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Configuration-Constrained Tube MPC for Tracking

Filippo Badalamenti, Sampath Kumar Mulagaleti, Alberto Bemporad, Boris Houska, Mario Eduardo Villanueva

TL;DR

This work addresses robust tracking for linear systems subject to additive and multiplicative uncertainties under time-varying references. It introduces a reference-tracking Configuration-Constrained Tube MPC (CCTMPC) that computes Robust Forward Invariant Tubes (RFITs) and an optimal invariant set via a single quadratic program, avoiding precomputation of terminal sets. The method handles LPV-like multiplicative uncertainty, removes the need for affine feedback in the tube, and guarantees recursive feasibility and Lyapunov stability for piecewise constant references, while allowing online shaping of the tube. Numerical examples, including a lane-change scenario for autonomous vehicles, demonstrate larger feasible regions and stable tracking compared to traditional rigid-tube approaches.

Abstract

This paper proposes a novel tube-based Model Predictive Control (MPC) framework for tracking varying setpoint references with linear systems subject to additive and multiplicative uncertainties. The MPC controllers designed using this framework exhibit recursively feasible for changing references, and robust asymptotic stability for piecewise constant references. The framework leverages configuration-constrained polytopes to parameterize the tubes, offering flexibility to optimize their shape. The efficacy of the approach is demonstrated through two numerical examples. The first example illustrates the theoretical results, and the second uses the framework to design a lane-change controller for an autonomous vehicle.

Configuration-Constrained Tube MPC for Tracking

TL;DR

This work addresses robust tracking for linear systems subject to additive and multiplicative uncertainties under time-varying references. It introduces a reference-tracking Configuration-Constrained Tube MPC (CCTMPC) that computes Robust Forward Invariant Tubes (RFITs) and an optimal invariant set via a single quadratic program, avoiding precomputation of terminal sets. The method handles LPV-like multiplicative uncertainty, removes the need for affine feedback in the tube, and guarantees recursive feasibility and Lyapunov stability for piecewise constant references, while allowing online shaping of the tube. Numerical examples, including a lane-change scenario for autonomous vehicles, demonstrate larger feasible regions and stable tracking compared to traditional rigid-tube approaches.

Abstract

This paper proposes a novel tube-based Model Predictive Control (MPC) framework for tracking varying setpoint references with linear systems subject to additive and multiplicative uncertainties. The MPC controllers designed using this framework exhibit recursively feasible for changing references, and robust asymptotic stability for piecewise constant references. The framework leverages configuration-constrained polytopes to parameterize the tubes, offering flexibility to optimize their shape. The efficacy of the approach is demonstrated through two numerical examples. The first example illustrates the theoretical results, and the second uses the framework to design a lane-change controller for an autonomous vehicle.
Paper Structure (16 sections, 3 theorems, 30 equations, 3 figures)

This paper contains 16 sections, 3 theorems, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Tube Model Predictive Control
  3. Uncertain Linear Systems
  4. Robust Forward Invariant Tubes
  5. Configuration-constrained polytopic tubes
  6. Optimal robust control invariant set
  7. Tube MPC with Stabilizing Initial and Terminal costs
  8. Tube MPC for reference tracking
  9. Reference-tracking CCTMPC
  10. Terminal ingredients design
  11. Closed-Loop Control Law
  12. Numerical Examples
  13. Illustrative example
  14. Lane change maneuver
  15. Computational aspects
  16. ...and 1 more sections

Key Result

Proposition 1

Let $(y_t\in\mathbb{R}^{m})_{t\in\mathbb{N}}$ be such that for all $t \in \mathbb{N}$, there exists some $u_t \in \mathbb{R}^{vn_u}$ such that $(y_t,u_t,y_{t+1}) \in \mathbb{S}$. Then, $(X(y_t))_{t\in\mathbb{N}}$ is an RFIT.

Figures (3)

  • Figure 1: Closed-loop results for System \ref{['eq:illustrative_system']}. Feasible regions of CCTMPC, RTMPC from Limon2010_tube, and ETMPC from RAKOVIC2013209, are compared. The reference is $r_t=5$ for $t \in [0,14]$ and $r_t=-5$ for $t\geq 15$. Closed-loop tubes and system trajectory are plotted for $x_0=(-3.12,2.95)$.
  • Figure 2: Closed-loop spatial evolution of \ref{['eq:lateral_model']}.
  • Figure 3: Lyapunov function $\mathcal{L}_t$ for system \ref{['eq:illustrative_system']} (left) and \ref{['eq:lateral_model']} (right) respectively. Changes in reference for the two systems are marked by vertical dashed lines.

Theorems & Definitions (7)

  • Proposition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Corollary 1
  • Remark 4