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Average Predictor-Feedback Control Design for Switched Linear Systems

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

TL;DR

This work addresses stabilizing switched linear systems with input delay when future switching is unknown. It introduces an average predictor-based feedback law using mean system matrices $\bar{A}$ and $\bar{B}$ and a fixed gain $\bar{K}$, analyzed via a backstepping-based Lyapunov functional to bound the predictor mismatch. The main result shows global uniform exponential stability for parameter deviations $\epsilon<\epsilon^*$, where $\epsilon^*$ decreases with larger delay $D$ and shorter dwell times. Numerical simulations corroborate the approach, confirming stabilization and highlighting the superiority of the average predictor over single-mode predictors, while also noting conservatism in the bound for larger mismatches. Overall, the method provides robust stabilization for uncertain switching with delays and offers practical guidelines for selecting the averaging parameters.

Abstract

We develop an input delay-compensating feedback law for linear switched systems with time-dependent switching. Because the future values of the switching signal, which are needed for constructing an exact predictor-feedback law, may be unavailable at current time, the key design challenge is how to construct a proper predictor state. We resolve this challenge constructing an average predictor-based feedback law, which may be viewed as an exact predictor-feedback law for a particular average system without switching. We establish that, under the predictor-based control law introduced, the closed-loop system is exponentially stable, provided that the plant's parameters are sufficiently close to the corresponding parameters of the average system. In particular, the allowable difference is inversely proportional to the size of delay and proportional to the dwell time of the switching signal. Since no restriction is imposed on the size of delay or dwell time themselves, such a limitation on the parameters of each mode is inherent to the problem considered (in which no a priori information on the switching signal is available), and thus, it cannot be removed. The stability proof relies on two main ingredients-a Lyapunov functional constructed via backstepping and derivation of solutions' estimates for the difference between the average and the exact predictor states. We present consistent, numerical simulation results, which illustrate the necessity of employing the average predictor-based law for achieving stabilization and desired performance of the closed-loop system.

Average Predictor-Feedback Control Design for Switched Linear Systems

TL;DR

This work addresses stabilizing switched linear systems with input delay when future switching is unknown. It introduces an average predictor-based feedback law using mean system matrices and and a fixed gain , analyzed via a backstepping-based Lyapunov functional to bound the predictor mismatch. The main result shows global uniform exponential stability for parameter deviations , where decreases with larger delay and shorter dwell times. Numerical simulations corroborate the approach, confirming stabilization and highlighting the superiority of the average predictor over single-mode predictors, while also noting conservatism in the bound for larger mismatches. Overall, the method provides robust stabilization for uncertain switching with delays and offers practical guidelines for selecting the averaging parameters.

Abstract

We develop an input delay-compensating feedback law for linear switched systems with time-dependent switching. Because the future values of the switching signal, which are needed for constructing an exact predictor-feedback law, may be unavailable at current time, the key design challenge is how to construct a proper predictor state. We resolve this challenge constructing an average predictor-based feedback law, which may be viewed as an exact predictor-feedback law for a particular average system without switching. We establish that, under the predictor-based control law introduced, the closed-loop system is exponentially stable, provided that the plant's parameters are sufficiently close to the corresponding parameters of the average system. In particular, the allowable difference is inversely proportional to the size of delay and proportional to the dwell time of the switching signal. Since no restriction is imposed on the size of delay or dwell time themselves, such a limitation on the parameters of each mode is inherent to the problem considered (in which no a priori information on the switching signal is available), and thus, it cannot be removed. The stability proof relies on two main ingredients-a Lyapunov functional constructed via backstepping and derivation of solutions' estimates for the difference between the average and the exact predictor states. We present consistent, numerical simulation results, which illustrate the necessity of employing the average predictor-based law for achieving stabilization and desired performance of the closed-loop system.

Paper Structure

This paper contains 9 sections, 7 theorems, 25 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Control Design
  3. Switched Linear Systems with Input Delay
  4. Average Predictor-Based Control Design
  5. Stability under average predictor-feedback
  6. Main Result
  7. Proof of Theorem \ref{['Exponential Stability']}
  8. Numerical Simulation Results
  9. Conclusions and Discussion

Key Result

Theorem 1

Consider the closed-loop system (1.1) with the controller (1.5), in which the pair $\left(\bar{A},\bar{B}\right)$ is controllable and choose $\bar{K}$ such that $\bar{A} + \bar{B}\bar{K}$ is Hurwitz. There exists $\epsilon^* > 0$ such that for any $\epsilon < \epsilon^*$, where the closed-loop system is exponentially stable in the sense that there exist positive constants $\rho$ and $\xi$ such th

Figures (5)

  • Figure 2: Evolution of state $X(t)$ and control input $U(t)$ for system (\ref{['1.1']}) with (\ref{['ex1']}), (\ref{['ex1_dyn']}), under controller (\ref{['1.5']}) with (\ref{['ex1_k']}).
  • Figure 3: Evolution of switching signal $\sigma(t)$ for all the case studies.
  • Figure 4: Evolution of state $X(t)$ and control input $U(t)$ for system (\ref{['1.1']}) with (\ref{['ex2']}), (\ref{['ex1_dyn']}), under controller (\ref{['1.5']}) with (\ref{['ex2_k']}).
  • Figure 5: Evolution of state $X(t)$ and control input $U(t)$ for system (\ref{['1.1']}) with (\ref{['ex2']}), (\ref{['ex1_dyn']}), under controller (\ref{['onecontrol']}) for $i=1$ with (\ref{['ex2_k']}).
  • Figure 6: Evolution of state $X(t)$ and control input $U(t)$ for system (\ref{['1.1']}) with (\ref{['ex2']}), (\ref{['ex1_dyn']}), under controller (\ref{['onecontrol']}) for $i=2$ with (\ref{['ex2_k']}).

Theorems & Definitions (8)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6