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Stabilization of Linear Switched Systems with Long Constant Input Delay via Average or Averaging Predictor Feedbacks

Andreas Katsanikakis, Nikolaos Bekiaris-Liberis

TL;DR

The paper tackles stabilization of linear switched systems with a long input delay $D$, where future switching is not fully observable. It develops two predictor-based delay-compensation strategies: an average predictor using a nominal average system with known minimum dwell time, and an averaging predictor feedback that combines exact per-mode predictors; both rely on backstepping and a suite of Lyapunov functionals to prove uniform exponential stability under small parameter variations and a dwell-time bound that grows with the delay. Theoretical guarantees are complemented by simulations showing stabilization and performance gains when dwell-time information is available, as well as robustness to modest delay perturbations. The work extends prior conference results by introducing new stability analyses and providing practical guidelines for predictor design under uncertain switching, with implications for networked control, transportation, and energy systems where delays and switching co-occur.

Abstract

We develop delay-compensating feedback laws 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 two alternative, average predictor-based feedback laws. The first is viewed as a predictor-feedback law for a particular average system, properly modified to provide exact state predictions over a horizon that depends on a minimum dwell time of the switching signal (when it is available). The second is, essentially, a modification of an average of predictor feedbacks, each one corresponding to the fixed-mode predictor-feedback law. We establish that under the control laws introduced, the closed-loop systems are (uniformly) exponentially stable, provided that the differences among system's matrices and among (nominal stabilizing) controller's gains are sufficiently small, with a size that is inversely proportional to the delay length. Since no restriction is imposed on the delay, such a limitation is inherent to the problem considered (in which the future switching signal values are unavailable), and thus, it cannot be removed. The stability proof relies on multiple Lyapunov functionals constructed via backstepping and derivation of solutions' estimates for quantifying the difference between average and exact predictor states. We present consistent numerical simulation results, which illustrate the necessity of employing the average predictor-based laws and demonstrate the performance improvement when the knowledge of a minimum dwell time is properly utilized for improving state prediction accuracy.

Stabilization of Linear Switched Systems with Long Constant Input Delay via Average or Averaging Predictor Feedbacks

TL;DR

The paper tackles stabilization of linear switched systems with a long input delay , where future switching is not fully observable. It develops two predictor-based delay-compensation strategies: an average predictor using a nominal average system with known minimum dwell time, and an averaging predictor feedback that combines exact per-mode predictors; both rely on backstepping and a suite of Lyapunov functionals to prove uniform exponential stability under small parameter variations and a dwell-time bound that grows with the delay. Theoretical guarantees are complemented by simulations showing stabilization and performance gains when dwell-time information is available, as well as robustness to modest delay perturbations. The work extends prior conference results by introducing new stability analyses and providing practical guidelines for predictor design under uncertain switching, with implications for networked control, transportation, and energy systems where delays and switching co-occur.

Abstract

We develop delay-compensating feedback laws 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 two alternative, average predictor-based feedback laws. The first is viewed as a predictor-feedback law for a particular average system, properly modified to provide exact state predictions over a horizon that depends on a minimum dwell time of the switching signal (when it is available). The second is, essentially, a modification of an average of predictor feedbacks, each one corresponding to the fixed-mode predictor-feedback law. We establish that under the control laws introduced, the closed-loop systems are (uniformly) exponentially stable, provided that the differences among system's matrices and among (nominal stabilizing) controller's gains are sufficiently small, with a size that is inversely proportional to the delay length. Since no restriction is imposed on the delay, such a limitation is inherent to the problem considered (in which the future switching signal values are unavailable), and thus, it cannot be removed. The stability proof relies on multiple Lyapunov functionals constructed via backstepping and derivation of solutions' estimates for quantifying the difference between average and exact predictor states. We present consistent numerical simulation results, which illustrate the necessity of employing the average predictor-based laws and demonstrate the performance improvement when the knowledge of a minimum dwell time is properly utilized for improving state prediction accuracy.

Paper Structure

This paper contains 22 sections, 8 theorems, 129 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation for Addressing Switched Systems with Input Delay
  3. Literature Related to Linear Switched Systems with Input Delay
  4. Contributions to Input Delay Compensation for Switched Systems
  5. Relation to Conference Papers my_ECC, my_TDS
  6. Organization of the Results Presented
  7. Problem Formulation and Control Design
  8. Switched Linear Systems with Input Delay
  9. Predictor-Based Design via Average Predictor
  10. Predictor-Based Design via Averaging Predictor Feedbacks
  11. Stability Analysis
  12. Stability Under (\ref{['U1']})
  13. Stability Under (\ref{['U2']})
  14. Proof of Theorem \ref{['main_theorem']}
  15. Technical Differences with the Stability Strategy in my_ECC, my_TDS
  16. ...and 7 more sections

Key Result

Theorem 1

Consider the closed-loop system consisting of plant (system) and controller (U1) under the assumption that the pairs $(A_i, \, B_i)$, for $i=1, \ldots,p$, are stabilizable. Let $K_i$ be such that $A_i+B_iK_i$, $i=1,...,p$, are Hurwitz, and thus, there exist some ${S_i= S_i^T} > 0$, $Q_i = Q_i^T > 0$ There exist $\tau_d ^\star >0$, $\epsilon^\star > 0$, such that for any $\tau_d >\tau_d ^\star$, $\

Figures (10)

  • Figure 1: Switching instants and respective modes in the $[t, t+D]$ time interval.
  • Figure 2: Behavior of function $\tau(t)$ across several switching instants. Immediately after each switching $\tau(t)$ resets to $\tau_d$. It then decays linearly to zero until the next switching occurs (or remains at zero if no further switching occurs).
  • Figure 3: Future switching instants and respective modes in interval $[t, t+D]$.
  • Figure 4: Evolution of switching signal $\sigma(t)$ for all the case studies.
  • Figure 5: Evolution of state $X(t)$ and control input $U(t)=U_1(t)$ for system (\ref{['system']}) with (\ref{['ex1']}), (\ref{['ex1_dyn']}), under controller (\ref{['U1']}) with (\ref{['exavg']}).
  • ...and 5 more figures

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 5 more