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Output-feedback adaptive model predictive control for ramp metering: a set-membership approach

Zhexian Li, Ketan Savla

TL;DR

An output-feedback adaptive approach to ramp metering that combines model predictive control with set-membership parameter and state estimation is introduced, and stability of the closed-loop system under the proposed controller with time-varying demand and few mainline measurements is illustrated.

Abstract

Ramp metering, which regulates the flow entering the freeway, is one of the most effective freeway traffic control methods. This paper introduces an output-feedback adaptive approach to ramp metering that combines model predictive control (MPC) with set-membership parameter and state estimation. The set-membership estimator is based on a mixed-monotone embedding of underlying traffic dynamics. The embedding is also used as the modeling basis for MPC optimization. For a freeway stretch with unknown parameters and partial measurement on the freeway mainline, we provide sufficient conditions on the control horizon, cost functions, terminal sets of MPC, and inflow demand at the ramps such that the queue lengths in the closed-loop system remain bounded. The sufficient condition on the demand matches the necessary condition, thereby proving maximal throughput under the proposed controller. The result is strengthened to input-to-state stability when model parameters and demand are known. The stability analysis is conducted for the case of constant demand and unbounded on-ramps. The closed-loop trajectory data generated by the proposed controller is shown to facilitate finite time estimation of free-flow model parameters, i.e., free-flow speed and turning ratios. Simulation results illustrate stability of the closed-loop system under the proposed controller with time-varying demand and few mainline measurements, for which the system becomes unstable under a well-known approach from the literature. This indicates that the proposed controller renders higher throughput than the well-known approach, possibly using more computing resources.

Output-feedback adaptive model predictive control for ramp metering: a set-membership approach

TL;DR

An output-feedback adaptive approach to ramp metering that combines model predictive control with set-membership parameter and state estimation is introduced, and stability of the closed-loop system under the proposed controller with time-varying demand and few mainline measurements is illustrated.

Abstract

Ramp metering, which regulates the flow entering the freeway, is one of the most effective freeway traffic control methods. This paper introduces an output-feedback adaptive approach to ramp metering that combines model predictive control (MPC) with set-membership parameter and state estimation. The set-membership estimator is based on a mixed-monotone embedding of underlying traffic dynamics. The embedding is also used as the modeling basis for MPC optimization. For a freeway stretch with unknown parameters and partial measurement on the freeway mainline, we provide sufficient conditions on the control horizon, cost functions, terminal sets of MPC, and inflow demand at the ramps such that the queue lengths in the closed-loop system remain bounded. The sufficient condition on the demand matches the necessary condition, thereby proving maximal throughput under the proposed controller. The result is strengthened to input-to-state stability when model parameters and demand are known. The stability analysis is conducted for the case of constant demand and unbounded on-ramps. The closed-loop trajectory data generated by the proposed controller is shown to facilitate finite time estimation of free-flow model parameters, i.e., free-flow speed and turning ratios. Simulation results illustrate stability of the closed-loop system under the proposed controller with time-varying demand and few mainline measurements, for which the system becomes unstable under a well-known approach from the literature. This indicates that the proposed controller renders higher throughput than the well-known approach, possibly using more computing resources.
Paper Structure (33 sections, 7 theorems, 38 equations, 10 figures, 2 tables)

This paper contains 33 sections, 7 theorems, 38 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Model Predictive Control
  4. Ramp metering
  5. Problem formulation
  6. Traffic flow dynamics
  7. Equilibrium analysis
  8. Output-feedback adaptive control
  9. The Set-PC controller
  10. General framework
  11. Mixed-monotone embedding
  12. State, demand, and parameter estimation
  13. MPC control law
  14. Properties of the Set-PC controller
  15. Guaranteed set-membership estimation
  16. ...and 18 more sections

Key Result

Proposition 1

The mixed-monotone mapping eq:mixed-monotone-decomposition-function satisfies properties $(\emph{M1})-(\emph{M3})$ for dynamics $f$ in eq:cell-mass-balance if $f^{\text{out}}_{I+i}(x,u,\lambda;\theta)=u_{i}, i\in[I]$.

Figures (10)

  • Figure 1: For a freeway stretch in (a) with complete parameter and state information, the blue region in (b) represents the throughput region of ALINEA papageorgiou1991alinea estimated from simulations. The region combining blue and orange is the throughput region of the proposed Set-PC controller. This combined region is maximal in the sense that no controller can stabilize the freeway under demands outside the region.
  • Figure 2: A freeway stretch
  • Figure 3: Illustrations of the parameters of the fundamental diagram
  • Figure 4: Set-PC Controller
  • Figure 5: Traffic state evolution under the Set-PC controller, for unknown model parameters. The numbering in the legends are the indices of the cells from which measurements are available.
  • ...and 5 more figures

Theorems & Definitions (29)

  • Remark 1
  • Remark 2
  • Remark 3
  • Example 1
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9
  • ...and 19 more