^{1}

^{1}

^{2}

^{1}

^{3}

^{1}

^{2}

^{3}

This paper studies the input-to-state stability (ISS) of nonlinear switched systems. By using Lyapunov method involving indefinite derivative and average dwell-time (ADT) method, some sufficient conditions for ISS are obtained. In our approach, the time-derivative of the Lyapunov function is not necessarily negative definite and that allows wider applications than existing results in the literature. Examples are provided to illustrate the applications and advantages of our general results and the proposed approach.

Switched systems are a special subclass of hybrid systems which consist of two components: a family of systems and a switching signal. The systems in the family are described by a collection of indexed differential or difference equations. The switching signal selects an active mode at every instant of time, that is, the system from the family that is currently being followed. As a special class of hybrid systems, switched systems arise in a variety of applications, such as biological systems [

When investigating stability of a system, it is important to characterize the effects of external inputs. The concepts of input-to-state stability (ISS) introduced by Sontag et al. in [

Motivated by the above discussions, in this paper, we shall study the ISS property for switched systems via Lyapunov method involving indefinite derivative. Some sufficient conditions based on ADT method are derived. It is worth mentioning that, although the method used in this paper is based on [

Consider the following switched system:

By the ideas proposed by Hespanha and Morse [

A function

Suppose that a switching signal

In this section, we shall present some ADT results for ISS of switched system (

Assume that there exist functions

Let

In particular, if system (

Assume that there exist functions

Recently, [

Next we consider the time-varying linear switched system in the form of

Assume that there exist constants

Let

In particular, if we choose

Assume that there exist constants

In addition, note that the ISS property guarantees the uniform asymptotic stability (UAS) of a system with a zero input. Consider the nonlinear switched system

Assume that there exist functions

In this section, we present two examples to illustrate our main results.

Consider the switched system (

Note that

Simulation results for Example

Note that, if we choose

Consider the time-varying switched system (

In this case, choose

Simulation results for Example

In this paper, we presented some new ADT-based sufficient conditions for ISS of switched systems via Lyapunov method involving indefinite derivative. The ISS property of the switched system can be guaranteed under the designed ADT scheme. Our results improved some recent work in the literature. Two examples were given to show the effectiveness and advantage of the obtained results. It should be pointed out that the main results of this paper are based on multiple Lyapunov functions, which are more general than existing results in some cases. Since complex factors such as nonlinearities, impulsive perturbations, and delays exist widely in various engineering systems [

The authors declare that they have no conflicts of interest.

This work was supported by National Natural Science Foundation of China (11301308, 61673247) and the Research Fund for Excellent Youth Scholars of Shandong Province (ZR2016JL024, JQ201719).

_{0}