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Data-Driven Min-Max MPC for LPV Systems with Unknown Scheduling Signal

Yifan Xie, Julian Berberich, Felix Brändle, Frank Allgöwer

TL;DR

This paper develops a novel data-driven characterization of the consistent system matrices using only input-state data and minimizes a tractable upper bound on the worst-case cost over the consistent system matrices set and over all scheduling signals satisfying the QMI.

Abstract

This paper presents a data-driven min-max model predictive control (MPC) scheme for linear parameter-varying (LPV) systems. Contrary to existing data-driven LPV control approaches, we assume that the scheduling signal is unknown during offline data collection and online system operation. Assuming a quadratic matrix inequality (QMI) description for the scheduling signal, we develop a novel data-driven characterization of the consistent system matrices using only input-state data. The proposed data-driven min-max MPC minimizes a tractable upper bound on the worst-case cost over the consistent system matrices set and over all scheduling signals satisfying the QMI. The proposed approach guarantees recursive feasibility, closed-loop exponential stability and constraint satisfaction if it is feasible at the initial time. We demonstrate the effectiveness of the proposed method in simulation.

Data-Driven Min-Max MPC for LPV Systems with Unknown Scheduling Signal

TL;DR

This paper develops a novel data-driven characterization of the consistent system matrices using only input-state data and minimizes a tractable upper bound on the worst-case cost over the consistent system matrices set and over all scheduling signals satisfying the QMI.

Abstract

This paper presents a data-driven min-max model predictive control (MPC) scheme for linear parameter-varying (LPV) systems. Contrary to existing data-driven LPV control approaches, we assume that the scheduling signal is unknown during offline data collection and online system operation. Assuming a quadratic matrix inequality (QMI) description for the scheduling signal, we develop a novel data-driven characterization of the consistent system matrices using only input-state data. The proposed data-driven min-max MPC minimizes a tractable upper bound on the worst-case cost over the consistent system matrices set and over all scheduling signals satisfying the QMI. The proposed approach guarantees recursive feasibility, closed-loop exponential stability and constraint satisfaction if it is feasible at the initial time. We demonstrate the effectiveness of the proposed method in simulation.

Paper Structure

This paper contains 8 sections, 4 theorems, 36 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Data-Driven System Characterization
  4. Data-Driven Min-Max MPC for LPV systems
  5. Data-driven min-max MPC problem
  6. Closed-loop guarantees
  7. Simulation
  8. Conclusion

Key Result

Lemma 1

Suppose $E\in\mathbb{R}^{n_E \times n_x}$ has full row rank and $\tilde{G}=$ satisfies $\tilde{G}_{22}\prec0$ and $\tilde{G}_{11}-\tilde{G}_{12} \tilde{G}_{22}^{-1}\tilde{G}_{12}^\top \succ 0$, then the following two sets are equal with

Figures (1)

  • Figure 1: Closed-loop cost over $100$ iterations under the proposed data-driven min-max MPC scheme with different values of the bound $c$ (subfigure (a)) and the data length $T$ (subfigure (b)).

Theorems & Definitions (12)

  • Remark 1
  • Example 1
  • Lemma 1
  • proof
  • Lemma 2: Data-driven characterization of $\Sigma$
  • proof
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 2 more