Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Predictor-Feedback Stabilization of Linear Switched Systems with State-Dependent Switching and Input Delay

Andreas Katsanikakis, Nikolaos Bekiaris-Liberis, Delphine Bresch-Pietri

Abstract

We develop a predictor-feedback control design for a class of linear systems with state-dependent switching. The main ingredient of our design is a novel construction of an exact predictor state. Such a construction is possible as for a given, state-dependent switching rule, an implementable formula for the predictor state can be derived in a way analogous to the case of nonlinear systems with input delay. We establish uniform exponential stability of the corresponding closed-loop system via a novel construction of multiple Lyapunov functionals, relying on a backstepping transformation that we introduce. We validate our design in simulation considering a switching rule motivated by communication networks.

Predictor-Feedback Stabilization of Linear Switched Systems with State-Dependent Switching and Input Delay

Abstract

We develop a predictor-feedback control design for a class of linear systems with state-dependent switching. The main ingredient of our design is a novel construction of an exact predictor state. Such a construction is possible as for a given, state-dependent switching rule, an implementable formula for the predictor state can be derived in a way analogous to the case of nonlinear systems with input delay. We establish uniform exponential stability of the corresponding closed-loop system via a novel construction of multiple Lyapunov functionals, relying on a backstepping transformation that we introduce. We validate our design in simulation considering a switching rule motivated by communication networks.
Paper Structure (7 sections, 5 theorems, 53 equations, 3 figures)

This paper contains 7 sections, 5 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Control Design
  3. Switched Linear Systems With Input Delay and State-Dependent Switching
  4. Predictor-Feedback Control Design And Computation
  5. Stability Analysis
  6. Simulation Example
  7. Conclusions and Future Works

Key Result

Theorem III.1

Consider the closed-loop system (1.1) with the controller (controller). Under Assumptions assumption1 and delay-free as, for all $X_0\in\mathbb{R}^n$, $U_0\in L^2[-D,0]$ there exist positive constants $\rho,\xi$ such that the following holds

Figures (3)

  • Figure 1: Evolution of the switching signal $\sigma(X(t))$.
  • Figure 2: Evolution of state $X(t)$ and control input $U(t)$.
  • Figure 3: Phase portrait of the system trajectories along with the switching regions $\Omega_1$ (blue), $\Omega_2$ (red), and hysteresis band (gray).

Theorems & Definitions (10)

  • Remark II.3
  • Theorem III.1
  • Lemma III.2: backstepping transformation
  • proof
  • Lemma III.3: inverse backstepping transformation
  • proof
  • Lemma III.4: norm equivalency
  • proof
  • Lemma III.5: stability of the target system
  • proof