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Flexible-step MPC for Switched Linear Systems with No Quadratic Common Lyapunov Function

Annika Fürnsinn, Christian Ebenbauer, Bahman Gharesifard

TL;DR

This work addresses stabilizing switched linear systems within flexible-step MPC when a quadratic common Lyapunov function may not exist. It develops an LMI-based method to certify a generalized discrete-time Lyapunov function (g-dclf) whose averaged descent constraint guarantees stability. The key contributions include a tractable LMI formulation for linear systems, a proof that a large horizon $m$ yields feasibility, and a numerical demonstration of a quadratic common g-dclf for a switched system that enables flexible-step MPC to outperform standard MPC with non-quadratic costs or absent common CLFs. The findings extend MPC applicability to switched and nonholonomic-like linear systems and suggest practical potential in robotics and power systems.

Abstract

In this paper, we develop a systematic method for constructing a generalized discrete-time control Lyapunov function for the flexible-step Model Predictive Control (MPC) scheme, recently introduced in [2], when restricted to the class of linear systems. Specifically, we show that a set of Linear Matrix Inequalities (LMIs) can be used for this purpose, demonstrating its tractability. The main consequence of this LMI formulation is that, when combined with flexible-step MPC, we can effectively stabilize switched control systems, for which no quadratic common Lyapunov function exists.

Flexible-step MPC for Switched Linear Systems with No Quadratic Common Lyapunov Function

TL;DR

This work addresses stabilizing switched linear systems within flexible-step MPC when a quadratic common Lyapunov function may not exist. It develops an LMI-based method to certify a generalized discrete-time Lyapunov function (g-dclf) whose averaged descent constraint guarantees stability. The key contributions include a tractable LMI formulation for linear systems, a proof that a large horizon yields feasibility, and a numerical demonstration of a quadratic common g-dclf for a switched system that enables flexible-step MPC to outperform standard MPC with non-quadratic costs or absent common CLFs. The findings extend MPC applicability to switched and nonholonomic-like linear systems and suggest practical potential in robotics and power systems.

Abstract

In this paper, we develop a systematic method for constructing a generalized discrete-time control Lyapunov function for the flexible-step Model Predictive Control (MPC) scheme, recently introduced in [2], when restricted to the class of linear systems. Specifically, we show that a set of Linear Matrix Inequalities (LMIs) can be used for this purpose, demonstrating its tractability. The main consequence of this LMI formulation is that, when combined with flexible-step MPC, we can effectively stabilize switched control systems, for which no quadratic common Lyapunov function exists.
Paper Structure (7 sections, 4 theorems, 39 equations, 6 figures, 1 algorithm)

This paper contains 7 sections, 4 theorems, 39 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Flexible-step MPC
  4. g-dclf s for linear systems
  5. Numerical Example
  6. Quadratic Common g-dclf s
  7. Conclusion

Key Result

Theorem 3.1

Consider system eqn:linsys and a stabilizing matrix $K \in \mathbb{R}^{p,n}$, i.e. the absolute value of each eigenvalue of $A+BK$ is strictly less than one. Then there exists a sufficiently large $m \in \mathbb{N}_{>0}$ such that $\sigma_1, \dots, \sigma_m \in \mathbb{R}_{\geq 0}$ satisfy the LMI d

Figures (6)

  • Figure 1: Illustration of proposed MPC scheme: Consider Problem \ref{['eq:display']} with $m=4$ and $q=0$. The initial state is ${\color{DarkBlue}x^0}=x(0)$, whose Lyapunov function value is depicted in (a) and highlighted in green. After solving the finite-horizon optimal control problem, we obtain four predicted states and their corresponding Lyapunov function values ($V(x^0),V(x^1),V(x^2),V(x^3),V(x^4)$). Since there are multiple time indices for which the Lyapunov function decreases, we choose $\ell_{\textup{decr}}$ here as the index where the greatest descent occurs. We implement $\ell_{\textup{decr}}=2$ components of the control sequence and, consequently, declare $x^2$ as the new initial state for the finite-horizon optimal control problem at time $k=2$. This problem is solved in (b) and we obtain again four states and their corresponding Lyapunov function values ($V(x^0),V(x^1),V(x^2),V(x^3),V(x^4)$). We repeat the scheme and after solving Problem \ref{['eq:display']} three times, we obtain the trajectory of the closed-loop states and their Lyapunov function values shown in (d).
  • Figure 2: State trajectories according to the solution of Problem \ref{['problem:simulation1']}
  • Figure 3: In each optimization instance, Problem \ref{['problem:simulation1']} is solved. The solution yields the number of implemented steps, depicted on the vertical axis. Note that the sum of implemented steps of the optimization instances corresponds to the time step of the implementation.
  • Figure 4: State trajectories according to the solution of Problem \ref{['problem:simulation']}
  • Figure 5: In each optimization instance, Problem \ref{['problem:simulation']} is solved. The solution yields the number of implemented steps, depicted on the vertical axis. Note that the sum of implemented steps of the optimization instances corresponds to the time step of the implementation.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Definition 2.1: Set of Feasible Controls
  • Definition 2.2: g-dclf
  • Theorem 3.1
  • proof
  • Definition 4.1
  • Definition 4.2
  • Theorem 4.3
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • ...and 1 more