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Data-Driven Robust Predictive Control with Interval Matrix Uncertainty Propagation

Renato Quartullo, Andrea Garulli, Mirko Leomanni

Abstract

This paper presents a new data-driven robust predictive control law, for linear systems affected by unknown-but-bounded process disturbances. A sequence of input-state data is used to construct a suitable uncertainty representation based on interval matrices. Then, the effect of uncertainty along the prediction horizon is bounded through an operator leveraging matrix zonotopes. This yields a tube that is exploited within a variable-horizon optimal control problem, to guarantee robust satisfaction of state and input constraints. The resulting data-driven predictive control scheme is shown to be recursively feasible and practically stable. A numerical example shows that the proposed approach compares favorably to existing methods based on zonotopic tubes and is competitive with an approach combining set-membership system identification and model-based predictive control.

Data-Driven Robust Predictive Control with Interval Matrix Uncertainty Propagation

Abstract

This paper presents a new data-driven robust predictive control law, for linear systems affected by unknown-but-bounded process disturbances. A sequence of input-state data is used to construct a suitable uncertainty representation based on interval matrices. Then, the effect of uncertainty along the prediction horizon is bounded through an operator leveraging matrix zonotopes. This yields a tube that is exploited within a variable-horizon optimal control problem, to guarantee robust satisfaction of state and input constraints. The resulting data-driven predictive control scheme is shown to be recursively feasible and practically stable. A numerical example shows that the proposed approach compares favorably to existing methods based on zonotopic tubes and is competitive with an approach combining set-membership system identification and model-based predictive control.
Paper Structure (14 sections, 7 theorems, 82 equations, 3 figures, 1 table)

This paper contains 14 sections, 7 theorems, 82 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Problem Formulation
  4. Data-driven Interval System Matrices
  5. Set-membership identification of Interval System Matrices
  6. Robust Predictive Control
  7. Data-driven Robust Predictive Control
  8. Numerical results
  9. Conclusions
  10. Appendix
  11. Auxiliary material
  12. Proof of Proposition \ref{['prop:overapprox']}
  13. Proof of Theorem \ref{['thm:RF']}
  14. Proof of Theorem \ref{['thm:convergence']}

Key Result

Proposition 1

Let $\mathcal{I} = C \oplus \left\llbracket \Delta \right\rrbracket \in \mathbb{R}^{l \times p}$ be an interval matrix and $\mathcal{M}=\langle M;\, G_1, \ldots, G_{n_g}\rangle\subseteq \mathbb{R}^{p \times q}$ be a matrix zonotope. Then where and $\{F^{(i)}\}_{i=1}^{lq} = \mathcal{E}(F)$.

Figures (3)

  • Figure 1: Feasible domain comparison.
  • Figure 2: Area of the feasible domains as a function of $\epsilon_W$. The curves show the average (solid lines), minimum and maximum values (dashed lines) over 50 runs.
  • Figure 3: Closed-loop trajectories of the state (top) and input (bottom), for different realization of process disturbance sequences. The set $\mathcal{X}_\infty$ is shown in red, while the state and input constraints are shown in green.

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2