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Adaptive Data-Driven Min-Max MPC for Linear Time-Varying Systems

Yifan Xie, Julian Berberich, Frank Allgöwer

TL;DR

This paper proposes an adaptive data-driven min-max model predictive control scheme for discrete-time linear time-varying (LTV) systems and proves that the resulting closed-loop system is exponentially stabilized and satisfies the constraints.

Abstract

In this paper, we propose an adaptive data-driven min-max model predictive control (MPC) scheme for discrete-time linear time-varying (LTV) systems. We assume that prior knowledge of the system dynamics and bounds on the variations are known, and that the states are measured online. Starting from an initial state-feedback gain derived from prior knowledge, the algorithm updates the state-feedback gain using online input-state data. To this end, a semidefinite program (SDP) is solved to minimize an upper bound on the infinite-horizon optimal cost and to derive a corresponding state-feedback gain. We prove that the resulting closed-loop system is exponentially stabilized and satisfies the constraints. Further, we extend the proposed scheme to LTV systems with process noise. The resulting closed-loop system is shown to be robustly stabilized to a robust positive invariant (RPI) set. Finally, the proposed methods are demonstrated by numerical simulations.

Adaptive Data-Driven Min-Max MPC for Linear Time-Varying Systems

TL;DR

This paper proposes an adaptive data-driven min-max model predictive control scheme for discrete-time linear time-varying (LTV) systems and proves that the resulting closed-loop system is exponentially stabilized and satisfies the constraints.

Abstract

In this paper, we propose an adaptive data-driven min-max model predictive control (MPC) scheme for discrete-time linear time-varying (LTV) systems. We assume that prior knowledge of the system dynamics and bounds on the variations are known, and that the states are measured online. Starting from an initial state-feedback gain derived from prior knowledge, the algorithm updates the state-feedback gain using online input-state data. To this end, a semidefinite program (SDP) is solved to minimize an upper bound on the infinite-horizon optimal cost and to derive a corresponding state-feedback gain. We prove that the resulting closed-loop system is exponentially stabilized and satisfies the constraints. Further, we extend the proposed scheme to LTV systems with process noise. The resulting closed-loop system is shown to be robustly stabilized to a robust positive invariant (RPI) set. Finally, the proposed methods are demonstrated by numerical simulations.
Paper Structure (12 sections, 7 theorems, 88 equations, 5 figures, 1 algorithm)

This paper contains 12 sections, 7 theorems, 88 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Data-Driven Characterization using Online Data
  4. Adaptive Data-Driven Min-Max MPC Scheme
  5. Controller Design Using Prior Information
  6. Adaptive Data-Driven Min-Max MPC
  7. Closed-loop Guarantees
  8. Adaptive data-driven min-max MPC scheme for noisy systems
  9. Simulations
  10. Lipschitz continuous dynamics
  11. Periodic System
  12. Conclusion

Key Result

Lemma 1

Suppose Assumption assumption2 holds, and $\neq 0$ for all $i\in\mathbb{I}_{[1, t]}$. Then, the set $S_t$ is equal to where and $N_i$ is defined as in Ni.

Figures (5)

  • Figure 1: Closed loop state and input trajectories under the proposed adaptive data-driven min-max MPC scheme and the static state-feedback control $F_p^\star$ for system \ref{['system_simulation']} .
  • Figure 2: Closed loop state and input trajectories under the proposed adaptive data-driven min-max MPC scheme and the static state-feedback control $F_p^\star$ for system \ref{['system_simulation']} with process noise.
  • Figure 3: Closed loop state and input trajectories under the proposed adaptive data-driven min-max MPC scheme when the SDP \ref{['sdp_initial']} is initially infeasible.
  • Figure 4: Closed loop state and input trajectories under the proposed adaptive data-driven min-max MPC scheme and the static state-feedback control $F_p^\star$ for system in Example \ref{['example1']}.
  • Figure 5: Closed loop state and input trajectories under the proposed adaptive data-driven min-max MPC scheme and the static state-feedback control $F_p^\star$ for system in Example \ref{['example1']} with process noise.

Theorems & Definitions (28)

  • Remark 1
  • Remark 2
  • Remark 3
  • Example 1
  • Lemma 1
  • proof
  • Remark 4
  • Theorem 1
  • proof
  • Remark 5
  • ...and 18 more