Abstract
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.
1. 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 [5–7] 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 . 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 [17–19]. 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 , , rather than . 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.
2. Preliminaries
Consider the following switched nonlinear system:where is the state vector and denotes the piecewise constant switching signal, which is continuous from the right; i.e., . , is a smooth function with . Let be the switching sequence. Then, it follows that is active in . Throughout, we adopt the following assumption.
Assumption 1. For the switching sequence , there exists positive constant such that , .
Definition 2. Let . A function is called homogeneous of degree with respect to if
Definition 3. Let . A vector field is called homogeneous with respect to if for each holds for some constant . The constant is called the degree of homogeneity. A time-invariant system is called homogeneous if its vector filed is homogeneous.
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 of the origin and a function , called the settling time, such that the following statements hold: (1)Lyapunov stability.(2)Finite-time convergence: for every , the solution of (1) denoted by with as initial condition is defined on , and . The origin is said to be a globally finite-time-stable equilibrium if it is a finite-time stable equilibrium with .
Assumption 5. For each of switched system (1), there exists homogeneous function of degree with respect to such thatwhere , , , and .
Remark 6. Condition (4) is a direct corollary of function , in view of the homogeneous property. In (2), we choose ; it follows thatwhere . Let . We can get . Therefore, we conclude that , which implies (4) holds. Similarly, we havewhich implies (5) holds.
Remark 7. In the existing literature, switched systems with for some 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., for some , 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 , , defined in Assumption 5, we havewhere , , and .
Assumption 9. For the homogeneous function , , defined in Assumption 5, we havewhere , , and .
Assumption 10. For the homogeneous function , , defined in Assumption 5, we havewhere , , and .
3. Main Results
3.1. Stability with (8)
Lemma 11. Let . If function satisfiesfor , , and , then one gets that
Proof. Multiplying (11) by , we get thatwhich impliesLet . Then, we have thatIntegrating (15) on the time interval , we get thatwhich implies
Theorem 12. Consider switched system (1). Assume that Assumptions 1–8 hold. Let and . Then, the origin of (1) is locally finite-time stable for initial value satisfyingMoreover, the setting time is , where .
Proof. Using the homogeneous function , , we construct the multiple Lyapunov function corresponding to switching law . We can see that is piecewise smooth.
For any , there exists time interval , , such that . Integrating differential inequalities (8) successively on the intervals , we can getTaking into account condition (5), we obtainfor any . Substituting (20) into (19) yieldsIt follows thatTherefore, we obtain thatIt follows thatWe can see that , which implies for . Therefore, we conclude that for .
3.2. Stability with (9)
Lemma 13. Let . If function satisfiesfor , , and , then one gets that
Proof. Multiplying (25) by , we get thatwhich impliesLet . Then, we have thatIntegrating (29) on the time interval , we get thatwhich implies
Theorem 14. Consider switched system (1). Assume that Assumptions 1, 5, and 9 hold. Let and . Then, one can have the following:
(1) If , then the states of (1) will converge to set globally as , where .
(2) If , then the origin of (1) is globally finite-time stable and the setting time is , where .
Proof. Using the homogeneous function , , we construct the multiple Lyapunov function corresponding to switching law . We can see that is piecewise smooth.
For any , there exists time interval , , such that . Integrating differential inequalities (9) successively on the intervals , we can getTaking into account condition (5), we obtainfor any . Substituting (33) into (32) yieldsIt follows that Therefore, we obtain that It follows that Then, we consider the following two cases.
Case 1 . We can see that converges to as .
Case 2 . We can see that system (1) is globally finite-time stable, and the setting time is .
3.3. Stability with (10)
Lemma 15. Let . If function satisfiesfor , , and , then one gets that
Proof. Multiplying (25) by , we get thatwhich impliesLet . Then, we have thatIntegrating (42) on the time interval , we get thatwhich implies
Theorem 16. Consider switched system (1). Assume that Assumptions 1, 5, and 10 hold. Let and . Then, one has the following:
(1) If , then the origin of (1) is globally finite-time stable for any initial values.
(2) If , then the origin of (1) is locally finite-time stable for initial value satisfying .
Moreover, the setting times for cases (1) and (2) are both , where .
Proof. Using the homogeneous function , , we construct the multiple Lyapunov function corresponding to switching law . We can see that is piecewise smooth.
For any , there exists time interval , , such that . Integrating differential inequalities (10) successively on the intervals , we can getTaking into account condition (5), we obtainfor any . Substituting (46) into (45) yieldsIt follows thatTherefore, we obtain thatIt follows thatThen, we consider the following two cases.
Case 1 . We can see that the switched system (1) is globally finite-time stable with any initial value and the setting time is .
Case 2 ). We can see that if , then switched system (1) is locally finite-time stable and the setting time also is .
4. Numerical Example
Consider the following switched system:The sequence of switching times is given by , . Choose and . We get that and are both homogeneous of degree with respect to . The derivatives of the homogeneous functions are and . Then, we obtain that , , and . According to Theorem 16, we get that switched system (51) is finite-time stable. Figure 1 shows the state responses, which demonstrate the exactness of the results.
5. 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.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (61603050) and the Fundamental Research Funds for the Central Universities (2017RC25).