On Finite-Time Stability of Switched Systems with Hybrid Homogeneous Degrees

The finite-time stability is investigated for switched nonlinear systems. It is assumed that each subsystem possesses a positive homogeneous Lyapunov-like function. The derivative of the function is with hybrid homogenous degrees. Three substantially different situations are considered and different sufficient conditions are provided, respectively.The utility of our result is illustrated through the study of a numerical example.


Introduction
Switched nonlinear systems are widely considered in engineering practice to represent a system with parameter jump and device conversion [1,2].A switched system is essentially a hybrid system that consists of a family of subsystems and a switching law.The stability of the switching system is determined by both the individual stability of each mode and the logic of the switching law.The research achievements on the stability problem are fruitful [3,4], especially on the switched systems; refer to the excellent works [5][6][7] and references therein.
Two general approaches to the stability problem of switched systems are common Lyapunov function (CLF) technique and multiple Lyapunov functions (MLFs) technique.The CLF technique has been effectively used in many situations [8,9].A switched system with a CLF remains stable for any switching laws.Therefore, the CLF technique is naturally used when there is no a priori hypothesis of the switching law.However, the constructive problem of a CLF for general switched systems has not been solved.
MLFs technique relaxes the constraint conditions of CLF.In [10], it is shown that if the Lyapunov function of each mode is decreasing and the energy is decreasing at switching times, then the switched system is asymptotically stable.In [11], the MLFs condition is relaxed by introducing the concept of weak Lyapunov functions (WLFs).An extension of the invariance principle is provided relative to dwell time switched solutions.
In [12], union/intersection WLFs techniques are presented, where more accurate convergence region is obtained.In these works, maximal ratio coefficient is required among the Lyapunov functions.More specifically, for any subsystems  and , it is assumed that   ≤   with  ≥ 1.However, it is not easy to get the estimation of .Particularly, the existence of  is not clear in many situations.
It is worth noting that homogeneous theory [13,14] can give simplified conditions for stability analysis of switched nonlinear systems, where the value of  is obtained accurately.In [15], stability problem of switched homogeneous systems is addressed using semitensor product of matrices and LMI conditions are achieved.In [16], homogeneous Lyapunov function is constructed and stability analysis via both CLF and MLFs is given.Some other results on this topic can be found in [17][18][19].In comparison with the existing results where single homogeneous degree is considered, in this paper, we consider switched homogeneous systems with hybrid homogeneous degrees.That is to say, we consider homogeneous switched systems with Lyapunov function Recently, nonlinear systems with hybrid homogeneous degrees have attracted a considerable attention [20,21].However, such systems under switched conditions have not been investigated.This problem is considered in this paper.We extend the homogeneous results to the case with hybrid homogeneous degrees and sufficient conditions are obtained for finite-time stability.
Finite-time stability [22,23] is considered in this paper and some definitions are provided as follows.
Definition 4. The origin is said to be a finite-time-stable equilibrium of (1) if there exists an open neighborhood  ∈ R  of the origin and a function  : \{0} → (0, ∞), called the settling time, such that the following statements hold: (1) Lyapunov stability.
The origin is said to be a globally finite-time-stable equilibrium if it is a finite-time stable equilibrium with  =   .
Remark 7. In the existing literature, switched systems with V  ≤ −   for some ,  ∈ R are widely considered.Different kinds of sufficient conditions for switched stability have been presented.In this paper, we consider switched systems with hybrid homogeneous degrees, i.e., V  ≤    +   for some , ,  ∈ R, and as far as we know, there are no results on this kind of switched systems.Specifically, the following two different situations are considered.Assumption 8.For the homogeneous function   (),  ∈ P, defined in Assumption 5, we have where  > 0,  > 0, and 0 <  < 1.
Case 1 ( −(1−) −1 < 0).We can see that the switched system (1) is globally finite-time stable with any initial value and the setting time is  Π .

Conclusion
In this paper, the finite-time stability problem of switched homogeneous systems has been studied.Three substantially different situations with hybrid homogeneous degrees have been introduced.Sufficient conditions and estimations of the setting time have been given under both situations.A numerical example has been presented to show the effectiveness of the results.