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Stable MPC with maximal terminal sets and quadratic terminal costs

Mikael Johansson, Hamed Taghavian

TL;DR

The paper tackles enlarging the feasible operating region of linear MPC with quadratic stage and terminal costs by anchoring the terminal penalty to the maximal control invariant set $\mathcal{C}_\infty$. It develops a three-step approach: compute $\mathcal{C}_\infty^\lambda$, recover an explicit vertex-based feedback, and bound the cost-to-go via a semidefinite program to obtain a quadratic terminal cost $P$ valid on the set. This yields a terminal set $\mathcal{X}_T = \mathcal{C}_\infty^\lambda$ with $Q_T = P$, ensuring asymptotic stability under the standard MPC framework and enabling stable operation with short horizons. Numerical examples demonstrate significantly larger operating regions, stability without terminal-set contractivity, and effective performance for higher-order systems.

Abstract

This paper develops a technique for computing a quadratic terminal cost for linear model predictive controllers that is valid for all states in the maximal control invariant set. This maximizes the set of recursively feasible states for the controller, ensures asymptotic stability using standard proofs, and allows for easy tuning of the controller in linear operation.

Stable MPC with maximal terminal sets and quadratic terminal costs

TL;DR

The paper tackles enlarging the feasible operating region of linear MPC with quadratic stage and terminal costs by anchoring the terminal penalty to the maximal control invariant set . It develops a three-step approach: compute , recover an explicit vertex-based feedback, and bound the cost-to-go via a semidefinite program to obtain a quadratic terminal cost valid on the set. This yields a terminal set with , ensuring asymptotic stability under the standard MPC framework and enabling stable operation with short horizons. Numerical examples demonstrate significantly larger operating regions, stability without terminal-set contractivity, and effective performance for higher-order systems.

Abstract

This paper develops a technique for computing a quadratic terminal cost for linear model predictive controllers that is valid for all states in the maximal control invariant set. This maximizes the set of recursively feasible states for the controller, ensures asymptotic stability using standard proofs, and allows for easy tuning of the controller in linear operation.
Paper Structure (13 sections, 3 theorems, 30 equations, 6 figures)

This paper contains 13 sections, 3 theorems, 30 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Linear Model Predictive Control
  3. Computing a quadratic terminal cost valid on the maximal invariant set
  4. Maximal control invariant and $\lambda$-contractive sets
  5. Recovering the implicit feedback policy
  6. A quadratic upper bound on the cost-to-go
  7. The complete algorithm
  8. Horizon length and linear performance
  9. Numerical examples
  10. Horizon length and performance
  11. Stability in absence of terminal set contractivity
  12. A higher-order system
  13. Conclusions

Key Result

Theorem 1

Consider the system (eqn:linsys) with constraints (eqn:csets) under the the RHC control law (eqn:planning_problem)-(eqn:rhc_control). Assume that $(A,B)$ is a reachable pair and let $\mathcal{X}_0$ be the set of states $x_t$ for which the planning problem (eqn:planning_problem) admits a feasible sol Then, every trajectory $\{x_t\}$ of the closed-loop system remains in $\mathcal{X}_0$ and $\lim_{t\

Figures (6)

  • Figure 1: Operating region $\mathcal{X}_0$ for MPC controller depending on the horizon length $T$. A small terminal set may force us to use an unnecessarily long horizon.
  • Figure 2: Triangulation of $\mathcal{C}_{\infty}$ (gray) and the associated piecewise linear control law (blue) for the system in Example \ref{['ex:horizon']}.
  • Figure 3: The MPC controller with the proposed terminal cost ensures a feasible and stable operation for all horizon lengths.
  • Figure 4: Partitions of explicit MPC policies for different $T$. Light colors indicate a local behavior close to $u_t=-L_{\infty}x_t$.
  • Figure 5: Maximal invariant set is much larger than the invariant set of the LQR controller.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Example 1
  • Definition 1
  • Definition 2
  • Theorem 2
  • Remark 1
  • Example 2
  • Theorem 3
  • Remark 2
  • Remark 3
  • ...and 2 more