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Exponential stability of data-driven nonlinear MPC based on input/output models

Lea Bold, Irene Schimperna, Karl Worthmann, Johannes Köhler

Abstract

We consider nonlinear model predictive control (MPC) schemes using surrogate models in the optimization step based on input-output data only. We establish exponential stability for sufficiently long prediction horizons assuming exponential stabilizability and a proportional error bound. Moreover, we verify the imposed condition on the approximation using kernel interpolation and demonstrate the practical applicability to nonlinear systems with a numerical example.

Exponential stability of data-driven nonlinear MPC based on input/output models

Abstract

We consider nonlinear model predictive control (MPC) schemes using surrogate models in the optimization step based on input-output data only. We establish exponential stability for sufficiently long prediction horizons assuming exponential stabilizability and a proportional error bound. Moreover, we verify the imposed condition on the approximation using kernel interpolation and demonstrate the practical applicability to nonlinear systems with a numerical example.
Paper Structure (6 sections, 4 theorems, 38 equations, 1 figure, 1 algorithm)

This paper contains 6 sections, 4 theorems, 38 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation and MPC algorithm
  3. Stability analysis
  4. Kernel interpolation: Data-driven models
  5. Numerical example
  6. Conclusions

Key Result

Proposition 1

(Cost detectability, adapted from kohler2021constrained) Consider the quadratic storage function $W(x)=\|x\|^2_{P}$, $P=\mathrm{diag}(Q,\dots,\frac{1}{\nu}Q,R,\dots, \frac{2}{\nu}R )$, satisfying For any $x\in {\Omega}, u\in\mathbb{U}$, $x^+ = F_\mathrm{x}(x,u)$ and $y^+ = F_\mathrm{y}(x,u)$ it holds that

Figures (1)

  • Figure 1: Trajectories $y(k)$ (left) and state errors $\|x(k) - \bar{x}\|$, $\bar{x} = [\bar{h}_1, \bar{h}_1, \bar{u}]^\top$, (right) of the data-driven MPC closed loop for $k \in [0:100]$, where the models are obtained with $D \in \{101, 2501\}$ data points.

Theorems & Definitions (6)

  • Proposition 1
  • Proposition 2
  • proof
  • Theorem 1
  • Theorem 2
  • proof