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

Authors: Bin Zhang
Journal: Mathematical Problems in Engineering
 (2018)

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## 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.


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## Body

## 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 i and j, it is assumed that Vi≤ϕVj with ϕ≥1. However, it is not easy to get the estimation of ϕ. Particularly, the existence of ϕ 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 ϕ 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 V˙i(x)≤-pViθ(x)+_qVi(x), 0<θ<1, rather than V˙i(x)≤-pViθ(x). 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:(1)x˙=fσx,t≥t0,t0∈Rwhere x∈Rn is the state vector and σ(t):[0,∞)→P={1,2,…,p} denotes the piecewise constant switching signal, which is continuous from the right; i.e., σ(t+)=lims↓tσ(s). ∀i∈P, fi is a smooth function with fi(0)=0. Let {t1,t2,…} be the switching sequence. Then, it follows that σ(ti) is active in [ti,ti+1). Throughout, we adopt the following assumption.Assumption 1.
For the switching sequence{t1,t2,…}, there exists positive constant τ such that ti-ti-1=τ, ∀i∈{1,2,…}.Definition 2.
Letx=(x1,…,xn)T∈Rn. A function V:Rn→R is called homogeneous of degree μ∈R with respect to r=(r1,…,rn)T∈R+n if(2)Vλr1x1,…,λrnxn=λμVx1,…,xn,∀x∈Rn,∀λ∈R+.Definition 3.
Letx=(x1,…,xn)T∈Rn. A vector field f(x)=f1(x),…,fn(x)T:Rn→Rn is called homogeneous with respect to r=(r1,…,rn)T∈R+n if for each i∈{1,…,n}(3)fiλr1x1,…,λrnxn=λκ+rifix1,…,xn,∀x∈Rn,∀λ∈R+,holds for some constant κ≥-min1≤i≤nri. The constant κ is called the degree of homogeneity. A time-invariant system x˙=f(x) is called homogeneous if its vector filed f(x) 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 N∈Rn of the origin and a function T:N\{0}→(0,∞), called the settling time, such that the following statements hold: (1)
Lyapunov stability.(2)
Finite-time convergence: for everyx0∈N\{0}, the solution of (1) denoted by ψ(x0) with x0 as initial condition is defined on [0,T(x0)), and limt→T(x0)ψ(x0)=0.
The origin is said to be a globally finite-time-stable equilibrium if it is a finite-time stable equilibrium withN=Rn.Assumption 5.
For eachi∈P of switched system (1), there exists C1 homogeneous function Vi(x):Rn→R of degree μ∈R with respect to r=(r1,…,rn)T∈R+n such that(4)αΛμx≤Vix≤βΛμx(5)Vix≤ϕVjx,∀i,j∈Pwhere Λ(x)=∑i=1n‍|xi|1/ri, α=mini∈P⁡minΛ(x)=1⁡Vi(x), β=maxi∈P⁡maxΛ(x)=1⁡Vi(x), and ϕ=maxi,j∈P⁡maxΛx=1⁡(Vi(x)/Vj(x)).Remark 6.
Condition (4) is a direct corollary of function Vi(x), in view of the homogeneous property. In (2), we choose λ=1/Λ(x); it follows that(6)Vix1,…,xn=1λμViλr1x1,…,λrnxn=ΛμxViξ1,…,ξnwhere ξi=λrixi. Let ξ=(ξ1,…,ξn)T. We can get Λ(ξ)=∑i=1n‍|ξi|1/ri=λ∑i=1n‍|xi|1/ri=λΛ(x)=1. Therefore, we conclude that α≤Vi(ξ1,…,ξn)≤β, which implies (4) holds. Similarly, we have(7)VixVjx=ΛμxViξ1,…,ξnΛμxVjξ1,…,ξn≤maxi,j∈PmaxΛx=1VixVjxwhich implies (5) holds.Remark 7.
In the existing literature, switched systems withV˙i≤-pViθ for some p,θ∈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˙i≤aViθ+bVi for some a,b,θ∈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 functionVi(x), i∈P, defined in Assumption 5, we have(8)∂Vi∂xfix≤-pViθx+qVixwhere p>0, q>0, and 0<θ<1.Assumption 9.
For the homogeneous functionVi(x), i∈P, defined in Assumption 5, we have(9)∂Vi∂xfix≤pViθx-qVixwhere p>0, q>0, and 0<θ<1.Assumption 10.
For the homogeneous functionVi(x), i∈P, defined in Assumption 5, we have(10)∂Vi∂xfix≤-pViθx-qVixwhere p>0, q>0, and 0<θ<1.


## 3. Main Results

### 3.1. Stability with (8)

Lemma 11.
Lett′′>t′≥t0. If function y(t):R→R satisfies(11)y˙t=-pyθt+qytfor p>0, q>0, and 0<θ<1, then one gets that(12)y1-θt′′=e1-θqt′′-t′y1-θt′+pq1-e1-θqt′′-t′.Proof.
Multiplying (11) by e-qt, we get that(13)e-qty˙y-qe-qtyt=-pe-qtyθtwhich implies(14)de-qtytdt=-pe-qtyθt.Let z(t)=e-qty(t). Then, we have that(15)dzdt=-pe-1-θqtzθ.Integrating (15) on the time interval t′,t′′, we get that(16)z1-θt′′=z1-θt′+pqe-1-θqt′′-e-1-θqt′which implies(17)y1-θt′′=e1-θqt′′-t′y1-θt′+pq1-e1-θqt′′-t′.Theorem 12.
Consider switched system (1). Assume that Assumptions 1–8 hold. Let Υ=(p/q)(ϕ(e(1-θ)qτ-1)/(ϕe(1-θ)qτ-1)) and Ψ=e(1-θ)qτVσ(t0)1-θ(x(t0))-Υ. Then, the origin of (1) is locally finite-time stable for initial value x(t0) satisfying(18)Vσt0xt0≤1eqτpqϕe1-θqτ-1ϕe1-θqτ-11/1-θ.Moreover, the setting time is T=tΠ, where Π=ln(-Υ/Ψ)/ln(ϕe(1-θ)qτ).Proof.
Using the homogeneous functionVi(x), i∈P, we construct the multiple Lyapunov function Vσ(t)(x) corresponding to switching law σ(t). We can see that Vσ(t)(x) is piecewise smooth.
For anyt≥t0, there exists time interval [tj-1,tj), j≥1, such that tj-1≤t<tj. Integrating differential inequalities (8) successively on the intervals [t0,t1),[t1,t2),…,[tj-1,t], we can get(19)Vσtj-11-θxt≤e1-θqt-tj-1Vσtj-11-θxtj-1+pq1-e1-θqt-tj-1Vσtj-21-θxtj-1≤e1-θqtj-1-tj-2Vσtj-21-θxtj-2+pq1-e1-θqtj-1-tj-2⋯Vσt01-θxt1≤e1-θqt1-t0Vσt01-θxt0+pq1-e1-θqt1-t0.Taking into account condition (5), we obtain(20)Vσti1-αxti≤ϕVσti-11-αxtifor any i=1,…,j-1. Substituting (20) into (19) yields(21)Vσtj-11-θxt≤ϕe1-θqt-tj-1Vσtj-21-θxtj-1+pq1-e1-θqt-tj-1Vσtj-21-θxtj-1≤ϕe1-θqtj-1-tj-2Vσtj-31-θxtj-2+pq1-e1-θqtj-1-tj-2⋯Vσt01-θxt1≤e1-θqt1-t0Vσt01-θxt0+pq1-e1-θqt1-t0.It follows that(22)Vσtj-11-θxt≤ϕj-1e1-θqt-t0Vσt01-θxt0+pq1-e1-θqt-tj-1+pqϕ1-e1-θqtj-1-tj-2e1-θqt-tj-1+pqϕ21-e1-θqtj-2-tj-3e1-θqt-tj-2⋯+pqϕj-21-e1-θqt2-t1e1-θqt-t2+pqϕj-11-e1-θqt1-t0e1-θqt-t1≤ϕj-1e1-θqjτVσt01-θxt0+pqϕe1-θqτ-1ϕe1-θqτ-11-ϕj-1e1-θqj-1τ.Therefore, we obtain that(23)0≤Vσtj-11-θxt≤Ψϕj-1e1-θqj-1τ+Υ.It follows that(24)0≤Λxt≤1α1/μΨϕj-1e1-θqj-1τ+Υ1/1-θμ.We can see that Ψ<0, which implies Λ(x(t))=0 for j>1+ln(-Υ/Ψ)/ln(ϕe(1-θ)qτ). Therefore, we conclude that x(t)=0 for t>tΠ.


### 3.2. Stability with (9)

Lemma 13.
Lett′′>t′≥t0. If function y(t):R→R satisfies(25)y˙t=pyθt-qytfor p>0, q>0, and 0<θ<1, then one gets that(26)y1-θt′′=e-1-θqt′′-t′y1-θt′+pq1-e-1-θqt′′-t′.Proof.
Multiplying (25) by e-qt, we get that(27)eqty˙y+qeqtyt=peqtyθtwhich implies(28)deqtytdt=peqtyθt.Let z(t)=eqty(t). Then, we have that(29)dzdt=pe1-θqtzθ.Integrating (29) on the time interval t′,t′′, we get that(30)z1-θt′′=z1-θt′+pqe1-θqt′′-e1-θqt′which implies(31)y1-θt′′=e-1-θqt′′-t′y1-θt′+pq1-e-1-θqt′′-t′.Theorem 14.
Consider switched system (1). Assume that Assumptions 1, 5, and 9 hold. Let Υ=(p/q)(e-1-θqτ-1)/(ϕe-1-θqτ-1) and Ψ=Vσ(t0)1-θ(x(t0))-Υϕ. Then, one can have the following:
(1) Ifϕe-(1-θ)qτ-1<0, then the states x(t) of (1) will converge to set Ω globally as t→∞, where Ω={x:Λx≤1+ϕ-ϕe-1-θqτ(p/q)(e-1-θqτ-1)/(ϕe-(1-θ)qτ-1)}.
(2) Ifϕe-(1-θ)qτ-1>0, then the origin of (1) is globally finite-time stable and the setting time is tΠ, where Π=ln(-Υ1+ϕ-ϕe-1-θqτ/Ψ)/ln(e-(1-θ)qτ).Proof.
Using the homogeneous functionVi(x), i∈P, we construct the multiple Lyapunov function Vσ(t)(x) corresponding to switching law σ(t). We can see that Vσ(t)(x) is piecewise smooth.
For anyt≥t0, there exists time interval [tj-1,tj), j≥1, such that tj-1≤t<tj. Integrating differential inequalities (9) successively on the intervals [t0,t1),[t1,t2),…,[tj-1,t], we can get(32)Vσtj-11-θxt≤e-1-θqt-tj-1Vσtj-11-θxtj-1+pq1-e-1-θqt-tj-1Vσtj-21-θxtj-1≤e-1-θqtj-1-tj-2Vσtj-21-θxtj-2+pq1-e-1-θqtj-1-tj-2⋯Vσt01-θxt1≤e-1-θqt1-t0Vσt01-θxt0+pq1-e-1-θqt1-t0.Taking into account condition (5), we obtain(33)Vσti1-αxti≤ϕVσti-11-αxtifor any i=1,…,j-1. Substituting (33) into (32) yields(34)Vσtj-11-θxt≤ϕe-1-θqt-tj-1Vσtj-21-θxtj-1+pq1-e-1-θqt-tj-1Vσtj-21-θxtj-1≤ϕe-1-θqtj-1-tj-2Vσtj-31-θxtj-2+pq1-e-1-θqtj-1-tj-2⋯Vσt01-θxt1≤e-1-θqt1-t0Vσt01-θxt0+pq1-e-1-θqt1-t0.It follows that(35)Vσtj-11-θxt≤ϕj-1e-1-θqt-t0Vσt01-θxt0+pq1-e-1-θqt-tj-1+pqϕ1-e-1-θqtj-1-tj-2e-1-θqt-tj-1+pqϕ21-e-1-θqtj-2-tj-3e-1-θqt-tj-2⋯+pqϕj-21-e-1-θqt2-t1e-1-θqt-t2+pqϕj-11-e-1-θqt1-t0e-1-θqt-t1≤ϕj-1e-1-θqj-1τVσt01-θxt0+pqe-1-θqτ-1ϕe-1-θqτ-11+ϕ-ϕe-1-θqτ-ϕje-1-θqj-1τ. Therefore, we obtain that(36)0≤Vσtj-11-θxt≤Ψϕj-1e-1-θqj-1τ+Υ1+ϕ-ϕe-1-θqτ. It follows that(37)0≤Λxt≤1α1/μΨϕj-1e-1-θqj-1τ+Υ1+ϕ-ϕe-1-θqτ1/1-θμ. Then, we consider the following two cases.
Case 1 ( ϕ e - ( 1 - θ ) q τ - 1 < 0 ). We can see that x(t) converges to Ω={x:Λ(x)≤Υ(1+ϕ-ϕe-(1-θ)qτ)} as t→∞.
Case 2 ( ϕ e - 1 - θ q τ - 1 > 0 ). We can see that system (1) is globally finite-time stable, and the setting time is tΠ.


### 3.3. Stability with (10)

Lemma 15.
Lett′′>t′≥t0. If function y(t):R→R satisfies(38)y˙t=-pyθt-qytfor p>0, q>0, and 0<θ<1, then one gets that(39)y1-θt′′=e-1-θqt′′-t′y1-θt′-pq1-e-1-θqt′′-t′.Proof.
Multiplying (25) by eqt, we get that(40)eqty˙y+qeqtyt=-peqtyθtwhich implies(41)deqtytdt=-peqtyθt.Let z(t)=eqty(t). Then, we have that(42)dzdt=-pe1-θqtzθ.Integrating (42) on the time interval t′,t′′, we get that(43)z1-θt′′=z1-θt′-pqe1-θqt′′-e1-θqt′which implies(44)y1-θt′′=e-1-θqt′′-t′y1-θt′-pq1-e-1-θqt′′-t′.Theorem 16.
Consider switched system (1). Assume that Assumptions 1, 5, and 10 hold. Let Υ=(p/q)(ϕe-1-θqτe-1-θqτ-1/ϕe-1-θqτ-1) and Ψ=Vσ(t0)1-θ(x(t0))+Υ. Then, one has the following:
(1) Ifϕe-(1-θ)qτ-1<0, then the origin of (1) is globally finite-time stable for any initial values.
(2) Ifϕe-(1-θ)qτ-1>0, then the origin of (1) is locally finite-time stable for initial value x(t0) satisfying Vσt01-θ(x(t0))≤-(p/q)(ϕe-1-θqτe-1-θqτ-1/(ϕe-1-θqτ-1)).
Moreover, the setting times for cases (1) and (2) are bothtΠ, where Π=lnΥ/Ψ/lnϕ/e(1-θ)qτ.Proof.
Using the homogeneous functionVi(x), i∈P, we construct the multiple Lyapunov function Vσ(t)(x) corresponding to switching law σ(t). We can see that Vσ(t)(x) is piecewise smooth.
For anyt≥t0, there exists time interval [tj-1,tj), j≥1, such that tj-1≤t<tj. Integrating differential inequalities (10) successively on the intervals [t0,t1),[t1,t2),…,[tj-1,t], we can get(45)Vσtj-11-θxt≤e-1-θqt-tj-1Vσtj-11-θxtj-1-pq1-e-1-θqt-tj-1Vσtj-21-θxtj-1≤e-1-θqtj-1-tj-2Vσtj-21-θxtj-2-pq1-e-1-θqtj-1-tj-2⋯Vσt01-θxt1≤e-1-θqt1-t0Vσt01-θxt0-pq1-e-1-θqt1-t0.Taking into account condition (5), we obtain(46)Vσti1-αxti≤ϕVσti-11-αxtifor any i=1,…,j-1. Substituting (46) into (45) yields(47)Vσtj-11-θxt≤ϕe-1-θqt-tj-1Vσtj-21-θxtj-1-pq1-e-1-θqt-tj-1Vσtj-21-θxtj-1≤ϕe-1-θqtj-1-tj-2Vσtj-31-θxtj-2-pq1-e-1-θqtj-1-tj-2⋯Vσt01-θxt1≤e-1-θqt1-t0Vσt01-θxt0-pq1-e-1-θqt1-t0.It follows that(48)Vσtj-11-θxt≤ϕj-1e-1-θqt-t0Vσt01-θxt0-pq1-e-1-θqt-tj-1-pqϕ1-e-1-θqtj-1-tj-2e-1-θqt-tj-1-pqϕ21-e-1-θqtj-2-tj-3e-1-θqt-tj-2⋯-pqϕj-21-e-1-θqt2-t1e-1-θqt-t2-pqϕj-11-e-1-θqt1-t0e-1-θqt-t1≤ϕj-1e-1-θqj-1τVσt01-θxt0-pqϕe-1-θqτe-1-θqτ-1ϕe-1-θqτ-11-ϕj-1e-1-θqj-1τ.Therefore, we obtain that(49)0≤Vσtj-11-θxt≤Ψϕj-1e-1-θqj-1τ-Υ.It follows that(50)0≤Λxt≤1α1/μΨϕj-1e-1-θqj-1τ-Υ1/1-θμ.Then, we consider the following two cases.
Case 1 ( ϕ e - 1 - θ q τ - 1 < 0 ). We can see that the switched system (1) is globally finite-time stable with any initial value and the setting time is tΠ.
Case 2 ( ϕ e - ( 1 - θ ) q τ - 1 > 0). We can see that if Ψ<0, then switched system (1) is locally finite-time stable and the setting time also is tΠ.


## 3.1. Stability with (8)

Lemma 11.
Lett′′>t′≥t0. If function y(t):R→R satisfies(11)y˙t=-pyθt+qytfor p>0, q>0, and 0<θ<1, then one gets that(12)y1-θt′′=e1-θqt′′-t′y1-θt′+pq1-e1-θqt′′-t′.Proof.
Multiplying (11) by e-qt, we get that(13)e-qty˙y-qe-qtyt=-pe-qtyθtwhich implies(14)de-qtytdt=-pe-qtyθt.Let z(t)=e-qty(t). Then, we have that(15)dzdt=-pe-1-θqtzθ.Integrating (15) on the time interval t′,t′′, we get that(16)z1-θt′′=z1-θt′+pqe-1-θqt′′-e-1-θqt′which implies(17)y1-θt′′=e1-θqt′′-t′y1-θt′+pq1-e1-θqt′′-t′.Theorem 12.
Consider switched system (1). Assume that Assumptions 1–8 hold. Let Υ=(p/q)(ϕ(e(1-θ)qτ-1)/(ϕe(1-θ)qτ-1)) and Ψ=e(1-θ)qτVσ(t0)1-θ(x(t0))-Υ. Then, the origin of (1) is locally finite-time stable for initial value x(t0) satisfying(18)Vσt0xt0≤1eqτpqϕe1-θqτ-1ϕe1-θqτ-11/1-θ.Moreover, the setting time is T=tΠ, where Π=ln(-Υ/Ψ)/ln(ϕe(1-θ)qτ).Proof.
Using the homogeneous functionVi(x), i∈P, we construct the multiple Lyapunov function Vσ(t)(x) corresponding to switching law σ(t). We can see that Vσ(t)(x) is piecewise smooth.
For anyt≥t0, there exists time interval [tj-1,tj), j≥1, such that tj-1≤t<tj. Integrating differential inequalities (8) successively on the intervals [t0,t1),[t1,t2),…,[tj-1,t], we can get(19)Vσtj-11-θxt≤e1-θqt-tj-1Vσtj-11-θxtj-1+pq1-e1-θqt-tj-1Vσtj-21-θxtj-1≤e1-θqtj-1-tj-2Vσtj-21-θxtj-2+pq1-e1-θqtj-1-tj-2⋯Vσt01-θxt1≤e1-θqt1-t0Vσt01-θxt0+pq1-e1-θqt1-t0.Taking into account condition (5), we obtain(20)Vσti1-αxti≤ϕVσti-11-αxtifor any i=1,…,j-1. Substituting (20) into (19) yields(21)Vσtj-11-θxt≤ϕe1-θqt-tj-1Vσtj-21-θxtj-1+pq1-e1-θqt-tj-1Vσtj-21-θxtj-1≤ϕe1-θqtj-1-tj-2Vσtj-31-θxtj-2+pq1-e1-θqtj-1-tj-2⋯Vσt01-θxt1≤e1-θqt1-t0Vσt01-θxt0+pq1-e1-θqt1-t0.It follows that(22)Vσtj-11-θxt≤ϕj-1e1-θqt-t0Vσt01-θxt0+pq1-e1-θqt-tj-1+pqϕ1-e1-θqtj-1-tj-2e1-θqt-tj-1+pqϕ21-e1-θqtj-2-tj-3e1-θqt-tj-2⋯+pqϕj-21-e1-θqt2-t1e1-θqt-t2+pqϕj-11-e1-θqt1-t0e1-θqt-t1≤ϕj-1e1-θqjτVσt01-θxt0+pqϕe1-θqτ-1ϕe1-θqτ-11-ϕj-1e1-θqj-1τ.Therefore, we obtain that(23)0≤Vσtj-11-θxt≤Ψϕj-1e1-θqj-1τ+Υ.It follows that(24)0≤Λxt≤1α1/μΨϕj-1e1-θqj-1τ+Υ1/1-θμ.We can see that Ψ<0, which implies Λ(x(t))=0 for j>1+ln(-Υ/Ψ)/ln(ϕe(1-θ)qτ). Therefore, we conclude that x(t)=0 for t>tΠ.


## 3.2. Stability with (9)

Lemma 13.
Lett′′>t′≥t0. If function y(t):R→R satisfies(25)y˙t=pyθt-qytfor p>0, q>0, and 0<θ<1, then one gets that(26)y1-θt′′=e-1-θqt′′-t′y1-θt′+pq1-e-1-θqt′′-t′.Proof.
Multiplying (25) by e-qt, we get that(27)eqty˙y+qeqtyt=peqtyθtwhich implies(28)deqtytdt=peqtyθt.Let z(t)=eqty(t). Then, we have that(29)dzdt=pe1-θqtzθ.Integrating (29) on the time interval t′,t′′, we get that(30)z1-θt′′=z1-θt′+pqe1-θqt′′-e1-θqt′which implies(31)y1-θt′′=e-1-θqt′′-t′y1-θt′+pq1-e-1-θqt′′-t′.Theorem 14.
Consider switched system (1). Assume that Assumptions 1, 5, and 9 hold. Let Υ=(p/q)(e-1-θqτ-1)/(ϕe-1-θqτ-1) and Ψ=Vσ(t0)1-θ(x(t0))-Υϕ. Then, one can have the following:
(1) Ifϕe-(1-θ)qτ-1<0, then the states x(t) of (1) will converge to set Ω globally as t→∞, where Ω={x:Λx≤1+ϕ-ϕe-1-θqτ(p/q)(e-1-θqτ-1)/(ϕe-(1-θ)qτ-1)}.
(2) Ifϕe-(1-θ)qτ-1>0, then the origin of (1) is globally finite-time stable and the setting time is tΠ, where Π=ln(-Υ1+ϕ-ϕe-1-θqτ/Ψ)/ln(e-(1-θ)qτ).Proof.
Using the homogeneous functionVi(x), i∈P, we construct the multiple Lyapunov function Vσ(t)(x) corresponding to switching law σ(t). We can see that Vσ(t)(x) is piecewise smooth.
For anyt≥t0, there exists time interval [tj-1,tj), j≥1, such that tj-1≤t<tj. Integrating differential inequalities (9) successively on the intervals [t0,t1),[t1,t2),…,[tj-1,t], we can get(32)Vσtj-11-θxt≤e-1-θqt-tj-1Vσtj-11-θxtj-1+pq1-e-1-θqt-tj-1Vσtj-21-θxtj-1≤e-1-θqtj-1-tj-2Vσtj-21-θxtj-2+pq1-e-1-θqtj-1-tj-2⋯Vσt01-θxt1≤e-1-θqt1-t0Vσt01-θxt0+pq1-e-1-θqt1-t0.Taking into account condition (5), we obtain(33)Vσti1-αxti≤ϕVσti-11-αxtifor any i=1,…,j-1. Substituting (33) into (32) yields(34)Vσtj-11-θxt≤ϕe-1-θqt-tj-1Vσtj-21-θxtj-1+pq1-e-1-θqt-tj-1Vσtj-21-θxtj-1≤ϕe-1-θqtj-1-tj-2Vσtj-31-θxtj-2+pq1-e-1-θqtj-1-tj-2⋯Vσt01-θxt1≤e-1-θqt1-t0Vσt01-θxt0+pq1-e-1-θqt1-t0.It follows that(35)Vσtj-11-θxt≤ϕj-1e-1-θqt-t0Vσt01-θxt0+pq1-e-1-θqt-tj-1+pqϕ1-e-1-θqtj-1-tj-2e-1-θqt-tj-1+pqϕ21-e-1-θqtj-2-tj-3e-1-θqt-tj-2⋯+pqϕj-21-e-1-θqt2-t1e-1-θqt-t2+pqϕj-11-e-1-θqt1-t0e-1-θqt-t1≤ϕj-1e-1-θqj-1τVσt01-θxt0+pqe-1-θqτ-1ϕe-1-θqτ-11+ϕ-ϕe-1-θqτ-ϕje-1-θqj-1τ. Therefore, we obtain that(36)0≤Vσtj-11-θxt≤Ψϕj-1e-1-θqj-1τ+Υ1+ϕ-ϕe-1-θqτ. It follows that(37)0≤Λxt≤1α1/μΨϕj-1e-1-θqj-1τ+Υ1+ϕ-ϕe-1-θqτ1/1-θμ. Then, we consider the following two cases.
Case 1 ( ϕ e - ( 1 - θ ) q τ - 1 < 0 ). We can see that x(t) converges to Ω={x:Λ(x)≤Υ(1+ϕ-ϕe-(1-θ)qτ)} as t→∞.
Case 2 ( ϕ e - 1 - θ q τ - 1 > 0 ). We can see that system (1) is globally finite-time stable, and the setting time is tΠ.


## 3.3. Stability with (10)

Lemma 15.
Lett′′>t′≥t0. If function y(t):R→R satisfies(38)y˙t=-pyθt-qytfor p>0, q>0, and 0<θ<1, then one gets that(39)y1-θt′′=e-1-θqt′′-t′y1-θt′-pq1-e-1-θqt′′-t′.Proof.
Multiplying (25) by eqt, we get that(40)eqty˙y+qeqtyt=-peqtyθtwhich implies(41)deqtytdt=-peqtyθt.Let z(t)=eqty(t). Then, we have that(42)dzdt=-pe1-θqtzθ.Integrating (42) on the time interval t′,t′′, we get that(43)z1-θt′′=z1-θt′-pqe1-θqt′′-e1-θqt′which implies(44)y1-θt′′=e-1-θqt′′-t′y1-θt′-pq1-e-1-θqt′′-t′.Theorem 16.
Consider switched system (1). Assume that Assumptions 1, 5, and 10 hold. Let Υ=(p/q)(ϕe-1-θqτe-1-θqτ-1/ϕe-1-θqτ-1) and Ψ=Vσ(t0)1-θ(x(t0))+Υ. Then, one has the following:
(1) Ifϕe-(1-θ)qτ-1<0, then the origin of (1) is globally finite-time stable for any initial values.
(2) Ifϕe-(1-θ)qτ-1>0, then the origin of (1) is locally finite-time stable for initial value x(t0) satisfying Vσt01-θ(x(t0))≤-(p/q)(ϕe-1-θqτe-1-θqτ-1/(ϕe-1-θqτ-1)).
Moreover, the setting times for cases (1) and (2) are bothtΠ, where Π=lnΥ/Ψ/lnϕ/e(1-θ)qτ.Proof.
Using the homogeneous functionVi(x), i∈P, we construct the multiple Lyapunov function Vσ(t)(x) corresponding to switching law σ(t). We can see that Vσ(t)(x) is piecewise smooth.
For anyt≥t0, there exists time interval [tj-1,tj), j≥1, such that tj-1≤t<tj. Integrating differential inequalities (10) successively on the intervals [t0,t1),[t1,t2),…,[tj-1,t], we can get(45)Vσtj-11-θxt≤e-1-θqt-tj-1Vσtj-11-θxtj-1-pq1-e-1-θqt-tj-1Vσtj-21-θxtj-1≤e-1-θqtj-1-tj-2Vσtj-21-θxtj-2-pq1-e-1-θqtj-1-tj-2⋯Vσt01-θxt1≤e-1-θqt1-t0Vσt01-θxt0-pq1-e-1-θqt1-t0.Taking into account condition (5), we obtain(46)Vσti1-αxti≤ϕVσti-11-αxtifor any i=1,…,j-1. Substituting (46) into (45) yields(47)Vσtj-11-θxt≤ϕe-1-θqt-tj-1Vσtj-21-θxtj-1-pq1-e-1-θqt-tj-1Vσtj-21-θxtj-1≤ϕe-1-θqtj-1-tj-2Vσtj-31-θxtj-2-pq1-e-1-θqtj-1-tj-2⋯Vσt01-θxt1≤e-1-θqt1-t0Vσt01-θxt0-pq1-e-1-θqt1-t0.It follows that(48)Vσtj-11-θxt≤ϕj-1e-1-θqt-t0Vσt01-θxt0-pq1-e-1-θqt-tj-1-pqϕ1-e-1-θqtj-1-tj-2e-1-θqt-tj-1-pqϕ21-e-1-θqtj-2-tj-3e-1-θqt-tj-2⋯-pqϕj-21-e-1-θqt2-t1e-1-θqt-t2-pqϕj-11-e-1-θqt1-t0e-1-θqt-t1≤ϕj-1e-1-θqj-1τVσt01-θxt0-pqϕe-1-θqτe-1-θqτ-1ϕe-1-θqτ-11-ϕj-1e-1-θqj-1τ.Therefore, we obtain that(49)0≤Vσtj-11-θxt≤Ψϕj-1e-1-θqj-1τ-Υ.It follows that(50)0≤Λxt≤1α1/μΨϕj-1e-1-θqj-1τ-Υ1/1-θμ.Then, we consider the following two cases.
Case 1 ( ϕ e - 1 - θ q τ - 1 < 0 ). We can see that the switched system (1) is globally finite-time stable with any initial value and the setting time is tΠ.
Case 2 ( ϕ e - ( 1 - θ ) q τ - 1 > 0). We can see that if Ψ<0, then switched system (1) is locally finite-time stable and the setting time also is tΠ.


## 4. Numerical Example

Consider the following switched system:(51)f1x=-x1-x17/9-x23x11/3-135/6x21/3-13x2,f2x=-3x23x12/3-6x15/7-3x1x1-2x21/7-x2.The sequence of switching times is given by t0=0, tk=tk-1+0.5. Choose V1(x)=3x14/3+x24 and V2(x)=x14/3+x24. We get that V1(x) and V2(x) are both homogeneous of degree μ=4 with respect to r=(3,1)T. The derivatives of the homogeneous functions are ∇V1(x)f1(x)≤-(4/3)V1(x)-(4/3)V15/6(x) and ∇V2(x)f2(x)≤-4V2(x)-8V25/6(x). Then, we obtain that p=q=4/3, θ=5/6, and ϕ=3. 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.Figure 1
State trajectories ofx.


## 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.



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*Source: 3096986-2018-08-08.xml*