# Stochastic Stabilization of Itô Stochastic Systems with Markov Jumping and Linear Fractional Uncertainty

Authors: Fei Long; Hongmei Huang; Adan Ding
Journal: Journal of Control Science and Engineering
 (2013)

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## Abstract

For a class of Itô stochastic linear systems with the Markov jumping and linear fractional uncertainty, the stochastic stabilization problem is investigated via state feedback and dynamic output feedback, respectively. In order to guarantee the stochastic stability of such uncertain systems, state feedback and dynamic output control law are, respectively, designed by using multiple Lyapunov function technique and LMI approach. Finally, two numerical examples are presented to illustrate our results.


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## Body

## 1. Introduction

Modeled by a set of linear systems with the transitions among the linear systems governed by the Markov chain, the Markov jumping linear system (MJLS) can characterize and model different types of systems [1], for instance, fault-tolerant systems, target tracking systems, manufactory processes, networked control systems, multiagent systems, and so on. In the past years, many important results on MJLS have been addressed in the literature, such as, the stability analysis and control design which were discussed in [2–8]. Besides the aforementioned theoretical studies, MJLS also found applications in practical systems, such as networked direct current motor systems [9].The interest of this topic lies in the fact that many practical systems, with random abrupt changes, sudden environmental changes, and so on, can be modeled by Itô stochastic linear systems with Markov jumping and configurable uncertainty [10]. Therefore, it is very important to investigate robustly stochastic stabilization problem for the aforementioned Itô stochastic MJLS with linear fractional uncertainties. Recently, some results for Itô stochastic MJLS have been addressed in [11–18].In this paper, motivated by the recent work [19], we consider robustly stochastic stabilization problem for Markov jumping Itô stochastic linear system subject to linear fractional uncertainty. An effective LMI-based approach for state feedback and dynamic output feedback controller is designed such that the closed-loop system is robust stable in the mean square sense, respectively.The reminder of this paper is organized as follows: the problem statement is described in Section2, while in Section 3 the stochastic stability for Itô stochastic MJLS with linear fractional uncertainty is addressed. In Section 4, the robustly stochastic stabilization of Itô stochastic MJLS with linear fractional uncertainty is investigated via state feedback and dynamical output feedback, respectively. Two numerical examples are presented in Section 5 to illustrate the obtained results. Finally, some conclusions are drawn in Section 6.


## 2. Problem Formulation

Consider the following unforced Itô stochastic MJLS with linear fractional uncertainties:(1)dx(t)=A(t,r(t))x(t)dt+W(r(t))x(t)dω(t),x(0)=x0,r(0)=r0,
where {r(t),t≥0} is a Markov process taking values in a finite space ℕ-={1,2,…,N} and x(t)∈ℝn is the state vector. ω(t)∈ℝ is a standard Wiener process that is supposed to be independent of the Markov process {r(t),t≥0}.x0∈ℝn denotes the initial state of system at t=0, and r0∈ℕ- stands for the initial mode in the Markov process at t=0. A(t,r(t)) and W(r(t)), r(t)=i∈ℕ- are system matrices with compatible dimensions that depend on r(t) and t, and
(2)A(t,i)=A(i)+MA(i)HA(t,i)NA(i),HA(t,i)=ΣA(t,i)[I-GA(i)ΣA(t,i)]-1,
where A(i), MA(i), NA(i), and GA(i) are known real constant matrices with appropriate dimensions. ΣA(t,i) is an unknown real matrix function with Lebesgue-measurable elements, satisfying ΣAT(t,i)ΣA(t,i)≤I with I being the identity matrix.The evolution of the Markov process{r(t),t≥0} is governed by the following probability transitions:
(3)P[r(t+h)=j∣r(t)=i]={λijh+o(h),r(t)jumpsfrommodeitomodej,1+λiih+o(h),r(t)staysatmodei,
where λij≥0 is the transition rate from mode i to mode j at time instant t when i≠j, and λii=-∑j=1,j≠iNλij·o(h) is the little-o notation defined by limh→0(o(h)/h)=0.For the stochastic stability, we adopt the following definition. For more details, please refer to [18] and the references therein.Definition 1.
The Itô stochastic MJLS in (1) with all modes and all t≥0 is said to achieve stochastic stability if there exists a finite positive constant T(x0,r0) such that the following inequality holds for any initial condition (x0,r0):
(4)E[∫0∞∥x(t)∥2dt∣(x0,r0)]≤T(x0,r0),
where E[·∣·] is the expectation conditioning on the initial value of (x0,r0).


## 3. Stochastic Stability

Before proceeding further, we recall the following lemma, which will be used in the proof of the stochastic stability of the Itô stochastic MJLS in (1).Lemma 2 (see [20]).
LetH,E be given constant matrices with appropriate dimensions and norm-bounded time-varying uncertain matrix Σ(t) with ΣT(t)Σ(t)≤I. For any ε>0, one has
(5)HΣ(t)E+ETΣT(t)HT≤εHHT+ε-1ETE.Lemma 3 (see [19]).
Let the matrix G satisfyGTG<I; define the following set:
(6)ξ={H(t)=Σ(t)[I-GΣ(t)]-1,ΣT(t)Σ(t)≤I}.
Then, the set ξ can be rewritten as
(7)ξ={H(t)=(I-GTG)-1GT+(I-GTG)-1/2Π(t);ΠT(t)Π(t)≤I+G(I-GTG)-1GT(I-GTG)-1/2Π(t)}.Theorem 4.
The Itô stochastic uncertain MJLS in (1) is stochastically stable if there exists a set of symmetric and positive-definite matrices {P(i)∣i=1,2,…,N} and a set of positive scalars {εA(i)∣i=1,2,…,N} such that the following sets of coupled inequalities hold for each i∈ℕ- and all admissible linear fractional uncertainties:
(8)(J0(i)***RA0T(i)-QA(i)00εA(i)NA(i)0-I0P(i)W(i)00-P(i))<0,
where * represents the blocks that are induced by symmetry and
(9)J0(i)=AT(i)P(i)+P(i)A(i)+∑j=1NλijP(j),QA(i)=I-GAT(i)GA(i),RA0(i)=εA(i)NAT(i)GA(i)+εA-1(i)P(i)MA(i).Proof.
Consider the following Lyapunov-like function:(10)V(x(t),r(t))=xT(t)P(r(t))x(t),
where P(r(t)) denotes the positive symmetric matrix. The infinitesimal generator ζ can be considered as a derivative of the Lyapunov function V(x(t),r(t)) along the trajectory of the Markov process {r(t),t≥0} in point (x(t),r(t)) at time t. We need to derive the infinitesimal generator ζ of V(x(t),r(t)) first of all. According to the definition of ζ, we have
(11)ζV(x(t),r(t))=limΔt→01Δt{E[V(x(t+Δt),r(t+Δt))∣(x(t),r(t))]-V(x(t),r(t))},
where Δt is a small positive number. Setting r(t)=i and applying the law of total probability and the probability transition of the Markov process {r(t),t≥0} yield
(12)ζV(x(t),i)=limΔt→01Δt[∑j=1,j≠iNPr{r(t+Δt)=j∣r(t)=i}×V(x(t+Δt),j)∑j=1,j≠iNPr{r(t+Δt)=j∣r(t)=i}]+limΔt→01Δt[Pr{r(t+Δt)=i∣r(t)=i}×V(x(t+Δt),i)-V(x(t),i)Pr{r(t+Δt)=i∣r(t)=i}]=limΔt→0[∑j=1NλijV(x(t+Δt),j)+V(x(t+Δt),i)-V(x(t),i)Δt∑j=1NλijV(x(t+Δt),j)].
According to the Itô stochastic differential coefficient theorem [18], we have
(13)ζV(x(t),i)=∑j=1NλijV(x(t),j)+[A(t,i)x(t)]T∂V(x,i)∂x+12tr[xT(t)WT(i)∂2V(x,i)∂x2W(i)x(t)].
Note that ∂V(x,i)/∂x=2P(i)x(t) and ∂2V(x,i)/∂x2=2P(i), then the infinitesimal generator ζ becomes
(14)ζV(x(t),i)=xT(t)Ω(t,i)x(t),
where Ω(t,i)=AT(t,i)P(i)+P(i)A(t,i)+∑j=1NλijP(j)+WT(i)P(i)W(i).
Using Lemmas2 and 3, we have
(15)NAT(i)HAT(t,i)MAT(i)P(i)+P(i)MA(i)HA(t,i)NA(i)=NAT(i)[(I-GAT(i)GA(i))-12(I-GAT(i)GA(i))-1GAT(i)+(I-GAT(i)GA(i))-1/2ΠA(t,i)]T×MAT(i)P(i)+P(i)MA(i)×[(I-GAT(i)GA(i))-1GAT(i)+(I-GAT(i)GA(i))-1/2ΠA(t,i)]NA(i)≤NAT(i)GA(i)QA-1(i)MAT(i)P(i)+P(i)MA(i)QA-1(i)GAT(i)NA(i)+εA2(i)NAT(i)[I+GA(i)QA-1(i)GAT(i)]NA(i)+εA-2(i)P(i)MA(i)QA-1(i)MAT(i)P(i)=RA0(i)QA-1(i)RA0T(i)+εA2(i)NAT(i)NA(i),
where RA0(i)=εA(i)NAT(i)GA(i)+εA-1(i)P(i)MA(i), QA(i)=I-GAT(i)GA(i).
HenceΩ(t,i)≤Ω⌢(i), where
(16)Ω⌢(i)=J0(i)+RA0(i)QA-1(i)RA0T(i)+εA2(i)NAT(i)NA(i)+WT(i)P(i)W(i),J0(i)=AT(i)P(i)+P(i)A(i)+∑j=1NλijP(j).
According to the Schur complement, Ω⌢(i)<0, i∈ℕ- implies
(17)(J0(i)***RA0T(i)-QA(i)00εA(i)NA(i)0-I0P(i)W(i)00-P(i))<0.
Therefore (x(t),i)≤xT(t)Ω⌢(i)x(t)≤maxi∈ℕ-{λmaxΩ⌢(i)}xT(t)x(t).
By the generalized Dynkin formula, we have(18)E[V(x(t),i)∣(x0,r0)]-V(x0,r0)=E[∫0tζV(x(s),i)ds∣(x0,r0)]≤maxi∈ℕ-{λmaxΩ⌢(i)}E[∫0txT(s)x(s)ds∣(x0,r0)].
The last inequality implies
(19)-maxi∈ℕ-{λmaxΩ⌢(i)}E[∫0txT(s)x(s)ds∣(x0,r0)]≤V(x0,r0)-E[V(x(t),i)∣(x0,r0)]≤V(x0,r0).
Furthermore, the condition in (8) indicates maxi∈ℕ-{λmaxΩ⌢(i)}<0, so
(20)E[∫0txT(s)x(s)ds∣(x0,r0)]≤-V(x0,r0)maxi∈ℕ-{λmaxΩ⌢(i)}=defT(x0,r0)
holds for any t>0. Letting t go to infinity, then, we know that
(21)E[∫0∞xT(s)x(s)ds∣(x0,r0)]≤T(x0,r0).
According to Definition1, the Itô stochastic uncertain MJLS in (1) is stochastically stable. This ends the proof.


## 4. Stochastic Stabilization

### 4.1. State Feedback Stochastic Stabilization

In this subsection, we discuss how to design the robust state feedback control law for the following Itô stochastic uncertain MJLS:(22)dx(t)=A(t,r(t))x(t)dt+B(t,r(t))u(t)dt+W(r(t))x(t)dω(t),x(0)=x0,r(0)=r0,
where B(t,r(t)), r(t)=i∈ℕ- are system matrices with compatible dimensions that depend on r(t) and t, and
(23)B(t,i)=B(i)+MB(i)HB(t,i)NB(i),HB(t,i)=ΣB(t,i)[I-GB(i)ΣB(t,i)]-1,
where B(i), MB(i), NB(i), and GB(i) are known real constant matrices with appropriate dimensions. ΣB(t,i) is an unknown real matrix function with Lebesgue-measurable elements, satisfying ΣBT(t,i)ΣB(t,i)≤I with I being the identity matrix. The robust state feedback control law to be designed is
(24)u(t)=K(r(t))x(t),
where K(r(t)) is a gain with appropriate dimension to be determined for each mode r(t)∈ℕ-. Notice that the gain of the above controller is model dependent, which requires the knowledge of the model r(t) at time t to choose the appropriate gain among {K(1),K(2),…,K(N)}. Thus when the model switches from mode i, which uses the gain K(i), to modej, the controller gains must be switched instantaneously to the gain K(j) to guarantee the desired performances.Theorem 5.
Suppose that there exists a set of symmetric and positive-definite matrices{X(i)∣i=1,2,…,N}, a set of matrices {Y(i)∣i=1,2,…,N}, and a set of positive scalars {εA(i)∣i=1,2,…,N} and {εB(i)∣i=1,2,…,N} such that the following sets of coupled inequalities hold for each i∈ℕ- and all admissible linear fractional uncertainties. Then the controller (24) with gain K(i)=Y(i)X-1(i) can stabilize stochastically the Itô stochastic uncertain MJLS in (22)
(25)[Ju(i)******RA1T(i)-QA(i)00000RB1T(i)0-QB(i)0000εA-1(i)NA(i)X(i)00-I000εB-1(i)NB(i)Y(i)000-I00W(i)X(i)0000-X(i)0𝕊T(X(i))00000-𝕏(i)]<0,
where * represents the blocks that are induced by symmetry and
(26)Ju(i)=X(i)AT(i)+A(i)X(i)+YT(i)BT(i)+B(i)Y(i)+λiiX(i),RA1(i)=εA(i)MA(i)+εA-1(i)X(i)NAT(i)GA(i),RB1(i)=εB(i)MB(i)+εB-1(i)YT(i)NBT(i)GB(i),𝕏(i)=diag{X(1),…,X(i-1),X(i+1),…,X(N)},𝕊(X(i))=X(i)[λi1I,…,λi,i-1I,λi,i+1I,…,λiNI],QA(i)=I-GAT(i)GA(i),QB(i)=I-GBT(i)GB(i).Proof.
With the use of the state feedback control law in (24) for system (22), the closed-loop system becomes
(27)dx(t)=(A(t,r(t))+B(t,r(t))K(r(t)))x(t)dt+W(r(t))x(t)dω(t).
In order to guarantee the stochastic stability of system (27), using the stochastically stable condition in Theorem 4, we know that the following inequality should be satisfied for each i∈ℕ-:
(28)[A(t,i)+B(t,i)K(i)]TP(i)+P(i)[A(t,i)+B(t,i)K(i)]+WT(i)P(i)W(i)+∑j=1NλijP(j)<0.
Let X(i)=P-1(i), and before and after multiplying the matrix inequality (28) by X(i), then, we have
(29)X(i)AT(t,i)+A(t,i)X(i)+X(i)KT(i)BT(t,i)+B(t,i)K(i)X(i)+X(i)WT(i)X-1(i)W(i)X(i)+∑j=1NλijX(i)X-1(j)X(i)<0.
Applying expressions (2) and (23) of the linear fractional uncertainties, the above inequality can be described as
(30)X(i)AT(i)+A(i)X(i)+X(i)KT(i)BT(i)+B(i)K(i)X(i)+X(i)NAT(i)HAT(t,i)MAT(i)+MA(i)HA(t,i)NA(i)X(i)+X(i)KT(i)NBT(i)HBT(t,i)MBT(i)+MB(i)HB(t,i)NB(i)K(i)X(i)+X(i)WT(i)X-1(i)W(i)X(i)+X(i)[∑j=1NλijX-1(j)]X(i)<0.
Again in view of Lemmas2 and 3, the following two matrix inequalities are obvious:
(31)MA(i)HA(t,i)NA(i)X(i)+X(i)NAT(i)HAT(t,i)MAT(i)≤RA1(i)QA-1(i)RA1T(i)+εA-2(i)X(i)NAT(i)NA(i)X(i),MB(i)HB(t,i)NB(i)K(i)X(i)+X(i)KT(i)NBT(i)HBT(t,i)MBT(i)≤RB1(i)QB-1(i)RB1T(i)+εB-2(i)X(i)KT(i)NBT(i)NB(i)K(i)X(i),
where
(32)RA1(i)=εA(i)MA(i)+εA-1(i)X(i)NAT(i)GA(i),QA(i)=I-GAT(i)GA(i),RB1(i)=εB(i)MB(i)+εB-1(i)X(i)KT(i)NBT(i)GB(i),QB(i)=I-GBT(i)GB(i).
Setting
(33)𝕏(i)=diag{X(1),…,X(i-1),X(i+1),…,X(N)},𝕊(X(i))=X(i)[λi1I,…,λi,i-1I,λi,i+1I,…,λiNI].
Then X(i)[∑j=1NλijX-1(j)]X(i) can be rewritten as follows:
(34)X(i)[∑j=1NλijX-1(j)]X(i)=λiiX(i)+𝕊(X(i))𝕏-1(i)𝕊T(X(i)).
Let Y(i)=K(i)X(i), and by means of the matrix inequalities (31) and (34), the closed-loop system (27) will be stochastically stable if the following matrix inequality holds:
(35)X(i)AT(i)+A(i)X(i)+YT(i)BT(i)+B(i)Y(i)+λiiX(i)+RA1(i)QA-1(i)RA1T(i)+εA-2(i)X(i)NAT(i)NA(i)X(i)+RB1(i)QB-1(i)RB1T(i)+εB-2(i)YT(i)NBT(i)NB(i)Y(i)+X(i)WT(i)X-1(i)W(i)X(i)+𝕊(X(i))𝕏-1(i)𝕊T(X(i))<0.
After using the Schur complement, we can get the LMIs (25), which end the proof of Theorem 5.


### 4.2. Dynamic Output Feedback Stochastic Stabilization

In Section4.1, the design method of state feedback controller had been given under that the state vector x(t) and the mode r(t) are available at each time t. However, some of the state variables are not measurable by their construction or the lack of appropriate sensors to give information. Alternatively we can use dynamic output feedback controller that uses the system’s measurement to design and to compute the control gains in this subsection.In this subsection, we discuss how to design the dynamic output feedback control law for the following Itô stochastic uncertain MJLS:(36)dx(t)=A(t,r(t))x(t)dt+B(t,r(t))u(t)dt+W(r(t))x(t)dω(t),y(t)=C(t,r(t))x(t),x(0)=x0,r(0)=r0,
where C(t,r(t)), r(t)=i∈ℕ- are system matrices with compatible dimensions that depend on r(t) and t, and
(37)C(t,i)=C(i)+MC(i)HC(t,i)NC(i),HC(t,i)=ΣC(t,i)[I-GC(i)ΣC(t,i)]-1,
where C(i),  MC(i),  NC(i), and GC(i) are known real constant matrices with appropriate dimensions. ΣC(t,i) is an unknown real matrix function with Lebesgue-measurable elements, satisfying ΣCT(t,i)ΣC(t,i)≤I with I being the identity matrix. The dynamic output feedback control law is to be described by the following structure:
(38)x˙c(t)=KA(r(t))xc(t)+KB(r(t))y(t),u(t)=KC(r(t))xc(t),xc(0)=0,r(0)=0,
where xc(t)∈ℝn is the controller state and KA(r(t)), KB(r(t)), and KC(r(t)) are the controller gains to be determined.Theorem 6.
If there exists a set of matrices{XT(i)=X(i)>0∣i=1,2,…,N}, {Y(i)>0∣i=1,2,…,N}, {𝕜B(i)∣i=1,2,…,N}, and {𝕜C(i)∣i=1,2,…,N} and a set of positive scalars {εA(i)∣i=1,2,…,N}, {εB(i)∣i=1,2,…,N}, and {εC(i)∣i=1,2,…,N} such that the following sets of inequalities (39)–(41) hold for each i∈ℕ- and all admissible linear fractional uncertainties, then the dynamic output feedback controller (38) with control gain (43) can stabilize stochastically the Itô stochastic uncertain MJLS in  (36)(39)
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0
,where* represents the blocks that are induced by symmetry and
(42)Jd1(i)=A(i)Y(i)+YT(i)A(i)+B(i)𝕜C(i)+𝕜CT(i)BT(i)+λiiY(i),Jd2(i)=X(i)A(i)+𝕜B(i)C(i)+AT(i)X(i)+CT(i)𝕜BT(i)+∑j=1NλijX(j)+εA-2(i)NAT(i)NA(i)+εC-2(i)NCT(i)NC(i),𝕊(Y(i))=Y(i)[λi1I,…,λi,i-1I,λi,i+1I,…,λiNI],𝕐(i)=diag{Y(1),…,Y(i-1),Y(i+1),…,Y(N)},RA2(i)=εA(i)MA(i)+εA-1(i)YT(i)NAT(i)GA(i),RB2(i)=εB(i)MB(i)+εB-1(i)𝕜CT(i)NBT(i)GB(i),QA(i)=I-GAT(i)GA(i),QB(i)=I-GBT(i)GB(i),QC(i)=I-GCT(i)GC(i).The dynamical output feedback controller gain is given by(43)KA(i)=[X(i)-Y-1(i)]-1[AT(i)Y-1(i)+X(i)A(i)+𝕜𝔹(i)C(i)+X(i)B(i)𝕜ℂ(i)Y-1(i)+∑j=1NλijY-1(j)+NAT(i)GA(i)QA-1(i)MAT(i)Y-1(i)+X(i)MA(i)QA-1(i)GAT(i)NA(i)+X(i)MB(i)QB-1(i)GBT(i)NB(i)𝕜ℂ(i)Y-1(i)+𝕜𝔹(i)MC(i)QC-1(i)GCT(i)NC(i)+εA2(i)X(i)MA(i)QA-1(i)MAT(i)Y-1(i)+εA-2(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)+εA-2(i)NAT(i)NA(i)+εB2(i)X(i)MB(i)QB-1(i)MBT(i)Y-1(i)+εC-2(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)+εC-2(i)NCT(i)NC(i)+WT(i)X(i)W(i)],KB(i)=[Y-1(i)-X(i)]-1𝕜B(i),KC(i)=𝕜C(i)Y-1(i).Proof.
Combining the dynamic system (36) with controller dynamics (38), we can obtain the following extended dynamics:
(44)dη(t)=A⌢(t,r(t))η(t)dt+W⌢(r(t))η(t)dω(t),
where A⌢(t,r(t)) and W⌢(r(t)),  r(t)=i∈ℕ- are system matrices with compatible dimensions that depend on r(t) and t, and
(45)η(t)=[xT(t),xcT(t)]T∈ℝ2n,A⌢(t,i)=A⌢(i)+ΔA⌢A(t,i)+ΔA⌢B(t,i)+ΔA⌢C(t,i),A⌢(i)=[A(i)B(i)KC(i)KB(i)C(i)KA(i)],ΔA⌢A(t,i)=[MA(i)HA(t,i)NA(i)000],W⌢(i)=[W(i)000],ΔA⌢B(t,i)=[0MB(i)HB(t,i)NB(i)KC(i)00],ΔA⌢C(t,i)=[00KB(i)MC(i)HC(t,i)(t)NC(i)0].
Applying Theorem4, the dynamics (44) is stochastically stable if there exists a set of symmetric and positive-definite matrices {P(i)∣i=1,2,…,N} such that the following inequalities hold for every i∈ℕ-:
(46)A⌢T(t,i)P(i)+P(i)A⌢(t,i)+W⌢T(i)P(i)W⌢(i)+∑j=1NλijP(j)<0.
LetQA(i)=I-GAT(i)GA(i), QB(i)=I-GBT(i)GB(i), and   QC(i)=I-GCT(i)GC(i), and using Lemma 3, we have
(47)ΔA⌢A(t,i)=[MA(i)000][QA-1(i)000][GAT(i)000][NA(i)000]+[MA(i)000][QA-1/2(i)000][ΠA(t,i)000][NA(i)000]=defM⌢A(i)Q⌢A-1(i)G⌢AT(i)N⌢A(i)+M⌢A(i)Q⌢A-1/2(i)Π⌢A(t,i)N⌢A(i),ΔA⌢B(t,i)=[0MB(i)00][000QB-1(i)][000GBT(i)][000NB(i)KC(i)]+[0MB(i)00][000QB-1/2(i)][000ΠB(t,i)][000NB(i)KC(i)]=defM⌢B(i)Q⌢B-1(i)G⌢BT(i)N⌢B(i)+M⌢B(i)Q⌢B-1/2(i)Π⌢B(t,i)N⌢B(i),ΔA⌢C(t,i)=[00KB(i)MC(i)0][QC-1(i)000][GCT(i)000][NC(i)000]+[00KB(i)MC(i)0][QC-1/2(i)000][ΠC(t,i)000][NC(i)000]=defM⌢C(i)Q⌢C-1(i)G⌢CT(i)N⌢C(i)+M⌢C(i)Q⌢C-1/2(i)Π⌢C(t,i)N⌢C(i).
In view of (47) and Lemma 2, we have
(48)P(i)ΔA⌢A(t,i)+ΔA⌢AT(t,i)P(i)≤P(i)M⌢A(i)Q⌢A-1(i)G⌢AT(i)N⌢A(i)+N⌢AT(i)G⌢A(i)Q⌢A-1(i)M⌢AT(i)P(i)+εA-2(i)N⌢AT(i)N⌢A(i)+εA2(i)P(i)M⌢A(i)Q⌢A-1(i)M⌢AT(i)P(i)+εA-2(i)N⌢AT(i)G⌢A(i)Q⌢A-1(i)G⌢AT(i)N⌢A(i),P(i)ΔA⌢B(t,i)+ΔA⌢BT(t,i)P(i)≤P(i)M⌢B(i)NQ⌢B-1(i)G⌢BT(i)N⌢B(i)+N⌢BT(i)G⌢B(i)Q⌢B-1(i)M⌢BT(i)P(i)+εB-2(i)N⌢BT(i)N⌢B(i)+εB2(i)P(i)M⌢B(i)Q⌢B-1(i)M⌢BT(i)P(i)+εB-2(i)N⌢BT(i)G⌢B(i)Q⌢B-1(i)G⌢BT(i)N⌢B(i),P(i)ΔA⌢C(t,i)+ΔA⌢CT(t,i)P(i)≤P(i)M⌢C(i)Q⌢C-1(i)G⌢CT(i)N⌢C(i)+N⌢CT(i)G⌢C(i)Q⌢C-1(i)M⌢CT(i)P(i)+εC-2(i)N⌢CT(i)N⌢C(i)+εC2(i)P(i)M⌢C(i)Q⌢C-1(i)M⌢CT(i)P(i)+εC-2(i)N⌢cT(i)G⌢C(i)Q⌢C-1(i)G⌢CT(i)N⌢C(i).
Based on the matrix inequality (48), the condition (46) will be satisfied if the following holds:
(49)A⌢T(i)P(i)+P(i)A⌢(i)+∑j=1NλijP(j)+W⌢T(i)P(i)W⌢(i)+P(i)M⌢A(i)Q⌢A-1(i)G⌢AT(i)N⌢A(i)+N⌢AT(i)G⌢A(i)Q⌢A-1(i)M⌢AT(i)P(i)+εA-2(i)N⌢AT(i)N⌢A(i)+εB-2(i)N⌢BT(i)N⌢B(i)+εA2(i)P(i)M⌢A(i)Q⌢A-1(i)M⌢AT(i)P(i)+εA-2(i)N⌢AT(i)G⌢A(i)Q⌢A-1(i)G⌢AT(i)N⌢A(i)+P(i)M⌢B(i)Q⌢B-1(i)G⌢BT(i)N⌢B(i)+N⌢BT(i)G⌢B(i)Q⌢B-1(i)M⌢BT(i)P(i)+εB2(i)P(i)M⌢B(i)Q⌢B-1(i)M⌢BT(i)P(i)+εB-2(i)N⌢BT(i)G⌢B(i)Q⌢B-1(i)G⌢BT(i)N⌢B(i)+P(i)M⌢C(i)Q⌢C-1(i)G⌢CT(i)N⌢C(i)+N⌢CT(i)G⌢C(i)Q⌢C-1(i)M⌢CT(i)P(i)+εC-2(i)N⌢CT(i)N⌢C(i)+εC2(i)P(i)M⌢C(i)Q⌢C-1(i)M⌢CT(i)P(i)+εC-2(i)N⌢cT(i)G⌢C(i)Q⌢C-1(i)G⌢CT(i)N⌢C(i)<0.
It is obvious that inequality (49) is nonlinear in the design parameters KA(i), KB(i), KC(i), and P(i), i∈ℕ-. In order to transform inequality (49) into an LMI, define P(i), L(i), U(i), and V(i) as follows:
(50)P(i)=[P1(i)P2(i)P2T(i)P3(i)],L(i)=(P1(i)-P2(i)P3-1(i)P2T(i))-1,U(i)=[L(i)IL(i)0],V(i)=[I00-P3-1(i)P2T(i)],
where P1(i), P3(i) are symmetric and positive-define matrices.
Before and after multiply inequality (49) by UT(i)VT(i) and V(i)U(i), respectively, we have(51)UT(i)VT(i)P(i)A⌢(i)V(i)U(i)=[A(i)L(i)-B(i)KC(i)P3-1(i)P2T(i)L(i)A(i)(P1(i)A(i)L(i)-P1(i)B(i)KC(i)P3-1(i)P2T(i)L(i)-P2(i)KA(i)P3-1(i)P2T(i)L(i)+P2(i)KB(i)C(i)L(i))P1(i)A(i)+P2(i)KB(i)C(i)],UT(i)VT(i)P(j)V(i)U(i)=[(LT(i)L-1(j)L(i)+LT(i)(P2(i)P3-1(i)P3(j)-P2(j))×P3-1(j)(P2(i)P3-1(i)P3(j)-P2(j))TL(i))LT(i)(P1(j)-P2(j)P3-1(i)P2T(i))T(P1(j)-P2(j)P3-1(i)P2T(i))L(i)P1(j)],UT(i)VT(i)P(i)M⌢A(i)Q⌢A-1(i)G⌢AT(i)N⌢A(i)V(i)U(i)=[MA(i)QA-1(i)GAT(i)NA(i)L(i)MA(i)QA-1(i)GAT(i)NA(i)P1(i)MA(i)QA-1(i)GAT(i)NA(i)L(i)P1(i)MA(i)QA-1(i)GAT(i)NA(i)],UT(i)VT(i)P(i)M⌢A(i)Q⌢A-1(i)M⌢AT(i)P(i)V(i)U(i)=[MA(i)QA-1(i)MAT(i)MA(i)QA-1(i)MAT(i)P1(i)P1(i)MA(i)QA-1(i)MAT(i)P1(i)MA(i)QA-1(i)MAT(i)P1(i)],UT(i)VT(i)N⌢AT(i)G⌢A(i)Q⌢A-1(i)G⌢AT(i)N⌢A(i)V(i)U(i)=[LT(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)L(i)LT(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)L(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)],UT(i)VT(i)N⌢AT(i)N⌢A(i)V(i)U(i)=[LT(i)NAT(i)NA(i)L(i)LT(i)NAT(i)NA(i)NAT(i)NA(i)L(i)NAT(i)NA(i)],UT(i)VT(i)P(i)M⌢B(i)Q⌢B-1(i)G⌢BT(i)N⌢BT(i)V(i)U(i)=[-MB(i)QB-1(i)GBT(i)NB(i)KC(i)P3-1(i)P2T(i)L(i)0-P1(i)MB(i)QB-1(i)GBT(i)NB(i)KC(i)P3-1(i)P2T(i)L(i)0],UT(i)VT(i)P(i)M⌢B(i)Q⌢B-1(i)M⌢BT(i)P(i)V(i)U(i)=[MB(i)QB-1(i)MBT(i)MB(i)QB-1(i)MBT(i)P1(i)P1(i)MB(i)QB-1(i)MBT(i)P1(i)MB(i)QB-1(i)MBT(i)P1(i)],UT(i)VT(i)N⌢BT(i)G⌢B(i)Q⌢B-1(i)G⌢BT(i)N⌢B(i)V(i)U(i)=[LT(i)P2(i)P3-1(i)KCT(i)NBT(i)GB(i)QB-1(i)GBT(i)NB(i)KC(i)P3-1(i)P2T(i)L(i)000],UT(i)VT(i)N⌢BT(i)N⌢B(i)V(i)U(i)=[LT(i)P2(i)P3-1(i)KCT(i)NBT(i)NB(i)KC(i)P3-1(i)P2T(i)L(i)000],UT(i)VT(i)P(i)M⌢C(i)Q⌢C-1(i)G⌢CT(i)N⌢CT(i)V(i)U(i)=[00P2(i)KB(i)MC(i)QC-1(i)GCT(i)NC(i)L(i)P2(i)KB(i)MC(i)QC-1(i)GCT(i)NC(i)],UT(i)VT(i)P(i)M⌢C(i)Q⌢C-1(i)M⌢CT(i)P(i)V(i)U(i)=[000P2(i)KB(i)MC(i)QC-1(i)MCT(i)KBT(i)P2T(i)],UT(i)VT(i)N⌢CT(i)G⌢C(i)Q⌢C-1(i)G⌢CT(i)N⌢C(i)V(i)U(i)=[LT(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)L(i)LT(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)L(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)],UT(i)VT(i)N⌢CT(i)N⌢C(i)V(i)U(i)=[LT(i)NCT(i)NC(i)L(i)LT(i)NCT(i)NC(i)NCT(i)NC(i)L(i)NCT(i)NC(i)],UT(i)VT(i)W⌢T(i)P(i)W⌢(i)V(i)U(i)=[LT(i)WT(i)P1(i)W(i)L(i)LT(i)WT(i)P1(i)W(i)WT(i)P1(i)W(i)L(i)WT(i)P1(i)W(i)].
Using all these transformation, the condition (49) becomes
(52)[Ξ^1(i)Ξ2(i)Ξ2T(i)Ξ3(i)]<0,
with
(53)Ξ^1(i)=Ξ1(i)+∑j=1NλijLT(i)×[P2(i)P3-1(i)P3(j)-P2(j)]P3-1(j)×[P2(i)P3-1(i)P3(j)-P2(j)]TL(i)+LT(i)WT(i)P1(i)W(i)L(i),Ξ1(i)=A(i)L(i)+LT(i)AT(i)-B(i)KC(i)P3-1(i)P2T(i)L(i)-LT(i)P2(i)P3-1(i)KCT(i)BT(i)+∑j=1NλijLT(i)L-1(j)L(i)+MA(i)QA-1(i)GAT(i)NA(i)L(i)+LT(i)NAT(i)GA(i)QA-1(i)MAT(i)-MB(i)QB-1(i)GBT(i)NB(i)×KC(i)P3-1(i)P2T(i)L(i)-LT(i)P2(i)P3-1(i)KCT(i)NBT(i)×GB(i)QB-1(i)MBT(i)+εA2(i)MA(i)QA-1(i)MAT(i)+εA-2(i)LT(i)NAT(i)NA(i)L(i)+εA-2(i)LT(i)NAT(i)GA(i)×QA-1(i)GAT(i)NA(i)L(i)+εB-2(i)LT(i)P2(i)P3-1(i)KCT(i)×NBT(i)GB(i)QB-1(i)×GBT(i)NB(i)KC(i)P3-1(i)P2T(i)L(i)+εB2(i)MB(i)QB-1(i)MBT(i)+εB-2(i)LT(i)P2(i)P3-1(i)KCT(i)×NBT(i)NB(i)KC(i)P3-1(i)P2T(i)L(i)+εC-2(i)LT(i)NCT(i)GC(i)×QC-1(i)GCT(i)NC(i)L(i)+εC-2(i)LT(i)NCT(i)NC(i)L(i),Ξ2(i)=A(i)+LT(i)AT(i)P1T(i)-LT(i)P2(i)P3-1(i)KCT(i)BT(i)P1(i)-LT(i)P2(i)P3-1(i)KAT(i)P2T(i)+LT(i)CT(i)KBT(i)P2T(i)+∑j=1NλijLT(i)(P1(j)-P2(j)P3-1(i)P2T(i))T+MA(i)QA-1(i)GAT(i)NA(i)+LT(i)NAT(i)GA(i)QA-1(i)MAT(i)P1(i)+εA2(i)MA(i)QA-1(i)MAT(i)P1(i)+LT(i)WT(i)P1(i)W(i)-LT(i)P2(i)P3-1(i)KCT(i)×NBT(i)GB(i)QB-1(i)MBT(i)P1(i)+LT(i)NCT(i)GC(i)QC-1(i)MCT(i)KBT(i)P2T(i)+εA-2(i)LT(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)+εA-2(i)LT(i)NAT(i)NA(i)+εB2(i)MB(i)QB-1(i)MBT(i)P1(i)+εC-2(i)LT(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)+εC-2(i)LT(i)NCT(i)NC(i),Ξ3(i)=P1(i)A(i)+AT(i)P1(i)+P2(i)KB(i)C(i)+CT(i)KBT(i)P2T(i)+∑j=1NλijP1(j)+P1(i)MA(i)QA-1(i)GAT(i)NA(i)+NAT(i)GA(i)QA-1(i)MAT(i)P1(i)+P2(i)KB(i)MC(i)QC-1(i)GCT(i)NC(i)+NCT(i)GC(i)QC-1(i)MCT(i)KBT(i)P2T(i)+εA2(i)P1(i)MA(i)QA-1(i)MAT(i)P1(i)+εA-2(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)+εA-2(i)NAT(i)NA(i)+εB2(i)P1(i)MB(i)QB-1(i)MBT(i)P1(i)+εC2(i)P2(i)KB(i)Mc(i)QC-1(i)MCT(i)KBT(i)P2T(i)+WT(i)P1(i)W(i)+εC-2(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)+εC-2(i)NCT(i)NC(i).
Since∑j=1NλijLT(i)[P2(i)P3-1(i)P3(j)-P2(j)]P3-1(j)[P2(i)P3-1(i)P3(j)-P2(j)]TL(i) ≥0 and LT(i)WT(i)P1(i)W(i)L(i)≥0, we can obtain the following equivalent condition of (52) as follows:
(54)[Ξ1(i)Ξ2(i)Ξ2T(i)Ξ3(i)]<0.
Let P1(i)=X(i), P2(i)=Y-1(i)-X(i), and P3(i)=X(i)-Y-1(i), then,
(55)L(i)=(P1(i)-P2(i)P3-1(i)P2T(i))-1=Y(i),P3-1(i)P2T(i)=-I.
Define 𝕜B(i) and 𝕜C(i) by
(56)𝕜B(i)=P2(i)KB(i)=(Y-1(i)-X(i))KB(i),𝕜C(i)=-KC(i)P3-1(i)P2T(i)L(i)=KC(i)Y(i),
then applying all the previous algebraic manipulations, we have
(57)Ξ1(i)=A(i)Y(i)+YT(i)AT(i)+B(i)𝕜ℂ(i)+𝕜CT(i)BT(i)+∑j=1NλijYT(i)Y-1(j)Y(i)+MA(i)QA-1(i)GAT(i)NA(i)Y(i)+YT(i)NAT(i)GA(i)QA-1(i)MAT(i)+MB(i)QB-1(i)GBT(i)NB(i)𝕜C(i)+𝕜CT(i)NBT(i)GB(i)QB-1(i)MBT(i)+εA2(i)MA(i)QA-1(i)MAT(i)+εA-2(i)YT(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)Y(i)+εA-2(i)YT(i)NAT(i)NA(i)Y(i)+εB-2(i)𝕜CT(i)NBT(i)GB(i)QB-1(i)GBT(i)NB(i)𝕜C(i)+εB2(i)MB(i)QB-1(i)MBT(i)+εB-2(i)𝕜CT(i)NBT(i)NB(i)𝕜C(i)+εC-2(i)YT(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)Y(i)+εC-2(i)YT(i)NCT(i)NC(i)Y(i),Ξ2(i)=A(i)+YT(i)AT(i)X(i)+YT(i)CT(i)𝕜BT(i)+𝕜CT(i)BT(i)X(i)+YT(i)KAT(i)(Y-1(i)-X(i))T+∑j=1NλijYT(i)Y-1(j)+YT(i)WT(i)X(i)W(i)+MA(i)QA-1(i)GAT(i)NA(i)+YT(i)NAT(i)GA(i)QA-1(i)MAT(i)X(i)+𝕜CT(i)NBT(i)GB(i)QB-1(i)MBT(i)X(i)+YT(i)NCT(i)GC(i)QC-1(i)MCT(i)𝕜BT(i)+εA2(i)MA(i)QA-1(i)MAT(i)X(i)+εA-2(i)YT(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)+εA-2(i)YT(i)NAT(i)NA(i)+εB2(i)MB(i)QB-1(i)MBT(i)X(i)+εC-2(i)YT(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)+εC-2(i)YT(i)NCT(i)NC(i),Ξ3(i)=X(i)A(i)+AT(i)X(i)+𝕜B(i)C(i)+CT(i)𝕜BT(i)+WT(i)X(i)W(i)+∑j=1NλijX(j)+X(i)MA(i)QA-1(i)GAT(i)NA(i)+𝕜B(i)MC(i)QC-1(i)GCT(i)NC(i)+NCT(i)GC(i)QC-1(i)MCT(i)𝕜BT(i)+εA2(i)X(i)MA(i)QA-1(i)MAT(i)X(i)+εA-2(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)+εA-2(i)NAT(i)NA(i)+εB2(i)X(i)MB(i)QB-1(i)MBT(i)X(i)+εC2(i)𝕜B(i)Mc(i)QC-1(i)MCT(i)𝕜BT(i)+NAT(i)GA(i)QA-1(i)MAT(i)X(i)+εC-2(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)+εC-2(i)NCT(i)NC(i).
Using the expression of the controller given by (43), we have Ξ2(i)=0. This implies that the stability condition (54) is equivalent to the following conditions:
(58)Ξ1(i)<0,Ξ3(i)<0.
Let
(59)𝕊(Y(i))=Y(i)[λi1I,…,λi,i-1I,λi,i+1I,…,λiNI],𝕐(i)=diag{Y(1),…,Y(i-1),Y(i+1),⋯,Y(N)}.
Then,
(60)∑j=1NλijYT(i)X-1(j)Y(i)=λiiY(i)+𝕊(Y(i))𝕐-1(i)𝕊T(Y(i)).
According to the Schur complement and(61)RA2(i)=εA(i)MA(i)+εA-1(i)YT(i)NAT(i)GA(i),RB2(i)=εB(i)MB(i)+εB-1(i)𝕜CT(i)NBT(i)GB(i),
the stability conditions (58) become (39) and (40). Finally, notice that
(62)UT(i)VT(i)P(i)V(i)U(i)=[Y(i)IIX(i)]>0.
We get (41), which ends the proof of Theorem 6.


## 4.1. State Feedback Stochastic Stabilization

In this subsection, we discuss how to design the robust state feedback control law for the following Itô stochastic uncertain MJLS:(22)dx(t)=A(t,r(t))x(t)dt+B(t,r(t))u(t)dt+W(r(t))x(t)dω(t),x(0)=x0,r(0)=r0,
where B(t,r(t)), r(t)=i∈ℕ- are system matrices with compatible dimensions that depend on r(t) and t, and
(23)B(t,i)=B(i)+MB(i)HB(t,i)NB(i),HB(t,i)=ΣB(t,i)[I-GB(i)ΣB(t,i)]-1,
where B(i), MB(i), NB(i), and GB(i) are known real constant matrices with appropriate dimensions. ΣB(t,i) is an unknown real matrix function with Lebesgue-measurable elements, satisfying ΣBT(t,i)ΣB(t,i)≤I with I being the identity matrix. The robust state feedback control law to be designed is
(24)u(t)=K(r(t))x(t),
where K(r(t)) is a gain with appropriate dimension to be determined for each mode r(t)∈ℕ-. Notice that the gain of the above controller is model dependent, which requires the knowledge of the model r(t) at time t to choose the appropriate gain among {K(1),K(2),…,K(N)}. Thus when the model switches from mode i, which uses the gain K(i), to modej, the controller gains must be switched instantaneously to the gain K(j) to guarantee the desired performances.Theorem 5.
Suppose that there exists a set of symmetric and positive-definite matrices{X(i)∣i=1,2,…,N}, a set of matrices {Y(i)∣i=1,2,…,N}, and a set of positive scalars {εA(i)∣i=1,2,…,N} and {εB(i)∣i=1,2,…,N} such that the following sets of coupled inequalities hold for each i∈ℕ- and all admissible linear fractional uncertainties. Then the controller (24) with gain K(i)=Y(i)X-1(i) can stabilize stochastically the Itô stochastic uncertain MJLS in (22)
(25)[Ju(i)******RA1T(i)-QA(i)00000RB1T(i)0-QB(i)0000εA-1(i)NA(i)X(i)00-I000εB-1(i)NB(i)Y(i)000-I00W(i)X(i)0000-X(i)0𝕊T(X(i))00000-𝕏(i)]<0,
where * represents the blocks that are induced by symmetry and
(26)Ju(i)=X(i)AT(i)+A(i)X(i)+YT(i)BT(i)+B(i)Y(i)+λiiX(i),RA1(i)=εA(i)MA(i)+εA-1(i)X(i)NAT(i)GA(i),RB1(i)=εB(i)MB(i)+εB-1(i)YT(i)NBT(i)GB(i),𝕏(i)=diag{X(1),…,X(i-1),X(i+1),…,X(N)},𝕊(X(i))=X(i)[λi1I,…,λi,i-1I,λi,i+1I,…,λiNI],QA(i)=I-GAT(i)GA(i),QB(i)=I-GBT(i)GB(i).Proof.
With the use of the state feedback control law in (24) for system (22), the closed-loop system becomes
(27)dx(t)=(A(t,r(t))+B(t,r(t))K(r(t)))x(t)dt+W(r(t))x(t)dω(t).
In order to guarantee the stochastic stability of system (27), using the stochastically stable condition in Theorem 4, we know that the following inequality should be satisfied for each i∈ℕ-:
(28)[A(t,i)+B(t,i)K(i)]TP(i)+P(i)[A(t,i)+B(t,i)K(i)]+WT(i)P(i)W(i)+∑j=1NλijP(j)<0.
Let X(i)=P-1(i), and before and after multiplying the matrix inequality (28) by X(i), then, we have
(29)X(i)AT(t,i)+A(t,i)X(i)+X(i)KT(i)BT(t,i)+B(t,i)K(i)X(i)+X(i)WT(i)X-1(i)W(i)X(i)+∑j=1NλijX(i)X-1(j)X(i)<0.
Applying expressions (2) and (23) of the linear fractional uncertainties, the above inequality can be described as
(30)X(i)AT(i)+A(i)X(i)+X(i)KT(i)BT(i)+B(i)K(i)X(i)+X(i)NAT(i)HAT(t,i)MAT(i)+MA(i)HA(t,i)NA(i)X(i)+X(i)KT(i)NBT(i)HBT(t,i)MBT(i)+MB(i)HB(t,i)NB(i)K(i)X(i)+X(i)WT(i)X-1(i)W(i)X(i)+X(i)[∑j=1NλijX-1(j)]X(i)<0.
Again in view of Lemmas2 and 3, the following two matrix inequalities are obvious:
(31)MA(i)HA(t,i)NA(i)X(i)+X(i)NAT(i)HAT(t,i)MAT(i)≤RA1(i)QA-1(i)RA1T(i)+εA-2(i)X(i)NAT(i)NA(i)X(i),MB(i)HB(t,i)NB(i)K(i)X(i)+X(i)KT(i)NBT(i)HBT(t,i)MBT(i)≤RB1(i)QB-1(i)RB1T(i)+εB-2(i)X(i)KT(i)NBT(i)NB(i)K(i)X(i),
where
(32)RA1(i)=εA(i)MA(i)+εA-1(i)X(i)NAT(i)GA(i),QA(i)=I-GAT(i)GA(i),RB1(i)=εB(i)MB(i)+εB-1(i)X(i)KT(i)NBT(i)GB(i),QB(i)=I-GBT(i)GB(i).
Setting
(33)𝕏(i)=diag{X(1),…,X(i-1),X(i+1),…,X(N)},𝕊(X(i))=X(i)[λi1I,…,λi,i-1I,λi,i+1I,…,λiNI].
Then X(i)[∑j=1NλijX-1(j)]X(i) can be rewritten as follows:
(34)X(i)[∑j=1NλijX-1(j)]X(i)=λiiX(i)+𝕊(X(i))𝕏-1(i)𝕊T(X(i)).
Let Y(i)=K(i)X(i), and by means of the matrix inequalities (31) and (34), the closed-loop system (27) will be stochastically stable if the following matrix inequality holds:
(35)X(i)AT(i)+A(i)X(i)+YT(i)BT(i)+B(i)Y(i)+λiiX(i)+RA1(i)QA-1(i)RA1T(i)+εA-2(i)X(i)NAT(i)NA(i)X(i)+RB1(i)QB-1(i)RB1T(i)+εB-2(i)YT(i)NBT(i)NB(i)Y(i)+X(i)WT(i)X-1(i)W(i)X(i)+𝕊(X(i))𝕏-1(i)𝕊T(X(i))<0.
After using the Schur complement, we can get the LMIs (25), which end the proof of Theorem 5.


## 4.2. Dynamic Output Feedback Stochastic Stabilization

In Section4.1, the design method of state feedback controller had been given under that the state vector x(t) and the mode r(t) are available at each time t. However, some of the state variables are not measurable by their construction or the lack of appropriate sensors to give information. Alternatively we can use dynamic output feedback controller that uses the system’s measurement to design and to compute the control gains in this subsection.In this subsection, we discuss how to design the dynamic output feedback control law for the following Itô stochastic uncertain MJLS:(36)dx(t)=A(t,r(t))x(t)dt+B(t,r(t))u(t)dt+W(r(t))x(t)dω(t),y(t)=C(t,r(t))x(t),x(0)=x0,r(0)=r0,
where C(t,r(t)), r(t)=i∈ℕ- are system matrices with compatible dimensions that depend on r(t) and t, and
(37)C(t,i)=C(i)+MC(i)HC(t,i)NC(i),HC(t,i)=ΣC(t,i)[I-GC(i)ΣC(t,i)]-1,
where C(i),  MC(i),  NC(i), and GC(i) are known real constant matrices with appropriate dimensions. ΣC(t,i) is an unknown real matrix function with Lebesgue-measurable elements, satisfying ΣCT(t,i)ΣC(t,i)≤I with I being the identity matrix. The dynamic output feedback control law is to be described by the following structure:
(38)x˙c(t)=KA(r(t))xc(t)+KB(r(t))y(t),u(t)=KC(r(t))xc(t),xc(0)=0,r(0)=0,
where xc(t)∈ℝn is the controller state and KA(r(t)), KB(r(t)), and KC(r(t)) are the controller gains to be determined.Theorem 6.
If there exists a set of matrices{XT(i)=X(i)>0∣i=1,2,…,N}, {Y(i)>0∣i=1,2,…,N}, {𝕜B(i)∣i=1,2,…,N}, and {𝕜C(i)∣i=1,2,…,N} and a set of positive scalars {εA(i)∣i=1,2,…,N}, {εB(i)∣i=1,2,…,N}, and {εC(i)∣i=1,2,…,N} such that the following sets of inequalities (39)–(41) hold for each i∈ℕ- and all admissible linear fractional uncertainties, then the dynamic output feedback controller (38) with control gain (43) can stabilize stochastically the Itô stochastic uncertain MJLS in  (36)(39)
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(42)Jd1(i)=A(i)Y(i)+YT(i)A(i)+B(i)𝕜C(i)+𝕜CT(i)BT(i)+λiiY(i),Jd2(i)=X(i)A(i)+𝕜B(i)C(i)+AT(i)X(i)+CT(i)𝕜BT(i)+∑j=1NλijX(j)+εA-2(i)NAT(i)NA(i)+εC-2(i)NCT(i)NC(i),𝕊(Y(i))=Y(i)[λi1I,…,λi,i-1I,λi,i+1I,…,λiNI],𝕐(i)=diag{Y(1),…,Y(i-1),Y(i+1),…,Y(N)},RA2(i)=εA(i)MA(i)+εA-1(i)YT(i)NAT(i)GA(i),RB2(i)=εB(i)MB(i)+εB-1(i)𝕜CT(i)NBT(i)GB(i),QA(i)=I-GAT(i)GA(i),QB(i)=I-GBT(i)GB(i),QC(i)=I-GCT(i)GC(i).The dynamical output feedback controller gain is given by(43)KA(i)=[X(i)-Y-1(i)]-1[AT(i)Y-1(i)+X(i)A(i)+𝕜𝔹(i)C(i)+X(i)B(i)𝕜ℂ(i)Y-1(i)+∑j=1NλijY-1(j)+NAT(i)GA(i)QA-1(i)MAT(i)Y-1(i)+X(i)MA(i)QA-1(i)GAT(i)NA(i)+X(i)MB(i)QB-1(i)GBT(i)NB(i)𝕜ℂ(i)Y-1(i)+𝕜𝔹(i)MC(i)QC-1(i)GCT(i)NC(i)+εA2(i)X(i)MA(i)QA-1(i)MAT(i)Y-1(i)+εA-2(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)+εA-2(i)NAT(i)NA(i)+εB2(i)X(i)MB(i)QB-1(i)MBT(i)Y-1(i)+εC-2(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)+εC-2(i)NCT(i)NC(i)+WT(i)X(i)W(i)],KB(i)=[Y-1(i)-X(i)]-1𝕜B(i),KC(i)=𝕜C(i)Y-1(i).Proof.
Combining the dynamic system (36) with controller dynamics (38), we can obtain the following extended dynamics:
(44)dη(t)=A⌢(t,r(t))η(t)dt+W⌢(r(t))η(t)dω(t),
where A⌢(t,r(t)) and W⌢(r(t)),  r(t)=i∈ℕ- are system matrices with compatible dimensions that depend on r(t) and t, and
(45)η(t)=[xT(t),xcT(t)]T∈ℝ2n,A⌢(t,i)=A⌢(i)+ΔA⌢A(t,i)+ΔA⌢B(t,i)+ΔA⌢C(t,i),A⌢(i)=[A(i)B(i)KC(i)KB(i)C(i)KA(i)],ΔA⌢A(t,i)=[MA(i)HA(t,i)NA(i)000],W⌢(i)=[W(i)000],ΔA⌢B(t,i)=[0MB(i)HB(t,i)NB(i)KC(i)00],ΔA⌢C(t,i)=[00KB(i)MC(i)HC(t,i)(t)NC(i)0].
Applying Theorem4, the dynamics (44) is stochastically stable if there exists a set of symmetric and positive-definite matrices {P(i)∣i=1,2,…,N} such that the following inequalities hold for every i∈ℕ-:
(46)A⌢T(t,i)P(i)+P(i)A⌢(t,i)+W⌢T(i)P(i)W⌢(i)+∑j=1NλijP(j)<0.
LetQA(i)=I-GAT(i)GA(i), QB(i)=I-GBT(i)GB(i), and   QC(i)=I-GCT(i)GC(i), and using Lemma 3, we have
(47)ΔA⌢A(t,i)=[MA(i)000][QA-1(i)000][GAT(i)000][NA(i)000]+[MA(i)000][QA-1/2(i)000][ΠA(t,i)000][NA(i)000]=defM⌢A(i)Q⌢A-1(i)G⌢AT(i)N⌢A(i)+M⌢A(i)Q⌢A-1/2(i)Π⌢A(t,i)N⌢A(i),ΔA⌢B(t,i)=[0MB(i)00][000QB-1(i)][000GBT(i)][000NB(i)KC(i)]+[0MB(i)00][000QB-1/2(i)][000ΠB(t,i)][000NB(i)KC(i)]=defM⌢B(i)Q⌢B-1(i)G⌢BT(i)N⌢B(i)+M⌢B(i)Q⌢B-1/2(i)Π⌢B(t,i)N⌢B(i),ΔA⌢C(t,i)=[00KB(i)MC(i)0][QC-1(i)000][GCT(i)000][NC(i)000]+[00KB(i)MC(i)0][QC-1/2(i)000][ΠC(t,i)000][NC(i)000]=defM⌢C(i)Q⌢C-1(i)G⌢CT(i)N⌢C(i)+M⌢C(i)Q⌢C-1/2(i)Π⌢C(t,i)N⌢C(i).
In view of (47) and Lemma 2, we have
(48)P(i)ΔA⌢A(t,i)+ΔA⌢AT(t,i)P(i)≤P(i)M⌢A(i)Q⌢A-1(i)G⌢AT(i)N⌢A(i)+N⌢AT(i)G⌢A(i)Q⌢A-1(i)M⌢AT(i)P(i)+εA-2(i)N⌢AT(i)N⌢A(i)+εA2(i)P(i)M⌢A(i)Q⌢A-1(i)M⌢AT(i)P(i)+εA-2(i)N⌢AT(i)G⌢A(i)Q⌢A-1(i)G⌢AT(i)N⌢A(i),P(i)ΔA⌢B(t,i)+ΔA⌢BT(t,i)P(i)≤P(i)M⌢B(i)NQ⌢B-1(i)G⌢BT(i)N⌢B(i)+N⌢BT(i)G⌢B(i)Q⌢B-1(i)M⌢BT(i)P(i)+εB-2(i)N⌢BT(i)N⌢B(i)+εB2(i)P(i)M⌢B(i)Q⌢B-1(i)M⌢BT(i)P(i)+εB-2(i)N⌢BT(i)G⌢B(i)Q⌢B-1(i)G⌢BT(i)N⌢B(i),P(i)ΔA⌢C(t,i)+ΔA⌢CT(t,i)P(i)≤P(i)M⌢C(i)Q⌢C-1(i)G⌢CT(i)N⌢C(i)+N⌢CT(i)G⌢C(i)Q⌢C-1(i)M⌢CT(i)P(i)+εC-2(i)N⌢CT(i)N⌢C(i)+εC2(i)P(i)M⌢C(i)Q⌢C-1(i)M⌢CT(i)P(i)+εC-2(i)N⌢cT(i)G⌢C(i)Q⌢C-1(i)G⌢CT(i)N⌢C(i).
Based on the matrix inequality (48), the condition (46) will be satisfied if the following holds:
(49)A⌢T(i)P(i)+P(i)A⌢(i)+∑j=1NλijP(j)+W⌢T(i)P(i)W⌢(i)+P(i)M⌢A(i)Q⌢A-1(i)G⌢AT(i)N⌢A(i)+N⌢AT(i)G⌢A(i)Q⌢A-1(i)M⌢AT(i)P(i)+εA-2(i)N⌢AT(i)N⌢A(i)+εB-2(i)N⌢BT(i)N⌢B(i)+εA2(i)P(i)M⌢A(i)Q⌢A-1(i)M⌢AT(i)P(i)+εA-2(i)N⌢AT(i)G⌢A(i)Q⌢A-1(i)G⌢AT(i)N⌢A(i)+P(i)M⌢B(i)Q⌢B-1(i)G⌢BT(i)N⌢B(i)+N⌢BT(i)G⌢B(i)Q⌢B-1(i)M⌢BT(i)P(i)+εB2(i)P(i)M⌢B(i)Q⌢B-1(i)M⌢BT(i)P(i)+εB-2(i)N⌢BT(i)G⌢B(i)Q⌢B-1(i)G⌢BT(i)N⌢B(i)+P(i)M⌢C(i)Q⌢C-1(i)G⌢CT(i)N⌢C(i)+N⌢CT(i)G⌢C(i)Q⌢C-1(i)M⌢CT(i)P(i)+εC-2(i)N⌢CT(i)N⌢C(i)+εC2(i)P(i)M⌢C(i)Q⌢C-1(i)M⌢CT(i)P(i)+εC-2(i)N⌢cT(i)G⌢C(i)Q⌢C-1(i)G⌢CT(i)N⌢C(i)<0.
It is obvious that inequality (49) is nonlinear in the design parameters KA(i), KB(i), KC(i), and P(i), i∈ℕ-. In order to transform inequality (49) into an LMI, define P(i), L(i), U(i), and V(i) as follows:
(50)P(i)=[P1(i)P2(i)P2T(i)P3(i)],L(i)=(P1(i)-P2(i)P3-1(i)P2T(i))-1,U(i)=[L(i)IL(i)0],V(i)=[I00-P3-1(i)P2T(i)],
where P1(i), P3(i) are symmetric and positive-define matrices.
Before and after multiply inequality (49) by UT(i)VT(i) and V(i)U(i), respectively, we have(51)UT(i)VT(i)P(i)A⌢(i)V(i)U(i)=[A(i)L(i)-B(i)KC(i)P3-1(i)P2T(i)L(i)A(i)(P1(i)A(i)L(i)-P1(i)B(i)KC(i)P3-1(i)P2T(i)L(i)-P2(i)KA(i)P3-1(i)P2T(i)L(i)+P2(i)KB(i)C(i)L(i))P1(i)A(i)+P2(i)KB(i)C(i)],UT(i)VT(i)P(j)V(i)U(i)=[(LT(i)L-1(j)L(i)+LT(i)(P2(i)P3-1(i)P3(j)-P2(j))×P3-1(j)(P2(i)P3-1(i)P3(j)-P2(j))TL(i))LT(i)(P1(j)-P2(j)P3-1(i)P2T(i))T(P1(j)-P2(j)P3-1(i)P2T(i))L(i)P1(j)],UT(i)VT(i)P(i)M⌢A(i)Q⌢A-1(i)G⌢AT(i)N⌢A(i)V(i)U(i)=[MA(i)QA-1(i)GAT(i)NA(i)L(i)MA(i)QA-1(i)GAT(i)NA(i)P1(i)MA(i)QA-1(i)GAT(i)NA(i)L(i)P1(i)MA(i)QA-1(i)GAT(i)NA(i)],UT(i)VT(i)P(i)M⌢A(i)Q⌢A-1(i)M⌢AT(i)P(i)V(i)U(i)=[MA(i)QA-1(i)MAT(i)MA(i)QA-1(i)MAT(i)P1(i)P1(i)MA(i)QA-1(i)MAT(i)P1(i)MA(i)QA-1(i)MAT(i)P1(i)],UT(i)VT(i)N⌢AT(i)G⌢A(i)Q⌢A-1(i)G⌢AT(i)N⌢A(i)V(i)U(i)=[LT(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)L(i)LT(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)L(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)],UT(i)VT(i)N⌢AT(i)N⌢A(i)V(i)U(i)=[LT(i)NAT(i)NA(i)L(i)LT(i)NAT(i)NA(i)NAT(i)NA(i)L(i)NAT(i)NA(i)],UT(i)VT(i)P(i)M⌢B(i)Q⌢B-1(i)G⌢BT(i)N⌢BT(i)V(i)U(i)=[-MB(i)QB-1(i)GBT(i)NB(i)KC(i)P3-1(i)P2T(i)L(i)0-P1(i)MB(i)QB-1(i)GBT(i)NB(i)KC(i)P3-1(i)P2T(i)L(i)0],UT(i)VT(i)P(i)M⌢B(i)Q⌢B-1(i)M⌢BT(i)P(i)V(i)U(i)=[MB(i)QB-1(i)MBT(i)MB(i)QB-1(i)MBT(i)P1(i)P1(i)MB(i)QB-1(i)MBT(i)P1(i)MB(i)QB-1(i)MBT(i)P1(i)],UT(i)VT(i)N⌢BT(i)G⌢B(i)Q⌢B-1(i)G⌢BT(i)N⌢B(i)V(i)U(i)=[LT(i)P2(i)P3-1(i)KCT(i)NBT(i)GB(i)QB-1(i)GBT(i)NB(i)KC(i)P3-1(i)P2T(i)L(i)000],UT(i)VT(i)N⌢BT(i)N⌢B(i)V(i)U(i)=[LT(i)P2(i)P3-1(i)KCT(i)NBT(i)NB(i)KC(i)P3-1(i)P2T(i)L(i)000],UT(i)VT(i)P(i)M⌢C(i)Q⌢C-1(i)G⌢CT(i)N⌢CT(i)V(i)U(i)=[00P2(i)KB(i)MC(i)QC-1(i)GCT(i)NC(i)L(i)P2(i)KB(i)MC(i)QC-1(i)GCT(i)NC(i)],UT(i)VT(i)P(i)M⌢C(i)Q⌢C-1(i)M⌢CT(i)P(i)V(i)U(i)=[000P2(i)KB(i)MC(i)QC-1(i)MCT(i)KBT(i)P2T(i)],UT(i)VT(i)N⌢CT(i)G⌢C(i)Q⌢C-1(i)G⌢CT(i)N⌢C(i)V(i)U(i)=[LT(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)L(i)LT(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)L(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)],UT(i)VT(i)N⌢CT(i)N⌢C(i)V(i)U(i)=[LT(i)NCT(i)NC(i)L(i)LT(i)NCT(i)NC(i)NCT(i)NC(i)L(i)NCT(i)NC(i)],UT(i)VT(i)W⌢T(i)P(i)W⌢(i)V(i)U(i)=[LT(i)WT(i)P1(i)W(i)L(i)LT(i)WT(i)P1(i)W(i)WT(i)P1(i)W(i)L(i)WT(i)P1(i)W(i)].
Using all these transformation, the condition (49) becomes
(52)[Ξ^1(i)Ξ2(i)Ξ2T(i)Ξ3(i)]<0,
with
(53)Ξ^1(i)=Ξ1(i)+∑j=1NλijLT(i)×[P2(i)P3-1(i)P3(j)-P2(j)]P3-1(j)×[P2(i)P3-1(i)P3(j)-P2(j)]TL(i)+LT(i)WT(i)P1(i)W(i)L(i),Ξ1(i)=A(i)L(i)+LT(i)AT(i)-B(i)KC(i)P3-1(i)P2T(i)L(i)-LT(i)P2(i)P3-1(i)KCT(i)BT(i)+∑j=1NλijLT(i)L-1(j)L(i)+MA(i)QA-1(i)GAT(i)NA(i)L(i)+LT(i)NAT(i)GA(i)QA-1(i)MAT(i)-MB(i)QB-1(i)GBT(i)NB(i)×KC(i)P3-1(i)P2T(i)L(i)-LT(i)P2(i)P3-1(i)KCT(i)NBT(i)×GB(i)QB-1(i)MBT(i)+εA2(i)MA(i)QA-1(i)MAT(i)+εA-2(i)LT(i)NAT(i)NA(i)L(i)+εA-2(i)LT(i)NAT(i)GA(i)×QA-1(i)GAT(i)NA(i)L(i)+εB-2(i)LT(i)P2(i)P3-1(i)KCT(i)×NBT(i)GB(i)QB-1(i)×GBT(i)NB(i)KC(i)P3-1(i)P2T(i)L(i)+εB2(i)MB(i)QB-1(i)MBT(i)+εB-2(i)LT(i)P2(i)P3-1(i)KCT(i)×NBT(i)NB(i)KC(i)P3-1(i)P2T(i)L(i)+εC-2(i)LT(i)NCT(i)GC(i)×QC-1(i)GCT(i)NC(i)L(i)+εC-2(i)LT(i)NCT(i)NC(i)L(i),Ξ2(i)=A(i)+LT(i)AT(i)P1T(i)-LT(i)P2(i)P3-1(i)KCT(i)BT(i)P1(i)-LT(i)P2(i)P3-1(i)KAT(i)P2T(i)+LT(i)CT(i)KBT(i)P2T(i)+∑j=1NλijLT(i)(P1(j)-P2(j)P3-1(i)P2T(i))T+MA(i)QA-1(i)GAT(i)NA(i)+LT(i)NAT(i)GA(i)QA-1(i)MAT(i)P1(i)+εA2(i)MA(i)QA-1(i)MAT(i)P1(i)+LT(i)WT(i)P1(i)W(i)-LT(i)P2(i)P3-1(i)KCT(i)×NBT(i)GB(i)QB-1(i)MBT(i)P1(i)+LT(i)NCT(i)GC(i)QC-1(i)MCT(i)KBT(i)P2T(i)+εA-2(i)LT(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)+εA-2(i)LT(i)NAT(i)NA(i)+εB2(i)MB(i)QB-1(i)MBT(i)P1(i)+εC-2(i)LT(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)+εC-2(i)LT(i)NCT(i)NC(i),Ξ3(i)=P1(i)A(i)+AT(i)P1(i)+P2(i)KB(i)C(i)+CT(i)KBT(i)P2T(i)+∑j=1NλijP1(j)+P1(i)MA(i)QA-1(i)GAT(i)NA(i)+NAT(i)GA(i)QA-1(i)MAT(i)P1(i)+P2(i)KB(i)MC(i)QC-1(i)GCT(i)NC(i)+NCT(i)GC(i)QC-1(i)MCT(i)KBT(i)P2T(i)+εA2(i)P1(i)MA(i)QA-1(i)MAT(i)P1(i)+εA-2(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)+εA-2(i)NAT(i)NA(i)+εB2(i)P1(i)MB(i)QB-1(i)MBT(i)P1(i)+εC2(i)P2(i)KB(i)Mc(i)QC-1(i)MCT(i)KBT(i)P2T(i)+WT(i)P1(i)W(i)+εC-2(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)+εC-2(i)NCT(i)NC(i).
Since∑j=1NλijLT(i)[P2(i)P3-1(i)P3(j)-P2(j)]P3-1(j)[P2(i)P3-1(i)P3(j)-P2(j)]TL(i) ≥0 and LT(i)WT(i)P1(i)W(i)L(i)≥0, we can obtain the following equivalent condition of (52) as follows:
(54)[Ξ1(i)Ξ2(i)Ξ2T(i)Ξ3(i)]<0.
Let P1(i)=X(i), P2(i)=Y-1(i)-X(i), and P3(i)=X(i)-Y-1(i), then,
(55)L(i)=(P1(i)-P2(i)P3-1(i)P2T(i))-1=Y(i),P3-1(i)P2T(i)=-I.
Define 𝕜B(i) and 𝕜C(i) by
(56)𝕜B(i)=P2(i)KB(i)=(Y-1(i)-X(i))KB(i),𝕜C(i)=-KC(i)P3-1(i)P2T(i)L(i)=KC(i)Y(i),
then applying all the previous algebraic manipulations, we have
(57)Ξ1(i)=A(i)Y(i)+YT(i)AT(i)+B(i)𝕜ℂ(i)+𝕜CT(i)BT(i)+∑j=1NλijYT(i)Y-1(j)Y(i)+MA(i)QA-1(i)GAT(i)NA(i)Y(i)+YT(i)NAT(i)GA(i)QA-1(i)MAT(i)+MB(i)QB-1(i)GBT(i)NB(i)𝕜C(i)+𝕜CT(i)NBT(i)GB(i)QB-1(i)MBT(i)+εA2(i)MA(i)QA-1(i)MAT(i)+εA-2(i)YT(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)Y(i)+εA-2(i)YT(i)NAT(i)NA(i)Y(i)+εB-2(i)𝕜CT(i)NBT(i)GB(i)QB-1(i)GBT(i)NB(i)𝕜C(i)+εB2(i)MB(i)QB-1(i)MBT(i)+εB-2(i)𝕜CT(i)NBT(i)NB(i)𝕜C(i)+εC-2(i)YT(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)Y(i)+εC-2(i)YT(i)NCT(i)NC(i)Y(i),Ξ2(i)=A(i)+YT(i)AT(i)X(i)+YT(i)CT(i)𝕜BT(i)+𝕜CT(i)BT(i)X(i)+YT(i)KAT(i)(Y-1(i)-X(i))T+∑j=1NλijYT(i)Y-1(j)+YT(i)WT(i)X(i)W(i)+MA(i)QA-1(i)GAT(i)NA(i)+YT(i)NAT(i)GA(i)QA-1(i)MAT(i)X(i)+𝕜CT(i)NBT(i)GB(i)QB-1(i)MBT(i)X(i)+YT(i)NCT(i)GC(i)QC-1(i)MCT(i)𝕜BT(i)+εA2(i)MA(i)QA-1(i)MAT(i)X(i)+εA-2(i)YT(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)+εA-2(i)YT(i)NAT(i)NA(i)+εB2(i)MB(i)QB-1(i)MBT(i)X(i)+εC-2(i)YT(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)+εC-2(i)YT(i)NCT(i)NC(i),Ξ3(i)=X(i)A(i)+AT(i)X(i)+𝕜B(i)C(i)+CT(i)𝕜BT(i)+WT(i)X(i)W(i)+∑j=1NλijX(j)+X(i)MA(i)QA-1(i)GAT(i)NA(i)+𝕜B(i)MC(i)QC-1(i)GCT(i)NC(i)+NCT(i)GC(i)QC-1(i)MCT(i)𝕜BT(i)+εA2(i)X(i)MA(i)QA-1(i)MAT(i)X(i)+εA-2(i)NAT(i)GA(i)QA-1(i)GAT(i)NA(i)+εA-2(i)NAT(i)NA(i)+εB2(i)X(i)MB(i)QB-1(i)MBT(i)X(i)+εC2(i)𝕜B(i)Mc(i)QC-1(i)MCT(i)𝕜BT(i)+NAT(i)GA(i)QA-1(i)MAT(i)X(i)+εC-2(i)NCT(i)GC(i)QC-1(i)GCT(i)NC(i)+εC-2(i)NCT(i)NC(i).
Using the expression of the controller given by (43), we have Ξ2(i)=0. This implies that the stability condition (54) is equivalent to the following conditions:
(58)Ξ1(i)<0,Ξ3(i)<0.
Let
(59)𝕊(Y(i))=Y(i)[λi1I,…,λi,i-1I,λi,i+1I,…,λiNI],𝕐(i)=diag{Y(1),…,Y(i-1),Y(i+1),⋯,Y(N)}.
Then,
(60)∑j=1NλijYT(i)X-1(j)Y(i)=λiiY(i)+𝕊(Y(i))𝕐-1(i)𝕊T(Y(i)).
According to the Schur complement and(61)RA2(i)=εA(i)MA(i)+εA-1(i)YT(i)NAT(i)GA(i),RB2(i)=εB(i)MB(i)+εB-1(i)𝕜CT(i)NBT(i)GB(i),
the stability conditions (58) become (39) and (40). Finally, notice that
(62)UT(i)VT(i)P(i)V(i)U(i)=[Y(i)IIX(i)]>0.
We get (41), which ends the proof of Theorem 6.


## 5. Numerical Example

The aim of the following two examples is used to illustrate the theoretical results in Section4.Example 7 (state feedback stochastic stabilization).
Consider the Itô stochastic uncertain MJLS in (22) with N=2,r(t):[0,∞)→{1,2} and
mode no. 1(63)A(1)=[1.0-0.50.11.0],MA(1)=[0.10.2],NA(1)=[0.20.1],GA(1)=[0.2],W(1)=[0.10.00.00.1],B(1)=[1.00.00.01.0],MB(1)=[0.10.2],NB(1)=[0.20.1],GB(1)=[0.2].
Mode no. 2
(64)A(2)=[-0.20.50.0-0.25],MA(2)=[0.130.1],NA(2)=[0.10.2],GA(2)=[0.2],W(2)=[0.20.00.00.2],B(2)=[1.00.00.01.0],MB(2)=[0.130.1],NB(2)=[0.10.2],GB(2)=[0.2].
The jumping between the different modes is described by the transition rate matrix:
(65)Λ=[-2.02.03.0-3.0].
Setting εA(1)=εA(2)=0.5 and εB(1)=εB(2)=0.6. By means of inequality (25), we have
(66)X(1)=[0.3445-0.0142-0.01420.3521],X(2)=[0.3061-0.0107-0.01070.3069],Y(1)=[-0.73330.08190.0819-0.7503],Y(2)=[-0.2168-0.0850-0.0850-0.1993].
Then applying Theorem 5, the state feedback controller gains are obtained as follows:
(67)K(1)=[-2.12270.14690.1500-2.1252],(68)K(2)=[-0.7197-0.3022-0.3009-0.6600].
Therefore, the designed state feedback controller (24) is feasible and can guarantee the stochastic stability of the Itô stochastic uncertain MJLS in (22).Example 8 (dynamic output feedback stochastic stabilization).
Consider the Itô stochastic uncertain MJLS in (36) with N=2,r(t):[0,∞)→{1,2} and
mode no. 1(69)A(1)=[0.2-0.60.31.2],MA(1)=[0.120.23],NA(1)=[0.20.1],GA(1)=[0.5],W(1)=[0.10.00.00.1],B(1)=[1.00.00.01.0],MB(1)=[0.120.23],NB(1)=[0.20.1],GB(1)=[0.5],C(1)=[1.00.00.01.0],MC(1)=[0.120.23],NC(1)=[0.20.1],GC(1)=[0.5].
Mode no. 2
(70)A(2)=[-0.30.40.2-0.15],MA(2)=[0.120.15],NA(2)=[0.130.23],GA(2)=[0.5],W(2)=[0.20.00.00.2],B(2)=[1.00.00.01.0],MB(2)=[0.120.15],NB(2)=[0.130.23],GB(2)=[0.5],C(2)=[1.00.00.01.0],MC(2)=[0.120.15],NC(2)=[0.130.23],GC(2)=[0.5].
The transition probability rate matrix between the two modes is given by
(71)Λ=[-2.02.03.0-3.0].
Let εA(1)=εA(2)=0.5, εB(1)=εB(2)=0.1 and εC(1)=εC(2)=0.1; in view of inequalities (39)–(41), we have
(72)X(1)=[5.0423-0.2399-0.23992.9182],X(2)=[4.84610.15590.15594.4064],Y(1)=[0.5467-0.7160-0.71601.3711],Y(2)=[1.0892-0.7109-0.71090.6812],𝕜B(1)=[-6.8119-0.4532-0.4532-8.6839],𝕜B(2)=[-4.1475-2.0622-2.0622-4.1055],𝕜C(1)=[-1.73263.22413.2241-6.7326],𝕜C(2)=[-5.17422.74342.7434-1.5899].
Then according to Theorem 6, the dynamic output feedback controller gains are obtained as follows:
(73)KA(1)=[1.80304.38604.3860-4.1137],KB(1)=[0.5083-2.5782-1.97250.4493],KC(1)=[-0.28292.2038-1.6886-5.7923],KA(2)=[-4.82952.08906.75846.3166],KB(2)=[-0.5943-1.3272-1.8660-1.6399],KC(2)=[-6.6548-2.91763.12180.9240].
Therefore, the designed dynamic output feedback controller (38) is feasible and guarantees the stochastic stability of the Itô stochastic uncertain MJLS in (36).


## 6. Conclusions

The robustly stochastic stabilization problem for the Markov jumping Itô stochastic linear systems with linear fractional uncertainties had been investigated via state feedback and dynamic output feedback. Sufficient conditions, by solving linear matrix inequalities, are presented to realize the design of controller. The robustH∞ control problem via state feedback and dynamic output feedback for such Itô stochastic MJLS with linear fractional uncertainties should be studied in future work.



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*Source: 697849-2013-02-20.xml*