# H

Authors: Xin-Gang Zhao; Dan Ye; Jian-Da Han
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

For a class of continuous-time Markovian jump linear uncertain
systems with partly known transition rates and input quantization, theH2 state-feedback control design is considered. The elements in
the transition rates matrix include completely known, boundary known,
and completely unknown ones. First, an H2 cost index for Markovian jump linear uncertain systems is introduced; then by introducing a new matrix
inequality condition, sufficient conditions are formulated in terms
of linear matrix inequalities (LMIs) for the H2 control of the Markovian jump linear uncertain systems. Less conservativeness  is achieved
than the result obtained with the existing technique. Finally, a
numerical example is given to verify the validity of the theoretical
results.


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## Body

## 1. Introduction

Recently, much attention has been devoted to the study of the stochastic stability for the Markovian jump systems, and many important results have been published [1–4]. This is because the Markovian jump systems have been widely employed to model many practical systems, such as manufacturing systems, the power systems, and the economic systems in which they may experience abrupt changes in their structures and parameters [5, 6]. It is worth noticing that these results require that the transition probabilities/rates must be knowna priori. However, in many practical engineering applications, the likelihood for obtaining the perfect information on all transition probabilities/rates elements is questionable, and the cost might be expensive in some cases. Therefore, the study of the stabilization of the Markovian jump systems with partly known transition probabilities/rates becomes interesting, and some well-known results have been published. The idea for the stochastic stability of the Markovian jump linear uncertain systems with partly known transition probabilities/rates is developed in Zhang et al. [7]. It is then applied to the H∞ control design in Zhang and Boukas [8]. In those papers, the feature of the information about the transition probabilities/rates matrix considered includes two kinds of elements: completely known and completely unknown ones. As a matter of fact, the transition probabilities matrix might involve completely known, completely unknown, and boundary known elements. In Shen and Yang [9], an H2 state-feedback controller design method is proposed for the continuous-time Markovian jump linear uncertain systems with the three kinds of transition rates matrix elements. In order to yield the design condition for analysis and synthesis, a matrix inequality is introduced to present LMIs conditions. In this paper, less conservative design conditions will be formulated by introducing a new relaxed matrix inequality condition.On the other hand, in many modern engineering practices, all kinds of information processing devices, such as analog-to-digital and digital-to-analog converters, have been widely used. By the utilization of such information processing devices, some advantages have been brought, for example, lower cost, reduced weight and power, simple installation, and maintenance. However, some new phenomena have also been induced, which might cause server deterioration of system performance or even lead to system instability. Signal quantization is one of the important aspects that should be fully considered in such cases, which always exists in computer-based control systems. Nowadays, many well-known results have been published on quantized feedback control. For example, the feedback stabilization problem is considered by utilizing dynamic quantizers [10–13] and static quantizers [14–17]. In addition, the filter design [18] and the H∞ control design [19] are also investigated. Specially, quantization errors have adverse effects on the network control systems which can often be modeled as Markovian jump systems. In Xiao et al. [20], the stabilization problem for single-input discrete Markovian jump linear uncertain systems via mode-dependent quantized state-feedback is addressed, but the transition rates are assumed to be completely known.To the best of our knowledge, no result has been presented for the control design of the continuous-time Markovian jump linear uncertain systems with partly known transition rates and input signal quantization. In this paper, theH2 control for a class of continuous-time Markovian jump linear uncertain systems with respect to partly known transition rates and input signal quantization is addressed. The structure of the controller consists of two parts: the nonlinear part is provided to eliminate the effect of input quantization, and the linear part is obtained by solving LMIs for achieving the H2 performance against unknown transition rates and model uncertainties. In comparison with the design utilizing the LMIs technique in Shen and Yang [9], the design method has less conservativeness by introducing a relaxed inequality condition.The rest of this paper is organized as follows. The problem statement and preliminaries are presented in Section2. The main results are given in Section 3. In Section 4, a numerical example is presented to illustrate the effectiveness of the results, and the conclusions are drawn in Section 5.Notations. Throughout this paper, the following notations are used. Rn denotes the n-dimensional Euclidean space; AT denotes the transpose of matrix A; I and 0 represent the identity matrix and a zero matrix in appropriate dimensions, respectively; 𝔼{·} denotes the mathematical expectation operator; X>Y(X≥Y), where X and Y are symmetric matrices, means that X-Y is positive definite (positive semi-definite); |x|p denotes the p-norm of the vector x; that is, |x|p=(|x1|p+|x2|p+⋯+|xn|p)1/p, where p≥1. When p=∞, |x|∞=max1≤i≤n|xi|. For matrix X∈Rm×n, |X|p is used to present the matrix p-norm: |X|p=supy≠0(|Xy|p/|y|p). The notation |·|, in particular denotes the absolute value of a scalar, the standard Euclidean norm of a vector, and the induced norm of a matrix, respectively. In symmetric block matrices, an * is used to represent a term that is induced by symmetry. Finally, the symbol He(X) is used to represent X+XT.


## 2. Problem Statement and Preliminaries

Consider a class of the continuous-time Markovian jump linear uncertain systems in the following probability space(Ω,ℱ,𝒫):
(1)x˙(t)=(A(rt)+ΔA(rt))x(t)+(B(rt)+ΔB(rt))q(u(t)),x0=x(0),r0=r(0),
where x(t)∈Rn is the system state and u(t)∈Rm is the control input. {rt,t≥0} is a continuous-time Markovian process with right continuous trajectories taking values in the finite set 𝒮={1,2,…,ℕ}. It governs the switching among the different system modes with the following mode transition probabilities:
(2)Pr{rt+h=j∣rt=i}={λijh+o(h),ifj≠i,1+λiih+o(h),ifj=i,whereh>0, limh→0(o(h)/h)=0, and λij≥0(i,j∈𝒮;j≠i) denote the switching rate from mode i at time t to mode j at time t+h and that λii=-∑j=1,j≠iℕ‍λij for each i∈𝒮.In general, the Markovian process transition rates matrixΛ is defined by:
(3)Λ=[λ11λ12⋯λ1ℕλ21λ22⋯λ2ℕ⋮⋯⋱⋮λℕ1⋯⋯λℕℕ].
In this paper, the transition rates of the jumping process are assumed to be partly available; that is, some elements in matrix Λ have been exactly known, some ones have been merely known with lower and upper bounds, and others may have no information to use. For instance, for the system (1) with four operation modes, the transition rates matrix might be described by
(4)Λ=[λ11??λ14??λ23???β?α?λ43λ44],where? represents the completely unknown element of the transition rates matrix and the parameters α and β represent the elements with known lower and upper bounds. That is, α_≤α≤α-, and β_≤β≤β-, where α_, α-, β_, and β- are known parameters; λij denotes the precisely known element.For clarity, we denote that𝒮=𝒮k1i∪𝒮k2i∪𝒮uki,  i=1,2,…,ℕ, with
(5)𝒮k1i={j:λijisexactlyknown},𝒮k2i={j:theboundsofλijareknown},𝒮uki={j:thereisnoinformationavailableforλij}.
Furthermore, let 𝒮ki={j:λ_ij≤λij≤λ-ij}; then one can obtain 𝒮ki=𝒮k1i∪𝒮k2i. If 𝒮ki≠Ø, it can be described as
(6)𝒮ki={𝒦1i,𝒦2i,…,𝒦mkii}.
Similarly, if 𝒮uki≠Ø, let us denote that
(7)𝒮uki={𝕂1i,𝕂2i,…,𝕂mukii},
where 𝒦l1i denotes the l1th element in 𝒮ki with the index 𝒦l1i in the ith row of the matrix Λ. 𝕂l2i denotes the l2th completely unknown element with the index 𝕂l2i in the ith row of the matrix Λ. mki and muki represent the number of elements in 𝒮ki and 𝒮uki, respectively. For example, considering the transition rates matrix (4), one can easily check that 𝒮k4={𝒦14,𝒦24,𝒦34} with 𝒦14=1, 𝒦24=3, 𝒦34=4, 𝒮uk4={𝕂14} with 𝕂14=2,  mk4=3, and muk4=1.Remark 1.
When the lower and upper bounds of the elements in𝒮k2i are equal, the transition rates matrix is reduced to the considered case in Zhang et al. [7]. It is obvious that the solving method there can only treat j∈𝒮k2i as the completely unknown case which can result in some conservativeness.For convenience, the notationsAi=A(rt), Bi=B(rt), ΔAi=ΔA(rt), and ΔBi=ΔB(rt) are used for each possible value ri=i, i∈𝒮, where Ai and Bi are known constant matrices with appropriate dimensions. Then, the system (1) can be described by
(8)x˙(t)=(Ai+ΔAi)x(t)+(Bi+ΔBi)q(u(t)),x0=x(0),r0=r(0).The following assumptions are assumed to be valid.Assumption
1. The pair (Ai,Bi) is controllable.Assumption
2. Consider the following: ΔAi=DiEi(t)Fi, ΔBi=BiMiΞi(t)Ni, and |MiΞi(t)Ni|∞≤ψi, where Di, Fi, Mi, and Ni are known constant matrices with appropriate dimensions, Ei(t) and Ξi(t) are time-varying uncertain matrices satisfying EiEiT≤I and ΞiΞiT≤I, and the parameter ψi satisfies 0≤ψi<1.In addition, the quantizerq(·) is defined by an operator function round(·) that rounds towards the nearest integer; that is,
(9)q(u(t))=μ·round(u(t)μ),whereμ(>0) is called a quantizing level of the quantizer. In computer-based control systems, the value of μ depends on the sampling accuracy and is known a priori. q(·) is the uniform quantizer with the fixed level μ. Define eμ=q(u(t))-u(t), since each component of eμ is bounded by the half of the quantizing level μ; thus, we have |eμ|∞≤μ/2.The objective of this paper is to design the state-feedback control law(10)u(t)=Kix+uic,Ki=K(rt),whenrt=i,
such that the resulting closed-loop system is stochastically stable and obtains as small value of the H2 cost index for the Markovian jump linear uncertain systems given in the following as possible
(11)𝒥(t)=𝔼[∫t∞‍{xT(τ)Q(rτ)x(τ)mmm+xT(τ)KT(rτ)R(rτ)K(rτ)x(τ)}dτ∣r0∫t∞‍],
where Q(rt) and R(rt) are positive definite matrices. The nonlinear part of the controller uic is designed against the effect of signal quantization, and the linear part Kix is proposed to deal with model uncertainties and to unknown transition rates and achieve optimal H2 performance.Remark 2.
WhenQ(rt)=Q>0, R(rt)=R>0, and K(rt)=K, the above H2 cost index is reduced to the following, (12) which is given in Yun et al. [17]:
(12)𝒥(t)=∫t∞{xT(τ)Qx(τ)+xT(τ)KTRKx(τ)}dτ.Some useful lemmas are firstly presented before formulating the main result.Lemma 3 (see [21]).
Given a symmetric matrixΠ and matrices M and N with appropriate dimensions, then Π+MF(t)N+NTFT(t)MT<0 for all F(t) satisfying FT(t)F(t)≤I, if and only if there exists a scalar ε>0 such that the following inequality holds:
(13)Π+εMMT+ε-1NTN<0.Lemma 4.
For any givenλj≥0 and matrices Pj>0  (Pj∈Rn×n,1≤j≤ℕ), if there exists Zi≥Pj, then the following inequality holds:
(14)∑j=1ℕ‍λjPj≤∑j=1ℕ‍λjZi.Proof.
According toZi≥Pj and λj≥0, one can obtain that λjZi≥λjPj, which further imply that inequality (14) holds.Remark 5.
In Shen and Yang [9], the inequality ∑j=1ℕλjPj≤(∑j=1ℕ‍λj)(∑j=1ℕ‍Pj) is introduced to obtain design conditions. It is clear that the utilization of Lemma 4 will result in less conservativeness since max{P1,P2,…,Pℕ}<∑j=1ℕ‍Pj for any Pi>0.Lemma 6.
Forα, β∈Rn, p≥1, and q≥1, the following inequality holds:
(15)|αTβ|≤|α|p|β|q,p-1+q-1=1.Lemma 7 (see [22]).
For the symmetric and positive definite matricesP and Q, if P≥Q>0, then Q-1≥P-1>0.


## 3. Main Results

Theorem 8.
For the system (1) subject to Assumptions 1 and 2, suppose that there exist the symmetric positive definite matrices Xi, general matrices Yi, and positive scalars εi, δi, and γ such that
(16)[Γ11XiFiTYiTNiTΓ14Γ15XiYi*-εiI00000**-δiI0000***Γ44000****-Zi00*****-Qi-10******-Ri-1]<0,Zi≤Xj,j∈𝒮uki,(17)[Φ11XiFiTYiTNiTΦ14XiYiT*-εiI0000**-δiI000***Φ4400****-Qi-10*****-Ri-1]<0,Xi≤Xj,j∈𝒮uki,(18)[γx0Tx0Xi]>0,
where
(19)Γ11=He(AiXi)+He(BiYi)+εiDiDiT+δiBiMiMiTBiT+λ-iiXi,Γ14=[λ-i𝒦1iXi,λ-i𝒦2iXi,…,λ-i𝒦mkiiXi],Γ44=diag{-X𝒦1i,-X𝒦2i,…,-X𝒦mkii},Γ15=-∑j∈𝒮ki‍λ_ijXi,Φ11=He(AiXi)+He(BiYi)+εiDiDiT+δiBiMiMiTBiT-∑j∈𝒮ki‍λ_ijXi,Φ14=[λ-i𝒦1iXi,λ-i𝒦2iXi,…,λ-i𝒦mkiiXi],Φ44=diag{-X𝒦1i,-X𝒦2i,…,-X𝒦mkii}.
Then, the controller designed as
(20)u(t)=Kix(t)+uic(t)
can drive the state trajectory to the origin asymptotically and can obtain the H2 cost by the minimum of r, where Ki=YiXi-1, Pi=Xi-1, and uic=(-(1+ψ)μ/2(1-ψ))sign(xTPiBi).Proof.
Take the Lyapunov function candidateV=xTPix; then, along the system trajectory of plant (1), the weak infinitesimal operator ℑax[·] of the process {x(t),rt,t≥0} for plant (8) at the point {t,x,i} is given by Kushner [23] as follows:
(21)ℑax[V]=x˙TPix+xTPix˙+xT∑j=1ℕ‍λijPjx=[(Ai+ΔAi)x+(Bi+ΔBi)(Kix+uic+eμ)]TPix+xTPi[(Ai+ΔAi)x+(Bi+ΔBi)(Kix+uic+eμ)]+xT∑j=1ℕ‍λijPjx=xT[(Ai+ΔAi)TPi+Pi(Ai+ΔAi)mmm+Pi(Bi+ΔBi)Ki+KiT(Bi+ΔBi)TPi]x+2xTPi(Bi+ΔBi)(uic+eμ)+xT∑j=1ℕ‍λijPjx.
According to Assumption 2, uic in Theorem 8, and Lemma 7, one can obtain that
(22)2xTPi(Bi+ΔBi)(uic+eμ)=2xTPiBiuic+2xTPiΔBi(uic+eμ)+2xTPiBieμ=2xTPiBiuic+2xTPiBiMiΞi(t)Ni(uic+eμ)+2xTPiBieμ≤2xTPiBiuic+2|xTPiBi|1|MiΞi(t)Ni|∞(|uic|∞+μ2)+2|xTPiBi|1μ2=2xTPiBiuic+2|xTPiBi|1ψ(|uic|∞+μ2)+|xTPiBi|1μ=2xTPiBiuic+(1+ψ)|xTPiBi|1μ+2ψ|xTPiBi|1|uic|∞=-(1+ψ)μ1-ψ|xTPiBi|1+(1+ψ)|xTPiBi|1μ+ψ(1+ψ)μ1-ψ|xTPiBi|1=0.
It follows from (21) and (22) that
(23)ℑax[V]≤xT[(Ai+ΔAi)TPi+Pi(Ai+ΔAi)mm+Pi(Bi+ΔBi)Ki+KiT(Bi+ΔBi)TPi]x+xT∑j=1ℕ‍λijPjx.
Furthermore, consider the following inequality:
(24)(Ai+ΔAi)TPi+Pi(Ai+ΔAi)+Pi(Bi+ΔBi)Ki+KiT(Bi+ΔBi)TP+∑j=1ℕ‍λijPj+Qi+KiTRiKi<0.
Since ΔBi=BiMiΞi(t)Ni and ΔAi=DiEi(t)Fi, the above inequality can be rewritten as
(25)AiTPi+PiAi+PiDiEi(t)Fi+FiTEiT(t)DiTPi+PiBiKi+KiTBiTPi+PiBiMiΞi(t)NiKi+KiTNiTΞiT(t)MiTBiTPi+∑j=1ℕ‍λijPj+Qi+KiTRiKi<0.
Using Lemma 3, it is equivalent to
(26)AiTPi+PiAi+εiPiDiDiTPi+1εiFiTFi+PiBiKi+KiTBiTPi+δiPiBiMiMiTBiTPi+1δiKiTNiTNiKi+∑j=1ℕ‍λijPj+Qi+KiTRiKi<0.
That is,
(27)He(AiTPi)+He(PiBiKi)+εiPiDiDiTPi+1εiFiTFi+δiPiBiMiMiTBiTPi+1δiKiTNiTNiKi+∑j∈𝒮ki,j≠i‍λijPj+∑j∈𝒮uki,j≠i‍λijPj+λiiPi+Qi+KiTRiKi<0.
Two cases will be considered.
Case
1 (λii∈𝒮ki). In this case, using Lemma 4, one can see that
(28)He(AiTPi)+He(PiBiKi)+εiPiDiDiTPi+1εiFiTFi+δiPiBiMiMiTBiTPi+1δiKiTNiTNiKi+∑j∈𝒮ki,j≠i‍λijPj+∑j∈𝒮uki,j≠i‍λijWi+λiiPi+Qi+KiTRiKi<0,Pj≤Wi,j∈𝒮uki.
Substituting ∑j∈𝒮uki,j≠iλij=-∑j∈𝒮ki,j≠iλij-λii into the first inequality in (22) and using the boundary information of the elements of the transition rates matrix, one can achieve that
(29)He(AiTPi)+He(PiBiKi)+εiPiDiDiTPi+1εiFiTFi+δiPiBiMiMiTBiTPi+1δiKiTNiTNiKi+∑j∈𝒮ki,j≠i‍λ-ijPj-(∑j∈𝒮ki,j≠i‍λ_ij+λ_ii)Wi+λ-iiPi+Qi+KiTRiKi<0,Pj≤Wi,j∈𝒮uki.
Case
2 (λii∈𝒮uki; namely, it is completely unknown). In such case, let us take λii=-∑j∈𝒮ki‍λij-∑j∈𝒮uki,j≠i‍λij into (21); then, one can see that
(30)AiTPi+PiAi+εiPiDiDiTPi+1εiFiTFi+PiBiKi+KiTBiTPi+δiPiBiMiMiTBiTPi+1δiKiTNiTNiKi+∑j∈𝒮ki‍λijPj+∑j∈𝒮uki,j≠i‍λijPj+(-∑j∈𝒮ki‍λij-∑j∈𝒮uki,j≠i‍λij)Pi+Qi+KiTRiKi<0.
Then, the stochastic stability can be guaranteed when
(31)He(AiTPi)+He(PiBiKi)+εiPiDiDiTPi+1εiFiTFi+δiPiBiMiMiTBiTPi+1δiKiTNiTNiKi+∑j∈𝒮ki‍λ-ijPj-∑j∈𝒮ki,j≠i‍λ_ijPi+Qi+KiTRiKi<0,Pj≤Pi,j∈𝒮uki.
For Case 1, pre- and postmultiplying Pi-1 in the first inequality in (27) and using Lemma 7 in the second inequality in (21), one can get that
(32)He(Pi-1AiT)+He(BiKiPi-1)+εiDiDiT+1εiPi-1FiTFiPi-1+δiBiMiMiTBiT+1δiPi-1KiTNiTNiKiPi-1+∑j∈𝒮ki,j≠i‍λ-ijPi-1PjPi-1-∑j∈𝒮ki‍λ_ijPi-1WiPi-1+λ-iiPi-1+Pi-1QiPi-1+Pi-1KiTRiKiPi-1<0,Wi-1≤Pj-1,j∈𝒮uki.
Let Xi=Pi-1, Yi=KiPi-1, and Zi=Wi-1; then, we have
(33)He(AiXi)+He(BiYi)+εiDiDiT+1εiXiFiTFiXi+δiBiMiMiTBiT+1δiYiTNiTNiYi+∑j∈𝒮ki,j≠i‍λ-ijXiXj-1Xi-∑j∈𝒮ki‍λ_ijXiZi-1Xi+λ-iiXi+XiQiXi+YiTRiYi<0,Zi≤Xj,j∈𝒮uki.
Applying the Schur complement formula, one can get (16).
For Case 2, pre- and postmultiplyingPi-1 in the first inequality in (28) and applying Lemma 7 to the second inequality in (28), one can obtain that
(34)He(Pi-1AiT)+He(BiKiPi-1)+εiDiDiT+1εiPi-1FiTFiPi-1+δiBiMiMiTBiT+1δiPi-1KiTNiTNiKiPi-1+∑j∈𝒮ki‍λ-ijPi-1PjPi-1-∑j∈𝒮ki‍λ_ijPi-1+Pi-1QiPi-1+Pi-1KiTRiKiPi-1<0,Pi-1≤Pj-1,j∈𝒮uki.
Let Xi=Pi-1 and Yi=KiPi-1; then, one can see that
(35)He(AiXi)+He(BiYi)+εiDiDiT+1εiXiFiTFiXi+δiBiMiMiTBiT+1δiYiTNiTNiYi+∑j∈𝒮ki‍λ-ijXiXj-1Xi-∑j∈𝒮ki‍λ_ijXi+XiQiXi+YiTRiYi<0,Xi≤Xj,j∈𝒮uki.
Thus, the LMIs in (17) are derived by using the Schur complement formula.
From the above proof, one can see that(36)ℑax[V]<-xTQix-xTKiTRiKix.
It follows from Kushner [23] that
(37)𝔼[V(t,x,i)∣r0]-V(x0,r0)≤-𝔼[∫0t‍{xT(τ)Qix(τ)-xT(τ)KiTRiKix(τ)}dτ].
Since 𝔼[V(t,x,i)∣r0]≥0, by some simple calculation, one can achieve that
(38)𝒥(t)≤𝒥(0)≤V(x0,r0)=x0TPix0≤γ.
Therefore, the minimum cost can be obtained by minimizing γ. Thus, the proof is achieved.Remark 9.
The merit of the proposed results lies in that the transition rates of the jumping process are assumed to be more general, which means that some elements in the transition rates matrix have been exactly known, some ones have been merely known with lower and upper bounds, and others may have no information to use. Dealing with the unknown transition rates, a less conservative method is used. At the same time, the impact of the input signal quantization on the system is also considered. Finally, the controller design conditions are presented in the framework of LMIs.In order to make comparison with the design method using the LMIs technique in Shen and Yang [9], we present the conditions designed by the utilization of Lemma 2 there.Proposition 10.
For the system (1) subject to Assumptions 1 and 2, suppose that there exist symmetric positive definite matrices Xi, general matrices Yi, and positive scalars εi, δi, and γ such that
(39)[Γ11XiFiTYiTNiTΓ14Γ15XiYi*-εiI00000**-δiI0000***Γ44000****-Γ5500*****-Qi-10******-Ri-1]<0,Zi≤Xj,j∈𝒮uki,(40)[Φ11XiFiTYiTNiTΦ14XiYiT*-εiI0000**-δiI000***Φ4400****-Qi-10*****-Ri-1]<0,Xi≤Xj,j∈𝒮uki,(41)[γx0Tx0Xi]>0,
where
(42)Γ11=He(AiXi)+He(BiYi)+εiDiDiT+δiBiMiMiTBiT+λ-iiXi,Γ14=[λ-i𝒦1iXi,λ-i𝒦2iXi,…,λ-i𝒦mkiiXi],Γ44=diag{-X𝒦1i,-X𝒦2i,…,-X𝒦mkii},Γ15=[-∑j∈𝒮ki‍λ_ijXi,-∑j∈𝒮ki‍λ_ijXi,…,-∑j∈𝒮ki‍λ_ijXi],Γ55=diag{-X𝕂1i,-X𝕂2i,…,-X𝕂mukii},Φ11=He(AiXi)+He(BiYi)+εiDiDiT+δiBiMiMiTBiT-∑j∈𝒮ki‍λ_ijXi,Φ14=[λ-i𝒦1iXi,λ-i𝒦2iXi,…,λ-i𝒦mkiiXi],Φ44=diag{-X𝒦1i,-X𝒦2i,…,-X𝒦mkii}.
Then, the controller designed as
(43)u(t)=Kix(t)+uic(t)
can drive the state trajectory to the origin asymptotically and can obtain the H2 cost by minimizing r, where Ki=YiXi-1, Pi=Xi-1, and uic=(-(1+ψ)μ/2(1-ψ))sign(xTPiBi).Proof.
The proof process is similar to that in Theorem8 but with replacing Pj≤Wi, j∈𝒮uki, and ∑j∈𝒮uki,j≠iλijWj with (∑j∈𝒮uki,j≠i‍λij)(∑j∈𝒮uki,j≠i‍Pj) in (22). The details are omitted here for space limitation.


## 4. Numerical Example

An example is presented to illustrate the effectiveness of the proposed method.Consider the MJLSs with four operation modes as follows:(44)A1=[10.20.4-4],A2=[-1.51.21.4-5],A3=[-132-2],A4=[0340],B1=[0.50.2],B2=[0.10.3],B3=[01],B4=[01],D1=[02],D2=[10],D3=[0.40.8],D4=[-0.50.5],F1=[10],F2=[0.51],F3=[00.3],F4=[0.50.8],M1=0.4,M2=0.6,M3=0.5,M4=1,N1=0.8,N2=0.2,N3=0.1,N4=0.5.The considered transition rates matrix is given as follows:(45)Λ=[-1.30.2???α10.30.30.6???0.4α2?α3],
where -1.2≤α1≤-0.8, 0.6≤α2≤0.8, and -2≤α3≤-1.5. Choosing x0=[1;-0.5] and solving (16)–(18) in Theorem 8 and (34)–(38) in Proposition 10, one can obtain the optimal H2 performance indices shown in Table 1. It can be seen that less conservativeness is obtained by the proposed method in Theorem 8.Table 1
Comparison of optimalγ.
Theorem8
Proposition10
1.4518
2.4567For simulation, one can obtain the controller gains by solving the LMIs in Theorem8 as follows:K
1
=
[
-
5.2949
-
0.5848
], K2=[-2.4332-1.6826], K3=[-4.0556-4.6580], and K4=[-14.2188-12.4624] and by considering the positive definite matrices
(46)P1=[1.51530.10520.10520.1670],P2=[1.16400.53370.53370.4595],P3=[0.77880.42690.42690.4903],P4=[3.67262.84382.84382.4925].The following parameters are used in the simulation:μ=0.3, E1(t)=0.2cos(t), E2(t)=0.05sin(t), E3(t)=0.8cos(3t), E4(t)=0.4sin(t), Ξ1(t)=0.5cos(3t), Ξ2(t)=0.6cos(2t), Ξ3(t)=0.2sin(t), and Ξ4(t)=0.05cos(2t). In order to reduce the chattering effects induced by uic, sign(xTPiBi) is substituted by sat(xTPiBi/δi), where δ1=0.0015, δ2=0.0004, δ3=0.0015, and δ4=0.0004.The switching mode, the control input, and the response curves of the system states are presented in Figures1, 2, and 3, respectively. Among them, Figure 1 shows a possible system modes evolution which meets the transition rates given in this example. As shown in Figure 1, the system has 4 modes and is in different modes at a different time. Figure 2 shows the curve of the control input u(t). With this controller, Figure 3 depicts the state response curves of the closed-loop system. It can be seen that the considered continuous-time Markovian jump linear uncertain system is stochastically stable in spite of mismatched uncertainty, the input signal quantization, and the partly known transition rates covering the completely known, the boundary unknown, and the completely unknown elements in the transition rates matrix.Figure 1
Evolution of the system mode.Figure 2
The curve of the control inputu(t).Figure 3
The response curve of the statex.


## 5. Conclusions

H
2 control design for a class of the continuous-time Markovian jump linear uncertain systems with partly unknown transition rates and input quantization has been investigated. The H2 cost index for the Markovian jump systems is first introduced, and then sufficient conditions of the H2 control for the Markovian jump linear uncertain systems with unknown transition rates have been presented. The controller is constructed by two parts. The nonlinear part is proposed to eliminate the effect of input quantization. The linear part is designed by solving the LMIs conditions for achieving the H2 performance against model uncertainties and unknown transition rates. In comparison with the existing result in the literature, less conservativeness has been obtained by introducing new relaxed inequality conditions. Finally, a numerical example is given to show the effectiveness of the proposed design method.



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*Source: 426271-2013-09-02.xml*