On construction of Lyapunov functions for scalar linear time-varying systems☆

https://doi.org/10.1016/j.sysconle.2019.104591Get rights and content

Abstract

It is known that the construction of Lyapunov functions for scalar linear time-varying systems is related with solutions to the scalar Lyapunov differential equation, whose solution involves both improper integrals and double integrals, and thus are not easy to compute in general. This paper establishes a systematic method for constructing Lyapunov functions for scalar linear time-varying systems. The constructed Lyapunov functions involve an integral of the system parameter with a weighting function over a finite interval. Explicit conditions are imposed on the weighting function and the integral interval such that the Lyapunov function is both positive definite and uniformly bounded, and its time-derivative is negative definite. As a result, constructive solutions to the associated scalar Lyapunov differential equations are also obtained. The established method includes some existing ones as special cases. Examples demonstrate the effectiveness of the proposed methods.

Section snippets

Introduction and motivation

We consider the following scalar linear time-varying (LTV) system ẏ(t)=12μ(t)y(t),∀t∈J=[0,∞),where μ:J→R is a given continuous function. In this paper, we are interested in the construction of Lyapunov functions for system (1), which is an important and a classic problem [1], [2], [3], [4]. Construction of Lyapunov functions and Lyapunov-Krasovskii functionals, especially for time-varying systems, has attracted much attention in the control

The main result

Let CkS denote the set of kth differentiable functions defined in the set S. We first present the following simple lemma, which is the starting point of our development.

Lemma 3

Let T>0 be a constant and ws∈C10,T be some function. Then q(t)=∫0Tμ(t+s)wsds,vt=∫0Tμ(t+s)w1sds, satisfy the following differential equation q̇(t)=−w0μt+wTμt+T−vt.If, moreover, ws satisfies the following two conditions w0=1,wT=0, then π(t),ω

Two illustrative examples

We provide two examples to illustrate the construction of strict Lyapunov functions.

Example 1

We consider the scalar function [14] μt=124lsin2(t+τ)−1τcos2(t),where τ>0 is a constant and l<14τ. Since −14τ−lt−t0−14τ−l≤∫t0tμ(s)ds≤−14τ−lt−t0+14τ+l, which corresponds to (4) with α=1∕(4τ)−l and β=1

Conclusion

This paper has established a systematic method for constructing Lyapunov functions for scalar linear time-varying systems, which are assumed to be uniformly exponentially stable and uniformly exponentially bounded. The constructed Lyapunov functions involve an integral of the system parameter with a weighting function over a finite interval. Conditions are imposed on the weighting function and the integral interval such that the Lyapunov function is both positive definite and uniformly bounded,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    This work was supported in part by the Natural Science Foundation of China under the grant number 61773140 and by the GRF HKSAR 17200918.

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