Input-to-State Stability and Stabilization of Sampled-Data Systems Under Aperiodic Sampling and Random Sampling

Ping Zhao,Wei Feng,Ben Niu,Zhiguo Yan

doi:10.1109/access.2021.3058153

Abstract

In this article, input-to-state stability (ISS) and stabilization are examined for sampled-data systems under deterministic aperiodic sampling and random sampling, respectively. Using the direct design method, the sampled-data systems are transformed into switched systems with switched time-varying delays. First, the ISS definition and criterion appropriate for these systems are provided. Based on this, the ISS criterion on sampled-data systems under deterministic aperiodic sampling is given. Second, after the stochastic ISS (SISS) definition and criterion are provided for switched nonlinear systems with randomly switching delays, the SISS criterion for the sampled-data systems under random sampling is provided. All of the ISS and SISS definitions are given in the form of $\mathcal {KL}$ function that is quite elegant and easy to work with. Then, sufficient conditions for input-to-state stabilization are obtained for sampled-data linear systems under deterministic aperiodic sampling and random sampling, respectively, via the Lyapunov-Krasovskii method. Finally, based on the criteria, a piecewise controller is designed by the matrix inequality approach for a sampled-data linear time invariant system, and simulation results are provided to illustrate our design method. The main conclusion of this article is that sampling intervals will affect the controller design of the systems, and the ISS properties are maintained using a piecewise controller.

Highlights

Since the performance of actual control system is affected by unmodeled dynamics, parameter perturbation, exogenous disturbance, measurement error and other uncertainties, research on the robustness of control system has been playing an important role in the development of control theory and technology
For nonlinear control system robustness analysis, a new method from the perspective of input-to-state stability (ISS), input-to-output stability (IOS) and integral input-tostate stability has been developed and a series of basic theoretical results focusing on ISS, IOS-lyapunov functions have been obtained by many researchers such as Angeli, Liberzon, Lin, Praly, and Sontag ( [2], [26]–[28], [34]–[40])
The remainder of this article is organized as follows: In Section II, after the aperiodic sampled-data systems are transformed into switched nonlinear systems with switched time-varying delays, the corresponding ISS definition and criterion will be provided

Summary

INTRODUCTION

Since the performance of actual control system is affected by unmodeled dynamics, parameter perturbation, exogenous disturbance, measurement error and other uncertainties, research on the robustness of control system has been playing an important role in the development of control theory and technology. For the ISS and stabilization of aperiodic sampled-data control systems, further study is necessary to accurately represent the influence of the length of the time-varying sampling interval. Sufficient conditions for input-to-state stabilization will be obtained for sampled-data linear systems under deterministic aperiodic sampling and random sampling via the Lyapunov-Krasovskii method and the matrix inequality approach. The remainder of this article is organized as follows: In Section II, after the aperiodic sampled-data systems are transformed into switched nonlinear systems with switched time-varying delays, the corresponding ISS definition and criterion will be provided.

ISS OF SAMPLED-DATA SYSTEMS UNDER DETERMINISTIC APERIODIC SAMPLING

CASE OF DETERMINISTIC APERIODIC SAMPLING

CASE OF RANDOM SAMPLING

SIMULATION EXAMPLE