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Adaptive tracking MPC for nonlinear systems via online linear system identification

Tatiana Strelnikova, Johannes Köhler, Julian Berberich

TL;DR

This work tackles controlling unknown nonlinear discrete-time systems by marrying online moving-window least-squares identification with an adaptive tracking MPC that uses a locally identified affine model for prediction. The method yields a practical exponential stability guarantee for the optimal reachable equilibrium via a Lyapunov-based analysis, even in the presence of identification error, and it demonstrates competitive performance against both model-based and direct data-driven MPC on a nonlinear CSTR. The key contribution is showing that indirect data-driven MPC can inherit similar guarantees to direct data-driven schemes while offering simpler tuning and reduced computational scaling with data length, at the cost of requiring state measurements. The results underscore the viability of online system identification to enable reliable, constrained MPC for unknown nonlinear systems, with practical implications for real-time control of complex processes.

Abstract

This paper presents an adaptive tracking model predictive control (MPC) scheme to control unknown nonlinear systems based on an adaptively estimated linear model. The model is determined based on linear system identification using a moving window of past measurements, and it serves as a local approximation of the underlying nonlinear dynamics. We prove that the presented scheme ensures practical exponential stability of the (unknown) optimal reachable equilibrium for a given output setpoint. Finally, we apply the proposed scheme in simulation and compare it to an alternative direct data-driven MPC scheme based on the Fundamental Lemma.

Adaptive tracking MPC for nonlinear systems via online linear system identification

TL;DR

This work tackles controlling unknown nonlinear discrete-time systems by marrying online moving-window least-squares identification with an adaptive tracking MPC that uses a locally identified affine model for prediction. The method yields a practical exponential stability guarantee for the optimal reachable equilibrium via a Lyapunov-based analysis, even in the presence of identification error, and it demonstrates competitive performance against both model-based and direct data-driven MPC on a nonlinear CSTR. The key contribution is showing that indirect data-driven MPC can inherit similar guarantees to direct data-driven schemes while offering simpler tuning and reduced computational scaling with data length, at the cost of requiring state measurements. The results underscore the viability of online system identification to enable reliable, constrained MPC for unknown nonlinear systems, with practical implications for real-time control of complex processes.

Abstract

This paper presents an adaptive tracking model predictive control (MPC) scheme to control unknown nonlinear systems based on an adaptively estimated linear model. The model is determined based on linear system identification using a moving window of past measurements, and it serves as a local approximation of the underlying nonlinear dynamics. We prove that the presented scheme ensures practical exponential stability of the (unknown) optimal reachable equilibrium for a given output setpoint. Finally, we apply the proposed scheme in simulation and compare it to an alternative direct data-driven MPC scheme based on the Fundamental Lemma.
Paper Structure (11 sections, 1 theorem, 33 equations, 2 figures, 1 algorithm)

This paper contains 11 sections, 1 theorem, 33 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem setup
  4. System identification
  5. Adaptive tracking MPC for nonlinear systems via linear system identification
  6. Theoretical analysis
  7. Technical assumptions and definitions
  8. Practical Exponential Stability
  9. Numerical Example
  10. Conclusion
  11. Acknowledgment

Key Result

theorem thmcountertheorem

Suppose $t \geq N \geq n$ and Assumptions ass:UniqueSS -- ass:PE hold. Then, there exists $\bar{\varepsilon}>0$ such that, for any $\varepsilon\in[0,\bar{\varepsilon}]$, there exist $V_{\max}, s_u , C>0$ and $0 < c_V < 1$, $\beta \in \mathcal{K}_{\infty}$, such that, if then, for any $t= N + ni$, $i \geq 0$, the problem eq:myMPC is feasible and the closed loop under Algorithm alg:MPC satisfies

Figures (2)

  • Figure 1: Closed-loop simulations under the MPC schemes from berberich2022linear1 (model-based MPC) and berberich2022linear2 (direct data-driven MPC) as well as the approach proposed in the present paper (indirect data-driven MPC).
  • Figure 2: Error \ref{['eq:error']} for indirect data-driven MPC (proposed approach).

Theorems & Definitions (2)

  • theorem thmcountertheorem
  • proof