On finite gain Lp stability of nonlinear sampled-data systems

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Abstract

It is shown that uniform global exponential stability of the input-free discrete-time model of a globally Lipschitz sampled-data time-varying nonlinear system with inputs implies finite gain Lp stability of the sampled-data system for all p∈[1,∞]. This result generalizes results on Lp stability of sampled-data linear systems and it is an important tool for analysis of robustness of sampled-data nonlinear systems with inputs.

Introduction

Prevalence of computer controlled systems strongly motivates investigation of sampled-data control systems. Moreover, due to the fact that the plant model and or control law are often nonlinear, we frequently need to consider nonlinear sampled-data systems. While the area of linear sampled-data systems has matured into a well understood and developed discipline (see [2]), a range of open problems still remains in the area of nonlinear sampled-data systems. In particular, a complete analysis of Lp stability properties of nonlinear sampled-data systems with inputs appears to be lacking in the literature.

One of the first results on L stability of linear sampled-data systems can be found in [3] and a complete characterization of Lp stability for any p∈[1,∞] of linear time-invariant and time-varying sampled-data systems can be found, respectively, in [1], [5]. Related results on integral stability properties with nonlinear gains, such as input-to-state stability (ISS) and integral input-to-state stability (iISS), for sampled-data systems with inputs were addressed, respectively, in [8], [9], [10], [13]. In particular, preservation of the ISS property under discretization (emulation) of the dynamic controllers for nonlinear sampled-data systems were presented in [9], [13]. Results on achieving iISS and ISS for nonlinear sampled-data systems via their approximate discrete-time models were considered respectively in [8], [10].

It is the purpose of this paper to present a result on Lp stability of globally Lipschitz nonlinear sampled-data systems with inputs. In particular, it is shown that if the discrete-time model of the input-free sampled-data system is uniformly globally exponentially stable, then the sampled-data nonlinear system with inputs is Lp stable for any p∈[1,∞]. This result generalizes similar results on Lp stability of linear time-invariant and time-varying sampled-data systems in [1], [5], respectively, and it is an important tool in analysis of robustness properties of sampled-data nonlinear systems. Moreover, our proof technique is based on Lyapunov arguments and it is different from the proof technique exploited in [1], [5]. We present detailed proofs only for global results and then comment on how the same proof technique applies to local results. We also apply our results to the case where the sampled-data system arises in feedback control schemes using discrete-time, dynamic controllers.

The paper is organized as follows. Preliminaries are presented in Section 2. Section 3 contains the main result and a discussion on how the same technique can be used to address several related problems. The proof of the main result is presented in Section 4 and proofs of some auxiliary results can be found in Appendix A.

We use Z⩾j to denote all integers greater than or equal to the integer j. For a function v:R⩾0Rm, we define the Lp norm of v(·) as follows:||v(·)||Lp0|v(t)|pdt1/pforp∈[1,∞)and ||v(·)||Less.sup.t⩾0|v(t)|, where the underlying vector norm is, without loss of generality, the Euclidean norm. Similarly, but in the discrete-time setting, given a sequence ν:Z⩾0Rm, we define the ℓp norm of ν(·) as||ν(·)||pk=0|ν(k)|p1/pforp∈[1,∞)and ||ν(·)||≔supk⩾0|ν(k)|.

Section snippets

Preliminaries

In this paper, we consider explicitly time-varying sampled-data systems with inputsẋ=f(x(t),x(⌊t⌋T),t,u(t)),⌊t⌋T=Tmaxj∈Z:j⩽tT.In this system, T is the sampling period, u is an exogenous input and x is the “state” (more precisely, values of x at the initial time t and at the possibly earlier time ⌊tT are needed to compute the solution forward in time) which, in a closed-loop control problem, may include some (possibly discrete-time) controller dynamics. The right-hand side's dependence on x(⌊

Main results

In this section, we present our main result, which states that uniform global exponential stability of the zero-input discrete-time model of (2) implies Lp stability of the sampled-data system for any p∈[1,∞]. The proof of this result is postponed until the next section. Then, in the second part of this section we discuss several possible generalizations of our results and relation to some existing results in the literature. The main result of this section is stated next.

Theorem 1

Suppose that Assumption

Proof of main results

We will make use of the following fact which is proved using Holder's inequality.

Fact 1

Let the sequence tk, k∈Z⩾−1 be such that t−1⩽0 and tk+1tk=T for all k∈Z⩾−1. Given a function u(·) defined on [t−1,∞) with u(t)=0 for all t∈[t−1,0), defineν̃(k)≔0T|u(tk+τ)|dτ∀k∈Z⩾−1.Then, for each p∈[1,∞] (where for p=∞ we let (p−1)/p=1),||ν̃(·−1)||p⩽T(p−1)/p||u(·)||Lp.

Proof

See Appendix A.1. 

The proof of our results will rely heavily on the input-to-state properties of the discrete-time systemξ+ϱ+=G(ξ,ϱ)+ν0,where G

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Cited by (0)

1

Supported in part by MIUR through project MISTRAL and ASI under grant I/R/152/00.

2

Supported in part by AFOSR under grant F49620-00-1-0106 and NSF under grant ECS-9988813.

3

This work was supported by the Australian Research Council under the Large Grants Scheme.

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