Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations

Masashi Wakaiki; Hideki Sano

doi:10.3934/mcrf.2021021

Article Contents

2022, Volume 12, Issue 1: 245-273. Doi: 10.3934/mcrf.2021021

This issue Previous Article A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order Next Article

Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations

Masashi Wakaiki^, and
Hideki Sano

Graduate School of System Informatics, Kobe University, Nada, Kobe, Hyogo 657-8501, Japan

^* Corresponding author: wakaiki@ruby.kobe-u.ac.jp
^* Corresponding author: wakaiki@ruby.kobe-u.ac.jp

Received: May 31, 2020

Revised: December 31, 2020

Early access: March 2021

Published: March 2022

The first author is supported by JSPS KAKENHI Grant Numbers JP20K14362

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Abstract

This paper addresses the following question: "Suppose that a state-feedback controller stabilizes an infinite-dimensional linear continuous-time system. If we choose the parameters of an event/self-triggering mechanism appropriately, is the event/self-triggered control system stable under all sufficiently small nonlinear Lipschitz perturbations?" We assume that the stabilizing feedback operator is compact. This assumption is used to guarantee the strict positiveness of inter-event times and the existence of the mild solution of evolution equations with unbounded control operators. First, for the case where the control operator is bounded, we show that the answer to the above question is positive, giving a sufficient condition for exponential stability, which can be employed for the design of event/self-triggering mechanisms. Next, we investigate the case where the control operator is unbounded and prove that the answer is still positive for periodic event-triggering mechanisms.

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Mathematics Subject Classification: Primary: 93C25, 93C57, 93D15; Secondary: 47D06.

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Figure 1. Event-triggered control system

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Figure 2. Self-triggered control system

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Figure 3. State norm $ \|x(t)\| $ of self-triggered control system

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Figure 4. Input $ u(t) $ of self-triggered control system

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Figure 5. Inter-event times $ t_{k+1} - t_k $ of self-triggered control system

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Figure 6. Bound on threshold $ \varepsilon $ and lower bound $ \tau_{\min} $ of inter-event times of event-triggering mechanism (27)

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Figure 7. Comparison of state norm $ \|x(t)\| $ between event-triggered control system and periodic sampled-data system

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Figure 8. Comparison of input $ u(t) $ between event-triggered control system and periodic sampled-data system

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Figure 9. Comparison of inter-event times $ t_{k+1}-t_k $ between event-triggered control system and periodic sampled-data system

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