# Output Feedback Stabilization for Stochastic Nonholonomic Systems under Arbitrary Switching

Authors: Xianjun Liu; Yinghui Zhao; Ye Yue; Dongkai Zhang
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

The output feedback controllers of stochastic nonholonomic systems under arbitrary switching are discussed. We adopt an observer which can simplify the design process. The designed control laws cause the calculation of the gain parameter to be very convenient since the denominator of virtual controllers does not contain the gain parameter. Finally, an example is given to show the effectiveness of controllers.


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## Body

## 1. Introduction

In recent years, switched system’s control, especially under arbitrary switching, has become an active field [1–3]. The global stabilization of switched systems based on arbitrary switching was given [4–6]. The stabilizing controllers of switched systems with arbitrary switching were given [7–10]. The output feedback controllers of nonlinear systems with arbitrary switching were designed [11, 12].In the past ten years, the problem of stabilization for stochastic nonholonomic systems (SNSs) received much attention. They mainly can be classified into two types. The first is state-feedback control: stabilization [13, 14], adaptive stabilization [15–18], finite-time stabilization [19], stabilization with time-varying delays [20], and stabilization of mobile robots [21, 22]. The second is the output feedback stabilization [23–25].Since sometimes part of the system states are unmeasurable, output feedback controllers are needed. Zhang et al. discussed the output feedback stabilizing controllers of SNSs whose virtual controlbi contains gain parameter L [25]. This will lead to a problem where the calculation of L is very difficult, especially for n≥3, since the inequalities about L were quintic. In addition, to the authors’ knowledge, there are some results about state-feedback stabilization of SNSs with Markovian switching [13, 14], with few available results for the output stabilization of SNSs under arbitrary switching. Based on the above analysis, there exists a problem, that is, how to choose a proper observer under arbitrary switching where the virtual control bi in controllers does not contain gain parameter L, which causes the calculation of L to be easier.Notations. R denotes the set of all real numbers. Rn denotes the real n-dimensional space. For a vector or matrix X∈Rn×m, XT denotes its transpose, X denotes the Euclidean norm, TrX is its trace when X is square, and L is a stochastic differential operator [26].


## 2. Problem Formulation

The stochastic nonholonomic nonlinear systems are given by(1a)dx0=u0dt+g0,σtTx0dω,(1b)dxi=u0xi+1dt+fi,σtx0,x¯idt+gi,σtTx0,x¯idω,i=1,…,n-1,dxn=udt+fn,σtx0,xdt+gi,σtTx0,xdω,y=x0,x1T,where u0 and u∈R are inputs, x0∈R and x1,…,xnT∈Rn are system states, x¯i=x1,…,xiT, fi≜fix0,x¯i:Ri+1→R are smooth functions named as nonlinear drifts with fi0,0=0, and g0x0:R→Rm and gi≜gix0,x¯i:Ri+1→Rm are smooth functions with g00=0 and gi0,0=0, 1≤i≤n. σt:0,+∞→M=1,2,…,m is a piecewise constant switching signal, and ω∈Rm is anm-dimensional independent standard Wiener process.Assumption 1.
If there exist smooth functionν0,σtx0:R→Rm and positive constant m¯, then(2)g0,σtx0=x0ν0,σtx0,ν0,σtx0≤m-,where ν0,σtx0 is a known bounded nonnegative function.Assumption 2.
Fori=1,…,n, one has known constants ν¯i,σt and γ¯i,σt satisfying (3)fi,σtx0,x¯i≤x1+⋯+xiν¯i,σt,gi,σtx0,x¯i≤x1+⋯+xiγ¯i,σt.Remark 3.
There are two main differences between systems (1a) and (1b) and those in [24, 25]. The first is the arbitrary switching mentioned in this paper. The second will be illustrated in Remark 7. In addition, Assumptions 1 and 2 are similar to those in [24, 25], but in fact they hold under arbitrary switching; for example, γ¯i,σt=max⁡γ¯i,1,γ¯i,2,γ¯i,3,γ¯i,4 when σt:0,+∞→M=1,2,3,4.


## 3. Output Feedback Stabilization

The controller design in Sections3.1 and 3.2 is under x0t0≠0. The other one is discussed in Section 3.3.


### 3.1. Controlleru0 Design

For subsystem (1a), one obtains(4)LV0x0≤-2λV0x0.with Lyapunov function V0x0=1/2x02 and controller u0 as follows:(5)u0=-η0x0,η0=λ+12m¯2,where λ>0 is a design real number.Theorem 4.
For system (1a), the closed-loop system with controller (4) is asymptotically stable in probability.
By (4) and (1a), one has(6)dx0=-η0x0dt+g0,σtTx0dω.Remark 5.
For anyx0t0≠0, one has x0t≠0 in (6) at the time interval t∈t0,+∞ a.s. with a similar proof of Proposition  1 in [25].


### 3.2. Controlleru Design

In order to design controlleru, let(7)zi=xiu0n-i,1≤i≤n.Remark 6.
For anyx0t0≠0, from Remark 5, we have that (7) is meaningful a.s.
By (1b) and (7), one has(8)dzi=zi+1dt+ϕi,σtx0,z¯idt+ψi,σtTx0,z¯idω,i=1,…,n-1,dzn=udt+ϕn,σtx0,xdt+ψn,σtTx0,xdω,y1=z1,where z¯i=z1,…,ziT and (9)ϕi,σtx0,z¯i=η0n-izi+fi,σtu0n-i+12n-in-i+1zig0,σtTg0,σtx02+η0n-igi,σtTg0,σtu0n-i+1,ψi,σtx0,z¯i=gi,σtu0n-i-n-izig0,σtx0,i=1,…,n.We adopt the following observer [27]:(10)z^˙i=z^i+1+Liaiy1-z^1,i=1,…,n-1,z^˙n=u+Lnany1-z^1,where L≥1 is a gain parameter and real numbers ai>0, i=1,…,n, such that ps=sn+a1sn-1+⋯+an is Hurwitz.Remark 7.
The second main difference between this manuscript and [25] is that the observer (10) we choose is the same as that in [27], but different from that in [25]. This observer has two advantages. The first is that it can simplify the process of designing controllers. The second is that bi in virtual control αi that we design in the following does not contain the gain parameter L.
Denoting(11)ξi=zi-z^iLi-1,i=1,…,n,
one has the error systems(12)dξ=LAξdt+Φσtdt+ΨσtTdω,where(13)A=-a1-a2In-1⋮-an0,Φσt=ϕ1,σtϕ2,σtL⋮ϕn-1,σtLn-2ϕn,σtLn-1,ΨσtT=ψ1,σtTψ2,σtTL⋮ψn-1,σtTLn-2ψn,σtTLn-1.Therefore, for positive definite matrix P, A is a Hurwitz matrix; that is, ATP+PA=-I. Now, one has(14)dξ=LAξdt+Φσtdt+ΨσtTdω,dy1=z2dt+ϕ1,σtdt+ψ1,σtTdω,z^˙i=z^i+1+Liaiy1-z^1,i=1,…,n-1,z^˙n=u+Lnany1-z^1.Proposition 8.
By Assumptions1 and 2, there exist constants τ1>0 and τ2>0 a.s., such that(15a)ϕi,σt≤z1+z2+⋯+ziτ1,ψi,σtT≤z1+z2+⋯+ziτ2,(15b)Φσtx0,z2≤nz1+z2L+⋯+znLn-12τ12,ΨσtTx0,z2≤nz1+z2L+⋯+znLn-12τ22,
wherez=z1,…,znT.Remark 9.
The proof of the above proposition is similar to that of Proposition  2 in [25]. In fact, we only need to let v¯¯i=η0+η0m¯i-1+⋯+1maxσt⁡v¯i,σt.
One can define variablesεi as follows in order to utilize the backstepping method:(16)ε1=z1,εiz1,z^2,…,z^i=z^i-αi-1z1,z^2,…,z^i-1,
with virtual smooth controllersαi-1(i=2,…,n).Step1. For positive parameter δ, letting V0=δξTPξ, with a similar method in [24], one has (17)LV0≤-δL-2P2-2nCeξ2+2nCez12+z^22L2+⋯+z^22L2n-2,where Ce=nδτ12+nδλmaxPτ22. Defining V1=V0+1/2ε12, we have(18)LV1≤-δL-2P2-2nCeξ2+2nCez12+z^22L2+⋯+z^n2L2n-2+ε1ε2+α1+Lξ2+ϕ1,σt+12Trψ1,σtψ1,σtT.The following inequalities hold with Lemma2.1 in [28]:(19)2nCez^2L22≤4nCeε22L+4nCeα12L2,ε1ε2≤14Lε22+Lε12,Lε1ξ2≤L4ε12+Lξ22,ε1ϕ1,σt≤Lτ12ε12,12Trψ1,σtψ1,σtT≤12Lτ22ε12,2nCez12≤2nCeLε12,where, together with (18), zi/Li-1≤ξi+z^i/Li-1. Choosing(20)α1=-Lb1ε1,b1=2n+1+2nCe+54+τ1+τ222,one has(21)LV1≤-δL-2P2-2nCe-Lξ2+2nCez^32L4+⋯+z^n2L2n-2-2n+1L-4nCeb12ε12+14Lε22+4nCeε22L.Stepi(2≤i≤n-1). From (16), we have(22)εi=z^i+Lbi-1z^i-1+L2bi-1bi-2z^i-2+⋯+Li-1bi-1bi-2⋯b1z1,dεi=z^i+1+Lid~iξ1+Lidi,1ε1+Li-1di,2ε2+⋯+L2di,i-1εi-1+Ldi,iεidt+Li-1bi-1bi-2⋯b1Lξ2dt+ϕ1,σtdt+ψ1,σtTdω,where (23)di,1=-bi-1⋯b2b1b1,di,j=bi-1⋯bjbj-1-bi-1⋯bjbj,2≤j≤i-1,di,i=bi-1,d~i=ai+bi-1bi-2⋯b2a2+bi-1⋯b3a3+⋯+bi-1ai-1.Now, we have finished the design step i-1, and αi-1 is chosen as follows:(24)αi-1=-Lbi-1εi-1,bi-1=n+4nCe+54+d~i-124+14di-1,12+⋯+14di-1,i-12+14bi-2⋯b12+14bi-2⋯b12τ12+di-1,i-1.Let Vi-1=Vi-2+1/2L2i-4εi-12, such that(25)LVi-1≤-δL-2δP2-2nCe-i-1Lξ2+2nCez^i+12L2i+⋯+z^n2L2n-2-2n-2i-1+3L-∑j=1i-2bj⋯b12τ222-4nCeb12ε12-∑j=2i-11L2j-2n+j-i+1L-4nCebj2εj2+4nceεi2L2i-3+εi24L2i-3.DefiningVi=Vi-1+1/2L2i-2εi2, we have(26)LVi≤-δL-2δP2-2nCe-i-1Lξ2+2nCez^i+12L2i+⋯+z^n2L2n-2-2n-2i-1+3L-∑j=1i-2bj⋯b12τ222-4nCeb12ε12-∑j=2i-11L2j-2n+j-i+1L-4nCebj2εj2+4nCeεi2L2i-3+14L2i-3εi2+1L2i-2εiεi+1+αi+Lid~iξ1+Lidi,1ε1+Li-1di,2ε2+⋯+L2di,i-1εi-1,p+Ldi,iεi+Li-1bi-1bi-2⋯b1Lξ2+ϕ1,σt+12L2i-2Li-1bi-1bi-2⋯b12Trψ1,σtψ1,σtT.By Lemma  2.1 in [28], one gets (27)2nCez^i+12L2i≤4nCeεi+12L2i-1+4nCeαi2L2i,1L2i-2εiεi+1≤14L2i-1εi+12+1L2i-3εi2,d~iLi-2εiξ1≤d~i24L2i-3εi2+Lξ12,di,1Li-2εiε1≤di,124L2i-3εi2+Lε12,⋮di,i-1L2i-4εiεi-1≤di,i-124L2i-3εi2+1L2i-5εi-12,1Li-2bi-1bi-2⋯b1εiξ2≤bi-1bi-2⋯b124L2i-3εi2+Lξ22,1Li-1bi-1bi-2⋯b1εiϕ1p≤bi-1bi-2⋯b1τ124L2i-3εi2+Lε12,12L2i-2Li-1bi-1bi-2⋯b12Trψ1,σtψ1,σtT≤bi-1bi-2⋯b122τ22ε12.Choosing αi as (28)αi=-Lbiεi,bi=n+4nCe+54+d~i24+14di,12+⋯+14di,i-12+14bi-1⋯b12+14bi-1⋯b12τ12+di,i,one has(29)LVi≤-δL-2δP2-2nCe-iLξ2+2nCez^i+22L2i+2+⋯+z^n2L2n-2-2n-2i+3L-∑j=1i-1bj⋯b12τ222-4nCeb12ε12-∑j=2i1L2j-2n+j-iL-4nCebj2εj2+4nC1εi+12L2i-1+14L2i-1εi+12.Stepn. Letting Vn=Vn-1+1/2L2n-2εn2 and choosing(30)u=-Lbnεn,bn=n+4nCe+54+d~n24+dn,124+⋯+dn,n-124+bn-1⋯b124+bn-1⋯b12τ124+dn,n,one gets(31)LVn≤-h0ξ2-h1ε12-∑j=2n-1hj1L2j-2εj2-hn1L2n-3εn2≤-h~Vn,where h~=min⁡h0/δ,2h1,…,2hn and(32)h0=δL-2δP2-2nCe-nL≥0,h1=3L-∑j=1n-1bj⋯b12τ222-4nCeb12≥0,hj=jL-4nCebj2≥0,j=2,…,n-1,hn=n≥0,δ>n.Remark 10.
From (20), one has that b1 do not contain L under arbitrary switching. From (28), one has that all bi do not contain L under the designed controllers.Remark 11.
For example, withn=3, by (41) in [25] and (32), we have(33)h0=δL-2P2-B2-6Ce-3L≥0,h1=6L-4L-2η02L-b2b12τ222-12Ceb12≥0,h2=2L-η02L-12Ceb22≥0,h3=3≥0,δ>3,(34)h0=δL-2P2-6Ce-3L≥0,h1=7L-4L-∑j=12bj⋯b12τ222-12Ceb12≥0,h2=3L-12Ceb22≥0,h3=3≥0,δ>3,respectively. It is easy to see that h1 and h2 in (33) are all quintic functions about L, but they are linear functions about L in (34). So, the calculation of δ and L will be more simple with the method in this manuscript.ChooseV=V0+Vn, which together with (5) and (31) result in (35)LV≤-hV,where h=minλ,h~.Theorem 12.
For system (1b), the closed-loop system with controller (30) is asymptotically stable in probability.


### 3.3. Switching Control

In the above two subsections, we give the controllersu0 and u with x0t0≠0 as (4) and (30). Now, we turn to the case of x0t0=0. If x0t0=0, we firstly choose constant control u0=u0∗≠0. Secondly, there will exist a time ts∗>0 such that x0ts∗≠0. After that, controllers u0 and u as (4) and (30) can be applied.Theorem 13.
If we apply the above switching procedure, systems (1a) and (1b) will be asymptotically stabilized in probability.


## 3.1. Controlleru0 Design

For subsystem (1a), one obtains(4)LV0x0≤-2λV0x0.with Lyapunov function V0x0=1/2x02 and controller u0 as follows:(5)u0=-η0x0,η0=λ+12m¯2,where λ>0 is a design real number.Theorem 4.
For system (1a), the closed-loop system with controller (4) is asymptotically stable in probability.
By (4) and (1a), one has(6)dx0=-η0x0dt+g0,σtTx0dω.Remark 5.
For anyx0t0≠0, one has x0t≠0 in (6) at the time interval t∈t0,+∞ a.s. with a similar proof of Proposition  1 in [25].


## 3.2. Controlleru Design

In order to design controlleru, let(7)zi=xiu0n-i,1≤i≤n.Remark 6.
For anyx0t0≠0, from Remark 5, we have that (7) is meaningful a.s.
By (1b) and (7), one has(8)dzi=zi+1dt+ϕi,σtx0,z¯idt+ψi,σtTx0,z¯idω,i=1,…,n-1,dzn=udt+ϕn,σtx0,xdt+ψn,σtTx0,xdω,y1=z1,where z¯i=z1,…,ziT and (9)ϕi,σtx0,z¯i=η0n-izi+fi,σtu0n-i+12n-in-i+1zig0,σtTg0,σtx02+η0n-igi,σtTg0,σtu0n-i+1,ψi,σtx0,z¯i=gi,σtu0n-i-n-izig0,σtx0,i=1,…,n.We adopt the following observer [27]:(10)z^˙i=z^i+1+Liaiy1-z^1,i=1,…,n-1,z^˙n=u+Lnany1-z^1,where L≥1 is a gain parameter and real numbers ai>0, i=1,…,n, such that ps=sn+a1sn-1+⋯+an is Hurwitz.Remark 7.
The second main difference between this manuscript and [25] is that the observer (10) we choose is the same as that in [27], but different from that in [25]. This observer has two advantages. The first is that it can simplify the process of designing controllers. The second is that bi in virtual control αi that we design in the following does not contain the gain parameter L.
Denoting(11)ξi=zi-z^iLi-1,i=1,…,n,
one has the error systems(12)dξ=LAξdt+Φσtdt+ΨσtTdω,where(13)A=-a1-a2In-1⋮-an0,Φσt=ϕ1,σtϕ2,σtL⋮ϕn-1,σtLn-2ϕn,σtLn-1,ΨσtT=ψ1,σtTψ2,σtTL⋮ψn-1,σtTLn-2ψn,σtTLn-1.Therefore, for positive definite matrix P, A is a Hurwitz matrix; that is, ATP+PA=-I. Now, one has(14)dξ=LAξdt+Φσtdt+ΨσtTdω,dy1=z2dt+ϕ1,σtdt+ψ1,σtTdω,z^˙i=z^i+1+Liaiy1-z^1,i=1,…,n-1,z^˙n=u+Lnany1-z^1.Proposition 8.
By Assumptions1 and 2, there exist constants τ1>0 and τ2>0 a.s., such that(15a)ϕi,σt≤z1+z2+⋯+ziτ1,ψi,σtT≤z1+z2+⋯+ziτ2,(15b)Φσtx0,z2≤nz1+z2L+⋯+znLn-12τ12,ΨσtTx0,z2≤nz1+z2L+⋯+znLn-12τ22,
wherez=z1,…,znT.Remark 9.
The proof of the above proposition is similar to that of Proposition  2 in [25]. In fact, we only need to let v¯¯i=η0+η0m¯i-1+⋯+1maxσt⁡v¯i,σt.
One can define variablesεi as follows in order to utilize the backstepping method:(16)ε1=z1,εiz1,z^2,…,z^i=z^i-αi-1z1,z^2,…,z^i-1,
with virtual smooth controllersαi-1(i=2,…,n).Step1. For positive parameter δ, letting V0=δξTPξ, with a similar method in [24], one has (17)LV0≤-δL-2P2-2nCeξ2+2nCez12+z^22L2+⋯+z^22L2n-2,where Ce=nδτ12+nδλmaxPτ22. Defining V1=V0+1/2ε12, we have(18)LV1≤-δL-2P2-2nCeξ2+2nCez12+z^22L2+⋯+z^n2L2n-2+ε1ε2+α1+Lξ2+ϕ1,σt+12Trψ1,σtψ1,σtT.The following inequalities hold with Lemma2.1 in [28]:(19)2nCez^2L22≤4nCeε22L+4nCeα12L2,ε1ε2≤14Lε22+Lε12,Lε1ξ2≤L4ε12+Lξ22,ε1ϕ1,σt≤Lτ12ε12,12Trψ1,σtψ1,σtT≤12Lτ22ε12,2nCez12≤2nCeLε12,where, together with (18), zi/Li-1≤ξi+z^i/Li-1. Choosing(20)α1=-Lb1ε1,b1=2n+1+2nCe+54+τ1+τ222,one has(21)LV1≤-δL-2P2-2nCe-Lξ2+2nCez^32L4+⋯+z^n2L2n-2-2n+1L-4nCeb12ε12+14Lε22+4nCeε22L.Stepi(2≤i≤n-1). From (16), we have(22)εi=z^i+Lbi-1z^i-1+L2bi-1bi-2z^i-2+⋯+Li-1bi-1bi-2⋯b1z1,dεi=z^i+1+Lid~iξ1+Lidi,1ε1+Li-1di,2ε2+⋯+L2di,i-1εi-1+Ldi,iεidt+Li-1bi-1bi-2⋯b1Lξ2dt+ϕ1,σtdt+ψ1,σtTdω,where (23)di,1=-bi-1⋯b2b1b1,di,j=bi-1⋯bjbj-1-bi-1⋯bjbj,2≤j≤i-1,di,i=bi-1,d~i=ai+bi-1bi-2⋯b2a2+bi-1⋯b3a3+⋯+bi-1ai-1.Now, we have finished the design step i-1, and αi-1 is chosen as follows:(24)αi-1=-Lbi-1εi-1,bi-1=n+4nCe+54+d~i-124+14di-1,12+⋯+14di-1,i-12+14bi-2⋯b12+14bi-2⋯b12τ12+di-1,i-1.Let Vi-1=Vi-2+1/2L2i-4εi-12, such that(25)LVi-1≤-δL-2δP2-2nCe-i-1Lξ2+2nCez^i+12L2i+⋯+z^n2L2n-2-2n-2i-1+3L-∑j=1i-2bj⋯b12τ222-4nCeb12ε12-∑j=2i-11L2j-2n+j-i+1L-4nCebj2εj2+4nceεi2L2i-3+εi24L2i-3.DefiningVi=Vi-1+1/2L2i-2εi2, we have(26)LVi≤-δL-2δP2-2nCe-i-1Lξ2+2nCez^i+12L2i+⋯+z^n2L2n-2-2n-2i-1+3L-∑j=1i-2bj⋯b12τ222-4nCeb12ε12-∑j=2i-11L2j-2n+j-i+1L-4nCebj2εj2+4nCeεi2L2i-3+14L2i-3εi2+1L2i-2εiεi+1+αi+Lid~iξ1+Lidi,1ε1+Li-1di,2ε2+⋯+L2di,i-1εi-1,p+Ldi,iεi+Li-1bi-1bi-2⋯b1Lξ2+ϕ1,σt+12L2i-2Li-1bi-1bi-2⋯b12Trψ1,σtψ1,σtT.By Lemma  2.1 in [28], one gets (27)2nCez^i+12L2i≤4nCeεi+12L2i-1+4nCeαi2L2i,1L2i-2εiεi+1≤14L2i-1εi+12+1L2i-3εi2,d~iLi-2εiξ1≤d~i24L2i-3εi2+Lξ12,di,1Li-2εiε1≤di,124L2i-3εi2+Lε12,⋮di,i-1L2i-4εiεi-1≤di,i-124L2i-3εi2+1L2i-5εi-12,1Li-2bi-1bi-2⋯b1εiξ2≤bi-1bi-2⋯b124L2i-3εi2+Lξ22,1Li-1bi-1bi-2⋯b1εiϕ1p≤bi-1bi-2⋯b1τ124L2i-3εi2+Lε12,12L2i-2Li-1bi-1bi-2⋯b12Trψ1,σtψ1,σtT≤bi-1bi-2⋯b122τ22ε12.Choosing αi as (28)αi=-Lbiεi,bi=n+4nCe+54+d~i24+14di,12+⋯+14di,i-12+14bi-1⋯b12+14bi-1⋯b12τ12+di,i,one has(29)LVi≤-δL-2δP2-2nCe-iLξ2+2nCez^i+22L2i+2+⋯+z^n2L2n-2-2n-2i+3L-∑j=1i-1bj⋯b12τ222-4nCeb12ε12-∑j=2i1L2j-2n+j-iL-4nCebj2εj2+4nC1εi+12L2i-1+14L2i-1εi+12.Stepn. Letting Vn=Vn-1+1/2L2n-2εn2 and choosing(30)u=-Lbnεn,bn=n+4nCe+54+d~n24+dn,124+⋯+dn,n-124+bn-1⋯b124+bn-1⋯b12τ124+dn,n,one gets(31)LVn≤-h0ξ2-h1ε12-∑j=2n-1hj1L2j-2εj2-hn1L2n-3εn2≤-h~Vn,where h~=min⁡h0/δ,2h1,…,2hn and(32)h0=δL-2δP2-2nCe-nL≥0,h1=3L-∑j=1n-1bj⋯b12τ222-4nCeb12≥0,hj=jL-4nCebj2≥0,j=2,…,n-1,hn=n≥0,δ>n.Remark 10.
From (20), one has that b1 do not contain L under arbitrary switching. From (28), one has that all bi do not contain L under the designed controllers.Remark 11.
For example, withn=3, by (41) in [25] and (32), we have(33)h0=δL-2P2-B2-6Ce-3L≥0,h1=6L-4L-2η02L-b2b12τ222-12Ceb12≥0,h2=2L-η02L-12Ceb22≥0,h3=3≥0,δ>3,(34)h0=δL-2P2-6Ce-3L≥0,h1=7L-4L-∑j=12bj⋯b12τ222-12Ceb12≥0,h2=3L-12Ceb22≥0,h3=3≥0,δ>3,respectively. It is easy to see that h1 and h2 in (33) are all quintic functions about L, but they are linear functions about L in (34). So, the calculation of δ and L will be more simple with the method in this manuscript.ChooseV=V0+Vn, which together with (5) and (31) result in (35)LV≤-hV,where h=minλ,h~.Theorem 12.
For system (1b), the closed-loop system with controller (30) is asymptotically stable in probability.


## 3.3. Switching Control

In the above two subsections, we give the controllersu0 and u with x0t0≠0 as (4) and (30). Now, we turn to the case of x0t0=0. If x0t0=0, we firstly choose constant control u0=u0∗≠0. Secondly, there will exist a time ts∗>0 such that x0ts∗≠0. After that, controllers u0 and u as (4) and (30) can be applied.Theorem 13.
If we apply the above switching procedure, systems (1a) and (1b) will be asymptotically stabilized in probability.


## 4. A Simulation Example

Consider systems (1a) and (1b) with σ:0,+∞→1,2 and g0,1=c1x0cos⁡x0, g0,2=c1x0sin⁡(x0), f1,1=c1x1cos2⁡x1, f1,2=c1x1cos⁡x1, g1,1=c1x1sin2⁡x1, g1,2=c1x1sin⁡x1, f2,1=f2,2=0, g2,1=c1x2sin2⁡x2, g2,2=c1x2sin⁡x2, and c1=0.001.From (8), we have ϕ1,1=η0z1+c1z1cos2⁡x1+c12z1cos2⁡x0-c12z1sin2⁡x1cos⁡(x0), ϕ2,1=0, ψ1,1=c1z1sin2⁡x1-c1z1cos⁡x0, ψ2,1=c1z2sin2⁡z2, ϕ1,2=η0z1+c1z1cos2⁡x1+c12z1sin2⁡x0-c12z1sin⁡x1sin⁡x0, ϕ2,2=0, ψ1,2=c1z1sin⁡x1-c1z1sin⁡x0, and ψ2,2=c1z2sin⁡z2.Lettingη0=0.3, by Proposition 8, we have τ1=0.3023 and τ2=0.001. Choosing a1=0.75 and a2=1.25, then λmaxP=2 and P=2.From (20) and (30), one gets (36)b1=5+4Ce+1+14+τ1+12τ22,b2=2+8Ce+1+14+a2+b1a124+b124+b1+b124+b12τ124,u=-Lb2z^2-L2b2b1z1,where Ce=0.4131δ. From (32) and (35), we have (37)h0=δ-2L-2δP2-4Ce≥0,h1=2L-b12τ222-8Ceb12≥0,h2=2≥0,δ>2.Solving the above inequalities, one has δ=2.26 and L=76, which means h0=0.0277, h1=5.5526, and LV≤0.If we choose initial valuesx00=0.1, x10=0.0003, x20=0.2, z^10=-0.001, and z^20=7.5, responses of systems are as in Figures 1, 2, and 3.Figure 1
The responses of statesx0, x1, and x2.Figure 2
The responses of controllersu0 and u.Figure 3
The responses of switching signalsσ(t).Remark 14.
From the above example, the observer we adopted is the same as that in [27], but it can simplify the calculation of δ and L compared with the observer in [25].


## 5. Conclusions

The output feedback stabilization for SNSs under arbitrary switching is discussed. We proposed an observer which is different from that in [25]. bi of the designed output feedback stabilizing control laws do not contain the gain parameter. We will give some new results, for example, how to design an adaptive controller with the method LMI based on results in [29, 30].



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