# Stability Analysis for Autonomous Dynamical Switched Systems through Nonconventional Lyapunov Functions

Authors: V. Nosov; J. A. Meda-Campaña; J. C. Gomez-Mancilla; J. O. Escobedo-Alva; R. G. Hernández-García
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

The stability of autonomous dynamical switched systems is analyzed by means of multiple Lyapunov functions. The stability theorems given in this paper have finite number of conditions to check. It is shown that linear functions can be used as Lyapunov functions. An example of an exponentially asymptotically stable switched system formed by four unstable systems is also given.


---

## Body

## 1. Introduction

Switched systems are present in different areas of science and technology as aeronautical and automotive control, telecommunications, traffic control, chemical process, and so forth [1–5].The switched system is a special class of hybrid or variable structure systems [1, 3, 6–10]. Similar to variable structure systems, the dynamics of switched systems is described by different differential equations in different space regions and the change of dynamics occurs when the trajectories pass through the boundaries between two regions. Variable structure systems may have special type of solutions, the so-called sliding mode solutions. The theory and different applications of sliding mode solutions in control are investigated in many books and papers (see, e.g., [9]). In contrast to sliding mode theory, the case when the variable structure system has no sliding mode solution is not sufficiently studied. To separate these two cases, we call variable structure systems without sliding mode solutions as switched systems. In the theory of hybrid systems the change of systems dynamics may occur by action of automata or by other reasons. In this paper, the stability of switched systems is studied. The problem of stability of switched systems is not simple. First of all, we give some examples illustrating different aspects of this problem and showing that stability of all subsystems is not sufficient to ensure the stability of the whole switched system. Namely, in one example the whole switched system formed by stable subsystems is unstable or asymptotically stable depending on the structure of the switched systems, that is, depending on the regions where these subsystems are acting. If some of subsystems are unstable and others are asymptotically stable, the switched system may be unstable or asymptotically stable or it may have periodic solutions. Also, in some cases the stability of the whole switched system depends neither on the structure of the switched system nor on the order of the switching.The papers [11, 12] were dedicated essentially to the case of linear systems. In paper [13] the stability of switched and hybrid systems is investigated. However, for each trajectory, the theorem on stability from [13] imposes certain conditions on Lyapunov functions at the moment of passing through the switching lines. These conditions are possible to check only if the trajectory is known. Furthermore, typical trajectory of switched systems has an infinite number of switchings and it would be necessary to check an infinite number of conditions. Our previous works [14, 15] investigate some stability problems in switched systems and also investigate possibilities of appearance of chaotic solutions in such systems.In present paper, some stability theorems using multiple Lyapunov functions are established. In contrast to [13] our conditions are imposed on values of Lyapunov functions in corresponding space regions and on switching lines, allowing in this way obtaining different stability conditions without knowing trajectories, while only a finite number of conditions depending only on Lyapunov functions are verified. The Lyapunov functions used in these theorems may be different from usual Lyapunov functions defined in the whole space, which allows extending the class of functions used as Lyapunov functions. For example, it is possible to use linear Lyapunov functions to investigate the stability of switched systems. The sum of quadratic and linear functions may be used also as Lyapunov functions. Some examples of such functions are also given. The using of unusual Lyapunov functions simplifies the search for convenient Lyapunov function. We present also one example (Example 15) where all subsystems are unstable by considering the whole space, but the switched system formed by these subsystems is exponentially asymptotically stable.Our theorems which give conditions of asymptotic stability or instability of whole switched system in case when all subsystems are only stable are especially interesting. These results may be considered as a special type of parametric excitation or parametric stabilization of stable systems.


## 2. Description of Autonomous Two-Dimensional Switched Systems

Suppose the phase spaceR2, X=(x,y)T∈R2, with norm X, is divided into a finite number p of open 1-connected regions Qi, 1≤i≤p, with smooth boundaries such that the origin {0} of rectangular coordinate systems pertains to closure of any region Qi. The boundary between two regions Qi and Qi+1 is noted by Li,i+1 and is called switching line. These switching lines are supposed to be smooth and let the normal Ni,i+1(X) to the switching lines existing in each point X∈Li,i+1 have direction from Qi to Qi+1. Suppose each region Qi has points X with norms such that X>H, where H is any number. The topological properties of plane conduce to the conclusion that each region Qi has only two boundaries which go from origin {0} to infinity without intersections.In each regionQi the dynamics of switched systems is described by a proper autonomous equation Ei with Lipschitz continuous function fi(X)=fi(x,y):(1)Ei:X˙=fiX,X∈Qi,1≤i≤p,with initial condition(2)Xt0=X0.The trajectory of switched system (1)-(2) may pass from region Qi to region Qi+1 only crossing the switching line Li,i+1. Suppose also the nonexistence of sliding modes in switched system (1)-(2). The sufficient condition for absence of sliding modes is the followingtransversality condition: the normal Ni,i+1(X) which goes from Qi to Qi+1 and the trajectory velocities fi(X) and fi+1(X) from one to another side of switching line form acute angles; that is,(3)sign⁡Ni,i+1X,fiX=sign⁡Ni,i+1X,fi+1X=1,X∈Li,i+1,for every point on switching line Li,i+1 and all lines Li,i+1, i=1,2,3,…,p (Figure 1). Transversality condition (3) guarantees the passage only from region Qi to Qi+1 and not in opposite direction.Figure 1
Autonomous two-dimensional switched system.If at instantti,fin the trajectory of switched system arrives from region Qi on the switching line Li,i+1, the trajectory passes across the switching line Li,i+1 and at the following instants t>ti,fin the system dynamic is described by equation Ei+1 acting in region Qi+1 with initial condition Xti+1,init for the new equation Ei+1 coinciding with the final condition Xti,fin of the previous equation Ei on the switching line Li,i+1, for i=1,2,3,…; that is, Xti+1,init=Xti,fin. Under this condition, all trajectories of switched system will be continuous for all t>t0. The regions Qi together with equations Ei completely define the switched system. The set of regions Qi defines the geometrical structure while the set of equations Ei defines the dynamical structure of switched system.Any switched system for a chosen initial conditionX(t0)=X0 generates a sequence of continuous dynamical subsystems S1(X0),S2(X0),…,Si(X0),… acting in regions Q1(X0)=Qk, 1≤k≤p, Q2(X0)=Qk+1,…,Qi(X0),… and switching on lines L1(X0),2(X0)=Lk,k+1,L2(X0),3(X0),…,Li(X0),(i+1)(X0),…; that is,(4)SWX0=S1X0,S2X0,…,SiX0,….The initial conditionX0 defines the first element in (4). Depending on X0, the first subsystem S1(X0) is acting from initial time t0=t1(X0),init=tk,init to the first switching instant t1(X0),fin=tk,fin in region Qk such that X(t0)∈Qk. The second subsystem S2(X0) is acting in Qk+1 from time t2,init=t1,fin to the second switching instant t2,fin=t3,init and so on. In other words, each subsystem Si is defined by three elements Si=Qi,Ei,X(ti,init):(a)
regionQi where this subsystem is acting,(b)
differential equationEi acting in Qi,(c)
initial valuesti,init and X(ti,init).This definition is slightly different from definition given in [1, 3, 13]. The sequence (4) may contain finite or infinite number of dynamical subsystems Si although the total number of different regions Qi is finite because trajectory can return to region Qi after the whole rotation around origin, and therefore Qp+i=Qi for all i. If condition (3) holds for all switching lines, then the sequence (4) may contain an infinite number of subsystems S1,S2,…,Sn,…. In case of infinite number of subsystems Si, in sequence (4) each region Qi has two boundary switching lines: by one of them the trajectories enter from the precedent region Qi-1, and by the other, after a finite time, the trajectories go out from region Qi to region Qi+1. After the whole rotation around origin the switched system returns to initial region Qi and Lp,p+1=Lp,1.This property has, as a consequence, the following fundamental property of autonomous two-dimensional switched systems: the order of terms in sequence (4) does not depend on initial conditions.In multidimensional phase spaceRn, n>2, each region may have more than two switching surfaces. For that reason, there does not exist an analogy of announced fundamental property in multidimensional space. So, in Rn the order of terms in (4) may depend on initial conditions. This complicates the stability investigation of switched systems in multidimensional space Rn, n>2.The number of dynamical subsystemsSi(X0) is finite, 1≤i≤m, if the switched system stays in the final region QN for all time after last switching tm-1,fin, t>tm-1,fin. The switched system may have more than one final region and the final region may depend on initial conditions X0. The sufficient condition for finiteness of sequence (4) is the existence of at least one line Lm,m+1 such that from one side of the line Lm,m+1 the normal Nm,m+1(X) and the trajectory velocities fm(X) and fm+1(X) form an obtuse and acute angle, respectively:(5)sign⁡Nm,m+1X,fmX=-1,sign⁡Nm,m+1X,fm+1X=1,for X∈Lm,m+1.In this case, the subsystemSm will be the last subsystem in the sequence (4). If there exist more than one switching line satisfying condition (5) then the switched system may have more than one final region and the final region may depend on initial conditions X0. Let us consider two examples.Example 1.
Suppose the switched system is described by two pendulum equationsE1 and E2 with different natural frequencies acting in regions Q1 and Q2 of phase plane R2:(6)E1:x˙=y,y˙=-x,∀X∈Q1=R2-Q2,while equation E2 is(7)E2:x˙=y,y˙=-ω2x,∀X∈Q2=x<0,y>0,
Every trajectory of switched system has an infinite number of switchings and it is represented by an infinite number of continuous dynamical subsystemsS1,S2,…,Si,….
The subsystemsS1,S3,…,S2j+1,… are described by the same equation E1 in the form of (6) acting in region Q1 and the subsystems S2,S4,…,S2j,… are described by equation E2 in the form of (7) acting in region Q2. But the initial conditions for S1,S3,…,S2j+1,… and S2,S4,…,S2j,… are different: X1 for S1, X3 for S3, and so on. Therefore all subsystems S1,S2,…,Si,… are different (see Figure 2).Figure 2
Switched system with infinite sequence (4) of subsystems Si.Example 2.
Consider now the switched system withE1 acting in the region Q1={x<0}∩{y>0}∩{y<(-2+3)x}, E2 acting in the lower half-space Q2={y<0}, and E3 acting in Q3=R2-Q1-Q2 with(8)E1:x¨+4x+x=0,∀X∈Q1,(9)E2:x¨+ω2x=0,∀X∈Q2,(10)E3:x¨+x=0,∀X∈Q3.
The equationE1 is overdamped and its general solution is (11)xt=A·exp⁡-2+3t+B·exp⁡-2-3t,A,B=const.
The trajectories of the whole switched system are presented in Figure3 with ω=3. The region denoted as Q1 in Figure 3 is a final region of the switched system (8)–(10). Depending on initial conditions, the trajectory may has two, one, or zero commutations. If X0 belongs to region Q3 then the trajectory has two commutations, if X0 belongs to region Q2 then the trajectory has one commutation, and if X0 belongs to region Q1 then the trajectory has no commutations.Figure 3
Switched system with finite sequence of subsystems (8)-(9).


## 3. Stability of Switched Systems

Suppose(12)fi0=0,1≤i≤p.Under condition (12) switched system (1)-(2) or (4) has the trivial solution X(t)≡0.In the following, the stability analysis of trivial solution (or origin) of switched system (1)-(2) will be carried out.Let us introduce some definitions of Lyapunov stability.Definition 3.
The trivial solutionX(t)≡0 (or origin) of switched system (1)-(2) is said to be stable if for any ϵ>0 there exists a δ=δ(ϵ) such that the inequality X(t,t0,X0)<ϵ is satisfied for any time t>t0 whenever X0<δ(ϵ).Definition 4.
The trivial solutionX(t)≡0 (or origin) of system (1)-(2) is said to be asymptotically stable if (a)
it is stable;(b)
there existsΔ>0 such that X(t,t0,X0)→0, t→∞ for X0<Δ.Clearly, the stability (asymptotic stability) is uniform with respect tot0 because the switched system is a stationary system with all elements independent of t.Obviously, in case of finite sequence (4) the stability or instability of the whole switched system origin depends only on stability of instability of origin for final subsystem Sm.In case of infinite sequence (4) the instability of one equation Ei does not conduce automatically to the instability of the whole switched system. Also, the stability of all equations Ei is not sufficient to conclude that the whole switched system is stable [15].Let us give other corresponding examples.Example 5.
Consider the switched system with two equationsE1 and E2 acting in two half-planes Q1={x˙=y>0} and Q2={x˙=y<0}, respectively, with(13)E1:x¨-0.2x˙+1.01x=0,x,x˙=y∈Q1,(14)E2:x¨+0.4x˙+1.04x=0,x,x˙=y∈Q2.
Switching linesL1,2 and L2,1 in this case are L1,2={x>0,y=0} and L2,1={x<0,y=0}. Equation E1 is unstable and E2 is asymptotically stable. The asymptotic stability of equation E2 can be called stronger than the instability of E1. It is easy to verify that for any X1={x1>0,y1=0} the subsequent points, X2={x2<0,y2=0}, X3={x3>0,y3=0},…, of intersection with the axis x are equal to(15)x2=-x1exp⁡-0.2π,x3=-x2exp⁡0.1π=x1exp⁡-0.1π<x1.
Therefore, the trivial solution of switched system (13)-(14) is asymptotically stable.
Replace (14) in region Q2 by(16)E3:x¨+0.2x˙+1.01x=0,x,x˙=y∈Q2.
Now, the same calculations show thatx3=x1 for all x1. Therefore all solutions of switched system formed by unstable equation (13) and by asymptotically stable equation (16) are periodic.Example 6.
Consider switched system of Example1:(17)E1:x¨+ω2x=0,x,x˙=y∈Q1=x<0,x˙>0,(18)E2:x¨+x=0,x,x˙=y∈Q2=R2-Q1.
Both equationsE1 and E2 are stable. Also, it is easy to calculate for any X1={x1=0,y1>0} the values of X2={x2=-y1<0,y2=0}, X3={x3=0,y3>0} and that(19)y3=ωy1.
Therefore, ifω<1 then the switched system origin is asymptotically stable, but if ω>1 the switched system origin is unstable. Suppose now Q1 coincides with the first quadrant; that is, Q1={x>0,y>0}; then analogous calculations show that the switched system origin is asymptotically stable if ω>1, but if ω<1 the switched system origin is unstable.Remark 7.
These two examples show that stability of switched systems depends not only on the stability of equationsEi but also on all other elements which define the switched system; that is, it depends also on the regions where different equations Ei are acting and on the order of the switching.Thus, the following definitions are justified.Definition 8.
The origin of switched system (4) formed by an infinite sequence of subsystems S1(X0),S2(X0),…,Si(X0),… is stable (asymptotically stable) independently of geometrical structure of switched system if the origin is stable (asymptotically stable) for any choice of equations E1,…,Ep acting in arbitrary chosen regions Q1,…,Qp.Definition 9.
The origin of switched system (4) formed by a sequence of subsystems S1,S2,…,Si,… is stable (asymptotically stable) for given geometrical and dynamical structure of switched system if the origin is stable (asymptotically stable) for a given choice of equations E1,…,Ep acting in corresponding regions Q1,…,Qp.


## 4. General Theorems

The previous examples show that the stability properties of switched system origin need special investigation and they do not follow directly from simple stability conditions imposed on subsystems, but also they depend on how the switching occurs within the whole system.The only interesting case when sequence (4) which determines the switched system is infinite is considered below.Suppose equationsEi are stable (asymptotically stable) and also suppose the existence of some smooth positive definite functions Vi(X) defined for X∈Qi. Each function Vi(X) is called Lyapunov function in region Qi, if it is continuously differentiable in Qi and fulfilling Vi(X)>0, X∈Qi, X≠0, and Vi(0)=0 [16, 17].Denote byωk(u) scalar continuous nondecreasing functions (also called wedges) defined and positive for u>0 such that ωk(0)=0.The symbolsV˙i(X(t)) denote the derivatives of the functions Vi(X(t)) along the trajectory of equation Ei:(20)V˙iXt=∇VX,fiX,X∈Qi.The existence of a common Lyapunov function for all equationsEi simplifies the stability analysis of switched system origin. For this reason, Theorem 10 has been explicitly formulated but not proven, because this theorem is a direct consequence of the more general Theorem 12.Theorem 10.
Suppose there exists a common Lyapunov functionV(X) defined in the whole space R2 satisfying the following: (a)
ω1X≤V(X)≤ω2X, X∈R2,(b)
for any equationEi the following conditions are fulfilled(21)V˙Xt=∇VX,fiX≤0,V˙iX≤-ω3X<0,forX∈Qi, 1≤i≤p.Then, the trivial solution X(t)≡0 of switched system (1)-(2) is stable (asymptotically stable) independently of geometrical structure of switched system.Example 11.
Consider three equations:(22)E1:x¨+x=0,E2:x¨+x+x2x˙3=0,E3:x¨+sin⁡x˙+x=0.
As a common Lyapunov functionV(X) we can take the function(23)Vx,x˙=x2+x˙2.
The derivatives of these functions along the trajectories of equationsE1, E2, and E3 are not positive:(24)V˙ec21x,x˙=2xx˙-2xx˙=0,V˙ec22x,x˙=2xx˙-2x˙-x-x2x˙3=-2x2x˙4≤0,Vec23x,x˙=2xx˙-2x˙-x-sin⁡x˙=-2x˙sin⁡x˙≤0,x˙<1.
Therefore, the trivial solution of the switched system formed by equationsE1, E2, and E3 is stable independently of geometrical structure of switched system determined by a choice of arbitrary regions Q1, Q2, and Q3 such that Q1∪Q2∪Q3=R2.Theorem 12.
Suppose conditions (3) hold for all switching lines Li,i+1 and suppose that for each equation Ei acting in Qi∈R2 there exists a Lyapunov function Vi(X) defined in Qi such that (a)
ω1,iX≤Vi(X)≤ω2,iX, X∈Qi, i=1,…,p,(b)
V˙i(X(t))=〈∇(Vi(X)),fi(X)〉≤0, V˙i(X(t))≤-ω3,iX(t)<0, X∈Qi, i=1,…,p,(c)
on all switching linesLi,i+1 where trajectories pass from region Qi to Qi+1, the following inequalities hold: Vi(X)≥Vi+1(X), X∈Li,i+1.Then the trivial solution X(t)≡0 of switched system (4) is stable (asymptotically stable) for given switched system.Proof.
Let us proof stability of trivial solutionX(t)≡0. Denote (25)ω1u=mini⁡ω1,iu,u>0,ω2u=maxi⁡ω2,iu,u>0,ω3u=mini⁡ω3,iu,u>0,ω10=ω20=ω30=0.
The wedge functionsωi(u), i=1,2,3, are scalar continuous nondecreasing functions positive for u>0 and satisfy (25). For a given ϵ>0 we define a number δ=δ(ϵ) such that ω2(δ)≤ω1(ϵ). Evidently, such δ does exist. Using conditions (b) and (c) we obtain that the sequence of Lyapunov functions Vk(X(t,t0,X0)) for arbitrary k and tk,init<t<tk,fin is nonincreasing on the trajectories of switched system:(26)ω1Xt,t0,X0≤ω1,kXt,t0,X0≤VkXt,t0,X0≤VkXtk,init,t0,X0≤Vk-1Xtk-1,fin,t0,X0≤Vk-1Xtk-1,init,t0,X0≤⋯≤V1X0X0.
Using now condition (a) of the theorem, we obtain for arbitraryk and t, tk,init<t<tk,fin the following inequalities:(27)ω1Xt,t0,X0≤V1X0X0≤ω2,1X0≤ω2X0≤ω2δ≤ω1ϵ,
Therefore,(28)Xt,t0,X0≤ϵ,t≥t0,X0≤δ.
Inequality (28) is equivalent to stability of trivial solution of switched system (1)-(2).
To proof asymptotic stability it is necessary to demonstrate that for anyγ>0 there exist numbers Δ>0 and T(γ) such that X(t,t0,X0)≤γ for t≥t0+T(γ) and X0≤Δ. Take any γ>0 and determine δ>0 which corresponds to γ in demonstration of stability; that is, ω2(δ)≤ω1(γ). Take also T(γ)=2w2(Δ)/ω3(δ)>0.
Let us demonstrate that on the interval[t0,t0+T(γ)] there exists an instant t1 such that X(t1,t0,X0)≤δ.
If this is not so, that is,X(t,t0,X0)>δ for all t∈[t0,t0+T(γ)], then using continuity of X(t,t0,X0) and conditions (b) and (c) of Theorem 12 we have(29)VkXt0+Tγ,t0,X0≤V1X0X0+∫t0t1,finV˙1X0Xt,t0,X0dt+∫t1,fint2,finV˙2X0Xt,t0,X0dt+⋯+∫tk,initt0+TγV˙kXt,t0,X0dt≤V1X0X0-∫t0t1,finω3,1Xt,t0,X0dt-∫t1,fint2,finω3,2Xt,t0,X0dt-⋯-∫tk,initt0+Tγω3,kXt,t0,X0dt≤V1X0X0-∫tk,initt0+Tγω3Xt,t0,X0dt≤V1X0X0-Tγω3δ≤ω2Δ-Tγω3δ=-ω2Δ<0.
Inequality (29) contradicts the positiveness of Lyapunov function Vk. Therefore, there exists an instant T(γ) and t1, t0≤t1≤t0+T(γ) such that X(t1,t0,X0)≤δ for X0≤Δ. From condition w2(δ)≤ω1(γ) and the stability of the trivial solution it follows that X(t,t0,X0)≤γ for all t≥t1, t1≤t0+T(γ). Therefore X(t,t0,X0)≤γ for all X0≤Δ and for all t≥t0+T(γ). The asymptotic stability of the origin is proven.Example 13.
Consider once again first switched system of Example6 where Q1 coincides with second quadrant, Q1={x<0,y>0}. Lyapunov functions for equations E1, E2 are (30)V1x,y=ω2x2+y2,V2x,y=x2+y2,respectively.
On switching lineL1,2 where x=0 and y>0, we have V1(x,y)=y2=V2(x,y). On switching line L2,1 where y=0 we have V1(x,y)=ω2x2 and V2(x,y)=x2. Therefore, condition (c) of Theorem 12 holds on both lines L1,2 and L2,1 if ω2<1. Under this condition, the origin of the switched system described by (17)-(18) of Example 6 is stable.
Consider now the second switched system of Example6, where Q1 coincides with the first quadrant Q1={x≥0,y≥0}. In this case, on line L2,1, where x=0 we have V1(x,y)=y2=V2(x,y) and on line L1,2 where y=0 we have V1(x,y)=ω2x2 and V2(x,y)=x2. Therefore condition (c) of Theorem 12 holds if ω2>1. In this case, the origin of the second switched system of Example 6 is stable under this situation. This conclusion coincides with results of Example 6 obtained by direct analytical computations. Analytical calculations show asymptotic stability of switched system origin (17)-(18), while by using Theorem 12 we can only establish stability but no asymptotic stability.
Replacing (18) in region Q2 by new equation(31)E2:x¨+x+x2x˙3=0,X∈Q2,and considering the same Lyapunov function V2(x,y), we can also establish stability of the trivial solution of this new switched system (17), (31), but analytical calculations cannot be used in this case.Example 14.
Consider the switched system with two equationsE1 and E2 acting in two half-planes Q1={y>0} and Q2={y<0}:(32)E1:x˙=y+x2,y˙=-2xy-2x3,x,y∈Q1,E2:x˙=y,y˙=-x,x,y∈Q2.
The switching lines areL1,2={x>0,y=0} and L2,1={x<0,y=0}. Normal N1,2 going from Q1 to Q2 is N1,2=0-1, and condition (3) on line L1,2 has the form (33)N1,2,f1=2x3>0,N1,2,f2=x>0.
On the other hand, normalN2,1 going from Q2 to Q1 is N2,1=01, and condition (3) on line L2,1 has the form(34)N2,1,f1=-2x3>0,N2,1,f2=-x>0.
As Lyapunov functions for equationsE1 and E2 in regions y>0 and y<0 take(35)V1x,y=x2+y>0,y>0,(36)V2x,y=x2+y2,y<0.
The strange function (35) satisfies condition (a) of Theorem 12 for unusual norm in R2 of the form X=x2+y. The switching lines are L1,2={y=0,x>0} and L2,1={y=0,x<0} and on these switching lines V1(x,y)=x2=V2(x,y). The derivatives of functions (35) and (36) in regions y>0 and y<0 are equal to(37)V˙1x,y=2xx˙+y˙=2xy+x2+-2xy-2x3=0,y>0;V˙2x,y=2xx˙+2yy˙=2xy-2xy=0,y<0.
All conditions of Theorem12 hold and therefore, the trivial solution of switched system (32) is stable.Example 15.
Consider a switched system with four equations acting in four quadrants of the plane as shown in Figure4:(38)E1:x˙=3x+2y,y˙=-4x-3y,x,y∈Q1=x>0,y>0,E2:x˙=-3x+5y,y˙=-2x+4y,x,y∈Q2=x>0,y<0,E3:x˙=3x+y,y˙=-4x-2y,x,y∈Q3=x<0,y<0,E4:x˙=-4x+3y,y˙=-2x+2y,x,y∈Q4=x<0,y>0,where Q1∪Q2∪Q3∪Q4=R2.
It is easy to observe that equationsE1, E2, E3, and E4 are unstable in the whole plane R2. To analyze the stability of the switched system origin consider the following four linear Lyapunov functions defined in corresponding regions:(39)V1x,y=x+y,V1x,y>0,x,y∈Q1=x>0,y>0,V2x,y=x-y,V2x,y>0,x,y∈Q2=x>0,y<0,V3x,y=-x-y,V3x,y>0,x,y∈Q3=x<0,y<0,V4x,y=-x+y,V4x,y>0,x,y∈Q4=x<0,y>0.
Clearly, functions (39) satisfy conditions (a) of Theorem 12 if the norm in R2 is equal to X1=x+y. The derivatives of functions (39) are(40)V˙1x,y=x˙+y˙=3x+2y-4x-3y=-x-y=-V1,x,y∈Q1,V˙2x,y=x˙-y˙=-3x+5y+2x-4y=-x+y=-V2,x,y∈Q2,V˙3x,y=-x˙-y˙=-3x-y+4x+2y=x+y=-V3,x,y∈Q3,V˙4x,y=-x˙+y˙=4x-3y-2x+2y=2x-y<-V4,x,y∈Q4.
The derivatives of Lyapunov functionsV1, V2, V3, and V4 are negative definite with respect to norm X1 in the corresponding regions Qi.
On switching linesL1,2,…,L4,1, where x=0 or y=0, the corresponding Lyapunov functions are(41)L1,2:V1x,y=V2x,y=x,N1,2=0-1,N1,2,f1=4x>0,N1,2,f2=2x>0,y=0,L2,3:V2x,y=V3x,y=-y,N2,3=-10,N2,3,f2=-5y>0,N2,3,f3=-y>0,x=0,L3,4:V3x,y=V4x,y=-x,N3,4=01,N3,4,f3=-4x>0,N3,4,f4=-2x>0,y=0,L4,1:V4x,y=V1x,y=y,N4,1=10,N4,1,f1=3y>0,N4,1,f1=2y>0,x=0.
Furthermore, all conditions of Theorem12 are fulfilled; therefore, the trivial solution of the switched system is asymptotically stable (Figure 4). Moreover, as V˙i≤-Vi the trivial solution is exponentially asymptotically stable.Figure 4
Trajectory of switched system (38).Example 16.
Consider now the switched systems formed by the following nonlinear subsystems:(42)E1:x˙=3x+2y-xy,y˙=-4x-3y-x2y2,x,y∈Q1,E2:x˙=-3x+5y+xy,y˙=-2x+4y+x2y,x,y∈Q2,E3:x˙=3x+y,y˙=-4x-2y+xy,x,y∈Q3,E4:x˙=-4x+3y-xy,y˙=-2x+2y+x3y,x,y∈Q4,with Q1, Q2, Q3, and Q4 as in Example 15. As before, functions (39) fulfill conditions of Theorem 12 if the norm in R2 is equal to X1=x+y. The derivatives of functions (39) are(43)V˙1x,y=x˙+y˙=-x-y-xy-x2y≤-V1x,y<0,x,y∈Q1,V˙2x,y=x˙-y˙=-x+y+xy-x2y≤-V2x,y<0,x,y∈Q2,V˙3x,y=-x˙-y˙=x+y-xy≤-V3x,y<0,x,y∈Q3,V˙4x,y=-x˙+y˙=2x-y+xy+x3y≤-V4x,y<0,x,y∈Q4.
The derivatives of Lyapunov functionsV1, V2, V3, and V4 are negative definite with respect to norm X1 in the corresponding regions Qi, and on switching lines L1,2,…,L4,1, where x=0 or y=0, the corresponding Lyapunov functions coincide with (41). Therefore, the trivial solution of the switched system (42) is exponentially asymptotically stable. The simulation results are shown in Figure 5.Figure 5
Trajectory of switched system (42).Theorem 17.
Suppose that sequence (4) is infinite and that for each equation Ei, i=1,…,p, there exists a Lyapunov function Vi(X) such that (a)
ω1,iX<Vi(X)<ω2,iX, X∈Qi, i=1,…,p,(b)
V˙i(X(t))=〈∇(Vi),fi(X)〉≤0, X∈Qi, i=1,…,p,(c)
on all switched linesLi,i+1, 1<i<p, where the trajectories pass from region Qi to Qi+1, the following inequalities hold:(44)ViX≥Vi+1X,X∈Li,i+1,1<i<p.Furthermore, there exists at least one line Lk,k+1, 1<k<p such that(45)VkX≥Vk+1X+ω3,kX,X∈Lk,k+1.Then, the trivial solution X(t)≡0 of the switched system (1)-(2) or (4) is asymptotically stable for a given switched system.Proof.
The trivial solution is stable because conditions of Theorem12 are satisfied. It means that any Lyapunov function Vk(X(t)) in region Qk as function of t does not increase. Moreover, the sequence of successive Lyapunov functions as function of t does not increase either. More exactly, for any numbers j, k and any instants tj, tk such that X(tj)∈Qj, X(tk)∈Qk, tj<tk, the following inequality holds: Vk(X(tk))≤Vj(X(tj)).
Suppose the trivial solutionX(t)≡0 of switched system (4) is not asymptotically stable. It means that there exists the solution X(t,t0,X0) with sufficiently small X0, X0<δ, such that it is bounded X(t,t0,X0)<H and does not tend to zero as t→∞. In this case there exists γ>0 and a sequence tj→∞ such that X(tj,t0,X0)>γ, γ<H. The solution X(t,t0,X0) must satisfy condition(46)Xt,t0,X0≥μ,t≥t1,where μ is a number corresponding to γ in definition of stability for solution X(t)≡0; that is, ω2(μ)≤ω1(γ). If (46) does not hold then there exists an instant T>t1 such that(47)XT,t0,X0<μ.
Taking nowT as a new initial moment and using stability of X(t,t0,X0)=X(t,T,X(T,t0,X0)) this solution must satisfy condition(48)Xt,t0,X0<γ,t>T.
Condition (48) contradicts condition (46) and this means that condition (46) holds. It follows from (46) that for an infinite number of moments ti(k)>T, i(k)→∞, when solution X(t,t0,X0) crosses line Lk,k+1 we have H>X(ti(k),t0,X0)≥μ>0, ω3,kX(ti(k),t0,X0)≥ω3,k(μ)>0, and(49)ω2H≥VkXtik,t0,X0≥Vk+1Xtik,t0,X0+ω3,kXtik,t0,X0≥VkXtik-1,t0,X0+ω3,kXtik-1,t0,X0+ω3,kXtik,t0,X0≥Vk+1Xtik-1,t0,X0+ω3,kXtik-1,t0,X0+ω3,kXtik,t0,X0≥⋯≥Vk+1Xt1,t0,X0+ω3,kXt1,t0,X0+⋯+ω3,kXtik-1,t0,X0+ω3,kXtik,t0,X0≥ikω3,kμ⟶∞,ik⟶∞.
Inequality (49) contradicts condition ω2(H)<∞ and our supposition that X(t,t0,X0) does not tend to zero conducts to a contradiction. Furthermore, the asymptotic stability of trivial solution X(t)≡0 of switched system (4) is proven.Theorem 18.
Suppose that sequence (4) is infinite and that for all switching lines Li,i+1 and that for each Ei, i=1,…,p, there exists a Lyapunov function Vi(X) such that (a)
ω1,iX<Vi(X)<ω2,iX, X∈Qi, i=1,…,p,(b)
V˙i(X(t))=〈∇(Vi),fi(X)〉≥0, X∈Qi, i=1,…,p,(c)
on all switched linesLi,i+1, 1≤i<p, where the trajectories pass from region Qi to Qi+1, the following inequalities hold:(50)ViX≤Vi+1X,X∈Li,i+1,1≤i<p,furthermore, there exists at least one line Lk,k+1, 1≤k<p, such that(51)VkX≤Vk+1X+ω3,kX,X∈Lk,k+1.Then, the trivial solution X(t)≡0 of the switched system (4) is unstable for a given switched system.Proof.
Theorem18 is a clear modification of Theorem 17. Therefore its proof is omitted.Example 19.
Consider the switched system formed by two nonlinear pendulums:(52)E1:x¨+sin⁡x=0,x,x˙=y∈Q1=x<0,y>0,E2:x¨+ω2sin⁡x=0,x,x˙=y∈R2-Q1.
Take as Lyapunov functions for (52) the following functions:(53)V1x,y=1-cos⁡x+12y2,x,x˙=y∈Q1=x<0,y>0,V2x,y=ω21-cos⁡x+12y2,x,x˙=y∈R2-Q1.
The derivatives of functions (53) along the trajectories of (52) are equal to zero: V˙1=V˙2=0. On switching line L1,2 where x=0, y>0 we have V1(x,y)=(1/2)y2=V2(x,y). On switching line L2,1 where x<0 and y=0 we have V1(x,y)=(1-cos⁡x) and V2(x,y)=ω2(1-cos⁡x). Therefore, condition (c) of Theorem 18 holds on both lines L1,2 and L2,1 if(54)ω2<1.
Condition (54) is a condition of instability of the trivial solution of switched system (52). Also, this condition may be considered as a condition of parametric discontinuous excitations for the nonlinear pendulum.


## 5. Autonomous Multidimensional Switched Systems

Consider multidimensional switched system acting inRn which is divided into a finite number p of open 1-connected regions Qk,1≤k≤p, Q1+Q2+⋯+Qp=Rn, such that the origin {0} of rectangular coordinates systems pertains to closure of any region Qk. The boundaries of all regions Qk are supposed to be smooth. Suppose, also, that each region Ql has points X fulfilling condition X>H, where H is any number. An equation Ek is defined in each region Qk satisfying conditions of existence and uniqueness of solutions. In multidimensional case, the switched system may have much more complicated behavior because region Qk may have more than two boundaries separating it from other regions. To eliminate topological complications, consider only the case when all boundaries (switching surfaces) Bk,k+l separating regions Qk and Qk+l are planes in Rn. Suppose the trajectories of switched system pass through the boundary Bk,k+l in all points only in direction from Qk to Qk+l and on all lines Lk,k+l where Qk touches other regions all trajectories pass from Qk only to one determined region Qk+j. Suppose also the existence of some Lyapunov functions Vk(X) for all equation Ek, 1≤k≤p. In this case theorems similar to Theorems 10, 12, 17, and 18 may be established. The definitions of stability for switched systems in Rn are the same as for the two-dimensional case. Now, consider the case of a switched system with infinite number of switchings. Let us formulate one of the theorems on stability.Theorem 20.
Suppose that for each equationEk acting in Qk∈Rn there exists a Lyapunov function Vk(X) defined in Qk such that(a)
ω1,kX<Vk(X)<ω2,kX, X∈Qk, k=1,…,p,(b)
V˙k(X(t))=〈∇(Vk),fk(X)〉<0, X∈Qk, k=1,…,p,(c)
on all switched surfacesBk,k+j and lines Lk,k+j where the trajectories pass from region Qk to Qk+j, the following inequalities hold:(55)VkX>Vk+jX,X∈Bk,k+j,VkX>Vk+lX,X∈Lk,k+l.Then, the trivial solution X(t)≡0 of the switched system acting in Rn is stable (asymptotically stable) for a given switched structure.Proof.
The demonstration of this theorem is similar to proof of Theorem12. Therefore it is omitted.Example 21.
In spaceR3 consider two regions Q1={z>0} and Q2={z<0}. Suppose equations E1 and E2 are defined in Q1 and Q2:(56)E1:x˙=-x+y+y2,y˙=-x-y,z˙=-2z-zx2-2y2x,x,y,z∈Q1,E2:x˙=-x+y-y2,y˙=-x-y-xy,z˙=-2z-4y2x,x,y,z∈Q2.
NormalsN1,2 and N2,1 to the surface B=z=0 separating Q1 and Q2 are N1,2T=[0,0,-1]T and N2,1T=[0,0,1]T. Therefore, condition (3) holds on surface B=z=0: (57)N1,2,fE1=2y2x>0,N1,2,fE2=4y2x>0,x>0,N2,1,fE1=-2y2x>0,N2,1,fE2=-4y2x>0,x<0.
Therefore, surfacesB1,2 and B2,1 where the trajectories pass from E1 to E2 or vice versa are B1,2={z=0,x>0,y=arbitrary} and B2,1={z=0,x<0,y=arbitrary}. On line {z=0,x=0,y=arbitrary} the derivatives x˙ from one and another side of this line are positive: x˙=y2>0. Therefore, the trajectory which arrives on this line passes throughout it and then enters into the region Q2. Consider two Lyapunov functions V1 and V2 defined in Q1 and Q2, respectively: (58)V1x,y,z=x2+y2+z,x,y,z∈Q1,V2x,y,z=x2+y2-z,x,y,z∈Q2.
DerivativesV˙1,E1 and V˙2,E2 computed on the trajectories of equations E1 and E2, respectively, are (59)V˙1,E1=2xx˙+2yy˙+z˙=2x-x+y+y2+2y-x-y+-2z-zx2-2y2x=-2x2-2y2-2z-zx2<-2V1,x,y,z∈Q1,V˙2,E2=2xx˙+2yy˙-z˙=2x-x+y-y2+2y-x-y-xy+-2z-4y2x=-2x2-2y2+2z=-2V2,x,y,z∈Q2.
All conditions of Theorem20 are fulfilled and therefore the considered switching system (56) is asymptotically stable and, moreover, exponentially asymptotically stable (see Figure 6).Figure 6
Trajectory of switched system (56).


## 6. Conclusions

The stability of autonomous switched systems is investigated. Some theorems giving sufficient conditions of stability, asymptotic stability, or instability of switched systems are established. Some results are similar to conditions of parametric stabilization or excitation of stable systems.



---

*Source: 502475-2015-10-11.xml*