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Robust Model Predictive Control for Linear Systems with Interval Matrix Model Uncertainty

Renato Quartullo, Andrea Garulli, Mirko Leomanni

TL;DR

Numerical simulations demonstrate that the proposed MPC scheme is able to match the feasibility regions of the most effective state-of-the-art methods, while significantly reducing the computational burden, thereby enabling the treatment of nontrivial dimensional systems with multiple uncertain parameters.

Abstract

This paper proposes a novel robust Model Predictive Control (MPC) scheme for linear discrete-time systems affected by model uncertainty described by interval matrices. The key feature of the proposed method is a bound on the uncertainty propagation along the prediction horizon which exploits a set-theoretic over-approximation of each term of the uncertain system impulse response. Such an approximation is based on matrix zonotopes and leverages the interval matrix structure of the uncertainty model. Its main advantage is that all the relevant bounds are computed offline, thus making the online computational load independent of the number of uncertain parameters. A variable-horizon MPC formulation is adopted to guarantee recursive feasibility and to ensure robust asymptotic stability of the closed-loop system. Numerical simulations demonstrate that the proposed approach is able to match the feasibility regions of the most effective state-of-the-art methods, while significantly reducing the computational burden, thereby enabling the treatment of nontrivial dimensional systems with multiple uncertain parameters.

Robust Model Predictive Control for Linear Systems with Interval Matrix Model Uncertainty

TL;DR

Numerical simulations demonstrate that the proposed MPC scheme is able to match the feasibility regions of the most effective state-of-the-art methods, while significantly reducing the computational burden, thereby enabling the treatment of nontrivial dimensional systems with multiple uncertain parameters.

Abstract

This paper proposes a novel robust Model Predictive Control (MPC) scheme for linear discrete-time systems affected by model uncertainty described by interval matrices. The key feature of the proposed method is a bound on the uncertainty propagation along the prediction horizon which exploits a set-theoretic over-approximation of each term of the uncertain system impulse response. Such an approximation is based on matrix zonotopes and leverages the interval matrix structure of the uncertainty model. Its main advantage is that all the relevant bounds are computed offline, thus making the online computational load independent of the number of uncertain parameters. A variable-horizon MPC formulation is adopted to guarantee recursive feasibility and to ensure robust asymptotic stability of the closed-loop system. Numerical simulations demonstrate that the proposed approach is able to match the feasibility regions of the most effective state-of-the-art methods, while significantly reducing the computational burden, thereby enabling the treatment of nontrivial dimensional systems with multiple uncertain parameters.
Paper Structure (15 sections, 9 theorems, 75 equations, 4 figures)

This paper contains 15 sections, 9 theorems, 75 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Notation and set operations
  4. Set approximations
  5. Problem Formulation
  6. Robust MPC with Interval Matrices
  7. Numerical results
  8. Feasible domain evaluation
  9. Conservatism evaluation
  10. Runtime evaluation
  11. Conclusions
  12. Appendix
  13. Auxiliary lemmas
  14. Proof of Theorem \ref{['thm:RF']}
  15. Proof of Theorem \ref{['thm:convergence']}

Key Result

Proposition 1

The minimum interval matrix containing a matrix zonotope $\mathcal{M} = \langle M;\, G_1, \ldots, G_g \rangle$, denoted by $\mathbb{B}\left(\mathcal{M}\right)$, is given by $\mathbb{B}\left(\mathcal{M}\right) = M \oplus \left\llbracket \Delta \right\rrbracket$, where $\Delta = \sum_{i=1}^g |G_i|.$

Figures (4)

  • Figure 1: Feasible domain comparison for the double integrator example.
  • Figure 2: Coverage of the feasible domain for different uncertainty bounds $\delta$.
  • Figure 3: Average computation time required to solve the robust optimal control problem (left) and corresponding number of constraints (right, logarithmic scale) as functions of the number of uncertain entries in the matrix $\left[A \;\; B\right]$.
  • Figure 4: Closed-loop trajectories of states $x_1$ and $x_6$ (top) and control input (bottom), generated by IM-MPC, for different realizations of $\left[A\;\;B\right] \in \mathcal{I}_S$. Dashed lines represent constraints (green) and reference values (black).

Theorems & Definitions (16)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Proposition 2
  • Definition 3
  • Proposition 3
  • Definition 4
  • Proposition 4
  • Remark 1
  • Proposition 5
  • ...and 6 more