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

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.

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 Section 2, 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: (𝑡)((𝑡𝑥(𝑡)(𝑟(𝑡𝑥(𝑡)(𝑡),𝑥(0)𝑑𝑥=𝐴𝑡,𝑟))𝑑𝑡+𝑊))𝑑𝜔=𝑥₀,𝑟(0)=𝑟₀, (1) where {𝑟(𝑡),𝑡≥0} is a Markov process taking values in a finite space ℕ={1,2,…,𝑁} and 𝑥(𝑡)∈ℝ^𝑛 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∈ℝ^𝑛 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 𝐴(𝑡,𝑖)=𝐴(𝑖)+𝑀_𝐴(𝑖)𝐻_𝐴(𝑡,𝑖)𝑁_𝐴(𝑖),𝐻_𝐴(𝑡,𝑖)=Σ_𝐴(𝑡,𝑖)𝐼−𝐺_𝐴(𝑖)Σ_𝐴(𝑡,𝑖)⁻¹, (2) where 𝐴(𝑖), 𝑀_𝐴(𝑖), 𝑁_𝐴(𝑖), and 𝐺_𝐴(𝑖) are known real constant matrices with appropriate dimensions. Σ_𝐴(𝑡,𝑖) is an unknown real matrix function with Lebesgue-measurable elements, satisfying Σ^𝑇_𝐴(𝑡,𝑖)Σ_𝐴(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: 𝑃𝑟()(𝑡)=𝜆𝑡+ℎ=𝑗∣𝑟=𝑖_𝑖𝑗ℎ+𝑜(ℎ),𝑟(𝑡)jumpsfrommode𝑖tomode𝑗,1+𝜆_𝑖𝑖ℎ+𝑜(ℎ),𝑟(𝑡)staysatmode𝑖, (3) where 𝜆_𝑖𝑗≥0 is the transition rate from mode 𝑖 to mode 𝑗 at time instant 𝑡 when 𝑖≠𝑗, and 𝜆_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗⋅𝑜(ℎ) is the little-𝑜 notation defined by lim_ℎ→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): 𝐸^∞₀‖𝑥(𝑡)‖²𝑑𝑡∣𝑥₀,𝑟₀≤𝑇𝑥₀,𝑟₀, (4) 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]). Let 𝐻,𝐸 be given constant matrices with appropriate dimensions and norm-bounded time-varying uncertain matrix Σ(𝑡) with Σ^𝑇(𝑡)Σ(𝑡)≤𝐼. For any 𝜀>0, one has 𝐻Σ(𝑡)𝐸+𝐸^𝑇Σ^𝑇(𝑡)𝐻^𝑇≤𝜀𝐻𝐻^𝑇+𝜀⁻¹𝐸^𝑇𝐸. (5)

Lemma 3 (see [19]). Let the matrix G satisfy 𝐺^𝑇𝐺<𝐼; define the following set: 𝜉=𝐻(𝑡)=Σ(𝑡)[𝐼−𝐺Σ(𝑡)]⁻¹,Σ^𝑇(𝑡)Σ(𝑡)≤𝐼. (6) Then, the set 𝜉 can be rewritten as 𝜉=𝐻(𝑡)=𝐼−𝐺^𝑇𝐺⁻¹𝐺^𝑇+𝐼−𝐺^𝑇𝐺^−1/2Π(𝑡);Π^𝑇(𝑡)Π(𝑡)≤𝐼+𝐺𝐼−𝐺^𝑇𝐺⁻¹𝐺^𝑇. (7)

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 {𝜀_𝐴(𝑖)∣𝑖=1,2,…,𝑁} such that the following sets of coupled inequalities hold for each 𝑖∈ℕ and all admissible linear fractional uncertainties: 𝐽₀(𝑖)𝑅∗∗∗_𝑇𝐴0(𝑖)−𝑄_𝐴(𝑖)𝜀00_𝐴(𝑖)𝑁_𝐴(𝑖)𝑃(𝑖)𝑊(𝑖)(𝑖)0−𝐼000−𝑃<0, (8) where ∗ represents the blocks that are induced by symmetry and 𝐽₀(𝑖)=𝐴^𝑇(𝑖)𝑃(𝑖)+𝑃(𝑖)𝐴(𝑖)+^𝑁_𝑗=1𝜆_𝑖𝑗𝑃𝑗,𝑄_𝐴(𝑖)=𝐼−𝐺^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖),𝑅_𝐴0(𝑖)=𝜀_𝐴(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)+𝜀_𝐴⁻¹(𝑖)𝑃(𝑖)𝑀_𝐴(𝑖). (9)

Proof. Consider the following Lyapunov-like function: 𝑉(𝑥(𝑡),𝑟(𝑡))=𝑥^𝑇(𝑡)𝑃(𝑟(𝑡))𝑥(𝑡), (10) 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 𝜁𝑉(𝑥(𝑡),𝑟(𝑡))=lim_Δ𝑡→01{𝐸[𝑉(𝑥()(∣(𝑥(𝑡)(𝑡(𝑥(𝑡)(𝑡,Δ𝑡𝑡+Δ𝑡,𝑟𝑡+Δ𝑡)),𝑟))]−𝑉,𝑟))} (11) 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 𝜁𝑉(𝑥(𝑡),𝑖)=lim_Δ𝑡→01Δ𝑡^𝑁_{𝑗=1,𝑗≠𝑖}𝑟()(𝑡)𝑥()Pr𝑡+Δ𝑡=𝑗∣𝑟=𝑖×𝑉𝑡+Δ𝑡,𝑗+lim_Δ𝑡→01[{𝑟()(𝑡)}(𝑥())(𝑥(𝑡)Δ𝑡Pr𝑡+Δ𝑡=𝑖∣𝑟=𝑖×𝑉𝑡+Δ𝑡,𝑖−𝑉,𝑖)]=lim_Δ𝑡→0^𝑁_𝑗=1𝜆_𝑖𝑗𝑉𝑥()+𝑉(𝑥())(𝑥(𝑡))𝑡+Δ𝑡,𝑗𝑡+Δ𝑡,𝑖−𝑉,𝑖.Δ𝑡 (12)
According to the Itô stochastic differential coefficient theorem [18], we have 𝜁𝑉(𝑥(𝑡),𝑖)=^𝑁_𝑗=1𝜆_𝑖𝑗𝑉𝑥(𝑡),𝑗+[𝐴(𝑡,𝑖)𝑥(𝑡)]^𝑇𝜕𝑉(𝑥,𝑖)+1𝜕𝑥2tr𝑥^𝑇(𝑡)𝑊^𝑇(𝑖)𝜕²𝑉(𝑥,𝑖)𝜕𝑥²𝑊(𝑖)𝑥(𝑡). (13) Note that 𝜕𝑉(𝑥,𝑖)/𝜕𝑥=2𝑃(𝑖)𝑥(𝑡) and 𝜕²𝑉(𝑥,𝑖)/𝜕𝑥²=2𝑃(𝑖), then the infinitesimal generator 𝜁 becomes 𝜁𝑉(𝑥(𝑡),𝑖)=𝑥^𝑇(𝑡)Ω(𝑡,𝑖)𝑥(𝑡), (14) where Ω(𝑡,𝑖)=𝐴^𝑇(𝑡,𝑖)𝑃(𝑖)+𝑃(𝑖)𝐴(𝑡,𝑖)+∑^𝑁_𝑗=1𝜆_𝑖𝑗𝑃(𝑗)+𝑊^𝑇(𝑖)𝑃(𝑖)𝑊(𝑖).
Using Lemmas 2 and 3, we have 𝑁^𝑇_𝐴(𝑖)𝐻^𝑇_𝐴(𝑡,𝑖)𝑀^𝑇_𝐴(𝑖)𝑃(𝑖)+𝑃(𝑖)𝑀_𝐴(𝑖)𝐻_𝐴(𝑡,𝑖)𝑁_𝐴(𝑖)=𝑁^𝑇_𝐴(𝑖)𝐼−𝐺^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)⁻¹𝐺^𝑇_𝐴(𝑖)+𝐼−𝐺^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)^−1/2Π_𝐴(𝑡,𝑖)^𝑇×𝑀^𝑇_𝐴(𝑖)𝑃(𝑖)+𝑃(𝑖)𝑀_𝐴(𝑖)×𝐼−𝐺^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)⁻¹𝐺^𝑇_𝐴(𝑖)+𝐼−𝐺^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)^−1/2Π_𝐴(𝑡,𝑖)𝑁_𝐴(𝑖)≤𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)𝑃(𝑖)+𝑃(𝑖)𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝜀²_𝐴(𝑖)𝑁^𝑇_𝐴(𝑖)𝐼+𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝜀_𝐴⁻²(𝑖)𝑃(𝑖)𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)𝑃(𝑖)=𝑅_𝐴0(𝑖)𝑄_𝐴⁻¹(𝑖)𝑅^𝑇_𝐴0(𝑖)+𝜀²_𝐴(𝑖)𝑁^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖), (15) where 𝑅_𝐴0(𝑖)=𝜀_𝐴(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)+𝜀_𝐴⁻¹(𝑖)𝑃(𝑖)𝑀_𝐴(𝑖), 𝑄_𝐴(𝑖)=𝐼−𝐺^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖).
Hence Ω(𝑡,𝑖)≤Ω^⌣(𝑖), where Ω^⌣(𝑖)=𝐽₀(𝑖)+𝑅_𝐴0(𝑖)𝑄_𝐴⁻¹(𝑖)𝑅^𝑇_𝐴0(𝑖)+𝜀²_𝐴(𝑖)𝑁^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝑊^𝑇(𝑖)𝑃(𝑖)𝑊(𝑖),𝐽₀(𝑖)=𝐴^𝑇(𝑖)𝑃(𝑖)+𝑃(𝑖)𝐴(𝑖)+^𝑁_𝑗=1𝜆_𝑖𝑗𝑃𝑗. (16) According to the Schur complement, Ω^⌣(𝑖)<0, 𝑖∈ℕ implies 𝐽₀(𝑖)𝑅∗∗∗_𝑇𝐴0(𝑖)−𝑄_𝐴(𝑖)𝜀00_𝐴(𝑖)𝑁_𝐴(𝑖)𝑃(𝑖)𝑊(𝑖)(𝑖)0−𝐼000−𝑃<0. (17) Therefore (𝑥(𝑡),𝑖)≤𝑥^𝑇(𝑡)Ω^⌣(𝑖)𝑥(𝑡)≤max_𝑖∈ℕ{𝜆^⌣_maxΩ(𝑖)}𝑥^𝑇(𝑡)𝑥(𝑡).
By the generalized Dynkin formula, we have 𝐸𝑉(𝑥(𝑡),𝑖)∣𝑥₀,𝑟₀−𝑉𝑥₀,𝑟₀=𝐸^𝑡₀𝜁𝑉(𝑥(𝑠),𝑖)𝑑𝑠∣𝑥₀,𝑟₀≤max_𝑖∈ℕ𝜆^⌣_maxΩ(𝑖)𝐸^𝑡₀𝑥^𝑇(𝑠)𝑥(𝑠)𝑑𝑠∣𝑥₀,𝑟₀. (18) The last inequality implies −max_𝑖∈ℕ𝜆^⌣_maxΩ(𝑖)𝐸^𝑡₀𝑥^𝑇(𝑠)𝑥(𝑠)𝑑𝑠∣𝑥₀,𝑟₀≤𝑉𝑥₀,𝑟₀−𝐸𝑉(𝑥(𝑡),𝑖)∣𝑥₀,𝑟₀≤𝑉𝑥₀,𝑟₀. (19) Furthermore, the condition in (8) indicates max_𝑖∈ℕ{𝜆^⌣_maxΩ(𝑖)}<0, so 𝐸^𝑡₀𝑥^𝑇(𝑠)𝑥(𝑠)𝑑𝑠∣𝑥₀,𝑟₀𝑉𝑥≤−⁰,𝑟₀max_𝑖∈ℕ𝜆^⌣_maxΩ(𝑖)^def=𝑇𝑥₀,𝑟₀ (20) holds for any 𝑡>0. Letting 𝑡 go to infinity, then, we know that 𝐸^∞₀𝑥^𝑇(𝑠)𝑥(𝑠)𝑑𝑠∣𝑥₀,𝑟₀≤𝑇𝑥₀,𝑟₀. (21)
According to Definition 1, 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: (𝑡)((𝑡𝑥(𝑡)((𝑡𝑢(𝑡)(𝑟(𝑡𝑥(𝑡)(𝑡),𝑥(0)𝑑𝑥=𝐴𝑡,𝑟))𝑑𝑡+𝐵𝑡,𝑟))𝑑𝑡+𝑊))𝑑𝜔=𝑥₀,𝑟(0)=𝑟₀, (22) where 𝐵(𝑡,𝑟(𝑡)), 𝑟(𝑡)=𝑖∈ℕ are system matrices with compatible dimensions that depend on 𝑟(𝑡) and 𝑡, and 𝐵(𝑡,𝑖)=𝐵(𝑖)+𝑀_𝐵(𝑖)𝐻_𝐵(𝑡,𝑖)𝑁_𝐵(𝑖),𝐻_𝐵(𝑡,𝑖)=Σ_𝐵(𝑡,𝑖)𝐼−𝐺_𝐵(𝑖)Σ_𝐵(𝑡,𝑖)⁻¹, (23) where 𝐵(𝑖), 𝑀_𝐵(𝑖), 𝑁_𝐵(𝑖), and 𝐺_𝐵(𝑖) are known real constant matrices with appropriate dimensions. Σ_𝐵(𝑡,𝑖) is an unknown real matrix function with Lebesgue-measurable elements, satisfying Σ^𝑇_𝐵(𝑡,𝑖)Σ_𝐵(𝑡,𝑖)≤𝐼 with 𝐼 being the identity matrix. The robust state feedback control law to be designed is 𝑢(𝑡)=𝐾(𝑟(𝑡))𝑥(𝑡), (24) 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 {𝜀_𝐴(𝑖)∣𝑖=1,2,…,𝑁} and {𝜀_𝐵(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) 𝐽_𝑢(𝑖)𝑅∗∗∗∗∗∗_𝑇𝐴1(𝑖)−𝑄_𝐴(𝑖)𝑅00000_𝑇𝐵1(𝑖)0−𝑄_𝐵(𝑖)𝜀0000_𝐴−1(𝑖)𝑁_𝐴(𝑖)𝑋(𝑖)𝜀00−𝐼000_𝐵−1(𝑖)𝑁_𝐵(𝑖)𝑌(𝑖)𝑊(𝑖)𝑋(𝑖)(𝑖)0𝕊000−𝐼000000−𝑋_𝑇(𝑋(𝑖))00000−𝕏(𝑖)<0, (25) where ∗ represents the blocks that are induced by symmetry and 𝐽_𝑢(𝑖)=𝑋(𝑖)𝐴^𝑇(𝑖)+𝐴(𝑖)𝑋(𝑖)+𝑌^𝑇(𝑖)𝐵^𝑇(𝑖)+𝐵(𝑖)𝑌(𝑖)+𝜆_𝑖𝑖𝑋(𝑖),𝑅_𝐴1(𝑖)=𝜀_𝐴(𝑖)𝑀_𝐴(𝑖)+𝜀_𝐴⁻¹(𝑖)𝑋(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖),𝑅_𝐵1(𝑖)=𝜀_𝐵(𝑖)𝑀_𝐵(𝑖)+𝜀_𝐵⁻¹(𝑖)𝑌^𝑇(𝑖)𝑁^𝑇_𝐵(𝑖)𝐺_𝐵(𝑖),𝕏(𝑖){𝑋(1)()()(𝑁,𝕊(𝑋(𝑖(𝑖=diag,…,𝑋𝑖−1,𝑋𝑖+1,…,𝑋)}))=𝑋)𝜆_𝑖1𝐼,…,𝜆_{𝑖,𝑖−1}𝐼,𝜆_𝑖,𝑖+1𝐼,…,𝜆_𝑖𝑁𝐼,𝑄_𝐴(𝑖)=𝐼−𝐺^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖),𝑄_𝐵(𝑖)=𝐼−𝐺^𝑇_𝐵(𝑖)𝐺_𝐵(𝑖). (26)

Proof. With the use of the state feedback control law in (24) for system (22), the closed-loop system becomes (𝑡)=(𝐴((𝑡((𝑡𝐾(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥(𝑡)(𝑡).𝑑𝑥𝑡,𝑟))+𝐵𝑡,𝑟)))))𝑑𝑡+𝑊))𝑑𝜔 (27)
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: [𝐴(𝑡,𝑖)+𝐵(𝑡,𝑖)𝐾(𝑖)]^𝑇𝑃(𝑖)+𝑃(𝑖)[𝐴(𝑡,𝑖)+𝐵(𝑡,𝑖)𝐾(𝑖)]+𝑊^𝑇(𝑖)𝑃(𝑖)𝑊(𝑖)+^𝑁_𝑗=1𝜆_𝑖𝑗𝑃𝑗<0. (28) Let 𝑋(𝑖)=𝑃⁻¹(𝑖), and before and after multiplying the matrix inequality (28) by 𝑋(𝑖), then, we have 𝑋(𝑖)𝐴^𝑇(𝑡,𝑖)+𝐴(𝑡,𝑖)𝑋(𝑖)+𝑋(𝑖)𝐾^𝑇(𝑖)𝐵^𝑇()()𝐾(𝑖)𝑋(𝑖)(𝑖)𝑊𝑡,𝑖+𝐵𝑡,𝑖+𝑋^𝑇(𝑖)𝑋⁻¹(𝑖)𝑊(𝑖)𝑋(𝑖)+^𝑁_𝑗=1𝜆_𝑖𝑗𝑋(𝑖)𝑋⁻¹𝑗𝑋(𝑖)<0. (29)
Applying expressions (2) and (23) of the linear fractional uncertainties, the above inequality can be described as 𝑋(𝑖)𝐴^𝑇(𝑖)+𝐴(𝑖)𝑋(𝑖)+𝑋(𝑖)𝐾^𝑇(𝑖)𝐵^𝑇(𝑖)(𝑖)𝐾(𝑖)𝑋(𝑖)(𝑖)𝑁+𝐵+𝑋^𝑇_𝐴(𝑖)𝐻^𝑇_𝐴(𝑡,𝑖)𝑀^𝑇_𝐴(𝑖)+𝑀_𝐴(𝑖)𝐻_𝐴(𝑡,𝑖)𝑁_𝐴(𝑖)𝑋(𝑖)+𝑋(𝑖)𝐾^𝑇(𝑖)𝑁^𝑇_𝐵(𝑖)𝐻^𝑇_𝐵(𝑡,𝑖)𝑀^𝑇_𝐵(𝑖)+𝑀_𝐵(𝑖)𝐻_𝐵(𝑡,𝑖)𝑁_𝐵(𝑖)𝐾(𝑖)𝑋(𝑖)+𝑋(𝑖)𝑊^𝑇(𝑖)𝑋⁻¹(𝑖)𝑊(𝑖)𝑋(𝑖)+𝑋(𝑖)^𝑁_𝑗=1𝜆_𝑖𝑗𝑋⁻¹𝑗𝑋(𝑖)<0. (30)
Again in view of Lemmas 2 and 3, the following two matrix inequalities are obvious: 𝑀_𝐴(𝑖)𝐻_𝐴(𝑡,𝑖)𝑁_𝐴(𝑖)𝑋(𝑖)+𝑋(𝑖)𝑁^𝑇_𝐴(𝑖)𝐻^𝑇_𝐴(𝑡,𝑖)𝑀^𝑇_𝐴(𝑖)≤𝑅_𝐴1(𝑖)𝑄_𝐴⁻¹(𝑖)𝑅^𝑇_𝐴1(𝑖)+𝜀_𝐴⁻²(𝑖)𝑋(𝑖)𝑁^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)𝑋(𝑖),𝑀_𝐵(𝑖)𝐻_𝐵(𝑡,𝑖)𝑁_𝐵(𝑖)𝐾(𝑖)𝑋(𝑖)+𝑋(𝑖)𝐾^𝑇(𝑖)𝑁^𝑇_𝐵(𝑖)𝐻^𝑇_𝐵(𝑡,𝑖)𝑀^𝑇_𝐵(𝑖)≤𝑅_𝐵1(𝑖)𝑄_𝐵⁻¹(𝑖)𝑅^𝑇_𝐵1(𝑖)+𝜀_𝐵⁻²(𝑖)𝑋(𝑖)𝐾^𝑇(𝑖)𝑁^𝑇_𝐵(𝑖)𝑁_𝐵(𝑖)𝐾(𝑖)𝑋(𝑖), (31) where 𝑅_𝐴1(𝑖)=𝜀_𝐴(𝑖)𝑀_𝐴(𝑖)+𝜀_𝐴⁻¹(𝑖)𝑋(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖),𝑄_𝐴(𝑖)=𝐼−𝐺^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖),𝑅_𝐵1(𝑖)=𝜀_𝐵(𝑖)𝑀_𝐵(𝑖)+𝜀_𝐵⁻¹(𝑖)𝑋(𝑖)𝐾^𝑇(𝑖)𝑁^𝑇_𝐵(𝑖)𝐺_𝐵(𝑖),𝑄_𝐵(𝑖)=𝐼−𝐺^𝑇_𝐵(𝑖)𝐺_𝐵(𝑖). (32) Setting 𝕏(𝑖){𝑋(1)()()(𝑁,𝕊(𝑋(𝑖(𝑖=diag,…,𝑋𝑖−1,𝑋𝑖+1,…,𝑋)}))=𝑋)𝜆_𝑖1𝐼,…,𝜆_{𝑖,𝑖−1}𝐼,𝜆_𝑖,𝑖+1𝐼,…,𝜆_𝑖𝑁𝐼. (33) Then 𝑋(𝑖)[∑^𝑁_𝑗=1𝜆_𝑖𝑗𝑋⁻¹(𝑗)]𝑋(𝑖) can be rewritten as follows: 𝑋(𝑖)^𝑁_𝑗=1𝜆_𝑖𝑗𝑋⁻¹𝑗𝑋(𝑖)=𝜆_𝑖𝑖𝑋(𝑖)+𝕊(𝑋(𝑖))𝕏⁻¹(𝑖)𝕊^𝑇(𝑋(𝑖)). (34) 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: 𝑋(𝑖)𝐴^𝑇(𝑖)+𝐴(𝑖)𝑋(𝑖)+𝑌^𝑇(𝑖)𝐵^𝑇(𝑖)+𝐵(𝑖)𝑌(𝑖)+𝜆_𝑖𝑖𝑋(𝑖)+𝑅_𝐴1(𝑖)𝑄_𝐴⁻¹(𝑖)𝑅^𝑇_𝐴1(𝑖)+𝜀_𝐴⁻²(𝑖)𝑋(𝑖)𝑁^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)𝑋(𝑖)+𝑅_𝐵1(𝑖)𝑄_𝐵⁻¹(𝑖)𝑅^𝑇_𝐵1(𝑖)+𝜀_𝐵⁻²(𝑖)𝑌^𝑇(𝑖)𝑁^𝑇_𝐵(𝑖)𝑁_𝐵(𝑖)𝑌(𝑖)+𝑋(𝑖)𝑊^𝑇(𝑖)𝑋⁻¹(𝑖)𝑊(𝑖)𝑋(𝑖)+𝕊(𝑋(𝑖))𝕏⁻¹(𝑖)𝕊^𝑇(𝑋(𝑖))<0. (35) 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 Section 4.1, 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: (𝑡)((𝑡𝑥(𝑡)((𝑡𝑢(𝑡)(𝑟(𝑡𝑥(𝑡)(𝑡),𝑦(𝑡)((𝑡𝑥(𝑡)(0)𝑑𝑥=𝐴𝑡,𝑟))𝑑𝑡+𝐵𝑡,𝑟))𝑑𝑡+𝑊))𝑑𝜔=𝐶𝑡,𝑟)),𝑥=𝑥₀,𝑟(0)=𝑟₀, (36) where 𝐶(𝑡,𝑟(𝑡)), 𝑟(𝑡)=𝑖∈ℕ are system matrices with compatible dimensions that depend on 𝑟(𝑡) and 𝑡, and 𝐶(𝑡,𝑖)=𝐶(𝑖)+𝑀_𝐶(𝑖)𝐻_𝐶(𝑡,𝑖)𝑁_𝐶(𝑖),𝐻_𝐶(𝑡,𝑖)=Σ_𝐶(𝑡,𝑖)𝐼−𝐺_𝐶(𝑖)Σ_𝐶(𝑡,𝑖)⁻¹, (37) where 𝐶(𝑖), 𝑀_𝐶(𝑖), 𝑁_𝐶(𝑖), and 𝐺_𝐶(𝑖) are known real constant matrices with appropriate dimensions. Σ_𝐶(𝑡,𝑖) is an unknown real matrix function with Lebesgue-measurable elements, satisfying Σ^𝑇_𝐶(𝑡,𝑖)Σ_𝐶(𝑡,𝑖)≤𝐼 with 𝐼 being the identity matrix. The dynamic output feedback control law is to be described by the following structure: 𝑥̇_𝑐(𝑡)=𝐾_𝐴(𝑟(𝑡))𝑥_𝑐(𝑡)+𝐾_𝐵(𝑟(𝑡𝑦(𝑡),𝑢(𝑡)))=𝐾_𝐶(𝑟(𝑡))𝑥_𝑐(𝑡),𝑥_𝑐(0)=0,𝑟(0)=0, (38) where 𝑥_𝑐(𝑡)∈ℝ^𝑛 is the controller state and 𝐾_𝐴(𝑟(𝑡)), 𝐾_𝐵(𝑟(𝑡)), and 𝐾_𝐶(𝑟(𝑡)) are the controller gains to be determined.

Theorem 6. If there exists a set of matrices {𝑋^𝑇(𝑖)=𝑋(𝑖)>0∣𝑖=1,2,…,𝑁}, {𝑌(𝑖)>0∣𝑖=1,2,…,𝑁}, {𝕜_𝐵(𝑖)∣𝑖=1,2,…,𝑁}, and {𝕜_𝐶(𝑖)∣𝑖=1,2,…,𝑁} and a set of positive scalars {𝜀_𝐴(𝑖)∣𝑖=1,2,…,𝑁}, {𝜀_𝐵(𝑖)∣𝑖=1,2,…,𝑁}, and {𝜀_𝐶(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)

𝐽_𝑑1(𝑖)𝑅∗∗∗∗∗∗∗_𝑇𝐴2(𝑖)−𝑄_𝐴(𝑖)𝑅000000_𝑇𝐵2(𝑖)0−𝑄_𝐵(𝑖)𝜀00000_𝐶−1(𝑖)𝐺^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)𝑌(𝑖)00−𝑄_𝐶(𝑖)𝜀0000_𝐴−1(𝑖)𝑁_𝐴(𝑖)𝑌(𝑖)𝜀000−𝐼000_𝐵−1(𝑖)𝑁_𝐵(𝑖)𝕜_𝐶(𝑖)𝜀0000−𝐼00_𝐶−1(𝑖)𝑁_𝐶(𝑖)𝑌(𝑖)𝕊00000−𝐼0_𝑇(𝑌(𝑖(𝑖)𝐽))000000−𝕐<0,_𝑑2(𝑖)𝜀∗∗∗∗_𝐴(𝑖)𝑀^𝑇_𝐴(𝑖)𝑋(𝑖)+𝜀_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)−𝑄_𝐴(𝑖)𝜀000_𝐵(𝑖)𝑀^𝑇_𝐵(𝑖)𝑋(𝑖)0−𝑄_𝐵(𝑖)𝜀00_𝐶(𝑖)𝑀^𝑇_𝐶(𝑖)𝕜^𝑇_𝐵(𝑖)+𝜀_𝐶⁻¹(𝑖)𝐺^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)00−𝑄_𝐶(𝑖)0𝑊(𝑖)𝑋(𝑖)(𝑖)𝑌(𝑖)𝐼(𝑖)000−𝑋<0,𝐼𝑋>0, (39)(40)(41)

where ∗ represents the blocks that are induced by symmetry and 𝐽_𝑑1(𝑖)=𝐴(𝑖)𝑌(𝑖)+𝑌^𝑇(𝑖)𝐴(𝑖)+𝐵(𝑖)𝕜_𝐶(𝑖)+𝕜^𝑇_𝐶(𝑖)𝐵^𝑇(𝑖)+𝜆_𝑖𝑖𝑌(𝑖),𝐽_𝑑2(𝑖)=𝑋(𝑖)𝐴(𝑖)+𝕜_𝐵(𝑖)𝐶(𝑖)+𝐴^𝑇(𝑖)𝑋(𝑖)+𝐶^𝑇(𝑖)𝕜^𝑇_𝐵(𝑖)+^𝑁_𝑗=1𝜆_𝑖𝑗𝑋𝑗+𝜀_𝐴⁻²(𝑖)𝑁^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝜀_𝐶⁻²(𝑖)𝑁^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖),𝕊(𝑌(𝑖(𝑖))=𝑌)𝜆_𝑖1𝐼,…,𝜆_{𝑖,𝑖−1}𝐼,𝜆_𝑖,𝑖+1𝐼,…,𝜆_𝑖𝑁𝐼,𝕐(𝑖){𝑌(1)()()(𝑁,𝑅=diag,…,𝑌𝑖−1,𝑌𝑖+1,…,𝑌)}_𝐴2(𝑖)=𝜀_𝐴(𝑖)𝑀_𝐴(𝑖)+𝜀_𝐴⁻¹(𝑖)𝑌^𝑇(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖),𝑅_𝐵2(𝑖)=𝜀_𝐵(𝑖)𝑀_𝐵(𝑖)+𝜀_𝐵⁻¹(𝑖)𝕜^𝑇_𝐶(𝑖)𝑁^𝑇_𝐵(𝑖)𝐺_𝐵(𝑖),𝑄_𝐴(𝑖)=𝐼−𝐺^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖),𝑄_𝐵(𝑖)=𝐼−𝐺^𝑇_𝐵(𝑖)𝐺_𝐵(𝑖),𝑄_𝐶(𝑖)=𝐼−𝐺^𝑇_𝐶(𝑖)𝐺_𝐶(𝑖). (42)

The dynamical output feedback controller gain is given by𝐾_𝐴(𝑖)=𝑋(𝑖)−𝑌⁻¹(𝑖)⁻¹𝐴^𝑇(𝑖)𝑌⁻¹(𝑖)+𝑋(𝑖)𝐴(𝑖)+𝕜_𝔹(𝑖)𝐶(𝑖)+𝑋(𝑖)𝐵(𝑖)𝕜_ℂ(𝑖)𝑌⁻¹(𝑖)+∑^𝑁_𝑗=1𝜆_𝑖𝑗𝑌⁻¹𝑗+𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)𝑌⁻¹(𝑖)+𝑋(𝑖)𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝑋(𝑖)𝑀_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝐺^𝑇_𝐵(𝑖)𝑁_𝐵(𝑖)𝕜_ℂ(𝑖)𝑌⁻¹(𝑖)+𝕜_𝔹(𝑖)𝑀_𝐶(𝑖)𝑄_𝐶⁻¹(𝑖)𝐺^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)+𝜀²_𝐴(𝑖)𝑋(𝑖)𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)𝑌⁻¹(𝑖)+𝜀_𝐴⁻²(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝜀_𝐴⁻²(𝑖)𝑁^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝜀²_𝐵(𝑖)𝑋(𝑖)𝑀_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝑀^𝑇_𝐵(𝑖)𝑌⁻¹(𝑖)+𝜀_𝐶⁻²(𝑖)𝑁^𝑇_𝐶(𝑖)𝐺_𝐶(𝑖)𝑄_𝐶⁻¹(𝑖)𝐺^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)+𝜀_𝐶⁻²(𝑖)𝑁^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)+𝑊^𝑇(𝑖)𝑋(𝑖)𝑊(𝑖),𝐾_𝐵(𝑖)=𝑌⁻¹(𝑖)−𝑋(𝑖)⁻¹𝕜_𝐵(𝑖),𝐾_𝐶(𝑖)=𝕜_𝐶(𝑖)𝑌⁻¹(𝑖). (43)

Proof. Combining the dynamic system (36) with controller dynamics (38), we can obtain the following extended dynamics: 𝑑𝜂(𝑡)=𝐴^⌣(𝑡,𝑟(𝑡))𝜂(𝑡)𝑑𝑡+𝑊^⌣(𝑟(𝑡))𝜂(𝑡)𝑑𝜔(𝑡), (44) 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 𝜂(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇_𝑐(𝑡)^𝑇∈ℝ^2𝑛,_⌣𝐴(𝑡,𝑖)=𝐴^⌣(𝑖)+Δ𝐴^⌣_𝐴(𝑡,𝑖)+Δ𝐴^⌣_𝐵(𝑡,𝑖)+Δ𝐴^⌣_𝐶(𝑡,𝑖_⌣),𝐴(𝑖)=𝐴(𝑖)𝐵(𝑖)𝐾_𝐶(𝑖)𝐾_𝐵(𝑖)𝐶(𝑖)𝐾_𝐴(𝑖),Δ𝐴^⌣_𝐴()=𝑀𝑡,𝑖^𝐴(𝑖)𝐻_𝐴(𝑡,𝑖)𝑁_𝐴(𝑖)0,00_⌣𝑊(𝑖)=𝑊(𝑖)0,Δ00_⌣𝐴_𝐵(𝑡,𝑖)=0𝑀_𝐵(𝑖)𝐻_𝐵(𝑡,𝑖)𝑁_𝐵(𝑖)𝐾_𝐶(𝑖),Δ00_⌣𝐴_𝐶()=𝐾𝑡,𝑖00_𝐵(𝑖)𝑀_𝐶(𝑖)𝐻_𝐶(𝑡,𝑖)(𝑡)𝑁_𝐶(𝑖)0. (45)
Applying Theorem 4, 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: 𝐴^⌣𝑇(𝑡,𝑖)𝑃(𝑖)+𝑃(𝑖)𝐴^⌣(𝑡,𝑖)+𝑊^⌣𝑇(𝑖)𝑃(𝑖)𝑊^⌣(𝑖)+^𝑁_𝑗=1𝜆_𝑖𝑗𝑃𝑗<0. (46)
Let 𝑄_𝐴(𝑖)=𝐼−𝐺^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖), 𝑄_𝐵(𝑖)=𝐼−𝐺^𝑇_𝐵(𝑖)𝐺_𝐵(𝑖), and   𝑄_𝐶(𝑖)=𝐼−𝐺^𝑇_𝐶(𝑖)𝐺_𝐶(𝑖), and using Lemma 3, we have Δ𝐴^⌣_𝐴()=𝑀𝑡,𝑖_𝐴(𝑖)0𝑄00^𝐴−1(𝑖)0𝐺00^𝑇𝐴(𝑖)0𝑁00^𝐴(𝑖)0+𝑀00_𝐴(𝑖)0𝑄00^𝐴−1/2(𝑖)0Π00^𝐴(𝑡,𝑖)0𝑁00^𝐴(𝑖)000_def=𝑀^⌣_𝐴(𝑖)𝑄^⌣_𝐴⁻¹(𝑖)𝐺^⌣𝑇_𝐴(𝑖)𝑁^⌣_𝐴(𝑖)+𝑀^⌣_𝐴(𝑖)𝑄^⌣_𝐴^−1/2(𝑖)Π^⌣_𝐴(𝑡,𝑖)𝑁^⌣_𝐴(𝑖),Δ𝐴^⌣_𝐵()=𝑡,𝑖0𝑀_𝐵(𝑖)00000𝑄_𝐵⁻¹(𝑖)000𝐺^𝑇_𝐵(𝑖)000𝑁_𝐵(𝑖)𝐾_𝐶(𝑖)+0𝑀_𝐵(𝑖)00000𝑄_𝐵^−1/2(𝑖)000Π_𝐵(𝑡,𝑖)000𝑁_𝐵(𝑖)𝐾_𝐶(𝑖)_def=𝑀^⌣_𝐵(𝑖)𝑄^⌣_𝐵⁻¹(𝑖)𝐺^⌣𝑇_𝐵(𝑖)𝑁^⌣_𝐵(𝑖)+𝑀^⌣_𝐵(𝑖)𝑄^⌣_𝐵^−1/2(𝑖)Π^⌣_𝐵(𝑡,𝑖)𝑁^⌣_𝐵(𝑖),Δ𝐴^⌣_𝐶()=𝐾𝑡,𝑖00_𝐵(𝑖)𝑀_𝐶(𝑖)0𝑄_𝐶⁻¹(𝑖)0𝐺00^𝑇𝐶(𝑖)0𝑁00^𝐶(𝑖)0+𝐾0000_𝐵(𝑖)𝑀_𝐶(𝑖)0𝑄_𝐶^−1/2(𝑖)0Π00^𝐶(𝑡,𝑖)0𝑁00^𝐶(𝑖)000_def=𝑀^⌣_𝐶(𝑖)𝑄^⌣_𝐶⁻¹(𝑖)𝐺^⌣𝑇_𝐶(𝑖)𝑁^⌣_𝐶(𝑖)+𝑀^⌣_𝐶(𝑖)𝑄^⌣_𝐶^−1/2(𝑖)Π^⌣_𝐶(𝑡,𝑖)𝑁^⌣_𝐶(𝑖). (47) In view of (47) and Lemma 2, we have 𝑃(𝑖)Δ𝐴^⌣_𝐴(𝑡,𝑖)+Δ𝐴^⌣𝑇_𝐴()𝑃(𝑖)(𝑖)𝑡,𝑖≤𝑃𝑀^⌣_𝐴(𝑖)𝑄^⌣_𝐴⁻¹(𝑖)𝐺^⌣𝑇_𝐴(𝑖)𝑁^⌣_𝐴(𝑖)+𝑁^⌣𝑇_𝐴(𝑖)𝐺^⌣_𝐴(𝑖)𝑄^⌣_𝐴⁻¹(𝑖)𝑀^⌣𝑇_𝐴(𝑖)𝑃(𝑖)+𝜀_𝐴⁻²(𝑖)𝑁^⌣𝑇_𝐴(𝑖)𝑁^⌣_𝐴(𝑖)+𝜀²_𝐴(𝑖)𝑃(𝑖)𝑀^⌣_𝐴(𝑖)𝑄^⌣_𝐴⁻¹(𝑖)𝑀^⌣𝑇_𝐴(𝑖)𝑃(𝑖)+𝜀_𝐴⁻²(𝑖)𝑁^⌣𝑇_𝐴(𝑖)𝐺^⌣_𝐴(𝑖)𝑄^⌣_𝐴⁻¹(𝑖)𝐺^⌣𝑇_𝐴(𝑖)𝑁^⌣_𝐴(𝑖),𝑃(𝑖)Δ𝐴^⌣_𝐵(𝑡,𝑖)+Δ𝐴^⌣𝑇_𝐵()𝑃(𝑖)(𝑖)𝑡,𝑖≤𝑃𝑀^⌣_𝐵(𝑖)𝑁^⌣𝑄_𝐵⁻¹(𝑖)𝐺^⌣𝑇_𝐵(𝑖)𝑁^⌣_𝐵(𝑖)+𝑁^⌣𝑇_𝐵(𝑖)𝐺^⌣_𝐵(𝑖)𝑄^⌣_𝐵⁻¹(𝑖)𝑀^⌣𝑇_𝐵(𝑖)𝑃(𝑖)+𝜀_𝐵⁻²(𝑖)𝑁^⌣𝑇_𝐵(𝑖)𝑁^⌣_𝐵(𝑖)+𝜀²_𝐵(𝑖)𝑃(𝑖)𝑀^⌣_𝐵(𝑖)𝑄^⌣_𝐵⁻¹(𝑖)𝑀^⌣𝑇_𝐵(𝑖)𝑃(𝑖)+𝜀_𝐵⁻²(𝑖)𝑁^⌣𝑇_𝐵(𝑖)𝐺^⌣_𝐵(𝑖)𝑄^⌣_𝐵⁻¹(𝑖)𝐺^⌣𝑇_𝐵(𝑖)𝑁^⌣_𝐵(𝑖),𝑃(𝑖)Δ𝐴^⌣_𝐶(𝑡,𝑖)+Δ𝐴^⌣𝑇_𝐶()𝑃(𝑖)(𝑖)𝑡,𝑖≤𝑃𝑀^⌣_𝐶(𝑖)𝑄^⌣_𝐶⁻¹(𝑖)𝐺^⌣𝑇_𝐶(𝑖)𝑁^⌣_𝐶(𝑖)+𝑁^⌣𝑇_𝐶(𝑖)𝐺^⌣_𝐶(𝑖)𝑄^⌣_𝐶⁻¹(𝑖)𝑀^⌣𝑇_𝐶(𝑖)𝑃(𝑖)+𝜀_𝐶⁻²(𝑖)𝑁^⌣𝑇_𝐶(𝑖)𝑁^⌣_𝐶(𝑖)+𝜀²_𝐶(𝑖)𝑃(𝑖)𝑀^⌣_𝐶(𝑖)𝑄^⌣_𝐶⁻¹(𝑖)𝑀^⌣𝑇_𝐶(𝑖)𝑃(𝑖)+𝜀_𝐶⁻²(𝑖)𝑁^⌣𝑇_𝑐(𝑖)𝐺^⌣_𝐶(𝑖)𝑄^⌣_𝐶⁻¹(𝑖)𝐺^⌣𝑇_𝐶(𝑖)𝑁^⌣_𝐶(𝑖). (48)
Based on the matrix inequality (48), the condition (46) will be satisfied if the following holds: 𝐴^⌣𝑇(𝑖)𝑃(𝑖)+𝑃(𝑖)𝐴^⌣(𝑖)+^𝑁_𝑗=1𝜆_𝑖𝑗𝑃𝑗+𝑊^⌣𝑇(𝑖)𝑃(𝑖)𝑊^⌣(𝑖)+𝑃(𝑖)𝑀^⌣_𝐴(𝑖)𝑄^⌣_𝐴⁻¹(𝑖)𝐺^⌣𝑇_𝐴(𝑖)𝑁^⌣_𝐴(𝑖)+𝑁^⌣𝑇_𝐴(𝑖)𝐺^⌣_𝐴(𝑖)𝑄^⌣_𝐴⁻¹(𝑖)𝑀^⌣𝑇_𝐴(𝑖)𝑃(𝑖)+𝜀_𝐴⁻²(𝑖)𝑁^⌣𝑇_𝐴(𝑖)𝑁^⌣_𝐴(𝑖)+𝜀_𝐵⁻²(𝑖)𝑁^⌣𝑇_𝐵(𝑖)𝑁^⌣_𝐵(𝑖)+𝜀²_𝐴(𝑖)𝑃(𝑖)𝑀^⌣_𝐴(𝑖)𝑄^⌣_𝐴⁻¹(𝑖)𝑀^⌣𝑇_𝐴(𝑖)𝑃(𝑖)+𝜀_𝐴⁻²(𝑖)𝑁^⌣𝑇_𝐴(𝑖)𝐺^⌣_𝐴(𝑖)𝑄^⌣_𝐴⁻¹(𝑖)𝐺^⌣𝑇_𝐴(𝑖)𝑁^⌣_𝐴(𝑖)+𝑃(𝑖)𝑀^⌣_𝐵(𝑖)𝑄^⌣_𝐵⁻¹(𝑖)𝐺^⌣𝑇_𝐵(𝑖)𝑁^⌣_𝐵(𝑖)+𝑁^⌣𝑇_𝐵(𝑖)𝐺^⌣_𝐵(𝑖)𝑄^⌣_𝐵⁻¹(𝑖)𝑀^⌣𝑇_𝐵(𝑖)𝑃(𝑖)+𝜀²_𝐵(𝑖)𝑃(𝑖)𝑀^⌣_𝐵(𝑖)𝑄^⌣_𝐵⁻¹(𝑖)𝑀^⌣𝑇_𝐵(𝑖)𝑃(𝑖)+𝜀_𝐵⁻²(𝑖)𝑁^⌣𝑇_𝐵(𝑖)𝐺^⌣_𝐵(𝑖)𝑄^⌣_𝐵⁻¹(𝑖)𝐺^⌣𝑇_𝐵(𝑖)𝑁^⌣_𝐵(𝑖)+𝑃(𝑖)𝑀^⌣_𝐶(𝑖)𝑄^⌣_𝐶⁻¹(𝑖)𝐺^⌣𝑇_𝐶(𝑖)𝑁^⌣_𝐶(𝑖)+𝑁^⌣𝑇_𝐶(𝑖)𝐺^⌣_𝐶(𝑖)𝑄^⌣_𝐶⁻¹(𝑖)𝑀^⌣𝑇_𝐶(𝑖)𝑃(𝑖)+𝜀_𝐶⁻²(𝑖)𝑁^⌣𝑇_𝐶(𝑖)𝑁^⌣_𝐶(𝑖)+𝜀²_𝐶(𝑖)𝑃(𝑖)𝑀^⌣_𝐶(𝑖)𝑄^⌣_𝐶⁻¹(𝑖)𝑀^⌣𝑇_𝐶(𝑖)𝑃(𝑖)+𝜀_𝐶⁻²(𝑖)𝑁^⌣𝑇_𝑐(𝑖)𝐺^⌣_𝐶(𝑖)𝑄^⌣_𝐶⁻¹(𝑖)𝐺^⌣𝑇_𝐶(𝑖)𝑁^⌣_𝐶(𝑖)<0. (49)
It is obvious that inequality (49) is nonlinear in the design parameters 𝐾_𝐴(𝑖), 𝐾_𝐵(𝑖), 𝐾_𝐶(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: 𝑃(𝑖)=𝑃₁(𝑖)𝑃₂(𝑖)𝑃^𝑇₂(𝑖)𝑃₃(𝑖),𝐿(𝑖)=𝑃₁(𝑖)−𝑃₂(𝑖)𝑃₃⁻¹(𝑖)𝑃^𝑇₂(𝑖)⁻¹,𝑈(𝑖)=𝐿(𝑖)𝐼𝐿(𝑖)0,𝑉(𝑖)=𝐼00−𝑃₃⁻¹(𝑖)𝑃^𝑇₂(𝑖), (50) where 𝑃₁(𝑖), 𝑃₃(𝑖) are symmetric and positive-define matrices.
Before and after multiply inequality (49) by 𝑈^𝑇(𝑖)𝑉^𝑇(𝑖) and 𝑉(𝑖)𝑈(𝑖), respectively, we have𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑃(𝑖)𝐴^⌣(𝑖)𝑉(𝑖)𝑈(𝑖)=^{𝐴(𝑖)𝐿(𝑖)(𝑖)𝐾−𝐵𝐶(𝑖)𝑃3−1(𝑖)𝑃𝑇2(𝑖)𝐿(𝑖)𝐴(𝑖)}_{𝑃1(𝑖)𝐴(𝑖)𝐿(𝑖)−𝑃1(𝑖)𝐵(𝑖)𝐾𝐶(𝑖)𝑃3}⁻¹_(𝑖)𝑃^𝑇_{2(𝑖)𝐿(𝑖)−𝑃2(𝑖)𝐾𝐴(𝑖)𝑃3−1(𝑖)𝑃𝑇2(𝑖)𝐿(𝑖)+𝑃2(𝑖)𝐾𝐵(𝑖)𝐶(𝑖)𝐿(𝑖)𝑃1(𝑖)𝐴(𝑖)+𝑃2(𝑖)𝐾𝐵(𝑖)𝐶(𝑖)},𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑃𝑗𝑉(𝑖)𝑈(𝑖)=^{𝐿𝑇(𝑖)𝐿−1𝑗𝐿(𝑖)+𝐿𝑇(𝑖𝑃)2(𝑖)𝑃3−1(𝑖)𝑃3𝑗−𝑃2𝑗}_×𝑃3⁻¹_{𝑗𝑃2(𝑖)𝑃3}⁻¹_{(𝑖)𝑃3𝑗−𝑃2𝑗}^𝑇_𝐿(𝑖)^{𝐿𝑇(𝑖𝑃)1𝑗−𝑃2𝑗𝑃3−1(𝑖)𝑃𝑇2(𝑖)𝑇}_{𝑃1𝑗−𝑃2𝑗𝑃3−1(𝑖)𝑃𝑇2(𝑖𝐿(𝑖)𝑃)1𝑗},𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑃(𝑖)𝑀^⌣_𝐴(𝑖)𝑄^⌣_𝐴⁻¹(𝑖)𝐺^⌣𝑇_𝐴(𝑖)𝑁^⌣_𝐴(𝑖)𝑉(𝑖)𝑈(𝑖)=^{𝑀𝐴(𝑖)𝑄𝐴−1(𝑖)𝐺𝑇𝐴(𝑖)𝑁𝐴(𝑖)𝐿(𝑖)𝑀𝐴(𝑖)𝑄𝐴−1(𝑖)𝐺𝑇𝐴(𝑖)𝑁𝐴(𝑖)}_{𝑃1(𝑖)𝑀𝐴(𝑖)𝑄𝐴−1(𝑖)𝐺𝑇𝐴(𝑖)𝑁𝐴(𝑖)𝐿(𝑖)𝑃1(𝑖)𝑀𝐴(𝑖)𝑄𝐴−1(𝑖)𝐺𝑇𝐴(𝑖)𝑁𝐴(𝑖)},𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑃(𝑖)𝑀^⌣_𝐴(𝑖)𝑄^⌣_𝐴⁻¹(𝑖)𝑀^⌣𝑇_𝐴(𝑖)𝑃(𝑖)𝑉(𝑖)𝑈(𝑖)=^{𝑀𝐴(𝑖)𝑄𝐴−1(𝑖)𝑀𝑇𝐴(𝑖)𝑀𝐴(𝑖)𝑄𝐴−1(𝑖)𝑀𝑇𝐴(𝑖)𝑃1(𝑖)}_{𝑃1(𝑖)𝑀𝐴(𝑖)𝑄𝐴−1(𝑖)𝑀𝑇𝐴(𝑖)𝑃1(𝑖)𝑀𝐴(𝑖)𝑄𝐴−1(𝑖)𝑀𝑇𝐴(𝑖)𝑃1(𝑖)},𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑁^⌣𝑇_𝐴(𝑖)𝐺^⌣_𝐴(𝑖)𝑄^⌣_𝐴⁻¹(𝑖)𝐺^⌣𝑇_𝐴(𝑖)𝑁^⌣_𝐴(𝑖)𝑉(𝑖)𝑈(𝑖)=^{𝐿𝑇(𝑖)𝑁𝑇𝐴(𝑖)𝐺𝐴(𝑖)𝑄𝐴−1(𝑖)𝐺𝑇𝐴(𝑖)𝑁𝐴(𝑖)𝐿(𝑖)𝐿𝑇(𝑖)𝑁𝑇𝐴(𝑖)𝐺𝐴(𝑖)𝑄𝐴−1(𝑖)𝐺𝑇𝐴(𝑖)𝑁𝐴(𝑖)}_{𝑁𝑇𝐴(𝑖)𝐺𝐴(𝑖)𝑄𝐴−1(𝑖)𝐺𝑇𝐴(𝑖)𝑁𝐴(𝑖)𝐿(𝑖)𝑁𝑇𝐴(𝑖)𝐺𝐴(𝑖)𝑄𝐴−1(𝑖)𝐺𝑇𝐴(𝑖)𝑁𝐴(𝑖)},𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑁^⌣𝑇_𝐴(𝑖)𝑁^⌣_𝐴(𝑖)𝑉(𝑖)𝑈(𝑖)=^{𝐿𝑇(𝑖)𝑁𝑇𝐴(𝑖)𝑁𝐴(𝑖)𝐿(𝑖)𝐿𝑇(𝑖)𝑁𝑇𝐴(𝑖)𝑁𝐴(𝑖)}_{𝑁𝑇𝐴(𝑖)𝑁𝐴(𝑖)𝐿(𝑖)𝑁𝑇𝐴(𝑖)𝑁𝐴(𝑖)},𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑃(𝑖)𝑀^⌣_𝐵(𝑖)𝑄^⌣_𝐵⁻¹(𝑖)𝐺^⌣𝑇_𝐵(𝑖)𝑁^⌣𝑇_𝐵(𝑖)𝑉(𝑖)𝑈(𝑖)=^{−𝑀𝐵(𝑖)𝑄𝐵−1(𝑖)𝐺𝑇𝐵(𝑖)𝑁𝐵(𝑖)𝐾𝐶(𝑖)𝑃3−1(𝑖)𝑃𝑇2(𝑖)𝐿(𝑖)0}_{−𝑃1(𝑖)𝑀𝐵(𝑖)𝑄𝐵−1(𝑖)𝐺𝑇𝐵(𝑖)𝑁𝐵(𝑖)𝐾𝐶(𝑖)𝑃3−1(𝑖)𝑃𝑇2(𝑖)𝐿(𝑖)0},𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑃(𝑖)𝑀^⌣_𝐵(𝑖)𝑄^⌣_𝐵⁻¹(𝑖)𝑀^⌣𝑇_𝐵(𝑖)𝑃(𝑖)𝑉(𝑖)𝑈(𝑖)=^{𝑀𝐵(𝑖)𝑄𝐵−1(𝑖)𝑀𝑇𝐵(𝑖)𝑀𝐵(𝑖)𝑄𝐵−1(𝑖)𝑀𝑇𝐵(𝑖)𝑃1(𝑖)}_{𝑃1(𝑖)𝑀𝐵(𝑖)𝑄𝐵−1(𝑖)𝑀𝑇𝐵(𝑖)𝑃1(𝑖)𝑀𝐵(𝑖)𝑄𝐵−1(𝑖)𝑀𝑇𝐵(𝑖)𝑃1(𝑖)},𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑁^⌣𝑇_𝐵(𝑖)𝐺^⌣_𝐵(𝑖)𝑄^⌣_𝐵⁻¹(𝑖)𝐺^⌣𝑇_𝐵(𝑖)𝑁^⌣_𝐵(𝑖)𝑉(𝑖)𝑈(𝑖)=^{𝐿𝑇(𝑖)𝑃2(𝑖)𝑃3−1(𝑖)𝐾𝑇𝐶(𝑖)𝑁𝑇𝐵(𝑖)𝐺𝐵(𝑖)𝑄𝐵−1(𝑖)𝐺𝑇𝐵(𝑖)𝑁𝐵(𝑖)𝐾𝐶(𝑖)𝑃3−1(𝑖)𝑃𝑇2(𝑖)𝐿(𝑖)0}₀₀,𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑁^⌣𝑇_𝐵(𝑖)𝑁^⌣_𝐵(𝑖)𝑉(𝑖)𝑈(𝑖)=^{𝐿𝑇(𝑖)𝑃2(𝑖)𝑃3−1(𝑖)𝐾𝑇𝐶(𝑖)𝑁𝑇𝐵(𝑖)𝑁𝐵(𝑖)𝐾𝐶(𝑖)𝑃3−1(𝑖)𝑃𝑇2(𝑖)𝐿(𝑖)0}₀₀,𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑃(𝑖)𝑀^⌣_𝐶(𝑖)𝑄^⌣_𝐶⁻¹(𝑖)𝐺^⌣𝑇_𝐶(𝑖)𝑁^⌣𝑇_𝐶(𝑖)𝑉(𝑖)𝑈(𝑖)=_𝑃⁰⁰_{2(𝑖)𝐾𝐵(𝑖)𝑀𝐶(𝑖)𝑄𝐶−1(𝑖)𝐺𝑇𝐶(𝑖)𝑁𝐶(𝑖)𝐿(𝑖)𝑃2(𝑖)𝐾𝐵(𝑖)𝑀𝐶(𝑖)𝑄𝐶−1(𝑖)𝐺𝑇𝐶(𝑖)𝑁𝐶(𝑖)},𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑃(𝑖)𝑀^⌣_𝐶(𝑖)𝑄^⌣_𝐶⁻¹(𝑖)𝑀^⌣𝑇_𝐶(𝑖)𝑃(𝑖)𝑉(𝑖)𝑈(𝑖)=⁰⁰_{0𝑃2(𝑖)𝐾𝐵(𝑖)𝑀𝐶(𝑖)𝑄𝐶−1(𝑖)𝑀𝑇𝐶(𝑖)𝐾𝑇𝐵(𝑖)𝑃𝑇2(𝑖)},𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑁^⌣𝑇_𝐶(𝑖)𝐺^⌣_𝐶(𝑖)𝑄^⌣_𝐶⁻¹(𝑖)𝐺^⌣𝑇_𝐶(𝑖)𝑁^⌣_𝐶(𝑖)𝑉(𝑖)𝑈(𝑖)=^{𝐿𝑇(𝑖)𝑁𝑇𝐶(𝑖)𝐺𝐶(𝑖)𝑄𝐶−1(𝑖)𝐺𝑇𝐶(𝑖)𝑁𝐶(𝑖)𝐿(𝑖)𝐿𝑇(𝑖)𝑁𝑇𝐶(𝑖)𝐺𝐶(𝑖)𝑄𝐶−1(𝑖)𝐺𝑇𝐶(𝑖)𝑁𝐶(𝑖)}_{𝑁𝑇𝐶(𝑖)𝐺𝐶(𝑖)𝑄𝐶−1(𝑖)𝐺𝑇𝐶(𝑖)𝑁𝐶(𝑖)𝐿(𝑖)𝑁𝑇𝐶(𝑖)𝐺𝐶(𝑖)𝑄𝐶−1(𝑖)𝐺𝑇𝐶(𝑖)𝑁𝐶(𝑖)},𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑁^⌣𝑇_𝐶(𝑖)𝑁^⌣_𝐶(𝑖)𝑉(𝑖)𝑈(𝑖)=^{𝐿𝑇(𝑖)𝑁𝑇𝐶(𝑖)𝑁𝐶(𝑖)𝐿(𝑖)𝐿𝑇(𝑖)𝑁𝑇𝐶(𝑖)𝑁𝐶(𝑖)}_{𝑁𝑇𝐶(𝑖)𝑁𝐶(𝑖)𝐿(𝑖)𝑁𝑇𝐶(𝑖)𝑁𝐶(𝑖)},𝑈^𝑇(𝑖)𝑉^𝑇(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)}. (51)
Using all these transformation, the condition (49) becomes Ξ₁(𝑖)Ξ₂(𝑖)Ξ^𝑇₂(𝑖)Ξ₃(𝑖)<0, (52) with Ξ₁(𝑖)=Ξ₁(𝑖)+^𝑁_𝑗=1𝜆_𝑖𝑗𝐿^𝑇(𝑖)×𝑃₂(𝑖)𝑃₃⁻¹(𝑖)𝑃₃𝑗−𝑃₂𝑗𝑃₃⁻¹𝑗×𝑃₂(𝑖)𝑃₃⁻¹(𝑖)𝑃₃𝑗−𝑃₂𝑗^𝑇𝐿(𝑖)+𝐿^𝑇(𝑖)𝑊^𝑇(𝑖)𝑃₁(𝑖)𝑊(𝑖)𝐿(𝑖),Ξ₁(𝑖)=𝐴(𝑖)𝐿(𝑖)+𝐿^𝑇(𝑖)𝐴^𝑇(𝑖)−𝐵(𝑖)𝐾_𝐶(𝑖)𝑃₃⁻¹(𝑖)𝑃^𝑇₂(𝑖)𝐿(𝑖)−𝐿^𝑇(𝑖)𝑃₂(𝑖)𝑃₃⁻¹(𝑖)𝐾^𝑇_𝐶(𝑖)𝐵^𝑇(𝑖)+^𝑁_𝑗=1𝜆_𝑖𝑗𝐿^𝑇(𝑖)𝐿⁻¹𝑗𝐿(𝑖)+𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)𝐿(𝑖)+𝐿^𝑇(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)−𝑀_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝐺^𝑇_𝐵(𝑖)𝑁_𝐵(𝑖)×𝐾_𝐶(𝑖)𝑃₃⁻¹(𝑖)𝑃^𝑇₂(𝑖)𝐿(𝑖)−𝐿^𝑇(𝑖)𝑃₂(𝑖)𝑃₃⁻¹(𝑖)𝐾^𝑇_𝐶(𝑖)𝑁^𝑇_𝐵(𝑖)×𝐺_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝑀^𝑇_𝐵(𝑖)+𝜀²_𝐴(𝑖)𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)+𝜀_𝐴⁻²(𝑖)𝐿^𝑇(𝑖)𝑁^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)𝐿(𝑖)+𝜀_𝐴⁻²(𝑖)𝐿^𝑇(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)×𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)𝐿(𝑖)+𝜀_𝐵⁻²(𝑖)𝐿^𝑇(𝑖)𝑃₂(𝑖)𝑃₃⁻¹(𝑖)𝐾^𝑇_𝐶(𝑖)×𝑁^𝑇_𝐵(𝑖)𝐺_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)×𝐺^𝑇_𝐵(𝑖)𝑁_𝐵(𝑖)𝐾_𝐶(𝑖)𝑃₃⁻¹(𝑖)𝑃^𝑇₂(𝑖)𝐿(𝑖)+𝜀²_𝐵(𝑖)𝑀_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝑀^𝑇_𝐵(𝑖)+𝜀_𝐵⁻²(𝑖)𝐿^𝑇(𝑖)𝑃₂(𝑖)𝑃₃⁻¹(𝑖)𝐾^𝑇_𝐶(𝑖)×𝑁^𝑇_𝐵(𝑖)𝑁_𝐵(𝑖)𝐾_𝐶(𝑖)𝑃₃⁻¹(𝑖)𝑃^𝑇₂(𝑖)𝐿(𝑖)+𝜀_𝐶⁻²(𝑖)𝐿^𝑇(𝑖)𝑁^𝑇_𝐶(𝑖)𝐺_𝐶(𝑖)×𝑄_𝐶⁻¹(𝑖)𝐺^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)𝐿(𝑖)+𝜀_𝐶⁻²(𝑖)𝐿^𝑇(𝑖)𝑁^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)𝐿(𝑖),Ξ₂(𝑖)=𝐴(𝑖)+𝐿^𝑇(𝑖)𝐴^𝑇(𝑖)𝑃^𝑇₁(𝑖)−𝐿^𝑇(𝑖)𝑃₂(𝑖)𝑃₃⁻¹(𝑖)𝐾^𝑇_𝐶(𝑖)𝐵^𝑇(𝑖)𝑃₁(𝑖)−𝐿^𝑇(𝑖)𝑃₂(𝑖)𝑃₃⁻¹(𝑖)𝐾^𝑇_𝐴(𝑖)𝑃^𝑇₂(𝑖)+𝐿^𝑇(𝑖)𝐶^𝑇(𝑖)𝐾^𝑇_𝐵(𝑖)𝑃^𝑇₂(𝑖)+^𝑁_𝑗=1𝜆_𝑖𝑗𝐿^𝑇(𝑖)𝑃₁𝑗−𝑃₂𝑗𝑃₃⁻¹(𝑖)𝑃^𝑇₂(𝑖)^𝑇+𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝐿^𝑇(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)𝑃₁(𝑖)+𝜀²_𝐴(𝑖)𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)𝑃₁(𝑖)+𝐿^𝑇(𝑖)𝑊^𝑇(𝑖)𝑃₁(𝑖)𝑊(𝑖)−𝐿^𝑇(𝑖)𝑃₂(𝑖)𝑃₃⁻¹(𝑖)𝐾^𝑇_𝐶(𝑖)×𝑁^𝑇_𝐵(𝑖)𝐺_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝑀^𝑇_𝐵(𝑖)𝑃₁(𝑖)+𝐿^𝑇(𝑖)𝑁^𝑇_𝐶(𝑖)𝐺_𝐶(𝑖)𝑄_𝐶⁻¹(𝑖)𝑀^𝑇_𝐶(𝑖)𝐾^𝑇_𝐵(𝑖)𝑃^𝑇₂(𝑖)+𝜀_𝐴⁻²(𝑖)𝐿^𝑇(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝜀_𝐴⁻²(𝑖)𝐿^𝑇(𝑖)𝑁^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝜀²_𝐵(𝑖)𝑀_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝑀^𝑇_𝐵(𝑖)𝑃₁(𝑖)+𝜀_𝐶⁻²(𝑖)𝐿^𝑇(𝑖)𝑁^𝑇_𝐶(𝑖)𝐺_𝐶(𝑖)𝑄_𝐶⁻¹(𝑖)𝐺^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)+𝜀_𝐶⁻²(𝑖)𝐿^𝑇(𝑖)𝑁^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖),Ξ₃(𝑖)=𝑃₁(𝑖)𝐴(𝑖)+𝐴^𝑇(𝑖)𝑃₁(𝑖)+𝑃₂(𝑖)𝐾_𝐵(𝑖)𝐶(𝑖)+𝐶^𝑇(𝑖)𝐾^𝑇_𝐵(𝑖)𝑃^𝑇₂(𝑖)+^𝑁_𝑗=1𝜆_𝑖𝑗𝑃₁𝑗+𝑃₁(𝑖)𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)𝑃₁(𝑖)+𝑃₂(𝑖)𝐾_𝐵(𝑖)𝑀_𝐶(𝑖)𝑄_𝐶⁻¹(𝑖)𝐺^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)+𝑁^𝑇_𝐶(𝑖)𝐺_𝐶(𝑖)𝑄_𝐶⁻¹(𝑖)𝑀^𝑇_𝐶(𝑖)𝐾^𝑇_𝐵(𝑖)𝑃^𝑇₂(𝑖)+𝜀²_𝐴(𝑖)𝑃₁(𝑖)𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)𝑃₁(𝑖)+𝜀_𝐴⁻²(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝜀_𝐴⁻²(𝑖)𝑁^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝜀²_𝐵(𝑖)𝑃₁(𝑖)𝑀_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝑀^𝑇_𝐵(𝑖)𝑃₁(𝑖)+𝜀²_𝐶(𝑖)𝑃₂(𝑖)𝐾_𝐵(𝑖)𝑀_𝑐(𝑖)𝑄_𝐶⁻¹(𝑖)𝑀^𝑇_𝐶(𝑖)𝐾^𝑇_𝐵(𝑖)𝑃^𝑇₂(𝑖)+𝑊^𝑇(𝑖)𝑃₁(𝑖)𝑊(𝑖)+𝜀_𝐶⁻²(𝑖)𝑁^𝑇_𝐶(𝑖)𝐺_𝐶(𝑖)𝑄_𝐶⁻¹(𝑖)𝐺^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)+𝜀_𝐶⁻²(𝑖)𝑁^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖). (53)
Since ∑^𝑁_𝑗=1𝜆_𝑖𝑗𝐿^𝑇(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 𝐿^𝑇(𝑖)𝑊^𝑇(𝑖)𝑃₁(𝑖)𝑊(𝑖)𝐿(𝑖)≥0, we can obtain the following equivalent condition of (52) as follows: Ξ₁(𝑖)Ξ₂(𝑖)Ξ^𝑇₂(𝑖)Ξ₃(𝑖)<0. (54) Let 𝑃₁(𝑖)=𝑋(𝑖), 𝑃₂(𝑖)=𝑌⁻¹(𝑖)−𝑋(𝑖), and 𝑃₃(𝑖)=𝑋(𝑖)−𝑌⁻¹(𝑖), then, 𝐿(𝑖)=𝑃₁(𝑖)−𝑃₂(𝑖)𝑃₃⁻¹(𝑖)𝑃^𝑇₂(𝑖)⁻¹(𝑖),𝑃=𝑌₃₋₁(𝑖)𝑃^𝑇₂(𝑖)=−𝐼. (55) Define 𝕜_𝐵(𝑖) and 𝕜_𝐶(𝑖) by 𝕜_𝐵(𝑖)=𝑃₂(𝑖)𝐾_𝐵(𝑖)=𝑌⁻¹(𝑖)−𝑋(𝑖)𝐾_𝐵(𝑖),𝕜_𝐶(𝑖)=−𝐾_𝐶(𝑖)𝑃₃⁻¹(𝑖)𝑃^𝑇₂(𝑖)𝐿(𝑖)=𝐾_𝐶(𝑖)𝑌(𝑖), (56) then applying all the previous algebraic manipulations, we have Ξ₁(𝑖)=𝐴(𝑖)𝑌(𝑖)+𝑌^𝑇(𝑖)𝐴^𝑇(𝑖)+𝐵(𝑖)𝕜_ℂ(𝑖)+𝕜^𝑇_𝐶(𝑖)𝐵^𝑇(𝑖)+^𝑁_𝑗=1𝜆_𝑖𝑗𝑌^𝑇(𝑖)𝑌⁻¹𝑗𝑌(𝑖)+𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)𝑌(𝑖)+𝑌^𝑇(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)+𝑀_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝐺^𝑇_𝐵(𝑖)𝑁_𝐵(𝑖)𝕜_𝐶(𝑖)+𝕜^𝑇_𝐶(𝑖)𝑁^𝑇_𝐵(𝑖)𝐺_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝑀^𝑇_𝐵(𝑖)+𝜀²_𝐴(𝑖)𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)+𝜀_𝐴⁻²(𝑖)𝑌^𝑇(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)𝑌(𝑖)+𝜀_𝐴⁻²(𝑖)𝑌^𝑇(𝑖)𝑁^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)𝑌(𝑖)+𝜀_𝐵⁻²(𝑖)𝕜^𝑇_𝐶(𝑖)𝑁^𝑇_𝐵(𝑖)𝐺_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝐺^𝑇_𝐵(𝑖)𝑁_𝐵(𝑖)𝕜_𝐶(𝑖)+𝜀²_𝐵(𝑖)𝑀_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝑀^𝑇_𝐵(𝑖)+𝜀_𝐵⁻²(𝑖)𝕜^𝑇_𝐶(𝑖)𝑁^𝑇_𝐵(𝑖)𝑁_𝐵(𝑖)𝕜_𝐶(𝑖)+𝜀_𝐶⁻²(𝑖)𝑌^𝑇(𝑖)𝑁^𝑇_𝐶(𝑖)𝐺_𝐶(𝑖)𝑄_𝐶⁻¹(𝑖)𝐺^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)𝑌(𝑖)+𝜀_𝐶⁻²(𝑖)𝑌^𝑇(𝑖)𝑁^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)𝑌(𝑖),Ξ₂(𝑖)=𝐴(𝑖)+𝑌^𝑇(𝑖)𝐴^𝑇(𝑖)𝑋(𝑖)+𝑌^𝑇(𝑖)𝐶^𝑇(𝑖)𝕜^𝑇_𝐵(𝑖)+𝕜^𝑇_𝐶(𝑖)𝐵^𝑇(𝑖)𝑋(𝑖)+𝑌^𝑇(𝑖)𝐾^𝑇_𝐴(𝑖)𝑌⁻¹(𝑖)−𝑋(𝑖)^𝑇+^𝑁_𝑗=1𝜆_𝑖𝑗𝑌^𝑇(𝑖)𝑌⁻¹𝑗+𝑌^𝑇(𝑖)𝑊^𝑇(𝑖)𝑋(𝑖)𝑊(𝑖)+𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝑌^𝑇(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)𝑋(𝑖)+𝕜^𝑇_𝐶(𝑖)𝑁^𝑇_𝐵(𝑖)𝐺_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝑀^𝑇_𝐵(𝑖)𝑋(𝑖)+𝑌^𝑇(𝑖)𝑁^𝑇_𝐶(𝑖)𝐺_𝐶(𝑖)𝑄_𝐶⁻¹(𝑖)𝑀^𝑇_𝐶(𝑖)𝕜^𝑇_𝐵(𝑖)+𝜀²_𝐴(𝑖)𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)𝑋(𝑖)+𝜀_𝐴⁻²(𝑖)𝑌^𝑇(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝜀_𝐴⁻²(𝑖)𝑌^𝑇(𝑖)𝑁^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝜀²_𝐵(𝑖)𝑀_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝑀^𝑇_𝐵(𝑖)𝑋(𝑖)+𝜀_𝐶⁻²(𝑖)𝑌^𝑇(𝑖)𝑁^𝑇_𝐶(𝑖)𝐺_𝐶(𝑖)𝑄_𝐶⁻¹(𝑖)𝐺^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)+𝜀_𝐶⁻²(𝑖)𝑌^𝑇(𝑖)𝑁^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖),Ξ₃(𝑖)=𝑋(𝑖)𝐴(𝑖)+𝐴^𝑇(𝑖)𝑋(𝑖)+𝕜_𝐵(𝑖)𝐶(𝑖)+𝐶^𝑇(𝑖)𝕜^𝑇_𝐵(𝑖)+𝑊^𝑇(𝑖)𝑋(𝑖)𝑊(𝑖)+^𝑁_𝑗=1𝜆_𝑖𝑗𝑋𝑗+𝑋(𝑖)𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝕜_𝐵(𝑖)𝑀_𝐶(𝑖)𝑄_𝐶⁻¹(𝑖)𝐺^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)+𝑁^𝑇_𝐶(𝑖)𝐺_𝐶(𝑖)𝑄_𝐶⁻¹(𝑖)𝑀^𝑇_𝐶(𝑖)𝕜^𝑇_𝐵(𝑖)+𝜀²_𝐴(𝑖)𝑋(𝑖)𝑀_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)𝑋(𝑖)+𝜀_𝐴⁻²(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝐺^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝜀_𝐴⁻²(𝑖)𝑁^𝑇_𝐴(𝑖)𝑁_𝐴(𝑖)+𝜀²_𝐵(𝑖)𝑋(𝑖)𝑀_𝐵(𝑖)𝑄_𝐵⁻¹(𝑖)𝑀^𝑇_𝐵(𝑖)𝑋(𝑖)+𝜀²_𝐶(𝑖)𝕜_𝐵(𝑖)𝑀_𝑐(𝑖)𝑄_𝐶⁻¹(𝑖)𝑀^𝑇_𝐶(𝑖)𝕜^𝑇_𝐵(𝑖)+𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖)𝑄_𝐴⁻¹(𝑖)𝑀^𝑇_𝐴(𝑖)𝑋(𝑖)+𝜀_𝐶⁻²(𝑖)𝑁^𝑇_𝐶(𝑖)𝐺_𝐶(𝑖)𝑄_𝐶⁻¹(𝑖)𝐺^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖)+𝜀_𝐶⁻²(𝑖)𝑁^𝑇_𝐶(𝑖)𝑁_𝐶(𝑖). (57)
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: Ξ₁(𝑖)Ξ<0,₃(𝑖)<0. (58) Let 𝕊(𝑌(𝑖))=𝑌(𝑖)𝜆_𝑖1𝐼,…,𝜆_{𝑖,𝑖−1}𝐼,𝜆_𝑖,𝑖+1𝐼,…,𝜆_𝑖𝑁𝐼,𝕐(𝑖)=diag{𝑌(1),…,𝑌(𝑖−1),𝑌(𝑖+1),⋯,𝑌(𝑁)}. (59) Then, ^𝑁_𝑗=1𝜆_𝑖𝑗𝑌^𝑇(𝑖)𝑋⁻¹𝑗𝑌(𝑖)=𝜆_𝑖𝑖𝑌(𝑖)+𝕊(𝑌(𝑖))𝕐⁻¹(𝑖)𝕊^𝑇(𝑌(𝑖)). (60)
According to the Schur complement and 𝑅_𝐴2(𝑖)=𝜀_𝐴(𝑖)𝑀_𝐴(𝑖)+𝜀_𝐴⁻¹(𝑖)𝑌^𝑇(𝑖)𝑁^𝑇_𝐴(𝑖)𝐺_𝐴(𝑖),𝑅_𝐵2(𝑖)=𝜀_𝐵(𝑖)𝑀_𝐵(𝑖)+𝜀_𝐵⁻¹(𝑖)𝕜^𝑇_𝐶(𝑖)𝑁^𝑇_𝐵(𝑖)𝐺_𝐵(𝑖), (61) the stability conditions (58) become (39) and (40). Finally, notice that 𝑈^𝑇(𝑖)𝑉^𝑇(𝑖)𝑃(𝑖)𝑉(𝑖)𝑈(𝑖)=𝑌(𝑖)𝐼𝐼𝑋(𝑖)>0. (62) 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 Section 4.

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 𝐴(1)=1.0−0.50.11.0,𝑀_𝐴(1)=,𝑁0.10.2_𝐴(1)=0.20.1,𝐺_𝐴(1)=[],𝑊(1)=,𝐵(1)=0.20.10.00.00.11.00.00.01.0,𝑀_𝐵(1)=,𝑁0.10.2_𝐵(1)=0.20.1,𝐺_𝐵(1)=[0.2]. (63) Mode no. 2 𝐴(2)=−0.20.50.0−0.25,𝑀_𝐴(2)=,𝑁0.130.1_𝐴(2)=0.10.2,𝐺_𝐴(2)=[],𝑊(2)=,𝐵(2)=0.20.20.00.00.21.00.00.01.0,𝑀_𝐵(2)=,𝑁0.130.1_𝐵(2)=0.10.2,𝐺_𝐵(2)=[0.2]. (64) The jumping between the different modes is described by the transition rate matrix: Λ=−2.02.03.0−3.0. (65) Setting 𝜀_𝐴(1)=𝜀_𝐴(2)=0.5 and 𝜀_𝐵(1)=𝜀_𝐵(2)=0.6. By means of inequality (25), we have 𝑋(1)=,𝑋(2)=,𝑌(1)=,𝑌(2)=.0.3445−0.0142−0.01420.35210.3061−0.0107−0.01070.3069−0.73330.08190.0819−0.7503−0.2168−0.0850−0.0850−0.1993 (66) Then applying Theorem 5, the state feedback controller gains are obtained as follows: 𝐾(1)=𝐾(2)=−2.12270.14690.1500−2.1252,−0.7197−0.3022−0.3009−0.6600. (67)(68)
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 𝐴(1)=0.2−0.60.31.2,𝑀_𝐴(1)=,𝑁0.120.23_𝐴(1)=0.20.1,𝐺_𝐴(1)=[],𝑊(1)=,𝐵(1)=0.50.10.00.00.11.00.00.01.0,𝑀_𝐵(1)=,𝑁0.120.23_𝐵(1)=0.20.1,𝐺_𝐵(1)=[],𝐶(1)=0.51.00.00.01.0,𝑀_𝐶(1)=,𝑁0.120.23_𝐶(1)=0.20.1,𝐺_𝐶(1)=[0.5]. (69) Mode no. 2 𝐴(2)=−0.30.40.2−0.15,𝑀_𝐴(2)=,𝑁0.120.15_𝐴(2)=0.130.23,𝐺_𝐴(2)=[],𝑊(2)=,𝐵(2)=0.50.20.00.00.21.00.00.01.0,𝑀_𝐵(2)=,𝑁0.120.15_𝐵(2)=0.130.23,𝐺_𝐵(2)=[],𝐶(2)=0.51.00.00.01.0,𝑀_𝐶(2)=,𝑁0.120.15_𝐶(2)=0.130.23,𝐺_𝐶(2)=[0.5]. (70) The transition probability rate matrix between the two modes is given by Λ=−2.02.03.0−3.0. (71) Let 𝜀_𝐴(1)=𝜀_𝐴(2)=0.5, 𝜀_𝐵(1)=𝜀_𝐵(2)=0.1 and 𝜀_𝐶(1)=𝜀_𝐶(2)=0.1; in view of inequalities (39)–(41), we have 𝑋(1)=,𝑋(2)=,𝑌(1)=,𝑌(2)=,𝕜5.0423−0.2399−0.23992.91824.84610.15590.15594.40640.5467−0.7160−0.71601.37111.0892−0.7109−0.71090.6812_𝐵(1)=,𝕜−6.8119−0.4532−0.4532−8.6839_𝐵(2)=,𝕜−4.1475−2.0622−2.0622−4.1055_𝐶(1)=,𝕜−1.73263.22413.2241−6.7326_𝐶(2)=.−5.17422.74342.7434−1.5899 (72) Then according to Theorem 6, the dynamic output feedback controller gains are obtained as follows: 𝐾_𝐴(1)=,𝐾1.80304.38604.3860−4.1137_𝐵(1)=,𝐾0.5083−2.5782−1.97250.4493_𝐶(1)=,𝐾−0.28292.2038−1.6886−5.7923_𝐴(2)=,𝐾−4.82952.08906.75846.3166_𝐵(2)=,𝐾−0.5943−1.3272−1.8660−1.6399_𝐶(2)=.−6.6548−2.91763.12180.9240 (73)
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 robust 𝐻_∞ control problem via state feedback and dynamic output feedback for such Itô stochastic MJLS with linear fractional uncertainties should be studied in future work.

Acknowledgments

The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation. This work was partially supported by the Natural Science Foundation of China under Grant (nos. 61263005 and 61065010), the Research Fund for the Doctoral Program of Higher Education under Grant no. 20105201120003, the Program for New Century Excellent Talents in University, and the Graduate School Innovation Foundation of Guizhou University.