 First Uncontrollable Case and Corresponding Accessibility Graph : a = A=1l+; Fig. 2.3.8-4 Second Uncontrollable Case and Corresponding Accessibility Graph: a = 0; X = -1 -8- LIST OF FIGURES (con't) Fig. 2.3.8.-5 Exponential Decay of Downstream Perturbations Corresponding to an Initial Perturbation with Zero Total Flow Fig. 2.3.9-1 First Type of Subnetwork Fig. 2.3.9-2 Second Type of Subnetwork Fig. 2.3.9-3 Quasi-Stationary Network Fig. 2.3.9-4 Other Example of Quasi-Stationary Network Fig. 2.3.9-5 Typical Subnetwork for Equivalent Stationary Network Fig. 3.1.1 Upstream Perturbations Fig. 3.2.1-1 Incident Occurring in Subnetwork (N-k) Fig. 3.2.1-2 Relabeling for Upstream Perturbations Fig. 3.2.2 Downstream and Upstream Accessibility Graphs for One Same Subnetwork Fig. 5.2.1 Freeway Entrance Ramp Fig. 5.2.3 Signalized Intersection Fig. 5.2.7-1 Example of Nonstationary Network Fig. 5.2.7-2 Decomposition into Subnetworks Fig. 5.2.7-3 Revised Network, Taking the Binding Constraints into Account Fig. 5.2.7-4 Labeling of Links and Signalized Intersections in Subnetwork 2 Fig. 5.2.7-5 Labeling of Links and Siganlized Intersections in Subnetwork 3 Fig. 7.1 Standard Two-Dimensional Example and Corresponding Accessibility Graph Fig. 7.2 Limiting Case of Fig. 7.1; the Accessibility Graph is the same as in Fig.
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