# Observer Design for Delayed Markovian Jump Systems with Output State Saturation

Authors: Guoliang Wang; Bo Feng
Journal: Mathematical Problems in Engineering
 (2018)

---

## Abstract

This paper considers the observer design problem of continuous-time delayed Markovian jump systems with output state saturation. Different from the traditionally observer-based saturation control methods, a kind of system output state saturation with a partially delay-dependent property is proposed, where both nondelay and delay states exist at the same time but happen asynchronously. By exploiting the Bernoulli variable, the probability distributions of such two states are described and considered in the observer design. Based on an improved equality applied to deal with saturation terms, sufficient conditions for the designed observer with three kinds of output saturations are all provided with LMI forms. Finally, a numerical example is given to indicate the effectiveness of the obtained results.


---

## Body

## 1. Introduction

As we know, Markovian jump system (MJS) is a special kind of stochastic hybrid dynamical systems. Because of two kinds of mechanisms contained, it is very suitable to model such actual systems whose structures or parameters change [1, 2]. Over the past years, many research topics on MJSs have been extensively studied, like stability analysis [3–6], stabilization [7–11], robust control [12–15], adaptive control [16–19], H∞ filtering and control [20, 21], state estimation [22–25], synchronization [26–30], and so on.On the other hand, saturation problem is commonly encountered in many physical systems. It is mainly because the transmitted signal is usually affected by environmental factors, physical factors, and even technical factors. When signal occurs with saturation, the system performance will be reduced. Therefore, the effect of saturation is necessarily studied, as in [31–35]. By investigating the above results, it is found that they mainly focused on the estimate of domain of attraction, in which an equivalent description of saturation actuator is necessary. However, the introduction of this transformation will make the computation complexity very large, especially for an MJS with N operation modes. Recently, one inequality was introduced in [36] and applied to deal with saturation. In the modern control theory, it is obvious that the state feedback control has advantages in solving the problems of system stability, pole placement, stabilization, and optimal control. However, system state usually cannot be measured directly, or the measurement is limited by objective conditions. In this case, it is not easy to achieve the object by state feedback control. The better way is to make full use of system output. When output is saturated, some results were presented in [37, 38]. By investigating these references, it is found that there are still many problems to be considered. For example, there is no time delay in the saturated output, while time delay is usually associated with most system states and could lead to unstable system. Thus, it is necessary to consider them together. To our best knowledge, very few results are available to study the above problems. All observations motivate current research.In this paper, the observer design problem of continuous-time delayed Markovian jump systems with output state saturation is studied, where the output state saturation is partially delay-dependent. The main contributions of this paper are generalized as follows:(1) A kind of observer based on partially delay-dependent output is proposed. Here, both nondelay and delay states are contained in output saturation simultaneously, but their occurrences are asynchronous. (2) The probability distributions of such two states are embodied by the Bernoulli variable and fully considered in the observe design. (3) Sufficient conditions of the designed observer are obtained by applying an improved inequality to deal with the saturation terms. In order to make the computation solved easily and directly, the existence conditions are presented within LMI framework by introducing some additional variables and inequalities. (4) Based on the proposed methods and techniques, another two kinds of observers based on different output saturations are proposed, whose conditions are all LMIs.Notation 1.
R
n denotes the n-dimensional Euclidean space; Rq×n is the set of all q×n real matrices. · refers to the Euclidean vector norm or spectral matrix norm. Ω is the sample space, F is the σ-algebra of subsets of the sample space, and P is the probability measure on F. In symmetric block matrices, we use “∗” as an ellipsis for the terms induced by symmetry, diag{⋯} for a block-diagonal matrix, and (M)⋆≜M+MT.


## 2. Problem Formulation

Consider the following continuous-time delayed Markovian jump systems with output state saturation:(1)x˙t=Artxt,yt=Crtsatαtxt+1-αtxt-d,xt=ϕt,-d≤t≤0, where x(t)∈Rn is the state vector, y(t)∈Rq is the output, and ϕ(t)∈Rn is the initial condition. Matrices A(rt) and C(rt) are known matrices of compatible dimensions. {rt,t≥0} is a continuous-time Markov process taking values in a finite set S={1,2,…,N} with transition rate matrix (TRM) Π≜(πij)∈RN×N given by(2)Prrt+Δt=j∣rt=i=πijΔt+oΔt,i≠j1+πiiΔt+oΔt,i=j,where Δt>0,πij≥0, if i≠j, and πii=-∑j≠iπij for all i,j∈S. Time delay d satisfies d>0, and α(t) is the Bernoulli stochastic variable and defined as(3)αt=1,ifxtisavailable0,ifxt-disavailable. It is satisfied as(4)Pr⁡αt=1=Eαt=α,Prαt=0=1-α. Function sat(·): Rn→Rn is the standard saturation function and defined as(5)satui=μi,ui>μiui,-μi≤ui≤μi-μi,ui<-μi,i∈1,2,…,n,where (6)satu=satu1⋯satunT,μ=μ1⋯μnT.Remark 1.
It is worth mentioning that saturation in output is more general and has some advantages. Firstly, compared with saturations described in [36, 39–42] where nondelay state should be available online, system state x(t) in output is not necessary and could be replaced by delay state x(t-d). Secondly, compared with the saturated controllers designed for delayed systems without delay state [43, 44], both nondelay and delay states are included in output, whose distribution probabilities are also considered. Because all the delay terms could affect delayed systems, the obtained results are more conservative without considering them. In this sense, it is said that our formulation about output saturation has a larger application scope and less conservatism.By lettingη=x-sat(x) and ηd=xd-sat(xd) and considering the definition of α(t), it is known that(7)satαtxt+1-αtxt-d=αtxt+1-αtxt-d-αtη+1-αtηd=αxt+1-αxt-d+αt-αxt-xt-d-αη+1-αηd+αt-αη-ηd. Then, system (1) is rewritten to be(8)x˙t=Artxt,yt=Crtαxt+1-αxt-d+αt-αxt-xt-d-αη+1-αηd+αt-αη-ηd,xt=ϕt,-d≤t≤0.In this paper, the designed state observer system is described by(9)x^˙t=Artx^t+Grtyt-y^t,y^t=Crtx^t,where x^(t)∈Rn is the estimation of x(t), and y^(t)∈Rq is the corresponding output. Define e(t)=x(t)-x^(t); the resulting error system is rewritten to be(10)e˙t=Art-GrtCrtet+1-αGrtCrtxt-1-αGrtCrtxt-d+αGrtCrtη+1-αGrtCrtηd+αt-αGrtCrt×xt-d-xt+η-ηd.Here, it is assumed that stochastic variables α(t) and rt are independent.Proposition 2.
Letu=α(t)Kx(t)+(1-α(t))Kdx(t-d) and η=u-sat(u). Then, there exist real numbers ε∈(0,1) and εd∈(0,1) such that(11)ηTη≤αtεxTtKTKxt+1-αtεdxTt-dKdTKdxt-d,where η=η1⋯ηnT, and ηi is the dead-zone nonlinearity function, i={1,2,…,n}.Proof.
Based on the formulation ofu(t), it is obtained that(12)ηTη=αtKxt+1-αtKdxt-d-satαtKxt+1-αtKdxt-dTαtKxt+1-αt×Kdxt-d-satαtKxt+1-αtKdxt-d.On the one hand, if α(t)=1, one has u=Kx(t). Similar to [36], we have the following.(i)
Ifui>μi, it is obtained that(13)ηTη=Kxt-satKxtTKxt-satKxt=xTtKTKxt-2xTtKTsatKxt+satKxtTsatKxt=xTtKTKxt-2xTtKTμ+μTμ<xTtKTKxt-μTμ≤εxTtKTKxt,whereε∈(0,1) could be selected.(ii)
Ifui<-μi, it is found that(14)ηTη=Kxt-satKxtTKxt-satKxt=xTtKTKxt-2xTtKTsatKxt+satKxtTsatKxt=xTtKTKxt+2xTtKTμ+μTμ<xTtKTKxt-μTμ≤εxTtKTKxt,whereε∈(0,1) could be obtained too.(iii)
If-μi≤ui≤μi, we have ηTη=0. Then, there is always a scalar ε∈(0,1) such that(15)ηTη≤εxTtKTKxt.On the other hand, when α(t)=0, it is obtained that u=Kdx(t-d). Similarly, we have the following cases. (i)
Ifui>μi, it is obtained that(16)ηTη=Kdxt-d-satKdxt-dTKdxt-d-satKdxt-d=xTt-dKdTKdxt-d-2xTtKdTsatKdxt-d+satKdxt-dTsatKdxt-d=xTt-dKdTKdxt-d-2xTt-dKdTμ+μTμ<xTt-dKdTKdxt-d-μTμ≤εdxTt-dKdTKxt-d,whereεd∈(0,1).(ii)
Ifui<-μi, it is found that(17)ηTη=Kdxt-d-satKdxt-dTKdxt-d-satKdxt-d=xTt-dKdTKdxt-d-2xTtKdTsatKdxt-d+satKdxt-dTsatKdxt-d=xTt-dKdTKdxt-d+2xTt-dKdTμ+μTμ<xTt-dKdTKdxt-d-μTμ≤εdxTt-dKdTKxt-d,whereεd∈(0,1).(iii)
If-μi≤ui≤μi, we have ηTη=0. Then, there is always a scalar εd∈(0,1) such that(18)ηTη≤εdxTt-dKdTKdxt-d.This completes the proof.


## 3. Main Results

Theorem 3.
Given system (1) with d>0, there exists an observer (9) such that system (1) and error system (12) are asymptotically stable, if for given positive scalars α∈[0,1], εi∈(0,1), and εdi∈(0,1), there exist Pi>0,Q>0,Z>0,γi>0,λi>0, and Yi such that(19)Θi1Θi2Z0Θi3Θi4∗Θi5Θi6Z0Θi7∗∗Θi80Θi9Θi10∗∗∗Θ1100∗∗∗∗Θi120∗∗∗∗∗Θi12<0,(20)-γiICiTYiT∗-Pi≤0,(21)-λiICiTYiT∗Θi12≤0,where(22)Θi1=PiAi⋆+∑j=1NπijPj+αεiγiI+Q+d2AiTZAi-Z+2αd2εiλiI+4αα¯d2εiλiI,Θi2=α¯CiTYiT,Θi3=-2αα¯dCiTYiT,Θi4=2α¯dCiTYiT,Θi5=PiAi-YiCi⋆+Pi+∑j=1NπijPj+Q-Z,Θi6=-α¯YiCi,Θi7=2dAiTPi-CiTYiT,Θi8=α¯εdiγiI+2α¯d2εdiλiI+4αα¯d2εdiλiI-Q-Z,Θi9=2αα¯dCiTYiT,Θi10=-2α¯dCiTYiT,Θ11=-Q-Z,Θi12=-2Pi+Z,α¯=1-α.Then, the gain of observer (9) is computed by(23)Gi=YiPi-1.Proof.
For systems (1) and (12), choose the following Lyapunov function:(24)Vxt=V1xt+V2xt,where(25)V1xt,rt=xTtPrtxt+∫t-dtxTsQxsds+d∫-d0∫t+θtx˙TsZx˙sdsdθ,V2xt,rt=eTtPrtet+∫t-dteTsQesds+d∫-d0∫t+θte˙TsZe˙sdsdθ.Let L be the weak infinitesimal generator of random process {xt,rt}, for each rt=i∈S; it is defined as(26)LVxt,t,i=limΔt→0+1ΔtEVxt+Δt,rt+Δt,t+Δt∣xt,rt=i-Vxt,i,t.Then, it is further obtained that(27)LV1xt=xTtPiAi⋆xt+xTt∑j=1NπijPjxt+xTtQxt-xTt-dQxt-d+d2xTtAiTZAixt-∫t-dtx˙TθZx˙θdθ,LV2xt=eTtPiAi-GiCi⋆et+2α¯eTtPiGiCixt-2α¯eTtPiGiCixt-d+2αeTtPiGiCiη+2α¯eTtPiGiCiηd+eTt∑j=1NπijPjet-∫t-dte˙TθZe˙θdθ+eTtQet-eTt-dQet-d+d2fi1+fi2TZfi1+fi2+αα¯d2GiCixt-d-xt+η-ηdT×ZGiCixt-d-xt+η-ηd,where(28)fi1=Ai-GiCiet+α¯GiCixt-α¯GiCixt-d,fi2=αGiCiη+α¯GiCiηd.Based on the Jensen inequality, one has(29)-∫t-dtx˙TθZx˙θdθ≤-xTtZxt+2xTtZxt-d-xTt-dZxt-d,(30)-∫t-dte˙TθZe˙θdθ≤-eTtZet+2eTtZet-d-eTt-dZet-d.Moreover, it is concluded that(31)2αeTtPiGiCiη≤αeTtPiet+αηTCiTGiTPiGiCiη,(32)2α¯eTtPiGiCiηd≤α¯eTtPiet+α¯ηdTCiTGiTPiGiCiηd,(33)2d2αGiCiη+α¯GiCiηdTZαGiCiη+α¯GiCiηd≤2αd2ηTCiTGiTZGiCiη+2α¯d2ηdTCiTGiTZGiCiηd,(34)2αα¯d2GiCiη-GiCiηdTZGiCiη-GiCiηd≤4αα¯d2ηTCiTGiTZGiCiη+4αα¯d2ηdTCiTGiTZGiCiηd,(35)d2fi1+fi2TZfi1+fi2≤2d2fi1TZfi1+2d2αGiCiη+α¯GiCiηdTZαGiCiη+α¯GiCiηd,(36)αα¯d2GiCixt-d-xt+η-ηdTZGiCixt-d-xt+η-ηd≤2αα¯d2GiCixt-d-xtTZGiCixt-d-xt+2αα¯d2GiCiη-GiCiηdTZGiCiη-GiCiηd.Based on these conditions, the following is computed:(37)LVxt≤xTtPiAi⋆xt+xTt∑j=1NπijPjxt+eTtPiAi-GiCi⋆et+2α¯eTtPiGiCixt-2α¯eTtPiGiCixt-d+eTtPiet+αηTCiTGiTPiGiCiη+α¯ηdTCiTGiTPiGiCiηd+eTt∑j=1NπijPjet+xTtQxt-xTt-dQxt-d+eTtQet-eTt-dQet-d+d2xTtAiTZAixt-xTtZxt+2xTtZxt-d-xTt-dZxt-d+2αd2ηTCiTGiTZGiCiη+2α¯d2ηdTCiTGiTZGiCiηd+2d2Ai-GiCiet+α¯GiCixt-α¯GiCixt-dT×ZAi-GiCiet+α¯GiCixt-α¯GiCixt-d+2αα¯d2GiCixt-d-xtTZGiCixt-d-xt+4αα¯d2ηTCiTGiTZGiCiη+4αα¯d2ηdTCiTGiTZGiCiηd-eTtZet+2eTtZet-d-eTt-dZet-d.Moreover, the other conditions are assumed to be(38)CiTGiTPiGiCi≤γiI,(39)CiTGiTZGiCi≤λiI.Based on the above conditions and applying Proposition 2, it is obtained that (37) is implied by (40)LVxt≤ξTtΘi1Θi2Z0∗Θi5Θi6Z∗∗Θi80∗∗∗Θ11+2αα¯d2-CiTGiT0CiTGiT0Z-CiTGiT0CiTGiT0T+2d2α¯CiTGiTAi-CiTGiT-α¯CiTGiT0Zα¯CiTGiTAi-CiTGiT-α¯CiTGiT0Tξt,where(41)ξTt=xTteTtxTt-deTt-d.By using the Schur complement lemma, it is further obtained that(42)Θi1Θi2Z0Θ¯i3Θ¯i4∗Θi5Θi6Z0Θ¯i7∗∗Θi80Θ¯i9Θ¯i10∗∗∗Θ1100∗∗∗∗-Z-10∗∗∗∗∗-Z-1<0,where(43)Θ¯i3=-2αα¯dCiTGiT,Θ¯i4=2α¯dCiTGiT,Θ¯i9=2αα¯dCiTGiT,Θ¯i7=2dAiT-CiTGiT,Θ¯i10=-2α¯dCiTGiT.By pre- and postmultiplying both its sides with diag{I,I,I,I,Pi,Pi} and its transpose, respectively, one gets(44)Θi1Θi2Z0Θi3Θi4∗Θi5Θi6Z0Θi7∗∗Θi80Θi9Θi10∗∗∗Θ1100∗∗∗∗-Z¯0∗∗∗∗∗-Z¯<0, where Z¯=PiZ-1Pi. It is further obtained that Z¯>0 could be guaranteed by(45)-PiZ-1Pi≤-2Pi+Z<0.As for conditions (38) and (39), they are equivalent to(46)-γiI+CiTGiTPiGiCi≤0,-λiI+CiTGiTZGiCi≤0.By applying the Schur complement lemma and considering representation (23), it is known that condition (38) is equivalent to condition (20). As for condition (39), by pre- and postmultiplying both its sides with diag{I,Pi} and applying inequality (45), one could easily get condition (21) with representation (23) implying condition (39). This completes the proof.Remark 4.
Due toα included obviously, it plays important roles in analysis and synthesis of systems with output saturation. With two special cases that only system state with α=0 or delay state with α=1 is contained in output saturation, our results could be viewed as extension results on the saturation output problem from output without time delay to stochastic delay output. On the other hand, from the proof process, it is seen that some inequalities, like (31), (33), (35), and so on, have been used to get the LMI conditions and could lead to more conservatism. In order to reduce its conservatism, some additional variables could be introduced in these inequalities. However, such variables will lead to larger computation complexity. Moreover, the conservatism of inequality (45) could be further reduced by applying similar methods in [45]. Similarly, much larger computation complexity in terms of more variables and inequalities should be needed. Based on these facts, it is said that whether to choose these methods or not should be considered in concrete situations.Remark 5.
It is seen from this theorem that such a probability should be given exactly. It will be impossible or of high cost in some practical applications. Instead, only its estimation is available, even if it is totally unknown. In other words, there will be a uncertainty between the real and estimated values. As we know, such a uncertainty will degrade the system performance and even lead to unstable system. So, it is necessary and meaningful to consider this general case. Moreover, some necessarily additional variables and inequalities are introduced to obtain LMI conditions, where the variables are to be computed instead of being given beforehand. Because of the given results with LMI forms, the above general case could be handled by exploiting the methods in this paper and [46, 47] together.Next, we consider another state observer system described as(47)x^˙t=Artx^t+Grtyt-y^t,y^t=Crtαtx^t+1-αtx^t-d.Then, the resulting error system is rewritten to be(48)e˙t=Art-αGrtCrtet-1-αGrtCrtet-d+αGrtCrtη+1-αGrtCrtηd+αt-αGrtCrtxt-d-xt+η-ηd.Similarly, we have the following theorem.Theorem 6.
Given system (1) with d>0, there exists observer (47) such that system (1) and error system (48) are asymptotically stable, if for given positive scalars α∈[0,1], εi∈(0,1), and εdi∈(0,1), there exist Pi>0,  Q>0,  Z>0,  γi>0,  λi>0, and Yi satisfying conditions (20) and (21) and(49)Θi10Z000∗Ψi10Ψi2Θi3Ψi3∗∗Θi8000∗∗∗Θ11Θi9Θi10∗∗∗∗Θi120∗∗∗∗∗Θi12<0,where(50)Ψi1=PiAi-αYiCi⋆+Pi+∑j=1NπijPj+Q-Z,Ψi2=-α¯YiCi+Z,Ψi3=2dAiTPi-αCiTYiT.Then, the gain of observer (47) could be computed by (23).Proof.
For systems (1) and (48), choose the same Lyapunov function (24). Similar to (35) and (36), it is obtained that(51)d2fi3+fi2TZfi3+fi2≤2d2fi3TZfi3+2d2αGiCiη+α¯GiCiηdTZαGiCiη+α¯GiCiηd,where (52)fi3=Ai-αGiCiet-α¯GiCiet-d,(53)αα¯d2GiCiet-d-et+η-ηdTZGiCiet-d-et+η-ηd≤2αα¯d2GiCiet-d-etTZGiCiet-d-et+2αα¯d2GiCiη-GiCiηdTZGiCiη-GiCiηd.Then, it is obtained that(54)LVxt≤xTtPiAi⋆xt+xTt∑j=1NπijPjxt+eTtPiAi-GiCi⋆et-2α¯eTtPiGiCiet-d+eTtPiet+αηTCiTGiTPiGiCiη+α¯ηdTCiTGiTPiGiCiηd+eTt∑j=1NπijPjet+xTtQxt-xTt-dQxt-d+eTtQet-eTt-dQet-d+d2xTtAiTZAixt-xTtZxt+2xTtZxt-d-xTt-dZxt-d+2αd2ηTCiTGiTZGiCiη+2α¯d2ηdTCiTGiTZGiCiηd+2d2Ai-αGiCiet-α¯GiCiet-dT×ZAi-αGiCiet-α¯GiCiet-d+2αα¯d2GiCiet-d-etTZGiCiet-d-et+4αα¯d2ηTCiTGiTZGiCiη+4αα¯d2ηdTCiTGiTZGiCiηd-eTtZet+2eTtZet-d-eTt-dZet-d. Based on conditions (20), (21), (51), and (53), it is obtained that (54) is implied by (55)LVxt≤ξTtΘi10Z0∗Ψi10Ψi2∗∗Θi80∗∗∗Θ11+2αα¯d20-CiTGiT0CiTGiTZ0-CiTGiT0CiTGiTT+2d20Ai-αCiTGiT0-α¯CiTGiTZ0Ai-αCiTGiT0-α¯CiTGiTTξt.By using similar methods given in Theorem 3, it is known that condition (55) could be guaranteed by inequality (49). This completes the proof.Finally, we consider the following state observer system described by(56)x^˙t=Artx^t+Grtyt-y^t,y^t=Crtsatαtx^t+1-αtx^t-dwhich is equivalent to(57)x^˙t=Artx^t+Grtyt-y^t,y^t=Crtαx^t+α¯x^t-d+αt-αx^t-x^t-d-αη^+α¯η^d+αt-αη^-η^d. Then, the resulting error system is rewritten to be(58)e˙t=Art-αGrtCrtet-α¯GrtCrtet-d+αGrtCrtη-η^+α¯GrtCrtηd-η^d+αt-αGrtCrtxt-d-xt+η-η^-ηd+η^d.Theorem 7.
Given system (1) with d>0, there exists observer (56) such that system (1) and error system (58) are asymptotically stable, if for given positive scalars α∈[0,1],  εi∈(0,1), and εdi∈(0,1), there exist Pi>0,  Q>0,  Z>0,  γi>0,  λi>0, and Yi satisfying conditions (20) and (21) and(59)Ωi1Ωi2Z000∗Ωi30Ψi2Θi3Ψi3∗∗Ωi4Ωi500∗∗∗Ωi6Θi9Θi10∗∗∗∗Θi120∗∗∗∗∗Θi12<0,where(60)Ωi1=PiAi⋆+∑j=1NπijPj+2αεiγiI+Q+d2AiTZAi-Z+8αd2εiλiI+16αα¯d2εiλiI,Ωi2=-αεiγiI-4αd2εiλiI-8αα¯d2εiλiI,Ωi3=PiAi-αYiCi⋆+2Pi+∑j=1NπijPj+Q-Z+αεiγiI+4αd2εiλiI+8αα¯d2εiλiI,Ωi4=2α¯εdiγiI+8α¯d2εdiλiI+16αα¯d2εdiλiI-Q-Z,Ωi5=-α¯εdiγiI-4α¯d2εdiλiI-8αα¯d2εdiλiI,Ωi6=α¯εdiγiI+4α¯d2εdiλiI+8αα¯d2εdiλiI-Q-Z.Then, (23) could be used to compute the gain of observer (56).Proof.
Firstly, the same Lyapunov function is also selected for systems (1) and (56). Similar to computation process of Theorems 3 and 6, the corresponding terms are handled as(61)-2αeTtPiGiCiη^≤αeTtPiet+αη^TCiTGiTPiGiCiη^,-2α¯eTtPiGiCiη^d≤α¯eTtPiet+α¯η^dTCiTGiTPiGiCiη^d,d2fi3+fi4TZfi3+fi4≤2d2fi3TZfi3+2d2fi4TZfi4,where(62)fi4=αGiCiη-η^+α¯GiCiηd-ηd^,2d2fi4TZfi4≤2αd2η-η^TCiTGiTZGiCiη-η^+2α¯d2ηd-ηd^TCiTGiTZGiCiηd-ηd^≤4αd2ηTCiTGiTZCiTGiTη+4αd2η^TCiTGiTZGiCiη^+4α¯d2ηdTCiTGiTZGiCiηd+4α¯d2η^dTCiTGiTZGiCiη^d,d2fi3+fi4TZfi3+fi4≤2d2fi3TZfi3+2d2fi4TZfi4,αα¯d2GiCiet-d-et+η-η^-ηd-ηd^T×ZGiCiet-d-et+η-η^-ηd-ηd^≤2αα¯d2GiCiet-d-etTZGiCiet-d-et+2αα¯d2GiCiη-η^-GiCiηd-ηd^TZ×GiCiη-η^-GiCiηd-ηd^.Moreover, it is obtained that(63)2αα¯d2GiCiη-η^-GiCiηd-ηd^TZGiCiη-η^-GiCiηd-ηd^≤8αα¯d2ηTCiTGiTZCiTGiTη+8αα¯d2η^TCiTGiTZGiCiη^+8αα¯d2ηdTCiTGiTZGiCiηd+8αα¯d2η^dTCiTGiTZGiCiη^d.Based on these conditions, it is obtained that(64)LVxt≤xTtPiAi⋆xt+xTt∑j=1NπijPjxt+eTtPiAi-GiCi⋆et-2α¯eTtPiGiCiet-d+2eTtPiet+αηTCiTGiTPiGiCiη+α¯ηdTCiTGiTPiGiCiηd+αη^TCiTGiTPiGiCiη^+α¯η^dTCiTGiTPiGiCiη^d+eTt∑j=1NπijPjet+xTtQxt-xTt-dQxt-d+eTtQet-eTt-dQet-d+d2xTtAiTZAixt-xTtZxt+2xTtZxt-d-xTt-dZxt-d+4αd2ηTCiTGiTZGiCiη+4α¯d2ηdTCiTGiTZGiCiηd+4αd2η^TCiTGiTZGiCiη^+4α¯d2η^dTCiTGiTZGiCiη^d+2d2Ai-αGiCiet-α¯GiCiet-dT×ZAi-αGiCiet-α¯GiCiet-d+2αα¯d2GiCiet-d-etTZGiCiet-d-et+8αα¯d2ηTCiTGiTZGiCiη+8αα¯d2ηdTCiTGiTZGiCiηd+8αα¯d2η^TCiTGiTZGiCiη^+8αα¯d2η^dTCiTGiTZGiCiη^d-eTtZet+2eTtZet-d-eTt-dZet-d. Based on conditions (20) and (21) and above inequalities, it is obtained that (64) is implied by(65)LVxt≤ξTtΩi1Ωi2Z0∗Ωi30Ψi3∗∗Ωi4Ωi5∗∗∗Ωi6+2αα¯d20-CiTGiT0CiTGiTZ0-CiTGiT0CiTGiTT+2d20Ai-αCiTGiT0-α¯CiTGiTZ0Ai-αCiTGiT0-α¯CiTGiTTξt.By using similar methods given in Theorems 3 and 6, condition (65) is obtained easily from inequality (59) with representation (23). This completes the proof.


## 4. Numerical Examples

Example 1.
Consider a delayed Markovian jump system of form (1), whose parameters are described to be (66)A1=-1.50.11-0.8,C1=0.10.60.40.1,A2=-0.9-0.20.8-0.9,C2=-0.20.5-0.3-0.2.The transition rate matrix is given as (67)Π=-0.50.50.6-0.6.Firstly, we design an observer with form (9). Letting d=0.3, ε1=0.3, ε2=0.2, εd1=0.4, and εd2=0.1, by Theorem 3, the gains of observer (9) are computed as (68)G1=0.09020.28530.24430.2980,G2=-0.1308-0.0389-0.0230-0.4683.Applying the above designed observer, under the initial condition x0=1-1-11T, we have the state responses of systems (1) and (10) given in Figure 1, while the simulation of the corresponding operation mode is given in Figure 2.
When another observer (47) is designed, its gains could be computed by Theorem 6 and given as (69)G1=0.08550.38480.34150.2380,G2=-0.1079-0.0338-0.0148-0.5234,where the corresponding parameters are the same as the above ones. At the same time, by Theorem 7, the gains of observer (57) are computed to be (70)G1=0.28320.96480.22650.1486,G2=0.14890.8130-0.3814-2.0525.Under the same initial condition, the state curves of the corresponding systems are simulated in Figures 3 and 4, respectively. Based on these simulations, it is seen that the designed observers are all useful and also demonstrate the utility of the proposed methods.Figure 1
The state responses of systems (1) and (10).Figure 2
The simulation of operation mode.Figure 3
The state responses of systems (1) and (47).Figure 4
The state responses of systems (1) and (57).In order to further demonstrate the effects of output saturation existing in different forms, more comparisons among them will be done in the following. Firstly, we consider the relationship between probabilityα and allowable maximum d. Based on Theorems 3–7, the maximum allowable d along with α is given in Table 1, whose simulation is carried out in Figure 5. From this simulation, it is seen that observer (9) is the least conservative, while observer (56) has the largest conservatism.Table 1
The allowable maximumd for different α.
α
0
0.3
0.5
0.8
1
Theorem3
0.927
0.943
0.955
0.976
0.999
Theorem6
0.918
0.928
0.936
0.958
0.999
Theorem7
0.207
0.273
0.313
0.390
0.470Figure 5
The curves ofdmax along with α.Next, we consider the effects of parametersεi and εdi. In order to demonstrate their effects clearly, without loss of generality, matrix A1 is assumed to be (71)A1=-10.1+δ1-0.8,where δ is a positive scalar to be determined, and the other matrices are constant. Based on Theorem 3, the allowable maximum δ≜δmax along with different pair (ε1,ε2) is listed in Table 2. Similarly, Tables 3 and 4 are obtained by applying Theorems 6 and 7, respectively. Based on these tables, the simulation of correlation between pair (ε1,ε2) and δmax is shown Figures 6 and 7. From Figure 6, it is seen that Theorem 3 related to observer (10) is less conservative than Theorem 6 corresponding to observer (47). From their forms, it is seen in this example that the delay term in output has a negative effect. However, the differences between them are very small. To the contrary, there is an advantage that the nondelay state of observer (47) is not necessarily available online, which could be replaced by a delay one with some probability. In this sense, it is said that observer (47) is better in terms of having less application constraints. As for Figure 7, it is concluded that observer (57) is the most conservative. The main reason is that a more output saturation is included and has a large negative effect in reducing performance. Similarly, we could also make a correlation between pair (εd1,εd2) and δmax. Based on Theorems 3–7, the allowable δmax for different pair (εd1,εd2) are presented in Tables 5–7. From these tables, a similar conclusion could be obtained where Theorem 3 designed for observer (10) is superior over Theorems 6 and 7. Among them, it is found that the most conservative one is observer (57) obtained from Theorem 7. In addition, the corresponding simulations of such correlations between pair (εd1,εd2) and δmax could be obtained easily and will give the same conclusion, which are omitted here.Table 2
The allowableδmax for different pair (ε1,ε2).
ε
2
=
0.1
ε
2
=
0.4
ε
2
=
0.6
ε
2
=
0.9
ε1=0.1
0.9450
0.9370
0.9340
0.9300
ε1=0.4
0.9000
0.8930
0.8900
0.8860
ε1=0.6
0.8760
0.8690
0.8650
0.8610
ε1=0.9
0.8440
0.8370
0.8340
0.8300Table 3
The allowableδmax for different pair (ε1,ε2).
ε
2
=
0.1
ε
2
=
0.4
ε
2
=
0.6
ε
2
=
0.9
ε1=0.1
0.9390
0.9330
0.9300
0.9260
ε1=0.4
0.8950
0.8900
0.8870
0.8830
ε1=0.6
0.8710
0.8660
0.8630
0.8590
ε1=0.9
0.8400
0.8350
0.8320
0.8280Table 4
The allowableδmax for different pair (ε1,ε2).
ε
2
=
0.1
ε
2
=
0.4
ε
2
=
0.6
ε
2
=
0.9
ε1=0.01
0.1960
0.1220
0.0870
0.0430
ε1=0.05
0.1690
0.1020
0.0690
0.0260
ε1=0.10
0.1410
0.0800
0.0480
0.0050
ε1=0.40
0.0390
—
—
—Table 5
The allowableδmax for different pair (εd1,εd2).
ε
d
2
=
0.1
ε
d
2
=
0.4
ε
d
2
=
0.6
εd2=0.9
εd1=0.1
0.9470
0.9420
0.9390
0.9350
εd1=0.4
0.9110
0.9080
0.9060
0.9030
εd1=0.6
0.8890
0.8860
0.8840
0.8820
εd1=0.9
0.8580
0.8550
0.8540
0.8510Table 6
The allowableδmax for different pair (εd1,εd2).
ε
d
2
=
0.1
ε
d
2
=
0.4
ε
d
2
=
0.6
εd2=0.9
εd1=0.1
0.9440
0.9390
0.9370
0.9330
εd1=0.4
0.9060
0.9020
0.9000
0.8960
εd1=0.6
0.8830
0.8790
0.8770
0.8730
εd1=0.9
0.8510
0.8470
0.8450
0.8420Table 7
The allowableδmax for different pair (εd1,εd2).
ε
d
2
=
0.1
ε
d
2
=
0.4
ε
d
2
=
0.6
εd2=0.9
εd1=0.01
0.1950
0.1490
0.1210
0.0850
εd1=0.05
0.1830
0.1380
0.1100
0.0730
εd1=0.10
0.1660
0.1220
0.0940
0.0570
εd1=0.40
0.0470
0.0110
—
—Figure 6
The simulation of correlation between pair (ε1,ε2) and δmax.Figure 7
The simulation of correlation between pair (ϵ1,ϵ2) and δmax.


## 5. Conclusion

In this paper, we have studied observer design problem of continuous-time Markovian jump systems with saturated output. First of all, a kind of state observer with output state saturation is proposed to be partially dependent. More precisely, both nondelay and delay sates are contained but occur asynchronously, whose probability distributions are embodied by the Bernoulli variable and taken into account in the observer design. By applying an improved inequality to deal with saturation terms, the existence conditions for observers with three kinds of output state saturations have been proposed with LMIs. Finally, a numerical example is used to testify the effectiveness and advantages of the presented methods.



---

*Source: 9572927-2018-01-22.xml*