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Transient Performance of MPC for Tracking

Matthias Köhler, Lisa Krügel, Lars Grüne, Matthias A. Müller, Frank Allgöwer

TL;DR

This work analyzes transient performance of nonlinear MPC for tracking with an artificial reference variable. It establishes stability and recursive feasibility for tracking flexible references, and derives a transient performance bound that depends on horizon length and offset-cost structure. A key finding is that scaling the offset term with the horizon is essential to recover the infinite-horizon optimal cost as the horizon grows, with turnpike properties ensuring invariant feasible behavior. A CSTR example illustrates that short-horizon MPC for tracking can closely approximate long-horizon performance while reducing computation, indicating practical viability for constrained nonlinear control problems.

Abstract

We analyse the closed-loop performance of a model predictive control (MPC) for tracking formulation with artificial references. It has been shown that such a scheme guarantees closed-loop stability and recursive feasibility for any externally supplied reference, even if it is unreachable or time-varying. The basic idea is to consider an artificial reference as an additional decision variable and to formulate generalised terminal ingredients with respect to it. In addition, its offset is penalised in the MPC optimisation problem, leading to closed-loop convergence to the best reachable reference. In this paper, we provide a transient performance bound on the closed loop using MPC for tracking. We employ mild assumptions on the offset cost and scale it with the prediction horizon. In this case, an increasing horizon in MPC for tracking recovers the infinite horizon optimal solution.

Transient Performance of MPC for Tracking

TL;DR

This work analyzes transient performance of nonlinear MPC for tracking with an artificial reference variable. It establishes stability and recursive feasibility for tracking flexible references, and derives a transient performance bound that depends on horizon length and offset-cost structure. A key finding is that scaling the offset term with the horizon is essential to recover the infinite-horizon optimal cost as the horizon grows, with turnpike properties ensuring invariant feasible behavior. A CSTR example illustrates that short-horizon MPC for tracking can closely approximate long-horizon performance while reducing computation, indicating practical viability for constrained nonlinear control problems.

Abstract

We analyse the closed-loop performance of a model predictive control (MPC) for tracking formulation with artificial references. It has been shown that such a scheme guarantees closed-loop stability and recursive feasibility for any externally supplied reference, even if it is unreachable or time-varying. The basic idea is to consider an artificial reference as an additional decision variable and to formulate generalised terminal ingredients with respect to it. In addition, its offset is penalised in the MPC optimisation problem, leading to closed-loop convergence to the best reachable reference. In this paper, we provide a transient performance bound on the closed loop using MPC for tracking. We employ mild assumptions on the offset cost and scale it with the prediction horizon. In this case, an increasing horizon in MPC for tracking recovers the infinite horizon optimal solution.
Paper Structure (9 sections, 12 theorems, 26 equations, 1 figure)

This paper contains 9 sections, 12 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem formulation
  4. MPC for tracking
  5. MPC for tracking scheme
  6. Exponential stability
  7. Transient performance estimate
  8. Asymptotic performance estimate
  9. Example: Continuous stirred-tank reactor

Key Result

Lemma 1

Suppose Assumptions asm:stage_cost_lower_and_upper_bound and asm:terminal_ingredients hold. Consider the following optimisation problem where $\hat{r}$ is a fixed parameter: $\min_{u \in \mathbb{U}^N_{\mathcal{X}^{\mathrm{f}}(\hat{r})}(x)} J_N^{\mathrm{tc}}(x, u, \hat{r})$. Then, there exists $\vare

Figures (1)

  • Figure 1: Closed-loop evolution of MPC for tracking with different horizons. The infinite horizon solution is dashed; the solution using standard MPC ($N=515$) is indistinguishable from it. The steady state manifold is dotted.

Theorems & Definitions (26)

  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Remark 2
  • Proposition 1
  • Lemma 3
  • ...and 16 more