Elsevier

Systems & Control Letters

Volume 38, Issue 1, 15 September 1999, Pages 49-60
Systems & Control Letters

Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems

https://doi.org/10.1016/S0167-6911(99)00046-8Get rights and content

Abstract

We provide an explicit KL stability or input-to-state stability (ISS) estimate for a sampled-data nonlinear system in terms of the KL estimate for the corresponding discrete-time system and a K function describing inter-sample growth. It is quite obvious that a uniform inter-sample growth condition, plus an ISS property for the exact discrete-time model of a closed-loop system, implies uniform ISS of the sampled-data nonlinear system. Our results serve to quantify these facts by means of comparison functions. Our results can be used as an alternative to prove and extend results in [1] or extend some results in [4] to a class of nonlinear systems. Finally, the formulas we establish can be used as a tool for some other problems which we indicate.

Introduction

There is a strong motivation for the investigation of sampled-data systems due to the prevalence of computer controlled systems (see [4], [5], [6]). Moreover, very often linear theory does not suffice and we need to deal with nonlinear sampled-data systems (see [3], [9], [10]). Although the topic is very old there does not appear to be a comprehensive theory even for analysis of properties of sampled-data nonlinear systems. For instance, Lp stability properties of linear sampled-data systems were completely characterized recently in [4], whereas such a characterization is still lacking for nonlinear sampled-data systems.

The ideas of “sampling”, which are related to Poincare maps, can be used to analyze properties of continuous-time systems (see [1], [2], [16], [24], [18], [21]). Recently, a Lyapunov-type theorem was proved in [1] to show uniform local asymptotic stability of time-varying nonlinear systems by using a Lyapunov function whose value decreases along the solutions only at sampling instants. This result was used in [2], [16] to prove several new results on averaging of nonlinear systems. A generalization of the stability result from [1] was presented in [24] where it was shown that global asymptotic stability of the averaged system implies semi-global-practical stability of the original nonlinear system.

An important method in stability and ISS analysis of continuous-time systems is based on the use class-KL and class-K functions (for classical results on KL functions see [11, pp. 135–139] or [8, pp. 7–8 and 95–101]; for new results on KL functions see [20]; for ISS see [19]). We give precise definitions of these functions in the preliminaries section. By using this method, definitions and often some proofs are simplified and more obvious (for instance, see [12, Section 5.3] or [8], [18], [19]). However, the theory that would allow the use of class-KL and class-K functions in the context of sampled-data nonlinear systems seems to be lacking.

It is the purpose of this paper to relate discrete-time and sampled-data KL stability or ISS estimates. The explicit formulas that we present are a tool which allows us to prove new results as well as provide alternative proofs for some old results. A consequence of the established bounds is that stability or ISS of the exact discrete-time model of the system implies the same (equivalent) property of the sampled-data model, under a uniform inter-sample growth condition. Some of our results generalize the result on L stability of linear systems in [4] to a class of nonlinear systems. It is possible to use our method as an alternative to prove results of [1]. Moreover, in Section 4 we generalize the main result of [1] to cover the ISS property. The formulas we establish provide the last technical step in the proof of the main result in [15] where we presented conditions which guarantee that the controller that globally stabilizes an approximate discrete-time model of the plant also semi-globally practically stabilizes the sampled-data system. Also, the results in [13], [17] can be alternatively proved using the results of our paper and the approach in [15]. We introduce a new property of class-KL functions (the UIB property, defined below), that is very useful when relating discrete-time and sampled-data estimates. Finally, we prove a comparison theorem for discrete-time systems based on the use of an auxiliary scalar differential equation. We emphasize that our results hold for a large class of nonlinear systems and for arbitrary fixed sampling periods (this is not a fast sampling result).

As an example of the relationship we obtain, suppose that there exist rs,rb>0, β∈KL and γ̃K, such that: (1) the discrete-time model satisfies the estimate|x(k0)|⩽rs⇒|x(k)|⩽β(|x(k0)|,k−k0),k⩾k0⩾0,where k,k0N; and (2) the inter-sample behavior is characterized in the following way:∀t0⩾0,|x(t0)|⩽rb⇒|x(t)|⩽γ̃(|x(t0)|),t∈[t0,t0+T],where T>0 is the sampling period. Then the sampled-data model satisfies the bound|x(t0)|⩽rx⇒|x(t)|⩽β̄(|x(t0)|,t−t0),t⩾t0⩾0,where rxmin{γ̃−1(rs),γ̃−1∘β0−1(rb)},β0(s)=β(s,0) (it is safe to assume that β0K, see Remark 1 below) and β̄ is as follows: If γ̃(β(γ̃(s),τ))∈KL satisfies a property that we precisely define later (uniform incremental boundedness), thenβ̄(s,τ)≔maxγ̃(s)eT−τ,Pγ̃βγ̃(s),τT,where P>0. Otherwise, we show that in general β̄(s,τ)∈KL can be constructed as:β̄(s,τ)≔maxγ̃(s)eT−τ,4maxη∈[0,τ]2−ηγ̃βγ̃(s),τ−ηT.Similar formulas are derived for ISS and also under stronger hypotheses we obtain global estimates.

The paper is organized as follows. In Section 2 we introduce the class of systems we consider and present definitions and notation. In 3 Main results, 4 Applications of main results we present, respectively, main results and applications of main results. A summary is given in the last section. An important technical lemma is proved in the appendix.

Section snippets

Preliminaries

We concentrate on the class of nonlinear sampled-data systems (see, for example [9]). The model given below represents a continuous-time plant (Sct) controlled in a closed-loop by a digital controller (Sdt), the two being interconnected via the sampler (S) and the zero-order hold (H). The system is described by the equationsSct:ẋ1(t)=f1(t,x1(t),ỹ2(t),d1(t)),t⩾0,t∈R,y1(t)=h1(x1(t)),S:y1(k)=y1(kT),k⩾0,k∈N,Sdt:x2(k+1)=f2(k,y1(k),x2(k),d2(k)),y2(k)=h2(x2(k),y1(k)),H:ỹ2(t)=y2(k),t∈[kT,(k+1)T[.

Main results

We present below conditions that guarantee ULAS, UGAS, ULES, UGES, ULISS and UISS property for the sampled-data system. Moreover, we give the explicit formulas for computing the sampled-data KL estimates using such estimates for the discrete-time system, and the class-K functions given in Definition 2.

Theorem 1 sampled-data ULAS ⇔ discrete-time ULAS + UBT

The sampled-data system (1) is ULAS if and only if the following conditions hold:

(1) the discrete-time model is ULAS,

(2) the solutions are UBT.

In particular, if there exist rs,rb>0,β∈KL and γ̃K,

Applications of main results

In this section we show how our results can be applied to some problems. We indicate some results from the literature that either become corollaries of our results or for which our method provides an alternative proof.

Summary

We presented formulas that relate KL stability and ISS estimates between the discrete-time and sampled-data models for a large class of systems. The estimates are very important in the analysis of sampled-data nonlinear systems and they allowed us to recover or generalize some results from the literature. We showed that ULISS (total stability) and UISS results for the sampled-data system can be deduced from the corresponding results for the discrete-time model. A new result on ULISS and UISS

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    1

    This work was performed while the author was visiting CCEC, Electrical and Computer Engineering Department, University of California, Santa Barbara. Research supported by National Science Foundation under grant ECS 9528370.

    2

    This work was supported in part by AFOSR grant F49620-98-1-0087 and in part by National Science Foundation grant ECS-9896140.

    3

    This work was supported in part by US Air Force Grant F49620-98-1-0242.

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