# Positiveness and Observer-Based Finite-Time Control for a Class of Markov Jump Systems with Some Complex Environment Parameters

Authors: Chengcheng Ren; Shuping He
Journal: Complexity
 (2018)

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## Abstract

An observer-based finite-timeL2-L∞ control law is devised for a class of positive Markov jump systems in a complex environment. The complex environment parameters include bounded uncertainties, unknown nonlinearities, and external disturbances. The objective is to devise an appropriate observer-based control law that makes the corresponding augment error dynamic Markov jump systems be positive and finite-time stabilizable and satisfy the given L2-L∞ disturbance attenuation index. A sufficient condition is initially established on the existence of the observer-based finite-time controller by using proper stochastic Lyapunov-Krasovskii functional. The design criteria are presented by means of linear matrix inequalities. Finally, the feasibility and validity of the main results can be illustrated through a numerical example.


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## Body

## 1. Introduction

As a special kind of hybrid systems, Markov jump systems (MJSs) consist of two kinds of hybrid dynamic forms. One form is characterized by a discrete state and continuous-time Markov process, called mode; the other form is described by state space equations in each mode, called state. This kind of MJSs was firstly proposed by Krasovskii and Lidskii [1] in 1960s. Due to the stochastic Markov process, it is always considered as a special stochastic system. Moreover, MJSs tend to describe systems in which structures are subjected to abrupt stochastic variations. Such variations usually come from sudden failure of connection between system components, abrupt environmental changes, or changes in the operating point of nonlinear dynamics. MJSs are widely used in many applications, for example, economic systems [2], power electronic system [3], communication systems [4], and circuit network systems [5]. Due to the wide applications, the research of MJSs has been paid much attention in the past decades. For more details about this issue, we can refer to [6–15].On another research front, we notice that in some dynamic systems there always exist nonnegative characteristics, such as the number of animals, absolute temperature, density of matter, and the concentration of chemical reactions [16–18]. To describe these characteristics, we usually use a positive system to illustrate the nonnegative (positive, strictly) dynamic behavior of state variables. Positive system is a special system; the states and the outputs are both nonnegative (positive, strictly) for any nonnegative (positive, strictly) initial conditions. More recently, the research of positive system has become a heated topic and many publications about this issue have been developed; see, for example, [19–23] and the references therein. However, the main result in above references only considered that the states of the systems can be measurable case. It should be noted that the states always cannot be measurable in some practical application systems [24–26]. Up to present, the observer-based finite-time control problem of positive Markov jump systems (PMJSs) with some complex environment parameters has not been intensively studied, saying nothing of the simultaneous presence of uncertainties [27–32], unknown nonlinearities [21, 33–37], and external disturbances [38–40]. These motivate our research on observer-based finite-time controller design problem for MJSs.This paper analyzes the problems of observer-based finite-timeL2-L∞ control for a class of PMJSs in a complex environment. Compared with existing results related to Markovian jump systems, the main contributions and difficulties are addressed as follows.(i) The main contributions of this paper mainly consist of three aspects. Firstly, we aim to analyze the stabilizable problem of a class of PMJSs with some complex environment parameters. Secondly, we try to design an appropriate observer-based finite-timeL2-L∞ control law to ensure that closed-loop Markov jump systems be finite-time stabilizable when the states of the system cannot be measured. Thirdly, we attempt to give a sufficient condition to guarantee the positiveness of the augment error dynamic MJSs and we essay to illustrate the validity of the designed method through a numerical example.(ii) Different from the existing results in [3, 6, 12, 13], the main difficulties of the paper are how to design an appropriate control law for PMJSs in a complex environment with nonmeasurable states such that the corresponding augment error dynamic Markov jump systems be positive and finite-time stabilizable and satisfy the given L2-L∞ disturbance attenuation index. It is necessary to point out that the main results in [3, 6, 12, 13] only considering the states of the systems can be measurable case.In this paper, all matrices are assumed with proper dimensions and all notations are quite standard. The implication of the symbols is given in Table1.Table 1
Symbols throughout this paper.
Symbol Means Symbol Means A Τ matrix transpose d i a g A  B block-diagonal matrix ofA and B A - 1 matrix inverse λ m i n ( m a x ) ( D ) minimum (maximum) eigenvalue ofD ⋅ Euclidean vector norm 0 zero matrix I unit matrix R n × m n × m real matrices R n n-dimensional Euclidean space H ≻ ( ≺ , ⪰ , ⪯ ) 0 all elements of the matrixH are positive (negative, non-negative, non-positive) ∗ Symmetric matrix P > ( < , ≥ , ≤ ) 0 positive- (negative, non-negative, non-positive) definite matrix


## 2. Preliminaries

### 2.1. System Description

For a probability spaceΘs:(Φ,Λ,∏r) composed of sample space Φ, algebra of events Λ, and the probability measure Πr which is defined on Λ, assume that the stochastic process {rt,t≥0} is a continuous-time discrete-state Markov stochastic process in a finite set M={1,2,...,N} over the probability space Θs. The transition probability matrix ∏r={∏ij(t),i,j∈M} is defined as(1)∏ijt=∏rt+Δt=j∣rt=i=πijΔt+oΔt,i≠j,1+πiiΔt+oΔt,i=jwhere Δt>0 and limΔt→0⁡(o(Δt)/Δt)→0; πij(t)≥0 is the transition probability rate from mode i at time t to mode j at time t+Δt and satisfies ∑j=1,i≠jNπij=-πii.In this paper, we investigate a class of PMJSs in complex environments including uncertainties, unknown nonlinearities, and unknown external disturbances. The PMJSs in the probability spaceΘs are described by(2)∑S:x˙t=Art+AΔrt,txt+Brt+BΔrt,tut+F1rtfxt,t+W1rtwtyt=Crt+CΔrt,txtmt=Mrt+MΔrt,txt+W2rtwtx0=x0,r0=r0,t=0where x(t)∈Rn denote the state, m(t)∈Rl denote the measured output, y(t)∈Rm denote the controlled output, w(t)∈L2P[0,+∞] denote the unknown external disturbance, u(t)∈Rq denote the controlled input, and f(x(t),t) denote the unknown nonlinearities, which indicate the nonlinear disturbances related to the state. r0 and x0 are initial mode and initial state, respectively. [A(ri)+AΔ(ri,t)] is a mode-dependent Metzler matrix; B(ri), BΔ(ri,t), F(ri), W1(ri), W2(ri), C(ri), CΔ(ri,t), M(ri), and MΔ(ri,t) are mode-dependent positive matrices. For convenience, we use Ai, AΔi, Bi, BΔi, F1i, W1i, W2i, Ci, CΔi, Mi, and MΔi to denote the relevant parameter matrices with rt=i. Moreover, the time-varying uncertain matrices satisfy(3)AΔiBΔiCΔiMΔi=L1iΓitN1iN2iN3iN4iwhere Γi(t) is a mode-dependent Lebesgue norm measurable function and satisfies Γi(t)≤1; L1i, N1i, N2i, N3i, and N4i are known mode-dependent matrices.Remark 1.
In this paper, the uncertain matricesAΔi, BΔi, CΔi, and MΔi in (3) can be considered as admissible conditions. In actual applications, it is usually impossible directly to get the accurate mathematical model of realistic dynamics because of some complex environment including unknown nonlinearities, environmental noises, and time-varying parameters [10, 27]. Thus, the uncertain dynamics existing in PMJSs (2) reflect the inexactness in mathematical modeling of such Markov jump systems. Moreover, the mode-dependent Lebesgue norm measurable function Γi(t) is selected as a full row rank matrix and it also can be considered as state-dependent; that is, Γi(t)=Γi(t,x(t)) if Γi(t,x(t))≤1. For more results of this issue, we refer readers to [30–32].When the states of PMJSs (2) cannot be available, we can construct the following state observer and feedback control law: (4)Σ-s:x¯˙=Aix¯t+Bi+BΔiut+Eimt-m¯tm¯t=Mix¯tut=Kix¯ty¯t=Cix¯tx0=x0,r0=r0,t=0where x-(t)∈Rn denote the estimated state; m-(t)∈Rl denote the observer output; y-(t)∈Rm denote the estimation output. Ei and Ki are observer and control law gains to be devised, respectively. The state estimated error and the controlled output error of Σs are defined by e(t)=x(t)-x-(t) and z(t)=y(t)-y-(t), respectively. Therefore, we have the following MJSs by (2) and (4):(5)x˙t=Ai+AΔi+Bi+BΔiKixt-Bi+BΔiKiet+F1ifxt,t+W1iwte˙t=AΔi-EiMΔixt+Ai-EiMiet+F1ifxt,t+W1i-EiW2iwtLettingx~(t)=col[x(t)e(t)], the closed-loop augment error dynamic MJSs can be represented as(6)Σ~s:x~˙=A~i+A~Δix~t+F~ifx~t,t+W~iwtzt=C~ix~tx~0=x~0,t=0where(7)A~i=Ai+BiKi-BiKi0Ai-EiMi,A~Δi=AΔi+BΔiKi-BΔiKiAΔi-EiMΔi0,F~i=F1iF1i,W~i=W1iW1i-EiW2i,C~i=CΔiCi,fxt,t=fx~t,t.


### 2.2. Main Definitions, Lemmas, and Assumptions

The following main definitions, lemmas, and assumptions are important for analyzing and giving the main results of the paper.Definition 2.
For given constantsT>0 and μ1>0, the augment error dynamic MJSs (6) are finite-time stabilizable (FTS) with regard to (μ1,μ2,T,R), if there exist constants u2>u1>0 and positive-definite matrix R>0, such that(8)Εx~ΤtRx~t<μ2,∀t∈0T,ifx~Τ0Rx~0≤μ1.Definition 3.
For given constantsT>0 and μ1>0,Σ-s is said to be the finite-time L2-L∞ observer-based state feedback control law of MJSs (6) under the zero initial condition, if the augment error dynamic MJSs (6) are FTS with regard to (μ1,μ2,T,R) and satisfy(9)Εzt∞2-δ2wt22<0where d>0,δ>0,Εz(t)∞2=Esupt∈0T⁡z(t)Τz(t), andw(t)22=∫0Tw(t)Τw(t)dt.Definition 4.
The weak infinitesimal operator of stochastic Lyapunov-Krasovskii functional (SLKF)V[x~(t),i,t>0] is defined as(10)IVx~t,i,t=limΔt→0⁡1ΔtEVx~t+Δt,rt+Δt,t+Δt∣x~t,rt-Vx~t,rt,t.=∂∂tVx~t,i,t+∂∂x~Vx~t,i,tx~˙t+∑j=1NπijVx~t,j,tDefinition 5.
MJSs (2) are positive, if, for the initial conditions x0>0 and r0>0 and the controlled input u(t)>0, the relevant trajectories of Σs satisfy x(t)>0 and y(t)>0,∀t>0.Lemma 6 (see [41]).
A i + A Δ i is said to be a Metzler matrix, if there exists a mode-dependent constant εi satisfying Ai+AΔi+εiI⪰0, where Ai+AΔi is a real square matrix.Lemma 7 (see [41]).
MJSs (2) are said to be positive if and only if Ai+AΔi is a Metzler matrix and Bi+BΔi⪰0,Fi⪰0,W1i⪰0,Ci+CΔi⪰0,Mi+MΔi⪰0, andW2i⪰0.Lemma 8 (see [30]).
Suppose thatLi and Ni are mode-dependent real matrices and Γi(t) is a mode-dependent Lebesgue norm measurable function and satisfies Γi(t)≤1. There exists a mode-dependent constant αi>0, satisfying(11)LiΓitNi+LiΓitNiΤ<αi-1LiLiΤ+αiNiΤNi.Lemma 9 (see [30]).
Suppose thatFi and Ei are mode-dependent real matrices. There exists a positive-definite matrix M and a mode-dependent constant βi>0, satisfying(12)FiΤEi+EiΤFi≤βiFiΤMFi+β-1EiΤM-1Ei.Assumption 10.
The mode-dependent nonlinear functionf(x(t),t) satisfies the following Lipschitz condition:(13)fxt,t≤Gxtwhere G is a real matrix with proper dimension.Assumption 11.
For given constantd>0, w(t) is energy-bounded and satisfies(14)∫0ΤwΤtwtdt≤d.Remark 12.
Assumption10 guarantees that we can use linearization method to study the nonlinear systems by means of linear matrix inequalities [3, 5, 11]. In the design of observer-based finite-time L2-L∞ control law, Assumption 11 is given to assume that the unknown external disturbance is to be an arbitrary deterministic signal of bounded energy [20, 24, 27].


## 2.1. System Description

For a probability spaceΘs:(Φ,Λ,∏r) composed of sample space Φ, algebra of events Λ, and the probability measure Πr which is defined on Λ, assume that the stochastic process {rt,t≥0} is a continuous-time discrete-state Markov stochastic process in a finite set M={1,2,...,N} over the probability space Θs. The transition probability matrix ∏r={∏ij(t),i,j∈M} is defined as(1)∏ijt=∏rt+Δt=j∣rt=i=πijΔt+oΔt,i≠j,1+πiiΔt+oΔt,i=jwhere Δt>0 and limΔt→0⁡(o(Δt)/Δt)→0; πij(t)≥0 is the transition probability rate from mode i at time t to mode j at time t+Δt and satisfies ∑j=1,i≠jNπij=-πii.In this paper, we investigate a class of PMJSs in complex environments including uncertainties, unknown nonlinearities, and unknown external disturbances. The PMJSs in the probability spaceΘs are described by(2)∑S:x˙t=Art+AΔrt,txt+Brt+BΔrt,tut+F1rtfxt,t+W1rtwtyt=Crt+CΔrt,txtmt=Mrt+MΔrt,txt+W2rtwtx0=x0,r0=r0,t=0where x(t)∈Rn denote the state, m(t)∈Rl denote the measured output, y(t)∈Rm denote the controlled output, w(t)∈L2P[0,+∞] denote the unknown external disturbance, u(t)∈Rq denote the controlled input, and f(x(t),t) denote the unknown nonlinearities, which indicate the nonlinear disturbances related to the state. r0 and x0 are initial mode and initial state, respectively. [A(ri)+AΔ(ri,t)] is a mode-dependent Metzler matrix; B(ri), BΔ(ri,t), F(ri), W1(ri), W2(ri), C(ri), CΔ(ri,t), M(ri), and MΔ(ri,t) are mode-dependent positive matrices. For convenience, we use Ai, AΔi, Bi, BΔi, F1i, W1i, W2i, Ci, CΔi, Mi, and MΔi to denote the relevant parameter matrices with rt=i. Moreover, the time-varying uncertain matrices satisfy(3)AΔiBΔiCΔiMΔi=L1iΓitN1iN2iN3iN4iwhere Γi(t) is a mode-dependent Lebesgue norm measurable function and satisfies Γi(t)≤1; L1i, N1i, N2i, N3i, and N4i are known mode-dependent matrices.Remark 1.
In this paper, the uncertain matricesAΔi, BΔi, CΔi, and MΔi in (3) can be considered as admissible conditions. In actual applications, it is usually impossible directly to get the accurate mathematical model of realistic dynamics because of some complex environment including unknown nonlinearities, environmental noises, and time-varying parameters [10, 27]. Thus, the uncertain dynamics existing in PMJSs (2) reflect the inexactness in mathematical modeling of such Markov jump systems. Moreover, the mode-dependent Lebesgue norm measurable function Γi(t) is selected as a full row rank matrix and it also can be considered as state-dependent; that is, Γi(t)=Γi(t,x(t)) if Γi(t,x(t))≤1. For more results of this issue, we refer readers to [30–32].When the states of PMJSs (2) cannot be available, we can construct the following state observer and feedback control law: (4)Σ-s:x¯˙=Aix¯t+Bi+BΔiut+Eimt-m¯tm¯t=Mix¯tut=Kix¯ty¯t=Cix¯tx0=x0,r0=r0,t=0where x-(t)∈Rn denote the estimated state; m-(t)∈Rl denote the observer output; y-(t)∈Rm denote the estimation output. Ei and Ki are observer and control law gains to be devised, respectively. The state estimated error and the controlled output error of Σs are defined by e(t)=x(t)-x-(t) and z(t)=y(t)-y-(t), respectively. Therefore, we have the following MJSs by (2) and (4):(5)x˙t=Ai+AΔi+Bi+BΔiKixt-Bi+BΔiKiet+F1ifxt,t+W1iwte˙t=AΔi-EiMΔixt+Ai-EiMiet+F1ifxt,t+W1i-EiW2iwtLettingx~(t)=col[x(t)e(t)], the closed-loop augment error dynamic MJSs can be represented as(6)Σ~s:x~˙=A~i+A~Δix~t+F~ifx~t,t+W~iwtzt=C~ix~tx~0=x~0,t=0where(7)A~i=Ai+BiKi-BiKi0Ai-EiMi,A~Δi=AΔi+BΔiKi-BΔiKiAΔi-EiMΔi0,F~i=F1iF1i,W~i=W1iW1i-EiW2i,C~i=CΔiCi,fxt,t=fx~t,t.


## 2.2. Main Definitions, Lemmas, and Assumptions

The following main definitions, lemmas, and assumptions are important for analyzing and giving the main results of the paper.Definition 2.
For given constantsT>0 and μ1>0, the augment error dynamic MJSs (6) are finite-time stabilizable (FTS) with regard to (μ1,μ2,T,R), if there exist constants u2>u1>0 and positive-definite matrix R>0, such that(8)Εx~ΤtRx~t<μ2,∀t∈0T,ifx~Τ0Rx~0≤μ1.Definition 3.
For given constantsT>0 and μ1>0,Σ-s is said to be the finite-time L2-L∞ observer-based state feedback control law of MJSs (6) under the zero initial condition, if the augment error dynamic MJSs (6) are FTS with regard to (μ1,μ2,T,R) and satisfy(9)Εzt∞2-δ2wt22<0where d>0,δ>0,Εz(t)∞2=Esupt∈0T⁡z(t)Τz(t), andw(t)22=∫0Tw(t)Τw(t)dt.Definition 4.
The weak infinitesimal operator of stochastic Lyapunov-Krasovskii functional (SLKF)V[x~(t),i,t>0] is defined as(10)IVx~t,i,t=limΔt→0⁡1ΔtEVx~t+Δt,rt+Δt,t+Δt∣x~t,rt-Vx~t,rt,t.=∂∂tVx~t,i,t+∂∂x~Vx~t,i,tx~˙t+∑j=1NπijVx~t,j,tDefinition 5.
MJSs (2) are positive, if, for the initial conditions x0>0 and r0>0 and the controlled input u(t)>0, the relevant trajectories of Σs satisfy x(t)>0 and y(t)>0,∀t>0.Lemma 6 (see [41]).
A i + A Δ i is said to be a Metzler matrix, if there exists a mode-dependent constant εi satisfying Ai+AΔi+εiI⪰0, where Ai+AΔi is a real square matrix.Lemma 7 (see [41]).
MJSs (2) are said to be positive if and only if Ai+AΔi is a Metzler matrix and Bi+BΔi⪰0,Fi⪰0,W1i⪰0,Ci+CΔi⪰0,Mi+MΔi⪰0, andW2i⪰0.Lemma 8 (see [30]).
Suppose thatLi and Ni are mode-dependent real matrices and Γi(t) is a mode-dependent Lebesgue norm measurable function and satisfies Γi(t)≤1. There exists a mode-dependent constant αi>0, satisfying(11)LiΓitNi+LiΓitNiΤ<αi-1LiLiΤ+αiNiΤNi.Lemma 9 (see [30]).
Suppose thatFi and Ei are mode-dependent real matrices. There exists a positive-definite matrix M and a mode-dependent constant βi>0, satisfying(12)FiΤEi+EiΤFi≤βiFiΤMFi+β-1EiΤM-1Ei.Assumption 10.
The mode-dependent nonlinear functionf(x(t),t) satisfies the following Lipschitz condition:(13)fxt,t≤Gxtwhere G is a real matrix with proper dimension.Assumption 11.
For given constantd>0, w(t) is energy-bounded and satisfies(14)∫0ΤwΤtwtdt≤d.Remark 12.
Assumption10 guarantees that we can use linearization method to study the nonlinear systems by means of linear matrix inequalities [3, 5, 11]. In the design of observer-based finite-time L2-L∞ control law, Assumption 11 is given to assume that the unknown external disturbance is to be an arbitrary deterministic signal of bounded energy [20, 24, 27].


## 3. Main Results

### 3.1. FTS Analysis

In this subsection, the FTS analysis for the augment error dynamic MJSs (6) will be considered. Based on the SLKF approach and LMIs techniques, a sufficient condition of FTS will be given in Theorem 13.Theorem 13.
For given constantsT>0,μ1>0,γ>0, andd>0, the augment error dynamic MJSs (6) are FTS with regard to (μ1,μ2,T,R,γ), where μ2>μ1>0 and R>0, if there exists a set of mode-dependent positive-definite symmetric matrix P~i∈Rn×n, positive-definite symmetric matrix R>0, matrix G∈Rn×n, a set of mode-dependent constant βi>0, and constant γ>0, such that(15)ϑ1-γP~iP~iW~iP~iF~i∗-I0∗∗-βiI<0(16)eγΤλ1u1+d/r1-e-γΤλ2<u2where ϑ1=(A~i+A~Δi)ΤP~i+P~i(Ai+AΔi)+βiGΤG+∑j=1NπijP~j, λ1=maxr∈M⁡λmax[P-i], λ2=minr∈M⁡λmin[P-i], and P-i=R-1/2P~iR-1/2.Proof.
See AppendixA.


### 3.2. Finite-TimeL2-L∞ Disturbance Attenuation Index Analysis

Theorem13 gives a sufficient condition of FTS for augment error dynamic MJSs (6). Recalling Definition 3, we will give the following Theorem 14 to analyze the observer-based finite-time L2-L∞ control law design.Theorem 14.
For given constantsT>0,μ1>0,γ>0, andd>0, the augment error dynamic MJSs (6) are FTS with regard to (μ1,μ2,T,R,γ), where μ2>μ1>0 and satisfy the given L2-L∞ disturbance rejection disturbance attenuation index (10), if there exists a set of mode-dependent positive-definite symmetric matrix Pi∈Rn×n, positive-definite symmetric matrix R>0, matrix G∈Rn×n, a set of mode-dependent constant βi>0, and constant γ>0, such that inequalities (15)-(16) and the following relation hold:(17)P~iC~iΤ∗δ2I>0.Proof.
See AppendixB.


### 3.3. Positiveness and Observer-Based Control Law Gain Solution

Recalling Definition5, following sufficient condition will be given to ensure the positiveness of the augment error dynamic MJSs (6) in Theorem 15. It should be noted that there exist some time-varying uncertain matrices in Theorem 13. Therefore, it is difficult to obtain the observer gain Ei and the control law gain Ki from matrix inequalities (15)–(17). It is necessary for us to convert the nonlinear matrix inequalities (15)–(17) into the solvable inequalities, which can be directly solved by Matlab LMI toolbox.Theorem 15.
For given constantsT>0,μ1>0,γ>0, andd>0, there exists a finite-time L2-L∞ observer-based control law with Ki=SiVi-1 and Ei=UiMiΤ and the augment error dynamic MJSs (6) are positive and FTS and satisfy the given L2-L∞ disturbance rejection disturbance attenuation index (9) with regard to (μ1,μ2,T,R,γ), where μ2>μ1>0, if there exists a set of mode-dependent positive-definite symmetric matrix Ui∈Rn×n, positive-definite symmetric matrix R>0, a set of mode-dependent matrices Si∈Rp×n and Q1i∈R1×n, a set of mode-dependent constants α1i>0,α2i>0,α3i>0,α4i>0,βi>0,εi>0, andυi>0, and constants μ2>0,δ>0,λ1>0, andλ2>0γ>0, such that(18)∐11∐22∗∐22<0(19)-Ui00UiN3iΤ0∗-Ui-CiΤ00∗∗-δ2I0L1i∗∗∗-α4i-10∗∗∗∗α4i<0(20)d1-e-γΤ-γλ2μ2e-γΤγμ1γμ1-λ1<0(21)λ2Ri-1<Ui<Ri-1(22)AiUi+BiSi+εiI⪰0(23)Ai-EiMi+υiI⪰0(24)BiSi⪯0(25)W1iUi-MiΤW2iUi⪰0where(26)∐11=θ1θ2θ3θ4θ5∗θ6θ7θ8θ9∗∗θ10θ11θ12∗∗∗θ13θ14∗∗∗∗θ15,∐12=φ1φ2φ3φ4000φ70φ9000000000000000,θ5=θ11=θ12=θ14=0,∐22=diag⁡-βi-1I-α2i-1I-α1i-1I-α3i-1I-α3iI,θ1=AiUi+UiAiΤ+BiSi+SiΤBiΤ+πii-γUi+α1i-1L1iL1iΤ,θ2=-BiSi,θ3=W1iUi,θ4=F1iUi,θ6=AiUi+UiAiΤ+πii-γUi+α2i-1L1iL1iΤ,θ7=W1iUi-MiΤW2iUi,θ8=F1iUi,θ9=UiMiΤ,θ10=-I,θ13=-βiI,θ15=-2I,φ1=UiGiΤ,φ2=UiN1iΤ,φ3=UiN1iΤ+SiΤN2iΤ,φ4=UiN4iΤ,φ5=-SiΤN2iΤ,φ6=Q1i.Proof.
See AppendixC.Corollary 16.
The sufficient conditions to design the stochastic finite-timeL2-L∞ observer-based control law for a class of PMJSs in complex environments have been presented in Theorems 13–15. Considering that the coupling inequalities (18)–(25) are related to Ui, Si, α1i, α2i, α3i, α4i, βi, μ1, μ2, T, d, γ, and δ~=δeγT, we have the optimization algorithm by setting δ~ as an optimization variable value:(27)minUi,Si,α1i,α2i,α3i,α4i,βi,μ1,μ2,δ,T,d,γδ~s.t.inequalities18–25.Remark 17.
It should be pointed out that the optimization algorithm (27) is given to solve the unknown matrices and parameters through Matlab LMI toolbox. Recalling inequality (20), it is known that all of the parameters in d(1-e-γT)-γλ2μ2e-γT are linear and also can be solved through Matlab LMI toolbox by setting μ2 as an unknown parameter. Considering L2-L∞ disturbance attenuation index in Definition 3, we select δ~=δeγT as an optimization variable value in optimization algorithm (27).Remark 18.
For PMJSs (2) in complex environments without uncertainties in probability space Θs, we have the following dynamic systems:(28)Σs′x˙t=Artxt+Brtut+F1rtfxt,t+W1rtwtyt=Crtxtmt=Mrtxt+W2rtwtx0=x0,r0=r0,t=0.The state observer and feedback control law for PMJSs (28) can be designed as(29)Σ-s′:x¯˙=Aix¯t+Biut+Eimt-m¯tm¯t=Mix¯tut=Kix¯ty¯t=Cix¯tx0=x0,r0=r0,t=0Letting x~(t)=col[x(t)e(t)], we can obtain the following closed-loop augment error dynamic MJSs: (30)Σ~s:x~˙=A~ix~t+F~ifx~t,t+W~iwtzt=C~ix~tx~0=x~0,t=0where(31)A~i=Ai+BiKi-BiKi0Ai-EiMi,F~i=F1iF1i,W~i=W1iW1i-EiW2i,C~i=0Ci,fxt,t=fx~t,t.The main results in Theorem15 will reduce to the following Corollary 19.Corollary 19.
For given constantsT>0, μ1>0, γ>0, and d>0, there exists a finite-time L2-L∞ observer-based control law with Ki=SiVi-1 and Ei=UiMiΤ and the augment error dynamic MJSs (29) are positive and FTS and satisfy the given L2-L∞ disturbance rejection disturbance attenuation index (10) with regard to (μ1,μ2,T,R,γ), where μ2>μ1>0 and R>0, if there exists a set of mode-dependent positive-definite symmetric matrices Ui∈Rn×n, a set of mode-dependent matrix Si∈Rp×n, matrix G∈Rl×n, a set of mode-dependent constants υi>0 and εi>0, and constants μ2>0, δ>0, λ1>0, and λ2>0,γ>0, such that inequalities (20)–(25) and the following relations hold:(32)ν1-BiSiWiUiF1iUi0UiGiΤ∗ν2ν3F1iUiUiMiΤ0∗∗∗-I00∗∗∗∗-2I0∗∗∗∗∗-βi-1I<0(33)-Ui00∗-Ui-CiΤ∗∗-δ2I<0where ν1=AiUi+UiAiΤ+BiSi+SiΤBiΤ+(πii-γ)Ui, ν2=AiUi+UiAiΤ+(πii-γ)Ui, and ν3=W1iUi-MiΤW2iUi.


## 3.1. FTS Analysis

In this subsection, the FTS analysis for the augment error dynamic MJSs (6) will be considered. Based on the SLKF approach and LMIs techniques, a sufficient condition of FTS will be given in Theorem 13.Theorem 13.
For given constantsT>0,μ1>0,γ>0, andd>0, the augment error dynamic MJSs (6) are FTS with regard to (μ1,μ2,T,R,γ), where μ2>μ1>0 and R>0, if there exists a set of mode-dependent positive-definite symmetric matrix P~i∈Rn×n, positive-definite symmetric matrix R>0, matrix G∈Rn×n, a set of mode-dependent constant βi>0, and constant γ>0, such that(15)ϑ1-γP~iP~iW~iP~iF~i∗-I0∗∗-βiI<0(16)eγΤλ1u1+d/r1-e-γΤλ2<u2where ϑ1=(A~i+A~Δi)ΤP~i+P~i(Ai+AΔi)+βiGΤG+∑j=1NπijP~j, λ1=maxr∈M⁡λmax[P-i], λ2=minr∈M⁡λmin[P-i], and P-i=R-1/2P~iR-1/2.Proof.
See AppendixA.


## 3.2. Finite-TimeL2-L∞ Disturbance Attenuation Index Analysis

Theorem13 gives a sufficient condition of FTS for augment error dynamic MJSs (6). Recalling Definition 3, we will give the following Theorem 14 to analyze the observer-based finite-time L2-L∞ control law design.Theorem 14.
For given constantsT>0,μ1>0,γ>0, andd>0, the augment error dynamic MJSs (6) are FTS with regard to (μ1,μ2,T,R,γ), where μ2>μ1>0 and satisfy the given L2-L∞ disturbance rejection disturbance attenuation index (10), if there exists a set of mode-dependent positive-definite symmetric matrix Pi∈Rn×n, positive-definite symmetric matrix R>0, matrix G∈Rn×n, a set of mode-dependent constant βi>0, and constant γ>0, such that inequalities (15)-(16) and the following relation hold:(17)P~iC~iΤ∗δ2I>0.Proof.
See AppendixB.


## 3.3. Positiveness and Observer-Based Control Law Gain Solution

Recalling Definition5, following sufficient condition will be given to ensure the positiveness of the augment error dynamic MJSs (6) in Theorem 15. It should be noted that there exist some time-varying uncertain matrices in Theorem 13. Therefore, it is difficult to obtain the observer gain Ei and the control law gain Ki from matrix inequalities (15)–(17). It is necessary for us to convert the nonlinear matrix inequalities (15)–(17) into the solvable inequalities, which can be directly solved by Matlab LMI toolbox.Theorem 15.
For given constantsT>0,μ1>0,γ>0, andd>0, there exists a finite-time L2-L∞ observer-based control law with Ki=SiVi-1 and Ei=UiMiΤ and the augment error dynamic MJSs (6) are positive and FTS and satisfy the given L2-L∞ disturbance rejection disturbance attenuation index (9) with regard to (μ1,μ2,T,R,γ), where μ2>μ1>0, if there exists a set of mode-dependent positive-definite symmetric matrix Ui∈Rn×n, positive-definite symmetric matrix R>0, a set of mode-dependent matrices Si∈Rp×n and Q1i∈R1×n, a set of mode-dependent constants α1i>0,α2i>0,α3i>0,α4i>0,βi>0,εi>0, andυi>0, and constants μ2>0,δ>0,λ1>0, andλ2>0γ>0, such that(18)∐11∐22∗∐22<0(19)-Ui00UiN3iΤ0∗-Ui-CiΤ00∗∗-δ2I0L1i∗∗∗-α4i-10∗∗∗∗α4i<0(20)d1-e-γΤ-γλ2μ2e-γΤγμ1γμ1-λ1<0(21)λ2Ri-1<Ui<Ri-1(22)AiUi+BiSi+εiI⪰0(23)Ai-EiMi+υiI⪰0(24)BiSi⪯0(25)W1iUi-MiΤW2iUi⪰0where(26)∐11=θ1θ2θ3θ4θ5∗θ6θ7θ8θ9∗∗θ10θ11θ12∗∗∗θ13θ14∗∗∗∗θ15,∐12=φ1φ2φ3φ4000φ70φ9000000000000000,θ5=θ11=θ12=θ14=0,∐22=diag⁡-βi-1I-α2i-1I-α1i-1I-α3i-1I-α3iI,θ1=AiUi+UiAiΤ+BiSi+SiΤBiΤ+πii-γUi+α1i-1L1iL1iΤ,θ2=-BiSi,θ3=W1iUi,θ4=F1iUi,θ6=AiUi+UiAiΤ+πii-γUi+α2i-1L1iL1iΤ,θ7=W1iUi-MiΤW2iUi,θ8=F1iUi,θ9=UiMiΤ,θ10=-I,θ13=-βiI,θ15=-2I,φ1=UiGiΤ,φ2=UiN1iΤ,φ3=UiN1iΤ+SiΤN2iΤ,φ4=UiN4iΤ,φ5=-SiΤN2iΤ,φ6=Q1i.Proof.
See AppendixC.Corollary 16.
The sufficient conditions to design the stochastic finite-timeL2-L∞ observer-based control law for a class of PMJSs in complex environments have been presented in Theorems 13–15. Considering that the coupling inequalities (18)–(25) are related to Ui, Si, α1i, α2i, α3i, α4i, βi, μ1, μ2, T, d, γ, and δ~=δeγT, we have the optimization algorithm by setting δ~ as an optimization variable value:(27)minUi,Si,α1i,α2i,α3i,α4i,βi,μ1,μ2,δ,T,d,γδ~s.t.inequalities18–25.Remark 17.
It should be pointed out that the optimization algorithm (27) is given to solve the unknown matrices and parameters through Matlab LMI toolbox. Recalling inequality (20), it is known that all of the parameters in d(1-e-γT)-γλ2μ2e-γT are linear and also can be solved through Matlab LMI toolbox by setting μ2 as an unknown parameter. Considering L2-L∞ disturbance attenuation index in Definition 3, we select δ~=δeγT as an optimization variable value in optimization algorithm (27).Remark 18.
For PMJSs (2) in complex environments without uncertainties in probability space Θs, we have the following dynamic systems:(28)Σs′x˙t=Artxt+Brtut+F1rtfxt,t+W1rtwtyt=Crtxtmt=Mrtxt+W2rtwtx0=x0,r0=r0,t=0.The state observer and feedback control law for PMJSs (28) can be designed as(29)Σ-s′:x¯˙=Aix¯t+Biut+Eimt-m¯tm¯t=Mix¯tut=Kix¯ty¯t=Cix¯tx0=x0,r0=r0,t=0Letting x~(t)=col[x(t)e(t)], we can obtain the following closed-loop augment error dynamic MJSs: (30)Σ~s:x~˙=A~ix~t+F~ifx~t,t+W~iwtzt=C~ix~tx~0=x~0,t=0where(31)A~i=Ai+BiKi-BiKi0Ai-EiMi,F~i=F1iF1i,W~i=W1iW1i-EiW2i,C~i=0Ci,fxt,t=fx~t,t.The main results in Theorem15 will reduce to the following Corollary 19.Corollary 19.
For given constantsT>0, μ1>0, γ>0, and d>0, there exists a finite-time L2-L∞ observer-based control law with Ki=SiVi-1 and Ei=UiMiΤ and the augment error dynamic MJSs (29) are positive and FTS and satisfy the given L2-L∞ disturbance rejection disturbance attenuation index (10) with regard to (μ1,μ2,T,R,γ), where μ2>μ1>0 and R>0, if there exists a set of mode-dependent positive-definite symmetric matrices Ui∈Rn×n, a set of mode-dependent matrix Si∈Rp×n, matrix G∈Rl×n, a set of mode-dependent constants υi>0 and εi>0, and constants μ2>0, δ>0, λ1>0, and λ2>0,γ>0, such that inequalities (20)–(25) and the following relations hold:(32)ν1-BiSiWiUiF1iUi0UiGiΤ∗ν2ν3F1iUiUiMiΤ0∗∗∗-I00∗∗∗∗-2I0∗∗∗∗∗-βi-1I<0(33)-Ui00∗-Ui-CiΤ∗∗-δ2I<0where ν1=AiUi+UiAiΤ+BiSi+SiΤBiΤ+(πii-γ)Ui, ν2=AiUi+UiAiΤ+(πii-γ)Ui, and ν3=W1iUi-MiΤW2iUi.


## 4. A Numerical Example

Consider a class of PMJSs with two modes described as follows.Mode 1.
⁡(34)A1=-2017-5,B1=12,C1=0.70.5,L11=0.20.3,W11=0.1,W21=0.03,F11=0.2,M1=1,G1=0.20.3,N11=0.40.06,N21=0.5,N31=0.010.03,N41=0.20.4,α11=0.4,α21=0.2,α31=0.3,α41=0.1,β1=1;Mode 2.
⁡(35)A2=-1826-7,B2=21,C2=0.050.07,L12=0.40.1,W12=0.02,W22=0.05,F12=0.04,M2=2,G2=0.040.03,N12=0.10.4,N22=0.4,N32=0.070.04,N42=0.80.6,α12=0.5,α22=0.1,α32=0.2,α42=0.4,β2=2.The values of the relevant parameters are given asγ=0.01, T=2, λ1=1.1, λ2=0.6228,μ1=30, δ=2, and d=5. The mode of PMJSs (2) is converted according to the following Markov chain conversion rate matrix: π11π12π21π22=-221-1 and we select the unknown nonlinear function as fxt,t=0.6/1+2x2t.Solving LMIs (18)–(25) in Theorem 15, we can get the observer and the control law gain as E1=-1.2258-2.4435-2.4516-4.8871, E2=-7.8209-4.8137-3.9104-2.4068, K1=[-1.1075-2.0864], and K2=[-1.0552-1.8873] with μ2=37.254.The jumping modes, the response of the real states, the estimated statex-(t), the estimated error e(t), the evolution x~Τ(t)Rx~(t), and the controlled output error z(t) are shown in Figures 1–6.Figure 1
The plot of the jumping modes.Figure 2
The trajectories of the real statex(t) in open-loop case.Figure 3
The trajectories of the real states and the observer states.Figure 4
The trajectories of the state estimate errorse(t).Figure 5
The evolutionx~Τ(t)Rx~(t) for e(t) and x(t).Figure 6
The trajectories of the controlled output errorsz(t).Figure2 gives the real state trajectory in open case and it shows that the open-loop PMJSs are unstable. The responses of the real state and the observer state are depicted in Figure 3. Figure 4 shows the state estimated error e(t) and Figure 5 shows the state trajectory x~Τ(t)Rx~(t) of the closed-loop MJSs. From Figure 5, we know that the designed observer-based control law can ensure that the closed-loop MJSs are FTS in the given finite-time interval. Obviously, it can be seen from Figure 6 that the controlled output error of the closed-loop MJSs is positive and FTS.


## 5. Conclusions

In this paper, we studied the observer-based finite-timeL2-L∞ control law design problem of a class of PMJSs in a complex environment. Based on the designed SLKF methods and LMIs technique, sufficient conditions on the existence of the observer-based finite-time L2-L∞ control law are proposed and proven. The designed finite-time L2-L∞ control law makes the closed-loop augment error dynamic MJSs be positive and FTS and satisfy the given induced L2-L∞ disturbance attenuation index. A numerical example was delivered to demonstrate the contribution of the main results.



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*Source: 5365493-2018-11-08.xml*