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Robust contraction-based model predictive control for nonlinear systems

Marco Polver, Daniel Limon, Fabio Previdi, Antonio Ferramosca

TL;DR

This work introduces a robust contraction-based model predictive controller (MPC) for nonlinear, perturbed systems that achieves stability and recursive feasibility without terminal constraints and with an easily designable terminal cost. The approach leverages contractivity properties of stabilizable systems, using a contractive function $\\Gamma(x)$ and two sets of bound propagation sequences $\\mathcal{F}(j)$ and $\\mathcal{R}(j)$ to guarantee robustness under bounded disturbances. A two-stage optimization or a variant without integer decision variables selects a short, online horizon, ensuring stability via a robust ISpS-Lyapunov function $V_{N_p}^*(x(k),\\theta(k))$. Case studies on a perturbed nonholonomic system and a perturbed four-tank system demonstrate finite-time contraction, constraint satisfaction, and the potential to recycle standard terminal ingredients when available, highlighting practical impact for constrained nonlinear control with model mismatch or disturbances.

Abstract

Model Predictive Control (MPC) is a widely known control method that has proved to be particularly effective in multivariable and constrained control. Closed-loop stability and recursive feasibility can be guaranteed by employing accurate models in prediction and suitable terminal ingredients, i.e. the terminal cost function and the terminal constraint. Issues might arise in case of model mismatches or perturbed systems, as the state predictions could be inaccurate, and nonlinear systems for which the computation of the terminal ingredients can result challenging. In this manuscript, we exploit the properties of component-wise uniformly continuous and stabilizable systems to introduce a robust contraction-based MPC for the regulation of nonlinear perturbed systems, that employs an easy-to-design terminal cost function, does not make use of terminal constraints, and selects the shortest prediction horizon that guarantees the stability of the closed-loop system.

Robust contraction-based model predictive control for nonlinear systems

TL;DR

This work introduces a robust contraction-based model predictive controller (MPC) for nonlinear, perturbed systems that achieves stability and recursive feasibility without terminal constraints and with an easily designable terminal cost. The approach leverages contractivity properties of stabilizable systems, using a contractive function and two sets of bound propagation sequences and to guarantee robustness under bounded disturbances. A two-stage optimization or a variant without integer decision variables selects a short, online horizon, ensuring stability via a robust ISpS-Lyapunov function . Case studies on a perturbed nonholonomic system and a perturbed four-tank system demonstrate finite-time contraction, constraint satisfaction, and the potential to recycle standard terminal ingredients when available, highlighting practical impact for constrained nonlinear control with model mismatch or disturbances.

Abstract

Model Predictive Control (MPC) is a widely known control method that has proved to be particularly effective in multivariable and constrained control. Closed-loop stability and recursive feasibility can be guaranteed by employing accurate models in prediction and suitable terminal ingredients, i.e. the terminal cost function and the terminal constraint. Issues might arise in case of model mismatches or perturbed systems, as the state predictions could be inaccurate, and nonlinear systems for which the computation of the terminal ingredients can result challenging. In this manuscript, we exploit the properties of component-wise uniformly continuous and stabilizable systems to introduce a robust contraction-based MPC for the regulation of nonlinear perturbed systems, that employs an easy-to-design terminal cost function, does not make use of terminal constraints, and selects the shortest prediction horizon that guarantees the stability of the closed-loop system.

Paper Structure

This paper contains 32 sections, 7 theorems, 101 equations, 6 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Contractivity of stabilizable systems
  4. Problem formulation
  5. The proposed robust contraction-based MPC
  6. Robust design of the controller
  7. Controller formulation
  8. A robust contraction-based MPC formulation without integer decision variables
  9. Case studies
  10. Control of a perturbed nonholonomic system
  11. Computation of the sequences $\{\mathcal{F}(j)\}_{j \geq 0}$ and $\{\mathcal{R}(j)\}_{j \geq 0}$
  12. Computation of a robust controlled invariant set
  13. Choice of the contractive function and the maximum prediction horizon
  14. Controller settings
  15. Simulation results
  16. ...and 17 more sections

Key Result

Proposition 1

If the system eq:nonlinear_system is stabilizable, then for all the compact subsets $\mathcal{X}_\mathrm{sub} \subset \mathbb{R}^n$ and for all the positive definite functions $\Gamma(\cdot)$, there exist a contraction factor $\gamma \in (0,1)$ and a finite $N_p \in \mathbb{N}$ such that, for all $x where $\mathbf{u} \coloneqq \{ u(0),\hdots,u(q-1) \}$, is verified.

Figures (6)

  • Figure 1: Closed-loop state trajectories under our robust contraction-based MPC.
  • Figure 2: Control actions provided by our robust contraction-based MPC.
  • Figure 3: Evolution of the contractive function $\Gamma(x)$ during the simulations.
  • Figure 4: Closed-loop state trajectories under our robust contraction-based MPC.
  • Figure 5: Control actions provided by our robust contraction-based MPC.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Definition 1: Stabilizability
  • Proposition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1: Generic upper bound on the cost function value
  • proof
  • Lemma 2: Existence of an upper-bounding function for the cost function
  • proof
  • ...and 13 more