@article{33984ec9e8a6438fbe516569b4dd3459,
title = "Input-output stability of sampled-data linear time-varying systems",
abstract = "In this note we consider the input-output stability of feedback systems consisting of a continuous-time, linear, time-varying plant with a discrete-time, linear time-varying controller. These results generalize some of the results of Chen and Francis where the plant is restricted to be time-variant.",
author = "Iglesias, {Pablo A.}",
note = "Funding Information: The analysis and design of sampled-data systems has been an area of active research since the advent of digital computers in the 1950{\textquoteright}s. Conventionally, this research has been based on the study of discrete-time systems. This approach, however, does not take into account the fact that the underlying system that is being controlled is a linear time-invariant (LTI) continuous-time system and thus ignores the intersample behavior. Recently, there has been a renewed interest in analyzing these hybrid systems-so called because they combine discrete-time and continuous-time signals-as continuous-time, input-output operators. Francis and Georgiou showed that, under certain conditions, the stability (both exponential and L, input-output) of the hybrid system can be implied by exponential stability of a related discrete-time system [I]. These results were extended to general C, input-output stability by Chen and Francis who showed that &-stability will be guaranteed provided that a strictly causal stable continuous-time antialiasing filter is introduced prior to the sampler [2]. More recently, the optimal control of hybrid systems has been one of the most active areas of research in the robust control literature; see [3] and the references therein. The plant is usually assumed to be LTI, however, which gives rise to hybrid systems that are time-varying but periodic. As shown in [4], periodic control systems can be analyzed by using an isometric isomorphism of Fz onto a new space where the operators are Toeplitz. This procedure has been used in [5] where the periodic sampled-data LTI system is “lifted” to a corresponding discrete-time LTI system. In this note we present some new results related to the stability of sampled-data systems where the underlying plant is arbitrarily time-varying. In doing so, our goal is to lay a foundation for the study of optimal controllers for these systems. One motivation for this is that optimal controllers based on the 7-1, control paradigm are now being introduced to systems which vary with time [6]. A digital implementation of these controllers requires knowledge of the stability properties of sampled-data, time-varying systems. The rest of the note is organized as follows: In Section I1 we introduce some notation and collect some useful results. In Section 111 we consider the question of general input-output stability of the hybrid systems. This issue is complicated by the fact that, in contrast to LTI systems, C, stability does not imply C, stability for all 1 5 p 5 oz. Nevertheless, we will show that the input-output stability of the hybrid system can be associated with the input-output stability of a related time-varying discrete-time system. Section IV Manuscript received February 25, 1994; revised February 10, 1995 and April 5,1995. This work was supported in part by NSF Contract ECS-9309387 The author is with the Department of Electrical and Computer Engineering, The Johns Hokpins University, Baltimore, MD 21218 USA. IEEE Log Number 9413374.",
year = "1995",
doi = "10.1109/9.412638",
language = "English (US)",
volume = "40",
pages = "1646--1650",
journal = "IEEE Transactions on Automatic Control",
issn = "0018-9286",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
number = "9",
}