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Switched Systems Control via Discreteness-Promoting Regularization

Masaaki Nagahara, Takuya Ikeda, Ritsuki Hoshimoto

TL;DR

The paper addresses the challenge of designing finite-horizon discrete-valued switching signals for linear switched systems by replacing the discrete control with a continuous, non-convex optimization augmented with a discreteness-promoting regularizer $\Omega(u)=\int_0^T \psi(u(t))\,dt$. It proves that, under mild assumptions on $\psi$, any optimal solution to the relaxed problem is also optimal for the original discrete problem, and solves the relaxed problem via time discretization and DC programming. The approach is demonstrated on unstable two- and three-mode systems, where the computed switching signals stabilize the states and can be embedded in an MPC framework for feedback control. The work offers a scalable, regularization-based relaxation that integrates with standard constrained optimal control tools and enables real-time switching signal design for applications in engineering systems.

Abstract

This paper proposes a novel method for designing finite-horizon discrete-valued switching signals in linear switched systems based on discreteness-promoting regularization. The inherent combinatorial optimization problem is reformulated as a continuous optimization problem with a non-convex regularization term that promotes discreteness of the control. We prove that any solution obtained from the relaxed problem is also a solution to the original problem. The resulting non-convex optimization problem is efficiently solved through time discretization. Numerical examples demonstrate the effectiveness of the proposed method.

Switched Systems Control via Discreteness-Promoting Regularization

TL;DR

The paper addresses the challenge of designing finite-horizon discrete-valued switching signals for linear switched systems by replacing the discrete control with a continuous, non-convex optimization augmented with a discreteness-promoting regularizer . It proves that, under mild assumptions on , any optimal solution to the relaxed problem is also optimal for the original discrete problem, and solves the relaxed problem via time discretization and DC programming. The approach is demonstrated on unstable two- and three-mode systems, where the computed switching signals stabilize the states and can be embedded in an MPC framework for feedback control. The work offers a scalable, regularization-based relaxation that integrates with standard constrained optimal control tools and enables real-time switching signal design for applications in engineering systems.

Abstract

This paper proposes a novel method for designing finite-horizon discrete-valued switching signals in linear switched systems based on discreteness-promoting regularization. The inherent combinatorial optimization problem is reformulated as a continuous optimization problem with a non-convex regularization term that promotes discreteness of the control. We prove that any solution obtained from the relaxed problem is also a solution to the original problem. The resulting non-convex optimization problem is efficiently solved through time discretization. Numerical examples demonstrate the effectiveness of the proposed method.
Paper Structure (11 sections, 3 theorems, 40 equations, 7 figures)

This paper contains 11 sections, 3 theorems, 40 equations, 7 figures.

Table of Contents

  1. INTRODUCTION
  2. MATHEMATICAL FORMULATION
  3. RELAXATION TO CONTINUOUS OPTIMIZATION
  4. Relaxed Problem
  5. Functions that promote discreteness
  6. Validity theorem for the relaxed problem
  7. NUMERICAL COMPUTATION
  8. NUMERICAL EXAMPLES
  9. Example 1: Two-mode system
  10. Example 2: Three-mode system
  11. CONCLUSION

Key Result

Lemma 1

Suppose Assumption ass:psi holds, and consider the following optimization problem: where $\lambda>0$, and $\rho_1, \rho_2, \dots, \rho_N$ are given real numbers satisfying for some $m\in\{1,2,\dots,N\}$. Then, any optimal solution $u^\ast$ to problem prob:lemma satisfies $u^\ast\in\{e_1,\dots,e_m\}$, where $e_i$ is the $i$th canonical vector in $\mathbb{R}^N$.

Figures (7)

  • Figure 1: The graph of $\psi_i(u)=u(1-u)$.
  • Figure 2: Control signals $u_1(t)$ and $u_2(t)$.
  • Figure 3: The states $x_1(t)$ and $x_2(t)$.
  • Figure 4: The state trajectory $(x_1(t),x_2(t))$ in the state space $\mathbb{R}^2$.
  • Figure 5: Control signals $u_1(t)$, $u_2(t)$, and $u_3(t)$.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof