This monograph explores the design of controllers that suppress oscillations and instabilities in congested traffic flow using PDE backstepping methods. The first part of the text is concerned with basic backstepping control of freeway traffic using the Aw-Rascle-Zhang (ARZ) second-order PDE model. It begins by illustrating a basic control problem – suppressing traffic with stop-and-go oscillations downstream of ramp metering – before turning to the more challenging case for traffic upstream of ramp metering. The authors demonstrate how to design state observers for the purpose of stabilization using output-feedback control. Experimental traffic data are then used to calibrate the ARZ model and validate the boundary observer design. Because large uncertainties may arise in traffic models, adaptive control and reinforcement learning methods are also explored in detail.
Part II then extends the conventional ARZ model utilized until this point in order to address more complex traffic conditions: multi-lane traffic, multi-class traffic, networks of freeway segments, and driver use of routing apps. The final chapters demonstrate the use of the Lighthill-Whitham-Richards (LWR) first-order PDE model to regulate congestion in traffic flows and to optimize flow through a bottleneck. In order to make the text self-contained, an introduction to the PDE backstepping method for systems of coupled first-order hyperbolic PDEs is included.
Traffic Congestion Control by PDE Backstepping is ideal for control theorists working on control of systems modeled by PDEs and for traffic engineers and applied scientists working on unsteady traffic flows. It will also be a valuable resource for researchers interested in boundary control of coupled systems of first-order hyperbolic PDEs.
Author(s): Huan Yu, Miroslav Krstić
Series: Systems & Control: Foundations & Applications
Publisher: Birkhäuser
Year: 2022
Language: English
Pages: 362
City: Cham
Preface
What Does the Book Cover?
Who Is the Book For?
Acknowledgments
Contents
Acronyms
1 Introduction
1.1 Before Control—Models
1.2 The Basics of Traffic Flow Modeling
Macroscopic and Microscopic Models
LWR and ARZ Macroscopic Models
1.3 Macroscopic Traffic PDE Models
Lighthill–Whitham–Richards Model
Aw–Rascle–Zhang Model
1.4 Linearized Models and Free/Congested Regimes
Linearized LWR Model
Linearized ARZ Model
1.5 Traffic Actuation
Control Measures: Ramp Metering (RM) and Variable Speed Limits (VSL)
Distributed and In-Domain Actuation: CAVs
1.6 A Brief Review of Literature on Traffic Control
Control Employing PI and Backstepping Feedback
Optimal Control
Model Predictive Control
Other Control Problems and Strategies
1.7 Boundary Control by RM or VSL
Traffic Flow Boundary Control by Ramp Metering
Traffic Velocity Boundary Control by Varying Speed Limits
1.8 Open-Loop Stability
Linear Stability of LWR Model
Linear Stability of ARZ Model
1.9 Numerical Simulation
Numerical Simulation of the LWR PDE Model
Numerical Simulation of the ARZ PDE Model
1.10 Notes and References
2 Backstepping for Coupled Hyperbolic PDEs
2.1 A Brief History of PDE Backstepping
2.2 Coupled Hyperbolic PDEs
2.3 Backstepping Control for Coupled Hyperbolic PDEs
Target System
Backstepping Transformation
Full-State Feedback Design
2.4 Observer and Output-Feedback Design for General Hyperbolic PDEs
Boundary Sensing for State Estimation
Output Feedback Design
2.5 Backstepping Control for Second-Order Hyperbolic PDEs
Target System
Lyapunov Stability Analysis
Full-State Feedback Design
2.6 Observer and Collocated Output-Feedback Design for Second-Order Hyperbolic PDEs
Boundary Sensing for State Estimation
Output Feedback Design
2.7 Notes and References
Part I Basic Backstepping Control of Freeway Traffic
3 Stabilization of ARZ Model
3.1 What Can Be Controlled and Is It Worth Controlling with Ramp Metering?
3.2 Stop-and-Go Instabilities
3.3 Boundary Control Model
DORM Control
UORM Control
Spectrum Analysis of Control Models with Zero Input
3.4 DORM Control Design
3.5 UORM Control Designs
UORM Full-State Feedback Control Design
UORM Anti-Collocated Boundary Observer Design
UORM Collocated Boundary Observer Design
UORM Output-Feedback Control Design
3.6 Numerical Simulation
3.7 Notes and References
4 Observer Validation on Freeway Data
4.1 Testing PDE Backstepping Observers on Real Freeway Data
4.2 Introduction to Traffic State Estimation
4.3 Boundary Observer Design
Output Injection for the Linearized ARZ Model
4.4 Nonlinear Observer
4.5 Numerical Simulation
4.6 Model Calibration with NGSIM Data
Data Reconstruction
Calibration of Model Parameters
4.7 Data Validation of Observer with Calibrated Parameters
4.8 Notes and References
5 Adaptive Control of ARZ Traffic Model
5.1 Parametric Uncertainties in the ARZ Model
5.2 Adaptive Control for PDEs Enabled by Backstepping
5.3 Adaptive Output Feedback: Simultaneous Identification, Observer, and Control Design
5.4 Validation of Adaptive Design: Stability Proof and Simulations
5.5 The ARZ PDE Model with Parameter Uncertainty
Scaling the States
Observer Canonical Form
5.6 Parametric Model and Parameter Estimation
5.7 Filter-Based Observer Design
5.8 Adaptive Output-Feedback Control Design
5.9 Lyapunov Stability Analysis
L2 Boundedness
Convergence
5.10 Numerical Simulation
5.11 Notes and References
6 Event-Triggered Control of ARZ Model
6.1 Event-Triggered Control and Its Role in Controlled Traffic
6.2 VSL Full-State Feedback Control Design
6.3 Event-Triggered Strategies for Boundary Control
Static Triggering Condition
Dynamic Triggering Condition
6.4 Absence of the Zeno Phenomenon
Static Triggering Condition
Dynamic Triggering Condition
6.5 Stability Results
Static Triggering Condition
Dynamic Triggering Condition
6.6 Numerical Simulations
6.7 Notes and References
7 Comparison of Backstepping with Reinforcement Learning
7.1 From (Model-Based) Adaptive Control to (Less Model-Based) Reinforcement Learning
Adaptive Backstepping
Relative Merits of RL and Adaptive Backstepping
Learning Characteristics of RL and Adaptive Backstepping
Reliance on Full-State Measurement
Is RL Learning the Backstepping Feedback Law?
Comparison of RL with Proportional-Integral Controllers
7.2 RL Control Approach
7.3 Boundary Control Problem Reformulation
ARZ PDE Traffic Model
Boundary Control Design
Setpoint Control
PDE Backstepping Control
P Control
PI Control
7.4 Control of ARZ Model by Reinforcement Learning
Boundary Control of PDE as a MDP
Value Function and Q-Function
Actor-Critic
Critic
Actor
Proximal Policy Optimization
7.5 Comparative Simulation Study
Simulation Configuration
Comparative Study with Full Knowledge of System Dynamics
Learning Process of RL Controllers
State Evolution, Reward, and Control Inputs
Other Performance Measures
Comparison Study with Partial Knowledge of System
Scenario 1 of Lighter In-Domain Traffic
Scenario 2 of Denser In-Domain Traffic
7.6 Notes and References
Comparative Assessment of RL and Backstepping
RL Versus Extremum Seeking
Possible Advances with RL
Code Availability
Part II Advanced Backstepping for Traffic Flows
8 Two-Lane Traffic Control
8.1 Modeling and Controlling Two Lanes: By Four PDEs and Two VSL Inputs
8.2 Two-Lane Traffic ARZ Model
Driver's Preference Over Two Lanes
VSL Control of Linearized Two-Lane ARZ Model
8.3 Full-State Feedback Control Design
8.4 Collocated Observer and Output-Feedback Control
Collocated Observer Design
Output-Feedback Controller
8.5 Numerical Simulation
Output-Feedback Stabilization and Performance
Different Traffic Scenarios, One-Lane Backstepping and PI Controllers
8.6 Notes and References
9 Two-Class Traffic Control
9.1 Diverse Driver and Vehicle Classes: Additional PDEs Controlled by a Single Input
9.2 Two-Class ARZ Traffic Model
Linearized Two-Class PDE Model
Free/Congested Regime Analysis of Two-Class Traffic
9.3 Boundary Control Design Model
9.4 Full-State Feedback Control
9.5 Anti-collocated Boundary Observer Design
9.6 Output-Feedback Control Design
9.7 Numerical Simulation
Performance Indices
9.8 Notes and References
10 Control of Two Cascaded Freeway Segments
10.1 Taking the ARZ Control Design Beyond a Single Freeway Segment
10.2 Possible Control Configurations for a Cascade of Freeway Segments
Macroscopic Modeling of a Cascade of Freeway Segments
Boundary Control of a Cascade of Freeway Segments
10.3 ARZ PDE Model of a Cascade of Freeway Segments
Actuated Boundary at Two Different Locations
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Linearized Model in the Riemann Coordinates
10.4 State Feedback Control Designs
Feedback Law U0(t) with Flow Rate Control from x=0
Feedback Law UL(t) with Flow Rate Control from x=L
10.5 Boundary Observer Designs
Observer with Measurement Y0(t) at x=0
Observer with Measurement at Outlet
10.6 Output-Feedback Laws
10.7 Robustness to Input Delays
10.8 Simulation Results
Output-Feedback Stabilization
Robustness to Delays
Comparison with PI Controllers
10.9 Notes and References
11 Estimation of Freeway Diverge Flows
11.1 Traffic Flow Estimation Beyond a Single Road Segment
11.2 PDE Model of One Incoming and Two Outgoing Roads
11.3 Linearized Model in the Riemann Coordinates
11.4 Boundary Observer Design
11.5 Robustness to Disturbance and Noise
11.6 Notes and References
12 Control Under Routing-Induced Instability
12.1 ARZ Model with Routing Feedback
12.2 Feedback Design for the Linearized System
12.3 Closed-Loop Stability
12.4 Existence of Solutions to Kernel Equations
12.5 Notes and References
13 Bilateral Regulation of Moving Shock Position
13.1 Delay-Compensating Predictors for PDE-ODE Models of Traffic Shock Movement
13.2 Moving Shockwave Model
13.3 State-Dependent PDE-ODE Model
13.4 Predictor-Based Control Design
From Coupled PDE-ODE to Delay System Representation
Predictor-Based Backstepping Transformation
13.5 Lyapunov Analysis
13.6 Numerical Simulation
13.7 Notes and References
14 Extremum Seeking for Flow Maximization at Downstream Bottleneck
14.1 Bottleneck: Unknown Fundamental Diagram and Maximizing the Flow
14.2 Lane-Drop Bottleneck Control Problem
Lane-Drop Bottleneck Model
Linearized Reference Error System
14.3 Online Optimization by Extremum Seeking Control
14.4 Stability Analysis (Averaging, Backstepping, and Lyapunov)
Closed-Loop System
Average System
Backstepping Transformation
Lyapunov Functional
Averaging Theorem
Asymptotic Convergence to a Neighborhood of the Extremum (ρ, q)
14.5 Numerical Simulation
14.6 Notes and References
Bibliography
Index
