Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Beyond inherent robustness: strong stability of MPC despite plant-model mismatch

Steven J. Kuntz, James B. Rawlings

Abstract

In this technical report, we establish the asymptotic stability of MPC under plant-model mismatch for problems where the origin remains a steady state despite mismatch. This class of problems includes, but is not limited to, inventory management, path-planning, and control of systems in deviation variables. Our results differ from prior results on the inherent robustness of MPC, which guarantee only convergence to a neighborhood of the origin, the size of which scales with the magnitude of the mismatch. For MPC with quadratic costs, continuous differentiability of the system dynamics is sufficient to demonstrate exponential stability of the closed-loop system despite mismatch. For MPC with general costs, a joint comparison function bound and scaling condition guarantee asymptotic stability despite mismatch. The results are illustrated in numerical simulations, including the classic upright pendulum problem. The tools developed to establish these results can address the stability of offset-free MPC, an open and interesting question in the MPC research literature.

Beyond inherent robustness: strong stability of MPC despite plant-model mismatch

Abstract

In this technical report, we establish the asymptotic stability of MPC under plant-model mismatch for problems where the origin remains a steady state despite mismatch. This class of problems includes, but is not limited to, inventory management, path-planning, and control of systems in deviation variables. Our results differ from prior results on the inherent robustness of MPC, which guarantee only convergence to a neighborhood of the origin, the size of which scales with the magnitude of the mismatch. For MPC with quadratic costs, continuous differentiability of the system dynamics is sufficient to demonstrate exponential stability of the closed-loop system despite mismatch. For MPC with general costs, a joint comparison function bound and scaling condition guarantee asymptotic stability despite mismatch. The results are illustrated in numerical simulations, including the classic upright pendulum problem. The tools developed to establish these results can address the stability of offset-free MPC, an open and interesting question in the MPC research literature.

Paper Structure

This paper contains 47 sections, 26 theorems, 141 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Nominal MPC and basic assumptions
  4. Motivating example
  5. Robust and strong stability
  6. Robust stability
  7. Strong stability
  8. Inherent robustness of MPC
  9. Stability of MPC despite mismatch
  10. Maximum Lyapunov increase
  11. Asymptotic case
  12. Exponential case
  13. Error bounds
  14. Model error bounds
  15. Suboptimal cost error bounds
  16. ...and 32 more sections

Key Result

Theorem 1

Suppose assum:contassum:consassum:stabilizabilityassum:posdef hold. Then, the following holds:

Figures (9)

  • Figure 1: Contours of the cost difference as a function of the initial state $x$ and the parameter $\theta$.
  • Figure 2: For (left) positive and (right) negative values of $\theta$, the (top) closed-loop trajectories with initial state $x=3$, and (bottom) cost differences as a function of $x$, along with the nominal values.
  • Figure 3: Contours of the cost difference for the MPC of \ref{['eq:plant:sqrt']}.
  • Figure 4: For (left) nonnegative and (right) nonpositive values of $\theta$, the (top) closed-loop trajectories for the MPC of \ref{['eq:plant:sqrt']} with initial state $x=2$, and (bottom) cost differences of the same MPC as a function of $x$.
  • Figure 5: Contours of the cost difference for the MPC of \ref{['eq:plant:sin']}.
  • ...and 4 more figures

Theorems & Definitions (57)

  • Remark 1
  • Theorem 1: Thm. 2.19 of rawlings:mayne:diehl:2020
  • Theorem 2
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Definition 1: Robust positive invariance
  • Definition 2: Robust stability
  • Definition 3: ISS/ISES Lyapunov function
  • ...and 47 more