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Input Delay Compensation for a Class of Switched Linear Systems via Averaging Exact Predictor Feedbacks

Andreas Katsanikakis, Nikolaos Bekiaris-Liberis

TL;DR

This paper tackles predictor-based control for switched linear systems with long input delay and unknown future switching. It introduces a delay-compensating controller that averages the exact predictor feedbacks across all modes, avoiding reliance on future switching values. Uniform stability is established via a Lyapunov functional, under a bound on differences between mode matrices and controller gains, with a constructive estimate of the predictor-mismatch guiding the stability proof. Simulations demonstrate that the averaging predictor-feedback law delivers favorable transient performance and robustness compared with single-mode predictors and mean-based designs. The work advances practical delay compensation in systems with arbitrary switching, providing a framework that trades off mode-difference magnitude for allowable delay and dwell-time characteristics.

Abstract

The key challenges in design of predictor-based control laws for switched systems with arbitrary switching and long input delay are the potential unavailability of the future values of the switching signal (at current time) and the fact that dwell time may be arbitrary. In the present paper, we resolve these challenges developing a new predictor-based control law that is, essentially, an average of exact predictor feedbacks, each one corresponding to an exact predictor-feedback law for a system that operates only in a single mode. Because the predictor state in our control design does not correspond to an exact predictor, stability can be guaranteed under a restriction on the differences among the system's matrices and controller's gains. This is an unavoidable limitation, for a switching signal whose future values may be unavailable, when no constraint is imposed on the values of delay and dwell time (as it is the case here). We establish (uniform) stability of the closed-loop system employing a Lyapunov functional. The key step in the stability proof is constructive derivation of an estimate of the mismatch between an exact predictor feedback and the average of predictor feedbacks constructed. We illustrate the performance of the proposed predictor-based control law in simulation, including comparisons with alternative, predictor-based control laws.

Input Delay Compensation for a Class of Switched Linear Systems via Averaging Exact Predictor Feedbacks

TL;DR

This paper tackles predictor-based control for switched linear systems with long input delay and unknown future switching. It introduces a delay-compensating controller that averages the exact predictor feedbacks across all modes, avoiding reliance on future switching values. Uniform stability is established via a Lyapunov functional, under a bound on differences between mode matrices and controller gains, with a constructive estimate of the predictor-mismatch guiding the stability proof. Simulations demonstrate that the averaging predictor-feedback law delivers favorable transient performance and robustness compared with single-mode predictors and mean-based designs. The work advances practical delay compensation in systems with arbitrary switching, providing a framework that trades off mode-difference magnitude for allowable delay and dwell-time characteristics.

Abstract

The key challenges in design of predictor-based control laws for switched systems with arbitrary switching and long input delay are the potential unavailability of the future values of the switching signal (at current time) and the fact that dwell time may be arbitrary. In the present paper, we resolve these challenges developing a new predictor-based control law that is, essentially, an average of exact predictor feedbacks, each one corresponding to an exact predictor-feedback law for a system that operates only in a single mode. Because the predictor state in our control design does not correspond to an exact predictor, stability can be guaranteed under a restriction on the differences among the system's matrices and controller's gains. This is an unavoidable limitation, for a switching signal whose future values may be unavailable, when no constraint is imposed on the values of delay and dwell time (as it is the case here). We establish (uniform) stability of the closed-loop system employing a Lyapunov functional. The key step in the stability proof is constructive derivation of an estimate of the mismatch between an exact predictor feedback and the average of predictor feedbacks constructed. We illustrate the performance of the proposed predictor-based control law in simulation, including comparisons with alternative, predictor-based control laws.

Paper Structure

This paper contains 7 sections, 8 theorems, 78 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Control Design and Stability Analysis
  3. Switched Linear Systems with Input Delay
  4. Predictor-Based Design via Averaging Predictor Feedbacks
  5. Stability Statement and Proof
  6. Simulation Results
  7. Conclusions and Discussion

Key Result

Theorem II.1

Consider the closed-loop system (1.1) with the controller (1.5), (subprediction). Let the pairs $\left(A_i,B_i\right)$ be controllable and choose $K_i$ such that $A_i + B_i K_i$ are Hurwitz, for $i=0,\ldots,l,$ and such that there exist common $P = P^T > 0$, $Q = Q^T > 0$, satisfying There exists $\epsilon^* > 0$ such that for any $\epsilon < \epsilon^*$, where and for all $X_0\in\mathbb{R}^n$,

Figures (6)

  • Figure 1: Switching instants and respective modes in the $[t, t+D]$ time interval.
  • Figure 2: Evolution of switching signal $\sigma(t)$ for all the case studies.
  • Figure 3: 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 4: Evolution of state $X(t)$ and control input $U(t)$ for system (\ref{['1.1']}) with (\ref{['ex1']}), (\ref{['ex1_dyn']}), under controller (\ref{['onecontrol']}) for $i=1$ with $K_1$ as in (\ref{['ex1_k']}).
  • Figure 5: Evolution of state $X(t)$ and control input $U(t)$ for system (\ref{['1.1']}) with (\ref{['ex1']}), (\ref{['ex1_dyn']}), under controller (\ref{['onecontrol']}) for $i=2$ with $K_2$ as in (\ref{['ex1_k']}).
  • ...and 1 more figures

Theorems & Definitions (16)

  • Theorem II.1
  • Remark II.2
  • Lemma II.3: exact predictor construction
  • proof
  • Lemma II.4: backstepping transformation
  • proof
  • Lemma II.5: inverse backstepping transformation
  • proof
  • Lemma II.6: bound on error due to predictor mismatch
  • proof
  • ...and 6 more