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An adaptive extension to robust data-driven predictive control under parametric uncertainty

Ignacio Sanchez, Filiberto Fele, Daniel Limon

TL;DR

This work tackles stabilizing time-varying linear systems whose parameters lie in a bounded polytope by blending offline vertex data with a rolling online data window. It uses the data informativity framework to design a Lyapunov-based controller via LMIs, avoiding strict persistent excitation requirements. The main contribution is an adaptive data-driven predictive controller that updates the optimization variables through a data-driven epigraph formulation, guaranteeing robust stability and providing an upper bound on the cost-to-go for online data-consistent systems. Numerical results on a classic LPV/LDI example show performance gains over purely robust designs, with fixed online computational complexity due to the rolling window.

Abstract

Robust data-driven controllers typically rely on datasets from previous experiments, which embed information on the variability of the system parameters across past operational conditions. Complementarily, data collected online can contribute to improving the feedback performance relative to the current system's conditions, but are unable to account for the overall -- possibly time-varying -- system operation. With this in mind, we consider the problem of stabilizing a time-varying linear system, whose parameters are only known to lie within a bounded polytopic set. Taking a robust data-driven approach, we synthesize the control law by simultaneously leveraging two sets of historical state and input measures: an offline dataset -- which covers the extreme variations of the system parameters -- and an online dataset consisting of a rolling window of the latest state and input samples. Our approach relies on the data informativity framework: we thus relax persistent excitation requirements (i.e., the collected samples need not be sufficient for system identification), while still allowing for the design of a stabilizing controller. The state feedback law is obtained from standard Lyapunov arguments, implemented via semi-definite optimization: this also yields an upper bound on the cost-to-go for the class of systems that are consistent with the online data, while guaranteeing a decreasing cost for all systems compatible with the offline data. Numerical experiments are presented to illustrate the effectiveness of the proposed controller.

An adaptive extension to robust data-driven predictive control under parametric uncertainty

TL;DR

This work tackles stabilizing time-varying linear systems whose parameters lie in a bounded polytope by blending offline vertex data with a rolling online data window. It uses the data informativity framework to design a Lyapunov-based controller via LMIs, avoiding strict persistent excitation requirements. The main contribution is an adaptive data-driven predictive controller that updates the optimization variables through a data-driven epigraph formulation, guaranteeing robust stability and providing an upper bound on the cost-to-go for online data-consistent systems. Numerical results on a classic LPV/LDI example show performance gains over purely robust designs, with fixed online computational complexity due to the rolling window.

Abstract

Robust data-driven controllers typically rely on datasets from previous experiments, which embed information on the variability of the system parameters across past operational conditions. Complementarily, data collected online can contribute to improving the feedback performance relative to the current system's conditions, but are unable to account for the overall -- possibly time-varying -- system operation. With this in mind, we consider the problem of stabilizing a time-varying linear system, whose parameters are only known to lie within a bounded polytopic set. Taking a robust data-driven approach, we synthesize the control law by simultaneously leveraging two sets of historical state and input measures: an offline dataset -- which covers the extreme variations of the system parameters -- and an online dataset consisting of a rolling window of the latest state and input samples. Our approach relies on the data informativity framework: we thus relax persistent excitation requirements (i.e., the collected samples need not be sufficient for system identification), while still allowing for the design of a stabilizing controller. The state feedback law is obtained from standard Lyapunov arguments, implemented via semi-definite optimization: this also yields an upper bound on the cost-to-go for the class of systems that are consistent with the online data, while guaranteeing a decreasing cost for all systems compatible with the offline data. Numerical experiments are presented to illustrate the effectiveness of the proposed controller.

Paper Structure

This paper contains 8 sections, 4 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Data-driven approach
  4. Data informativity for robust stabilization
  5. Data-consistent optimal control
  6. Adaptive data-driven control for systems under polytopic parametric uncertainty
  7. Numerical evaluation
  8. Conclusions

Key Result

Lemma 1

Consider symmetric matrices $M, N$, both in $\mathbb{R}^{(k+l)\times (k+l)}$, partitioned as Assume that i)$M_{12} = 0$ and $M_{22} \preceq 0$, ii)$N_{22} \preceq 0$ and $N_{11} - N_{12}N^{\dagger}_{22}N^T_{12} = 0$$N^{\dagger}$ denotes the Moore-Penrose pseudo-inverse of $N$., iii)$\exists G$ such that $M_{11} + G^T M_{22} G \succ 0$ and $N_{22}G = N^T_{12}$. Then if and only if there exists $\

Figures (2)

  • Figure 1: States and input trajectories for robust (blue) and adaptive (orange) controllers. During the first $T^o = 5$ time steps the robust controller is active for both simulations, with $\delta = 0.15$. At time 1.5 s, $\delta$ changes to $0.3$.
  • Figure 2: Normalized cost difference $\tfrac{J_R - J_A}{J_R}$ for $\delta = 0, 0.1, \ldots, 1$, with random initial values $0 \leq \|x_0\|_\infty \leq 1$.

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Lemma 1: Finsler's lemma
  • Lemma 2
  • proof
  • Definition 3
  • Lemma 3
  • proof
  • Remark 1
  • Theorem 1
  • ...and 1 more