# Finite-Time Stabilization for Discrete-Time Delayed Markovian Jump Systems with Partially Delayed Actuator Saturation

Authors: Guoliang Wang; Bo Feng
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

The finite-time control problem of discrete-time delayed Markovian jump systems with partially delayed actuator saturation is considered by a mode-dependent parameter approach. Different from the traditionally saturated actuators, a kind of saturated actuator being partially delay-dependent is firstly proposed, where both nondelay and delay states are included and occur asynchronously. Moreover, the probability distributions of such two terms are described by the Bernoulli variable and are taken into account in the controller design. Sufficient conditions for the existence of the desired controller are presented with LMIs. Finally, a numerical example is provided to show the effectiveness and superiority of the obtained results.


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## Body

## 1. Introduction

As we know, Markovian jump systems (MJSs) are very appropriate to describe dynamical systems experiencing changes in their structures or parameters randomly. Up to now, many important research topics of this kind of system, such as stability analysis [1–4], stabilization [5–7], H∞ control and filtering [8–11], output control [12, 13], state estimation [14, 15], adaptive control [16, 17], and synchronization [18, 19], have been considered. By investigating most results on the stability in the literature, it is seen that the classical Lyapunov stability guaranteeing the stability in an infinite-time interval was usually considered. Different from the classical Lyapunov stability concept, finite-time stability is defined as the system state that does not exceed a certain bound during a fixed finite-time interval. The introduction of such a stability concept is very necessary and important in many practical applications, where the related quantities need to fall into a range with specified bounds in a fixed-time interval. Up to now, many results, such as finite-time stability [20–23], stabilization [24, 25], finite-time H∞ control or filtering [26–28], and input-output finite-time stability and stabilization [29, 30], have emerged.On the other hand, many dynamical systems experience physical constraints, where actuator saturation is a common control problem because the amplitude of the actuator signal is limited by physical, technological, or even safety constraints. It is very known that actuator saturation could lead to poor performance of the closed-loop systems and even instability. Because of its practical and theoretical importance, stability analysis and synthesis for control systems with actuator or sensor saturation have received a lot of attention [31–36]. When the considered system is an MJS, some results have been proposed such as [37–41]. By investigating the above results on actuator saturation, it is found that they are mainly based on an estimate of the domain of attraction, where an equivalent description of saturation actuator is needed. However, the introduction of this transformation will make the complexity of computation very large, especially for an MJS with N operation modes. By introducing an inequality to enlarge some terms related to saturation [42], some delay-dependent stabilization results with less complexity were obtained in [43]. It is seen that the introduced inequality is mainly used in the derivative of Lyapunov function. Very recently, another different inequality was introduced in [44], which was based on an equivalent definition and was firstly used in the system description. By investigating such references, it is found that the designed state feedback controllers including delayed systems are only related to nondelay state. This means that the corresponding state should be available online. It will be impossible in some practical applications, such as systems communicating through an unreliable network, where the transmitted information is not obtained on time but has time delay sometimes. In other words, the event of data transmitted with or without time delay happens randomly. Moreover, it is known that one cannot say that either of the controllers only with nondelay or delay states is less conservative or not. It is natural and better to consider them together and design a kind of controller where both nondelay and delay states are contained but occur asynchronously. To the best of our knowledge, very few results are available to study the above problems. All the observations motivate the current research.In this paper, we address the finite-time stabilization for a class of discrete-time delayed Markovian jump systems, where the actuator saturation is partially delay-dependent. The main contributions of this paper are summarized as follows. (1) A new kind of saturated controller being partially delay-dependent is proposed, where both nondelay and delay states are included but happen randomly. Moreover, the related probability distribution is embodied by the Bernoulli variable and is taken into account in its design. (2) Sufficient conditions checking the finite-time stability of the underlying system are presented with LMI forms, where a mode-dependent parameter approach is exploited successfully. Particularly, some techniques in terms of additional inequalities and variables are proposed to obtain the LMI conditions ultimately. (3) Compared with the similar results about finite-time stabilization of MJSs with input saturation, the proposed methods have some advantages in terms of being solved easily and being less conservative. (4) The effects of time delay and actuator saturation are fully considered, which are important in system analysis and synthesis. Moreover, because of the results within LMI framework, they could be extended to other general cases directly.Notation. Rn denotes the n-dimensional Euclidean space, and Rm×n is the set of all m×n real matrices. E{·} means the mathematical expectation of {·}. · refers to the Euclidean vector norm or spectral matrix norm. Z+ is the set of positive integers. Ω is the sample space, F is the σ-algebras 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 discrete-time delayed Markovian jump system (DDMJS) subject to actuator saturation:(1)xk+1=Arkxk+Adrkxk-d+Brksatuk,xk=ϕk,where x(k)∈Rn is the state vector, u(k)∈Rq is the control input, and ϕ(k)∈Rn is the initial condition. Matrices A(rk), Ad(rk), and B(rk) are known matrices of compatible dimensions. {rk,k∈Z+} is a discrete-time Markov process taking values in a finite space S={1,2,…,N} with TRM Π≜(πij)∈RN×N given by(2)πij=Pr⁡rk+1=j∣rk=i,where 0≤πij≤1 and ∑j=1N‍πij=1 for all i∈S. Time delay d satisfies d>0. Function sat(·): Rq→Rq is the standard function defined as(3)satuik=satu1k⋯satuqkT,where satuik=signuikmin⁡{1,ui(k)}.In this paper, the designed state feedback controller is named to be a partially delay-dependent controller and is described by(4)uk=αkKrkxk+1-αkKdrkxk-d,where K(rk) and Kd(rk) are the control gains to be determined. Stochastic variable α(k) is an indicator and is defined as follows:(5)αk=1,ifxkisavailable0,ifxk-disavailable,which is satisfied as follows:(6)Prαk=1=Eαk=α,Prαk=0=1-α.Then, the resulting closed-loop system is rewritten to be(7)xk+1=Arkxk+Adrkxk-d+BrksatαkKrkxk+1-αkKdrkxk-d,xk=ϕk.Definition 1.
Given scalarsc2>0, D>0 and matrix Ri>0, DDMJS (1) with u(k)=0 is said to be stochastic finite-time stable with respect to Γi=(c1,c2,D,Ri), if there is a positive real scalar c1, possibly depending on c2, D, and Ri, such that (8)ExTk1Rixk1≤c1⟹ExTk2Rixk2≤c2,where k1∈{-d,-d+1,…,-1,0} and k2∈{1,2,…,D}.By lettingη=u-sat(u), it is known that(9)satuk=αkKrkxk+1-αkKdrkxk-d-αkηrk+1-αkηdrkwhich is equivalent to(10)satuk=αKrkxk+1-αKdrkxk-d+αk-αKrkxk-Kdrkxk-d-αηrk+1-αηdrk+αk-αηrk-ηdrk.Then, the resulting closed-loop system becomes(11)xk+1=Arkxk+Adrkxk-d+BrkαKrkxk+1-αKdrkxk-d+αk-αKrkxk-Kdrkxk-d-αηrk+1-αηdrk+αk-αηrk-ηdrk,xk=ϕk.Remark 2.
It is said that actuator saturation with description (4) is more general and has some advantages. Firstly, compared with input saturation [31, 36, 37, 39, 41, 44] where nondelay state should be available online, system state x(k) in description (4) is not necessary and could be replaced by delay state x(k-d). Secondly, compared with designed saturated controllers [35, 43] which were designed for delayed systems without delay states, both nondelay and delay states are included in the designed saturated controller. As we know, because all the delay terms affect the results, it is more conservative without considering them. Then, it is said that our formulation about the actuator saturation will have a larger application scope and less conservatism, since both related states are involved, and theirs distribution probabilities are also considered.Lemma 3 (see [44]).
Letη=u-sat(u). Then, there exists a real number ϵ∈(0,1) such that (12)ηTη≤ϵuTu,where η=η1⋯ηqT and ηi is the dead-zone nonlinearity function, i={1,2,…,q}.


## 3. Main Results

Theorem 4.
Given a saturation controller (9), the resulting closed-loop system (11) is finite-time stable with respect to Γi, if, for given positive scalars μi∈[1,∞), ϵi∈(0,1), and ϵdi∈(0,1), there exist Pi>0, Q>0, γi>0, δ1>0, δ2>0, and ν2>0, such that(13)Θi10Θi2Θi3Θi40∗-QΘi5Θi60Θi7∗∗-P¯i000∗∗∗-P¯i00∗∗∗∗-γiI0∗∗∗∗∗-γiI<0,(14)-γiIBiTP¯i∗-P¯i≤0,(15)δ1I<P^i<δ2I,(16)0<Q^i<ν2I,(17)μiDδ2+dν2c1≤δ1c2,where(18)Θi1=Q-μiPi,Θi2=2A¯iTP¯i,A¯i=Ai+αBiKi,P¯i=∑j=1NπijPjΘi3=2α1-αKiTBiTP¯i,Θi4=6α-4α2ϵiγiKiTΘi5=2A¯diTP¯i,A¯di=Adi+1-αBiKdi,Θi6=-2α1-αKdiTBiTP¯iΘi7=2α-4α2+2ϵdiγiKdiT,P^i=Ri-1/2PiRi-1/2,Q^i=Ri-1/2QRi-1/2.Proof.
Choose a stochastic Lyapunov function for system (11) as(19)Vk=xTkPrkxk+∑l=k-dk-1xTlQxl.Then, it is further obtained that(20)EVk+1=A¯ixk+A¯dixk-dTP¯iA¯ixk+A¯dixk-d+α1-αKixk-Kdixk-d-ηi-ηdiTBiTP¯iBiKixk-Kdixk-d-ηi-ηdi+αηi+1-αηdiTBiTP¯iBiαηi+1-αηdi-2A¯ixk+A¯dixk-dTP¯iBiαηi+1-αηdi+∑l=k+1-dkxTlQxl=fi1TP¯ifi1+α1-αfi2TBiTP¯iBifi2+fi3TBiTP¯iBifi3-2fi1TP¯iBifi3+∑l=k+1-dkxTlQxl≤2fi1TP¯ifi1+α1-αfi2TBiTP¯iBifi2+2fi3TBiTP¯iBifi3+∑l=k+1-dkxTlQxl,where(21)fi1=A¯ixk+A¯dixk-d,fi2=fi4-ηi-ηdi,fi3=αηi+1-αηdi,fi4=Kixk-Kdixk-d.Moreover, it is concluded that(22)α1-αfi2TBiTP¯iBifi2=α1-αfi4-ηi-ηdiTBiTP¯iBifi4-ηi-ηdi≤2α1-αfi4TBiTP¯iBifi4+ηi-ηdiTBiTP¯iBiηi-ηdi≤2α1-αfi4TBiTP¯iBifi4+4α1-αηiTBiTP¯iBiηi+ηdiTBiTP¯iBiηdi,2fi3TBiTP¯iBifi3=2αηi+1-αηdiTBiTP¯iBiαηi+1-αηdi≤2αηiTBiTP¯iBiηi+1-αηdiTBiTP¯iBiηdi.Based on this, it is computed that(23)ΔVk=EVk+1,rk+1=jrk=i-Vk≤2fi1TP¯ifi1+2α1-αfi4TBiTP¯iBifi4+6α-4α2ηiTBiTP¯iBiηi+2α-4α2+2ηdiTBiTP¯iBiηdi+∑l=k+1-dkxTlQxl-xTkPixk-∑l=k-dk-1xTlQxl.From condition (14), it is seen that(24)BiTP¯iBi≤γiI.By Lemma 3, we have(25)αkηiTηi+1-αkηdiTηdi≤ϵiαkKixkTKixk+ϵdi1-αkKdixk-dTKdixk-d.Based on these, it is got that (23) is implied by(26)ΔVk≤2A¯ixk+A¯dixk-dTP¯iA¯ixk+A¯dixk-d+2α1-αKixk-Kdixk-dTBiTP¯iBiKixk-Kdixk-d+6α-4α2γiϵixTkKiTKixk+2α-4α2+2γiϵdixTk-dKdiTKdixk-d+∑l=k+1-dkxTlQxl-xTkPixk-∑l=k-dk-1xTlQxl=ξTKQ-Pi00-Q+2A¯iTA¯diTP¯iA¯iTA¯diTT+2α1-αKiTBiT-KdiTBiTP-iKiTBiT-KdiTBiTT+6α-4α2ϵiγiKiTKi002α-4α2+2ϵdiγiKdiTKdiξk=ξTKQ-μiPi00-Q+2A¯iTA¯diTP¯iA¯iTA¯diTT+2α1-αKiTBiT-KdiTBiTP-iKiTBiT-KdiTBiTT+6α-4α2ϵiγiKiTKi002α-4α2+2ϵdiγiKdiTKdiξk+μi-1xTtPixt,where(27)ξTk=xTkxTk-d.After applying the Schur complementary lemma to (13), it is concluded that (26) implies(28)ΔVk=EVk+1-Vk≤μi-1xTkPixk≤μi-1Vk.Therefore, one has(29)EVk+1≤μiVk.It follows that(30)EVk≤μikEV0.Based on the definitions of P^i and Q^i, it is obtained that(31)EV0=ExT0Ri1/2P^iRi1/2x0+E∑-d-1xTlRi1/2Q^iRi1/2xl≤c1supi∈S⁡λmaxP^i+dc1supi∈S⁡λmaxQ^i,where d∈Z+. On the other hand, it is known from (19) that(32)EVk≥ExTkPixk=ExTkRi1/2P^iRi1/2xk≥infi∈S⁡λminP^iExTkRixk.Taking into account (30)–(32), we obtain that(33)infi∈S⁡λminP^iExTkRixk≤μikc1supi∈S⁡λmaxP^i+dsupi∈S⁡λmaxQ^i.Based on conditions (15)–(17), it is obtained that(34)ExTkRixk≤μiDδ2+dν2c1δ1≤c2,where k∈{1,2,…,D}. This completes the proof.Remark 5.
In this paper, it is seen that a mode-dependent parameter approach in terms ofΔV(k)≤(μi-1)V(k) is exploited to deal with the finite-time stability problem of DDMJS (1) with actuator saturation (9). Instead of choosing a common scalar μ for all system modes, the mode-dependent parameter μi is selected. Similar to the results in [23], it is known that the corresponding results will be less conservative. Moreover, in order to obtain LMI conditions, some necessarily additional variables and inequalities are introduced to be computed, which are not given beforehand.Remark 6.
From the proof of this theorem, it is seen that the obtained result is based on the constructed Lyapunov functional (19). It is said the selected Lyapunov functional is without loss of generality. This means that it could be replaced by an improved or another one, where more information of time delay or some double summable quadratic terms could be used in the construction of Lyapunov functional. It is common that less conservative results could be obtained, when more information about time delay is used. Moreover, in order to further reduce the conservatism, some improved techniques referred to as slack variable method and Jensen inequality approach may be applied here too. However, because of the so many terms coming from a complex Lyapunov functional in addition to exploiting improved techniques, the detailed steps should be revisited carefully. In other words, they cannot be applied directly or easily to study the proposed problems. For example, when a more complex Lyapunov functional and improved techniques are applied, more terms and variables will be introduced, which will make the obtained results have large computation complexity. Moreover, some additional problems will emerge. How to get the LMI conditions is one of them. In a word, all the factors will make the problems very complicated, which will be considered in our future work.Theorem 7.
Given a scalarα∈[0,1], there exists a saturation controller (9) such that the resulting system is finite-time stable with respect to Γi, if, for given positive scalars μi∈[1,∞), ϵi∈(0,1), and ϵdi∈(0,1), there exist Xi>0, Q¯>0, γ¯i>0, δ1>0, δ2>0, ν2>0, Ki and Kdi satisfying conditions (17) and(35)-μiXi0Ψi1Ψi2Ψi30Xi∗-2Xi+Q¯Ψi4Ψi50Ψi60∗∗-X¯0000∗∗∗-X¯000∗∗∗∗-γ¯iI00∗∗∗∗∗-γ¯iI0∗∗∗∗∗∗-Q¯<0,(36)-γ¯iIΦ∗-X¯≤0,(37)δ1I-2Ri-1/2+Xi<0,(38)-δ2IRi-1/2∗-Xi<0,(39)-ν2IRi-1/2∗-Q¯<0,where(40)Ψi1=2πi1XiAiT+α2πi1YiTBiT⋯2πiNXiAiT+α2πiNYiTBiT,Ψi4=2πi1XiAdiT+1-α2πi1YdiTBiT⋯2πiNXiAdiT+1-α2πiNYdiTBiT,Ψi2=2α-2α2πi1YiTBiT⋯2α-2α2πiNYiTBiT,Ψi5=-2α-2α2πi1YdiTBiT⋯-2α-2α2πiNYdiTBiT,Ψi3=6α-4α2ϵiYiT,Ψi6=2α-4α2+2ϵdiYdiT,Φ=πi1γ¯iBiT⋯πiNγ¯iBiT,X¯=diag⁡X1,…,XN.Then, the gains of saturation controller (9) are computed by(41)Ki=YiXi-1,Kdi=YdiXi-1.Proof.
From condition (26), it is known that(42)ΔVk≤ξTkQ-μiPi00-Q+2A¯iTA¯diTP¯i2A¯iTA¯diTT+2α1-αKiTBiT-KdiTBiTP-i2α1-αKiTBiT-KdiTBiTT+6α-4α2ϵiγiKiTKi002α-4α2+2ϵdiγiKdiTKdiξk+μi-1xTtPixt.By the Schur complement lemma, it is obtained that(43)Ψ¯i10Ψ¯i2Ψ¯i3Ψ¯i40∗-QΨ¯i5Ψ¯i60Ψ¯i7∗∗-X¯000∗∗∗-X¯00∗∗∗∗-γ¯iI0∗∗∗∗∗-γ¯iI<0,where(44)Ψ¯i1=Q-μiXi-1,Xi=Pi-1,Ψ¯i2=2πi1A¯iT⋯2πiNA¯iT,Ψ¯i5=2πi1A¯diT⋯2πiNA¯diT,Ψ¯i3=2α-2α2πi1KiTBiT⋯2α-2α2πiNKiTBiT,Ψ¯i6=-2α-2α2πi1KdiTBiT⋯-2α-2α2πiNKdiTBiT,Ψ¯i4=6α-4α2ϵiKiT,Ψ¯i7=2α-4α2+2ϵdiKdiT,γ¯i=γi-1.By pre- and postmultiplying its both sides with diag{Xi,Xi,I,I,I,I} and its transpose and taking into account (41), one gets(45)Ωi10Ψi1Ψi2Ψi30∗Ωi2Ψi4Ψi50Ψi6∗∗-X¯000∗∗∗-X¯00∗∗∗∗-γ¯iI0∗∗∗∗∗-γ¯iI<0,where(46)Ωi1=XiQXi-μiXi,Ωi2=-XiQXi.As for Ωi2, it is obvious that Ωi2<0 can be guaranteed by(47)-2Xi+Q¯<0,where Q¯=Q-1. By applying the Schur complement lemma again and considering condition (47), it is concluded that condition (35) implies (45). As for condition (24), it is equivalent to(48)-γ¯iI+γ¯i2BiT∑j=1NπijPjBi≤0,which is equal to (36). As for condition (15), it is equivalent to(49)δ1I-Ri-1/2PiRi-1/2<0,-δ2I+Ri-1/2PiRi-1/2<0,which could be implied by (37) and (38), respectively. Similarly, it is known that (16) will be guaranteed by (39). This completes the proof.Remark 8.
Because of probabilityα included obviously, it is said that it is important to the analysis and synthesis of systems with actuator saturation. When α=a, saturated controller (9) will be simplified to one in [44]. On the contrary, one will get a saturated controller only related to delay state. Based on these explanations, it is said that our results can be viewed as extension results on the saturation stabilization problem from deterministic systems without time delay to stochastic delay systems. However, it is seen that such a probability should be given exactly. It will be impossible or will be of high cost in some practical applications. Instead, we only have its estimation. In this case, there will be an uncertainty between the real and estimated values. As we know, such an uncertainty will degrade the system performance and even lead to system instability. Thus, it is necessary and meaningful to consider this general case. Because of the obtained results with LMI forms here, it is concluded that they could be extended to the above general case, where some existing methods such as [5, 10] will be used to deal with this uncertainty.Remark 9.
It is worth mentioning that the inequality in Lemma3 plays an essential role in dealing with the saturated terms introduced by the new partially delay-dependent controller. By investigating such an inequality, it is actually a nonlinear function satisfying a global Lipschitz condition. Recently, some new LMI-based sufficient conditions were proposed in [45, 46], which were obtained by applying LMI method and homeomorphism theory simultaneously. Moreover, it has been proved in these references that the extended global Lipschitz condition on the activation functions is less conservative than the traditionally global Lipschitz conditions. By considering the methods proposed in this paper and such two references together, some new LMI conditions may be proposed, which will have less conservatism. However, how to obtain the LMI conditions ultimately should be considered carefully. Many steps should be revisited, where some new problems may be encountered. All the related problems will be considered in our future work.


## 4. Numerical Examples

In this section, a numerical example is used to show the applicability and superiority of the proposed methods.Example 1.
Consider a discrete-time delayed Markovian jump system of form (1) having two modes, whose parameters are described as follows:
Mode 1:(50)A1=-0.60.30.20.1,Ad1=0.200.10.1,B1=-10.5,
Mode 2:(51)A2=0.20.40.1-0.3,Ad2=0.2-0.10.3-0.1,B2=0.21,where matrices R1>0 and R2>0 are given to be(52)R1=1001,R2=1001.The transition rate matrix is given by (53)Π=0.80.20.30.7.Without loss of generality, letting T=2, c1=1, c2=8, d=3, D=4, μ1=1.3, μ2=1.2, ϵ1=0.4, ϵ2=0.5, ϵd1=0.3, and ϵd2=0.4, by Theorem 7, we have the gains of saturated controller (9) with α=0.6 computed as(54)K1=-0.02980.0146,Kd1=0.0214-0.0013,K2=-0.0230-0.0026,Kd2=-0.01350.0031.Under the initial condition x0=0.2-0.2, after applying the above saturated controller, one has the simulation of the resulting closed-loop system given in Figure 1, where the smaller figure is the simulation of stochastic variable α(k) with α=0.6. Moreover, the curve of E{xT(k)x(k)} is simulated in Figure 2. Based on these simulations, it is said that the designed saturated controller with form (9) could make the resulting system stochastic finite-time stable with respect to Γi=(c1,c2,D,Ri), which also demonstrates the utility of the proposed methods.Figure 1
The simulation of the resulting closed-loop system.Figure 2
The curve ofE{xT(k)x(k)}.In order to further demonstrate the efficiency of the proposed methods and the correlation among such given parameters, without loss of generality, the above matrixA1 is replaced by (55)A1=-0.3+δ0.30.20.1,where δ is a scalar, and the other matrices are invariant. Firstly, we demonstrate the correlation between mode-dependent parameter μi and expectation α. Without loss of generality, by letting δ=-0.359 and from Theorem 7, one gets the allowable minimum α such that the resulting system could be stochastic finite-time stable by the saturated controller (9) under different mode-dependent pair (μ1,μ2), given in Table 1. By denoting the allowable minimum α as αmin and based on Table 1, we have the simulation of correlation between pair (μ1,μ2) and αmin given in Figure 3. In order to further demonstrate the correlation between pair (μ1,μ2) and α, the allowable maximum of μ2 along with μ1 under different given α is given in Table 2, where “—” denotes the notion that there is no solution to μ2. Then, the corresponding simulation is presented in Figure 4. As for this case, it is concluded that larger probability α will lead to less conservative results. This means that, for a given μ1≥1, the range of the allowable maximum of μ2 with larger α will be larger. Moreover, it is found that the difference of such an allowable range along with μ1 will be smaller even when α has different values. This phenomenon could be used to design a better saturated controller (9). Based on these simulations, it is found that mode-dependent parameters are important in system synthesis, which could make the results less conservative in terms of smaller αmin.Table 1
The allowable minimumα for different pair (μ1, μ2).
μ 2 = 1.0 μ 2 = 1.1 μ 2 = 1.2 μ 2 = 1.28 μ 1 = 1.0 0.84 0.83 0.88 0.98 μ 1 = 1.1 0.71 0.70 0.74 0.81 μ 1 = 1.2 0.60 0.60 0.60 0.67 μ 1 = 1.28 0.55 0.54 0.54 0.55Table 2
The allowable maximumμ2 along with μ1 for different α.
α = 0.6 α = 0.7 α = 0.8 α = 0.9 α = 1 μ 1 = 1.00 — — — 1.22 1.29 μ 1 = 1.03 — — 1.13 1.27 1.30 μ 1 = 1.06 — — 1.21 1.29 1.30 μ 1 = 1.10 — 1.15 1.29 1.30 1.30 μ 1 = 1.15 — 1.24 1.30 1.30 1.30 μ 1 = 1.20 1.20 1.30 1.29 1.30 1.30 μ 1 = 1.28 1.30 1.30 1.29 1.30 1.30Figure 3
The simulation of correlation between pair (μ1, μ2) and αmin.Figure 4
The curves ofμ2 along with μ1 under different α.Secondly, we illustrate the effect of expectationα on finite-time stabilization by a partially delay-dependent controller with saturation (3). Under the selected parameters same as the above ones, the allowable maximum and minimum of δ denoted as δmax and δmin, respectively, could be obtained from Theorem 7, which are given in Table 3. Based on this table, we could get the curves of parameters δmax, δmin, and δran≜δmax-δmin along with α simulated in Figure 5. From this simulation, it is seen that probability α has a positive effect on this case in terms of δran having a larger value.Table 3
The allowable maximum and minimumδ for different α.
α 0 0.3 0.5 0.8 1 δ m a x 0.779 0.769 0.758 0.768 0.798 δ m i n −0.245 −0.298 −0.350 −0.470 −0.565Figure 5
The curves ofδmax, δmin, and δran along with α.Now, we consider the effects of parametersϵi and ϵdi on δran. First of all, the value of δran along with different pair (ϵ1,ϵ2) is studied. Based on Theorem 7 where the other parameters are invariant, δran under different pair (ϵ1,ϵ2) could be got, listed in Table 4. Then, the correlation between pair (ϵ1,ϵ2) and δran is simulated in Figure 6. Similarly, the corresponding results and simulations about pair (ϵd1,ϵd2) and δran are presented in Table 5 and Figure 7. Based on these simulations, it is said that different values of such parameters have different effects and could make the results have different conservatism in terms of δran being larger or not. For this case, it is seen that smaller values of (ϵ1,ϵ2) and (ϵd1,ϵd2) will lead to less conservative results in terms of larger δran.Table 4
The allowable rangeδran for different pair (ϵ1,ϵ2).
ϵ 2 = 0.1 ϵ 2 = 0.4 ϵ 2 = 0.6 ϵ 2 = 0.9 ϵ 1 = 0.1 1.2300 1.2300 1.2300 1.3050 ϵ 1 = 0.4 1.1410 1.1410 1.1410 1.1410 ϵ 1 = 0.6 1.0900 1.0890 1.0890 1.0890 ϵ 1 = 0.9 1.0430 1.0430 1.0430 1.0430Table 5
The allowable rangeδran for different pair (ϵd1,ϵd2).
ϵ d 2 = 0.1 ϵ d 2 = 0.4 ϵ d 2 = 0.6 ϵ d 2 = 0.9 ϵ d 1 = 0.1 1.1750 1.1700 1.1740 1.1650 ϵ d 1 = 0.4 1.1300 1.1290 1.1290 1.1280 ϵ d 1 = 0.6 1.1110 1.1110 1.1110 1.1110 ϵ d 1 = 0.9 1.0880 1.0880 1.0880 1.0870Figure 6
The simulation of correlation between pair (ϵ1,ϵ2) and δran.Figure 7
The simulation of correlation between pair (ϵd1,ϵd2) and δran.Finally, in order to further demonstrate the utility and superiority of the proposed partially delayed saturated controller (9), some additional comparisons will be done. Based on the above results, it is seen that larger α could lead to less conservative results. In other words, time delay d in controller (9) has a negative effect. In this sense, it is said that the traditionally saturated controller without delay designed for delay systems such as [35, 43] will be better in terms of less conservatism. However, it is assumed that system state x(k) of such controllers should be available online. It will be impossible or will be of high cost to satisfy this assumption. On the contrary, saturated controller (9) proposed here can be realized without this assumption. In this sense, it is said that the established method has a larger application scope. On the other hand, it should be pointed out that for a delay system it is not a deterministic conclusion that a saturated controller without delay is better than one with delay. Without loss of generality, when matrix A2 is selected to be (56)A2=0.20.4620.1-0.3and the other parameters with δ=0 are invariant, by Theorem 7, it is concluded that there is a solution to partially delayed saturated controller (9) only if α∈[0,0.364]. For this case, it is seen that larger α will lead to more conservative results in terms of no solvable solution. Particularly, when α=1, partially delayed saturated controller (9) will be reduced to be a traditionally saturated controller which could be computed by [44] and is more conservative. However, from this example with different A2, it is said that time delay will have a positive effect, which will result in less conservative results. All the comparisons directly show the necessity and superiority of the proposed partially delayed saturated controller.


## 5. Conclusion

In this paper, we have studied the finite-time stabilization problem of discrete-time delayed Markovian jump systems, where the desired saturated controller is partially delay-dependent. Firstly, a finite-time stability condition is presented by exploiting a mode-dependent parameter approach. Then, sufficient existence conditions for the partially delay-dependent saturated controller are presented with LMIs, which could be solved easily and directly. The correlations between time delay and mode-dependent parameters are considered in detail. Finally, a numerical example is used to demonstrate the effectiveness and advantages of the proposed methods.



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*Source: 1304379-2016-11-27.xml*