Pedestrian Dynamics

Summary

Effective evacuation of people from closed spaces is an extremely important topic, since it can save real lives in emergency situations that can be brought about by natural and human made disasters. Usually there are static maps posted at various places at buildings that illustrate routes that should be taken during emergencies. However, when disasters happen, some of these routes might not be valid because of structural problems due to the disaster itself and more importantly because of the distribution of congestion of people spread over the area. The average flow of traffic depends on the traffic density. Therefore, if all the people follow the same route, or follow a route without knowing the congestion situation, they can end up being part of the congestion which results in very low flow rate or worse a traffic jam. Hence it becomes extremely important to design evacuations that inform people how fast and in which direction to move based on real-time information obtained about the people distribution using various sensors. The sensors used can include cameras, infra red sensors etc., and the technology used to inform people about the desired movement can be communicated using light matrix, small speakers, and in the future using wireless PDAs. This book provides mathematical models of pedestrian movements that can be used specifically for designing feed back control laws for effective evacuation. The book also provides various feedback control laws to accomplish the effective evacuation. The book uses the hydrodynamic hyperbolic PDE macroscopic pedestrian models since they are amenable to feedback control design. The control designs are obtained through different nonlinear techniques including Lyapunov functional techniques, feedback linearization in the distributed model, and some discretized techniques. Book jacket.

Introduction	p. 1
Traffic Flow Theory for 1-D	p. 5
Introduction	p. 5
Microscopic vs Macroscopic	p. 6
Car-Following Model	p. 7
Traffic Flow Theory	p. 8
Flow	p. 8
Conservation Law	p. 9
Velocity-Density Relationship(s)	p. 11
Traffic Flow Model 1-D	p. 13
LWR Model	p. 14
PW Model	p. 15
AR Model	p. 17
Zhang Model	p. 19
Models Summary	p. 22
Method of Characteristics	p. 23
LWR Model Classification	p. 23
Exact Solution	p. 23
Blowup of Smooth Solutions	p. 25
Weak Solution	p. 27
Crowd Models for 2-D	p. 33
Introduction	p. 33
Traffic Flow Theory in 2-D	p. 34
One Equation Crowd Model	p. 35
First System Crowd Dynamic Model	p. 36
Model Description	p. 36
Conservation Form and Eigenvalues	p. 38
Second Crowd Dynamic Model	p. 40
Model Description	p. 40
Conservation Form and Eigenvalues	p. 41
Third Crowd Dynamic Model	p. 44
Model Description	p. 44
Derivation of a Macroscopic Model from a Microscopic Model in 2-D	p. 45
Conservation Form and Eigenvalues	p. 46
Comparison Between the Models	p. 50
Linearization	p. 53
One Equation Crowd Model	p. 53
First System Model	p. 55
Second System Model	p. 56
Third System Model	p. 58
Numerical Methods	p. 61
Introduction	p. 61
Fundamentals of FVM	p. 62
Formulation of 2-D Numerical Schemes	p. 63
Numerical Schemes	p. 65
Lax-Friendrichs Scheme	p. 65
FORCE Scheme	p. 65
Roe's Scheme	p. 66
Simulation	p. 67
Initial and Boundary Conditions	p. 67
Simulation Results	p. 68
Matlab Program Code	p. 78
One-equation Model	p. 78
First System Model	p. 81
Second System Model	p. 87
Third System Model	p. 91
Feedback Linearization (1-D Patches)	p. 95
Introduction	p. 95
Theory	p. 96
Control Problem	p. 96
Characteristic Index	p. 97
State Feedback Control	p. 97
Closed-Loop Stability	p. 98
Application to the LWR (One patch)	p. 98
Application to the LWR (n=5 patches)	p. 101
Matlab Program Code	p. 103
One-patch Control	p. 103
Five-patch Control	p. 104
Intelligent Evacuation Systems	p. 107
Introduction	p. 107
IPES Functions	p. 109
IES Functions for Evacuation Scenarios	p. 111
Subway Station	p. 111
Airport	p. 112
Four-Layer System Architecture	p. 115
Four-Layer System: Scenarios	p. 117
Subway Station	p. 117
Airport	p. 118
IT Issues and Requirements	p. 118
Feedback Control and Dynamic Modeling	p. 119
Discretized Feedback Control	p. 121
Introduction	p. 121
Pedestrian Flow Modeling	p. 123
Feedback Linearization of State Equations	p. 125
Stability Under Feedback Linearizing Control	p. 126
Saturation of Control	p. 128
Simulation Results	p. 129
Code	p. 133
Exercises	p. 134
Computer Code	p. 135
main.m	p. 135
rhodot-nsec.m	p. 138
vfcntrl_nsec_try.m	p. 138
Discretized Optimal Control	p. 141
Optimal Control	p. 142
State Equations	p. 142
Cost Function	p. 143
Calculus of Variation	p. 144
The Method of Steepest Descent	p. 146
Numerical Results	p. 147
Code	p. 150
main.m	p. 150
optimal_cntrl_calc_var_nsec_odesol.m	p. 151
Exercises	p. 152
Computer Code	p. 152
optimal_corridor_evacuation/main	p. 152
optimal_cntrl_calc_var_nsec_odesol	p. 155
Distributed Feedback Control 1-D	p. 161
Introduction	p. 162
Modeling	p. 163
One Equation Model	p. 164
Two Equation Model	p. 165
Feedback Control for One-Equation Model	p. 166
Continuity Equation Control Model	p. 166
State Feedback Control	p. 167
Lyapunov Stability Analysis	p. 168
Simulation Results	p. 170
Control Saturation	p. 172
Feedback Control for Two Equation Model	p. 175
Two Equation Control Model	p. 176
State Feedback Control Using Backstepping	p. 176
Simulation	p. 179
Exercises	p. 182
Computer Code	p. 182
feedback_1d_mass	p. 182
feedback_1d_momentum	p. 185
Distributed Feedback Control 2-D	p. 189
Introduction	p. 189
Feedback Control of One-Equation Model	p. 191
One-Equation Model	p. 191
Control Model	p. 192
State Feedback Control	p. 193
Lyapunov Stability Analysis	p. 193
Simulation Results	p. 195
Feedback Control for Two-Equation Model	p. 197
Two Equation Model	p. 197
Control Model	p. 198
State Feedback Control Using Backstepping	p. 199
Simulation Results	p. 202
Exercises	p. 204
Computer Code	p. 204
feedback_2d	p. 204
Robust Feedback Control	p. 209
Introduction	p. 209
Feedback Control for Continuity Equation Model	p. 210
Input Uncertain Control Model	p. 211
Robust Control by Lyapunov Redesign Method	p. 212
Simulation Results	p. 216
Robust Control for Two-Equation Model	p. 220
Robust Backstepping: Unmatched Uncertainty	p. 221
Robust Control: Matched Uncertainty	p. 225
Robust Control: Both Matched and Unmatched Uncertainties	p. 227
Computer Code	p. 229
robust_1d	p. 229
Bibliography	p. 233
Index	p. 243
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