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Stabilization of Switched Affine Systems With Dwell-Time Constraint

Antonio Russo, Gian Paolo Incremona, Patrizio Colaneri

Abstract

This paper addresses the problem of stabilization of switched affine systems under dwell-time constraint, giving guarantees on the bound of the quadratic cost associated with the proposed state switching control law. Specifically, two switching rules are presented relying on the solution of differential Lyapunov inequalities and Lyapunov-Metzler inequalities, from which the stability conditions are expressed. The first one allows to regulate the state of linear switched systems to zero, whereas the second one is designed for switched affine systems proving practical stability of the origin. In both cases, the determination of a guaranteed cost associated with each control strategy is shown. In the cases of linear and affine systems, the existence of the solution for the Lyapunov-Metzler condition is discussed and guidelines for the selection of a solution ensuring suitable performance of the system evolution are provided. The theoretical results are finally assessed by means of three examples.

Stabilization of Switched Affine Systems With Dwell-Time Constraint

Abstract

This paper addresses the problem of stabilization of switched affine systems under dwell-time constraint, giving guarantees on the bound of the quadratic cost associated with the proposed state switching control law. Specifically, two switching rules are presented relying on the solution of differential Lyapunov inequalities and Lyapunov-Metzler inequalities, from which the stability conditions are expressed. The first one allows to regulate the state of linear switched systems to zero, whereas the second one is designed for switched affine systems proving practical stability of the origin. In both cases, the determination of a guaranteed cost associated with each control strategy is shown. In the cases of linear and affine systems, the existence of the solution for the Lyapunov-Metzler condition is discussed and guidelines for the selection of a solution ensuring suitable performance of the system evolution are provided. The theoretical results are finally assessed by means of three examples.

Paper Structure

This paper contains 16 sections, 5 theorems, 83 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. State of the art for switched affine systems
  3. Contributions with respect to the state of the art
  4. Structure of the paper
  5. Problem Setup
  6. dwell-time State-Feedback Switching Law for Linear Switched Systems
  7. Practical Stabilization for Switched Affine Systems
  8. Parameter tuning and limit cases
  9. Simplified solution to Lyapunov-Metzler inequalities
  10. Existence of a Hurwitz average system
  11. Existence and design of Metzler matrix
  12. Illustrative Examples
  13. Unstable subsystems
  14. Boost-boost converter
  15. Congestion control
  16. ...and 1 more sections

Key Result

Theorem 1

Consider the switched linear system eq:linear_sys and assume that there exist positive definite symmetric time-varying matrices ${P_i(t) \colon \mathbb{R}_{\geq 0} \to \mathbb{R}^{n\times n}}$, constant symmetric positive definite matrices $X_i \in \mathbb{R}^{n\times n}$ and $\Pi \in \mathcal{M}$ s with matrices $X_i$ being the solution of the Lyapunov-Metzler inequalities for any $i \in \Omega$

Figures (9)

  • Figure 1: Time-evolution of the state $(x_1(t),x_2(t))$ for the switched linear system.
  • Figure 2: Time-evolution of the state $(x_1(t),x_2(t))$ for the affine switched system.
  • Figure 3: Time-evolution of the switching signal $\sigma(t)$ for the affine switched system.
  • Figure 4: Boost-boost converter topology.
  • Figure 5: Time-evolution of the boost-boost converter states. Top left: current $x_1(t)$. Top right: first stage voltage $x_2(t)$. Bottom left: current $x_3(t)$. Bottom right: second stage voltage $x_4(t)$.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Remark 3.1: Applicability of the switching control law
  • Remark 3.2: Initialization of the switching signal
  • Corollary 2
  • Remark 3.3: Relation with Theorem \ref{['thm:differential_P']}
  • Remark 3.4: Average cost ultimate bound
  • Theorem 3
  • Remark 4.1: Initialization of the switching signal
  • Remark 4.2: Average cost ultimate bound
  • Remark 4.3: Interpretation of the switching law
  • ...and 2 more