# Finite-TimeH∞ Control for a Class of Discrete-Time Markov Jump Systems with Actuator Saturation via Dynamic Antiwindup Design

Authors: Junjie Zhao; Jing Wang; Bo Li
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We deal with the finite-time control problem for discrete-time Markov jump systems
subject to saturating actuators. A finite-state Markovian process is given to govern the transition of
the jumping parameters. A controller designed for unconstrained systems combined with a dynamic antiwindup
compensator is given to guarantee that the resulting system is mean-square locally asymptotically
finite-time stabilizable. The proposed conditions allow us to find dynamic anti-windup compensator which
stabilize the closed-loop systems in the finite-time sense. All these conditions can be expressed in the form
of linear matrix inequalities and therefore are numerically tractable, as shown in the example included in
the paper.


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## Body

## 1. Introduction

It is well known that more and more attention has been paid to the study of actuator saturation due to its practical and theoretical importance. Therefore, various approaches were investigated to handle systems with actuator saturation and dynamic antiwindup approach which is one of the most effective ways to deal with it. To this end, a great number of results have been reported in the literature; see, for example, [1, 2]. Furthermore, the stabilization problem of singular Markovian jump systems with discontinuities and saturation inputs was presented in [3]. Via dynamic antiwindup fuzzy design, the robust stabilization problem of state delayed T-S fuzzy systems with input saturation was proposed in [4].On the other hand, Markov jump is frequently encountered in many practical systems. Therefore, the study of Markov jump systems has been a hot research topic due to its importance, and many results have been proposed based on various control techniques, such as robust control [5–9], H∞ control [10, 11], Passivity-based control [12–14], fuzzy dissipative control [15], and neural networks control [14, 16]. Furthermore, observer based finite-time H∞ control problem of discrete-time Markov jump systems was studied [17].As it is well known, when dealing with the stability of s system, a distinction should have been made between classical Lyapunov stability and finite-time stability (FTS). Conversely, a system is said to be finite-time stable if, once we fix a time-interval, its state does not exceed some bounds during this time-interval. Some results on FTS have been carried out; see, [18, 19]. Furthermore, finite-time H∞ filtering problem of time-delay stochastic jump systems with unbiased estimation was proposed in [20]. By applying dynamic observer-based state feedback and the Lyapunov-Krasovskii functional approach, the finite-time H∞ control problem for time-delay nonlinear jump systems was addressed in the work of He and Liu [21]. However, to the best of our knowledge, the problem of finite-time stabilization of discrete-time stochastic systems subject to actuator saturation has not been fully investigated and it is the main purpose of our study.In this paper, the attention is focused on the finite-timeH∞ control problem of discrete-time Markov jump systems with actuator saturation based on dynamic antiwindup approach. A controller designed for unconstrained systems combined with a dynamic antiwindup compensator is given to ensure the stochastic finite-time boundedness and stochastic finite-time stabilization of the resulting closed-loop system for all admissible disturbances. The desired compensator can be designed via solving a convex optimization problem. Finally, a numerical example is employed to show the effectiveness of the proposed method.Notation. Throughout the paper, for symmetric matrices X and Y, the notation X≥Y (resp., X>Y) means that the matrix X-Y is positive semidefinite (resp., positive definite). I is the identity matrix with appropriate dimension. The notation NT represents the transpose of the matrix N;λmax⁡(M) (resp., λmin⁡(M)) means the largest (resp., smallest) eigenvalue of the matrix M; (Ω,ℱ,𝒫) is a probability space; Ω is the sample space; ℱ is the σ-algebra of subsets of the sample space and 𝒫 is the probability measure on ℱ;ℰ{·} denotes the expectation operator with respect to some probability measure 𝒫. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The symbol * is used to denote a matrix which can be inferred by symmetry. He{A}=AT+A.


## 2. Preliminaries and Problem Description

Consider the following discrete-time Markov jump system(Σ) in the probability space (Ω,ℱ,𝒫):
(1)xp(k+1)=Ap(r(k))xp(k)+Bp,u(r(k))sat⁡(u(k))+Bp,w(r(k))w(k),y(k)=Cp,y(r(k))xp(k)+Dp,yw(r(k))w(k),z(k)=Cp,z(r(k))xp(k)+Dp,zu(r(k))sat⁡(u(k))+Dp,zw(r(k))w(k),
where xp(k)∈ℝnp is the state vector, u(k)∈ℝnu is the control input, and sat⁡(u(k))∈ℝnu is the saturated control input. w(k)∈L2p[0+∞) is the external disturbances, y(k)∈ℝny is the measurement output, and z(k)∈ℝnz is the performance output. {r(k)} is a discrete-time Markov process and takes values from a finite set S={1,2,…,𝒩} with transition probabilities given by
(2)Pr(rk+1=j∣rk=i)=πij,
where πij≥0, for ∀j,i∈S, and ∑j∈S‍πij=1. Moreover, the transition rates matrix of the system (Σ) is defined by
(3)[π11π12⋯π1𝒩π21π22⋯π2𝒩⋮⋮⋱⋮π𝒩1π𝒩2⋯π𝒩𝒩].
The plant inputs are supposed to be bounded as follows:
(4)-u0(k)≤u(k)≤u0(k),u0(k)>0,k=1,…,m.
For the system (Σ), to simplify the notation, we denote Api=Ap(r(k)) for each r(k)=i∈S, and the other symbols are similarly denoted. Assume that a linear controller is designed for any r(k)=i∈S; then,
(5)xc(k+1)=Acixc(k)+Bcy,iy(k)+Bcw,iw(k)+v1,yc(k)=Ccixc(k)+Dcy,iy(k)+Dcw,iw(k)+v2,
where xc(k)∈ℝnc is the controller state and yc(k)∈ℝnu is the controller output; v1 and v2 will be used for antiwindup augmentation. In absence of actuator saturation, the unconstrained closed-loop is formed by setting the following:
(6)u=yc,v1=0,v2=0.Assumption 1.
The unconstrained closed-loop system (1)–(5) is well posed and internally stable.In the presence of actuator saturation, the relation betweenu and yc is that u=sat⁡(yc). To minimize performance degradation caused by saturation, the following antiwindup compensator is designed for the closed-loop systems:
(7)xaw(k+1)=Aaw,ixaw(k)+Baw,iψ(yc(k)),v(k)=Caw,ixaw(k)+Daw,iψ(yc(k)),
where ψ(yc(k))=sat⁡(yc(k))-yc(k). The resulting nonlinear closed-loop system (1), (5), (7) is depicted in Figure 1 and can be represented in the following compact form:
(8)x(k+1)=Aix(k)+Bqiψ(yc(k))+Bwiw(k),yc(k)=Kix(k)+Kϕ,iψ(yc(k))+Kwiw(k),z(k)=Czix(k)+Dzq,iψ(yc(k))+Dzw,iw(k),
where
(9)Ai=[Api+Bpu,iDcy,iCpy,iBpu,iCciBpu,iI2Caw,iBcy,iCpy,iAciI1Caw,i00Aaw,i],Bqi=[Bpu,iI2Daw,i+Bpu,iI1Daw,iBaw,i],Bwi=[Bpw,i+Bpu,iDcy,iDp,yw,i+Bpu,iDcw,iBcy,iDp,yw,i+Bcw,i0],Czi=[Cpz,i+Dp,zu,iDciCpy,iDp,zu,iI2Caw,i],Dzq,i=I2Daw,i+Dp,zu,i,Dzw,i=Dp,zw,i+Dp,zu,iDcw,i+Dp,zu,iDciDp,yw,i,Ki=[Dcy,iCpy,iCciI2Caw,i],Kϕ,i=I2Daw,i,Kw,i=Dcw,i+Dp,yw,i,I1=[I0],I2=[0I].
For this system, we introduce the following definitions and assumption.Figure 1
The closed-loop systems with input saturation.Assumption 2 (see [17]).
The external disturbancew(k) is varying and satisfies the following constraint condition:
(10)∑k=0Tw(k)Tw(k)≤d,d≥0.Definition 3 (see [17]).
The resulting closed-loop system (8) is stochastic finite-time stable (SFTB) with respect to (δx,ϵ,Ri,N,d) with 0<δx<ϵ,Ri>0, and N∈Zk≥0, if
(11)E{xT(0)Rix(0)}≤δx2⟹E{xT(k)Rix(k)}<ϵ2,∀k∈{1,2,…,N}.Definition 4 (see [17]).
The resulting closed-loop system (8) is said to be stochastic H∞ finite-time stable with respect to (δx,ϵ,Ri,N,γ,d) with 0<δx<ϵ, Ri>0, γ>0, and N∈Zk≥0, if the system (8) is SFTB with respect to (δx,ϵ,Ri,N,γ,d), and under the zero-initial condition, the output z(k) satisfies
(12)E{∑j=0NzT(j)z(j)}≤γ2E{∑j=0NwT(j)w(j)},
for any nonzero w(k) which satisfies (10), where γ is a prescribed positive scalar.


## 3. Main Results

In this section, we investigate the stabilization analysis of the unconstrained systems and the antiwindup controller design of the resulting closed-loop system. Some sufficient conditions in terms of LMI are given. Before presenting the main results, we give some lemmas as follows.Lemma 5 (see [4]).
For the closed-loop systems (8) with the matrix Ki, the appropriate matrix Li∈ℝm×n is given, if x(k) is in the set D(uo), where D(uo) is defined as follows:
(13)D(uo)={x(k)∈ℝn;-u0(k)≤(Ki(k)+Li(k))x(t)≤u0(k),aaau0(k)>0,k=1,…,m},
then for any diagonal positive matrix T∈ℝm×m, one has the following:
(14)ψ(u(k))TT(ψ(u(k))-Lix(k)+Kϕ,iψ(yc(k))+Kwiw(k))≤0.Lemma 6 (see [12]).
For the given symmetric matrixS∈ℝ(n+m)×(n+m),
(15)S=[S11S12S12TS22],
where S11∈ℝn×n, S12∈ℝn×m, and S22∈ℝm×m, the following conditions are equivalent: (1)
S<0;(2)
S11<0,S22-S12TS11-1S12<0;(3)
S22<0,S11-S12S22-1S12T<0.


### 3.1. Design of Controller

In this section, we design the controller for the unconstrained systems withv1=0 and v2=0. Combining system (1) with controller (5), we have
(16)x(k+1)=Aix(k)+Bwiw(k),z(k)=Czix(k)+Dzw,iw(k),
where
(17)Ai=[Api+Bpu,iDcy,iCpy,iBpu,iCciBcy,iCpy,iAci],Bwi=[Bpw,i+Bpu,iDcy,iDp,yw,i+Bpu,iDcw,iBcy,iDp,yw,i+Bcw,i].Theorem 7.
For eachr(k)=i∈S, the unconstrained system (16) is SFTB with respect to (δx,ϵ,Ri,N,d) with 0<δx<ϵ, if there exist scalars μ≥0, σ1≥0, σ2≥0, and the given λ>0, two sets of mode-dependent symmetric positive-defined matrices {Xi,i∈S} and {Qi,i∈S}, such that the following conditions hold:
(18)[-μλI0L1iT*-QiL2iT**-W]<0,(19)[σ2d2-μ-Nϵ2*δx-σ1]<0,(20)λXi<I,(21)σ1Ri-1<Xi<Ri-1,(22)0<Qi<σ2I,
where
(23)W=diag⁡{X1,X2,…,Xn},L-1iT=[πi1AiTπi2AiT⋯πinAiT],L2iT=[πi1BwiTπi2BwiT⋯πinBwiT].Proof.
Define the following Lyapunov function for eachδ(t)=i∈S:
(24)V(k)=x(k)TPix(k).
It is readily obtained that
(25)E{V(k+1)}=E{∑j=1nπijx(k+1)TPjx(k+1)}=ξ(k)T[L1iL2i]TW-[L1iL2i]ξ(k),
where
(26)ξ(k)=[x(k)Tw(k)T],W-=diag⁡{P1,P2,…,Ph}.
By using of Schur complement lemma to (18), and note that Pi-1=Xi and λXi<I, we derive λI<Pi; then, we have
(27)ξ(k)T[-μλI0*-Qi]ξ(k)+ξ(k)T[L1iL2i]TW¯[L1iL2i]ξ(k)<0.
It follows that
(28)E{V(k+1)}<μx(k)TPix(k)+w(k)TQiw(k).
It is shown that
(29)E{V(k+1)}<μV(k)+sup⁡(i∈S)⁡{λmax⁡(Qi)}w(k)Tw(k).
Then we have
(30)E{V(k+1)}<μE{V(k)}+sup⁡(i∈S)⁡{λmax⁡(Qi)}E{w(k)Tw(k)}.
Since μ≥1, it is easily found that
(31)E{V(k+1)}<μE{V(0)}+sup⁡(i∈S)⁡{λmax⁡(Qi)}×E{∑j=0k-1‍μk-j-1w(j)Tw(j)}≤μkE{V(0)}+sup⁡(i∈S)⁡{λmax⁡(Qi)}μkd2.
Letting
(32)P-i=Ri-1/2PiRi-1/2,
and noting that
(33)E{xT(0)Rix(0)}≤δx2,
it can be verified that
(34)E{V(0)}=E{xT(0)Pix(0)}=E{xT(0)Ri1/2P-iRi1/2x(0)}≤sup⁡i∈S⁡{λmax⁡(P-i)}E{xT(0)Rix(0)}≤sup⁡i∈S⁡{λmax⁡(P-i)}δx2.
Similarly, for all i∈S, we can obtain
(35)E{V(k)}=E{xT(k)Pix(k)}=E{xT(k)Ri1/2P-iRi1/2x(k)}≥inf⁡i∈S{λmin⁡}(P-i)E{xT(k)Rix(k)}.
Then, it is not difficult to find that
(36)E{xT(k)Rix(k)}<sup⁡i∈S{λmax⁡(P-i)}μkδx2+sup⁡(i∈S){λmax⁡(Qi)}μkd2inf⁡i∈S{λmin⁡}(P-i),
which implies that
(37)sup⁡i∈S{λmax⁡(P-i)}μkδx2+sup⁡(i∈S){λmax⁡(Qi)}μkd2inf⁡i∈S{λmin⁡}(P-i)<ϵ2.
Then, one can obtain that
(38)sup⁡i∈S{λmax⁡(P-i)}δx2+sup⁡(i∈S){λmax⁡(Qi)}d2<inf⁡i∈S{λmin⁡}(P-i)μ-Nϵ2.
Setting
(39)Xi=Pi-1,σ1Ri-1<Xi<Ri-1,0<Qi<σ2I,
it is easy to see that
(40)σ1-1δx2+σ2d2<μ-Nϵ2.
It is obvious that (40) is equivalent to (19).
This completes the proof.


### 3.2. Design of Dynamic Antiwindup Compensator

Theorem 8.
For eachr(k)=i∈S, with antiwindup compensator (7), such that the resulting closed-loop system (10) is SFTB with respect to (δx,ϵ,Ri,N,d) with 0<δx<ϵ, if there exist scalars μ≥0, σ1≥0, and σ2≥0, three sets of mode-dependent symmetric positive-defined matrices {Xi,i∈S}, {Qi,i∈S} and diag positive-defined matrices {Si,i∈S}, and two sets of mode-dependent matrices {Yi,i∈S} and {L-i=LiXi,A-aw,i=Aaw,iXi,C-aw,i=Caw,iXi,B-aw,i=Baw,iSi,D-aw,i=Daw,iSii∈S}, such that the following conditions hold:
(41)[-μXi0L-iTL-1iT*-QiKwiTL2iT**-2Si-He(K-φ,i)L-3iT***-W]<0,(42)[σ2d2-μ-Nϵ2*δx-σ1]<0,(43)[Xi*K-i+L-iu0(k)2]>0,k=1,…,m,(44)σ1Ri-1<Xi<Ri-1,(45)0<Qi<σ2I,
where
(46)W=diag⁡{X1,X2,…,Xn},L-1iT=[πi1A-iTπi2A-iT⋯πinA-iT],L2iT=[πi1BwiTπi2BwiT⋯πinBwiT],L3iT=[πi1B-qiTπi2B-qiT⋯πinB-qiT],
with
(47)A-i=[ApiXi+Bpu,iDcy,iCpy,iXiBpu,iCciXiBpu,iI2C-aw,iBcy,iCpy,iXiAciXiI1C-aw,i00A-aw,i],B-qi=[Bpu,iI2D-aw,i+Bpu,iXiI1D-aw,iB-aw,i],K-i=[Dcy,iCpy,iXiCciXiI2C-aw,i],K-ϕ,i=I2D-aw,i.Proof.
Define the following Lyapunov function for eachδ(t)=i∈S:
(48)V(k)=x(k)TPix(k).
It is readily obtained that
(49)E{V(k+1)}=E{∑j=1nπijx(k+1)TPjx(k+1)}=ξ(k)T[L1iL2iL3i]TW-[L1iL2iL3i]ξ(k),
where
(50)ξ(k)=[x(k)Tw(k)Tψ(k)T],W-=diag⁡{P1,P2,…,Ph},L1iT=[πi1AiTπi2AiT⋯πinAiT],L2iT=[πi1BwiTπi2BwiT⋯πinBwiT],L3iT=[πi1BqiTπi2BqiT⋯πinBqiT].
Then, by pre- and postmultiplying (41) by diag⁡{Pi,I,Ti,I} with Pi=Xi-1,Ti=Si-1, we have
(51)[-μPi0LiTTiL1iT*-QiKwiTL2iT**-2Ti-He(TiKϕ,i)L3iT***-W]<0.
By using of Schur complement lemma, we derive
(52)ξ(k)T[-μPi0LiTTi*-QiKwiT**-2Ti-He(TiKϕ,i)]ξ(k)+ξ(k)T[L1iL2iL3i]TW-[L1iL2iL3i]ξ(k)<0.
It follows that
(53)E{V(k+1)}<μx(k)TPix(k)+w(k)TQiw(k)+ψ(k)T(2Ti+He{TiKϕ,i})ψ(k)-2ψ(k)TTiLix(k)+2ψ(k)TTiKwiw(k).
Since ψ(k)T(2Ti+He{TiKϕ,i})ψ(k)-2ψ(k)TTiLix(k)+2ψ(k)TTiKwiw(k)≤0, we get
(54)E{V(k+1)}<μx(k)TPix(k)+w(k)TQiw(k).
It is shown that
(55)E{V(k+1)}<μV(k)+sup⁡(i∈S)⁡{λmax⁡(Qi)}w(k)Tw(k).
Then, we have
(56)E{V(k+1)}<μE{V(k)}+sup⁡(i∈S){λmax⁡(Qi)}E{w(k)Tw(k)}.
Since μ≥1, it is easily found that
(57)E{V(k+1)}<μE{V(0)}+sup⁡(i∈S){λmax⁡(Qi)}×E{∑j=0k-1‍μk-j-1w(j)Tw(j)}≤μkE{V(0)}+sup⁡(i∈S){λmax⁡(Qi)}μkd2.
The following proof is similar to the process of Theorem 7.
Based on Lemma5, it is easy to obtain that
(58)[Pi*Ki+Liu0(k)2]>0,k=1,…,m,
then pre- and post-multiply (58) by diag⁡{Xi,I} which implies (43). This completes the proof.Theorem 9.
For eachr(k)=i∈S, with antiwindup compensator (7), such that the resulting closed-loop system (10) is said to be Stochastic H∞ finite-time stable via state feedback with respect to (δx,ϵ,Ri,N,γ,d), if there exist three scalars μ≥0, σ1≥0, and γ≥0, two sets of mode-dependent symmetric positive-defined matrices {Xi,i∈S} and diag matrices {Si,i∈S}, and two sets of mode-dependent matrices {Yi,i∈S} and {L-i=LiXi,i∈S}, such that the following conditions hold:
(59)[-μXi0L-iTL-1iTC-ziT*-γ2μ-NIKwiTL2iTDzw,iT**-2Si-He(K-ϕ,i)L-3iTD-zq,iT***-W0****-I]<0,(60)[μ-N(d2γ2-ϵ2)*δx-σ1]<0,(61)[Xi*K-i+L-iu0(k)2]>0,k=1,…,m,(62)σ1Ri-1<Xi<Ri-1,
with
(63)C-zi=[(Cpz,i+Dp,zu,iDciCpy,i)XiDp,zu,iXiI2C-aw,i],D-zq,iT=I2D-aw,i+Dp,zu,i.Proof.
Choose the similar Lyapunov function as Theorem7 and denote
(64)Π(x(k),w(k),r(k)=i)=E{V(k+1)}-μV(k)+z(k)Tz(k)-γ2μ-Nw(k)Tw(k).
Thus, in light of Lemma 5, we have
(65)Π(x(k),w(k),rk=i)≤ξ(k)T[L1iL2iL3i]TW-[L1iL2iL3i]ξ(k)+ξ(k)T[CziDzw,iDzq,i]T[CziDzw,iDzq,i]ξ(k)+ξT(k)[-μXi0LiTTi*-γ2μ-NIKwiT**-2Ti-He{TiKϕ,i}]ξ(k).
Then pre- and postmultiply (59) by diag⁡{Pi,I,Ti,I}, and considering Schur complement lemma and (65), we derive that
(66)Π(x(k),w(k),r(k)=i)<0
holds for all rk=i∈S. According to (66), one can obtain that
(67)E{V(k+1)}<μE{V(k)}-E{z(k)Tz(k)}+γ2μ-NE{w(k)Tw(k)}.
Then, we have
(68)E{V(k)}<μkE{V(0)}-∑j=0k-1μk-j-1E{z(j)Tz(j)}+γ2μ-NE{∑j=0k-1μk-j-1w(j)Tw(j)}.
Under the zero-value initial condition and noting that V(k)≥0, for all K∈Zk≥0, it is shown that
(69)∑j=0k-1μk-j-1E{z(j)Tz(j)}<γ2μ-NE{∑j=0k-1μk-j-1w(j)Tw(j)}.
Since μ≥1 and from (69), we have
(70)E{∑j=0Nz(j)Tz(j)}=∑j=0NE{z(j)Tz(j)}≤∑j=0NE{μN-jz(j)Tz(j)}≤γ2μ-NE{∑j=0NμN-jw(j)Tw(j)}≤γ2E{∑j=0Nw(j)Tw(j)}.
The following proof is similar to the process of Zhang and Liu [17].
Sinceε(Pi,1)⊂D(u0), it follows that
(71)[Pi*Ki+Liu0(k)2]>0,k=1,…,m,
and then pre- and post-multiply (71) by diag⁡(Xi,I) and its transpose, respectively; we derive condition (61). This completes the proof.


## 3.1. Design of Controller

In this section, we design the controller for the unconstrained systems withv1=0 and v2=0. Combining system (1) with controller (5), we have
(16)x(k+1)=Aix(k)+Bwiw(k),z(k)=Czix(k)+Dzw,iw(k),
where
(17)Ai=[Api+Bpu,iDcy,iCpy,iBpu,iCciBcy,iCpy,iAci],Bwi=[Bpw,i+Bpu,iDcy,iDp,yw,i+Bpu,iDcw,iBcy,iDp,yw,i+Bcw,i].Theorem 7.
For eachr(k)=i∈S, the unconstrained system (16) is SFTB with respect to (δx,ϵ,Ri,N,d) with 0<δx<ϵ, if there exist scalars μ≥0, σ1≥0, σ2≥0, and the given λ>0, two sets of mode-dependent symmetric positive-defined matrices {Xi,i∈S} and {Qi,i∈S}, such that the following conditions hold:
(18)[-μλI0L1iT*-QiL2iT**-W]<0,(19)[σ2d2-μ-Nϵ2*δx-σ1]<0,(20)λXi<I,(21)σ1Ri-1<Xi<Ri-1,(22)0<Qi<σ2I,
where
(23)W=diag⁡{X1,X2,…,Xn},L-1iT=[πi1AiTπi2AiT⋯πinAiT],L2iT=[πi1BwiTπi2BwiT⋯πinBwiT].Proof.
Define the following Lyapunov function for eachδ(t)=i∈S:
(24)V(k)=x(k)TPix(k).
It is readily obtained that
(25)E{V(k+1)}=E{∑j=1nπijx(k+1)TPjx(k+1)}=ξ(k)T[L1iL2i]TW-[L1iL2i]ξ(k),
where
(26)ξ(k)=[x(k)Tw(k)T],W-=diag⁡{P1,P2,…,Ph}.
By using of Schur complement lemma to (18), and note that Pi-1=Xi and λXi<I, we derive λI<Pi; then, we have
(27)ξ(k)T[-μλI0*-Qi]ξ(k)+ξ(k)T[L1iL2i]TW¯[L1iL2i]ξ(k)<0.
It follows that
(28)E{V(k+1)}<μx(k)TPix(k)+w(k)TQiw(k).
It is shown that
(29)E{V(k+1)}<μV(k)+sup⁡(i∈S)⁡{λmax⁡(Qi)}w(k)Tw(k).
Then we have
(30)E{V(k+1)}<μE{V(k)}+sup⁡(i∈S)⁡{λmax⁡(Qi)}E{w(k)Tw(k)}.
Since μ≥1, it is easily found that
(31)E{V(k+1)}<μE{V(0)}+sup⁡(i∈S)⁡{λmax⁡(Qi)}×E{∑j=0k-1‍μk-j-1w(j)Tw(j)}≤μkE{V(0)}+sup⁡(i∈S)⁡{λmax⁡(Qi)}μkd2.
Letting
(32)P-i=Ri-1/2PiRi-1/2,
and noting that
(33)E{xT(0)Rix(0)}≤δx2,
it can be verified that
(34)E{V(0)}=E{xT(0)Pix(0)}=E{xT(0)Ri1/2P-iRi1/2x(0)}≤sup⁡i∈S⁡{λmax⁡(P-i)}E{xT(0)Rix(0)}≤sup⁡i∈S⁡{λmax⁡(P-i)}δx2.
Similarly, for all i∈S, we can obtain
(35)E{V(k)}=E{xT(k)Pix(k)}=E{xT(k)Ri1/2P-iRi1/2x(k)}≥inf⁡i∈S{λmin⁡}(P-i)E{xT(k)Rix(k)}.
Then, it is not difficult to find that
(36)E{xT(k)Rix(k)}<sup⁡i∈S{λmax⁡(P-i)}μkδx2+sup⁡(i∈S){λmax⁡(Qi)}μkd2inf⁡i∈S{λmin⁡}(P-i),
which implies that
(37)sup⁡i∈S{λmax⁡(P-i)}μkδx2+sup⁡(i∈S){λmax⁡(Qi)}μkd2inf⁡i∈S{λmin⁡}(P-i)<ϵ2.
Then, one can obtain that
(38)sup⁡i∈S{λmax⁡(P-i)}δx2+sup⁡(i∈S){λmax⁡(Qi)}d2<inf⁡i∈S{λmin⁡}(P-i)μ-Nϵ2.
Setting
(39)Xi=Pi-1,σ1Ri-1<Xi<Ri-1,0<Qi<σ2I,
it is easy to see that
(40)σ1-1δx2+σ2d2<μ-Nϵ2.
It is obvious that (40) is equivalent to (19).
This completes the proof.


## 3.2. Design of Dynamic Antiwindup Compensator

Theorem 8.
For eachr(k)=i∈S, with antiwindup compensator (7), such that the resulting closed-loop system (10) is SFTB with respect to (δx,ϵ,Ri,N,d) with 0<δx<ϵ, if there exist scalars μ≥0, σ1≥0, and σ2≥0, three sets of mode-dependent symmetric positive-defined matrices {Xi,i∈S}, {Qi,i∈S} and diag positive-defined matrices {Si,i∈S}, and two sets of mode-dependent matrices {Yi,i∈S} and {L-i=LiXi,A-aw,i=Aaw,iXi,C-aw,i=Caw,iXi,B-aw,i=Baw,iSi,D-aw,i=Daw,iSii∈S}, such that the following conditions hold:
(41)[-μXi0L-iTL-1iT*-QiKwiTL2iT**-2Si-He(K-φ,i)L-3iT***-W]<0,(42)[σ2d2-μ-Nϵ2*δx-σ1]<0,(43)[Xi*K-i+L-iu0(k)2]>0,k=1,…,m,(44)σ1Ri-1<Xi<Ri-1,(45)0<Qi<σ2I,
where
(46)W=diag⁡{X1,X2,…,Xn},L-1iT=[πi1A-iTπi2A-iT⋯πinA-iT],L2iT=[πi1BwiTπi2BwiT⋯πinBwiT],L3iT=[πi1B-qiTπi2B-qiT⋯πinB-qiT],
with
(47)A-i=[ApiXi+Bpu,iDcy,iCpy,iXiBpu,iCciXiBpu,iI2C-aw,iBcy,iCpy,iXiAciXiI1C-aw,i00A-aw,i],B-qi=[Bpu,iI2D-aw,i+Bpu,iXiI1D-aw,iB-aw,i],K-i=[Dcy,iCpy,iXiCciXiI2C-aw,i],K-ϕ,i=I2D-aw,i.Proof.
Define the following Lyapunov function for eachδ(t)=i∈S:
(48)V(k)=x(k)TPix(k).
It is readily obtained that
(49)E{V(k+1)}=E{∑j=1nπijx(k+1)TPjx(k+1)}=ξ(k)T[L1iL2iL3i]TW-[L1iL2iL3i]ξ(k),
where
(50)ξ(k)=[x(k)Tw(k)Tψ(k)T],W-=diag⁡{P1,P2,…,Ph},L1iT=[πi1AiTπi2AiT⋯πinAiT],L2iT=[πi1BwiTπi2BwiT⋯πinBwiT],L3iT=[πi1BqiTπi2BqiT⋯πinBqiT].
Then, by pre- and postmultiplying (41) by diag⁡{Pi,I,Ti,I} with Pi=Xi-1,Ti=Si-1, we have
(51)[-μPi0LiTTiL1iT*-QiKwiTL2iT**-2Ti-He(TiKϕ,i)L3iT***-W]<0.
By using of Schur complement lemma, we derive
(52)ξ(k)T[-μPi0LiTTi*-QiKwiT**-2Ti-He(TiKϕ,i)]ξ(k)+ξ(k)T[L1iL2iL3i]TW-[L1iL2iL3i]ξ(k)<0.
It follows that
(53)E{V(k+1)}<μx(k)TPix(k)+w(k)TQiw(k)+ψ(k)T(2Ti+He{TiKϕ,i})ψ(k)-2ψ(k)TTiLix(k)+2ψ(k)TTiKwiw(k).
Since ψ(k)T(2Ti+He{TiKϕ,i})ψ(k)-2ψ(k)TTiLix(k)+2ψ(k)TTiKwiw(k)≤0, we get
(54)E{V(k+1)}<μx(k)TPix(k)+w(k)TQiw(k).
It is shown that
(55)E{V(k+1)}<μV(k)+sup⁡(i∈S)⁡{λmax⁡(Qi)}w(k)Tw(k).
Then, we have
(56)E{V(k+1)}<μE{V(k)}+sup⁡(i∈S){λmax⁡(Qi)}E{w(k)Tw(k)}.
Since μ≥1, it is easily found that
(57)E{V(k+1)}<μE{V(0)}+sup⁡(i∈S){λmax⁡(Qi)}×E{∑j=0k-1‍μk-j-1w(j)Tw(j)}≤μkE{V(0)}+sup⁡(i∈S){λmax⁡(Qi)}μkd2.
The following proof is similar to the process of Theorem 7.
Based on Lemma5, it is easy to obtain that
(58)[Pi*Ki+Liu0(k)2]>0,k=1,…,m,
then pre- and post-multiply (58) by diag⁡{Xi,I} which implies (43). This completes the proof.Theorem 9.
For eachr(k)=i∈S, with antiwindup compensator (7), such that the resulting closed-loop system (10) is said to be Stochastic H∞ finite-time stable via state feedback with respect to (δx,ϵ,Ri,N,γ,d), if there exist three scalars μ≥0, σ1≥0, and γ≥0, two sets of mode-dependent symmetric positive-defined matrices {Xi,i∈S} and diag matrices {Si,i∈S}, and two sets of mode-dependent matrices {Yi,i∈S} and {L-i=LiXi,i∈S}, such that the following conditions hold:
(59)[-μXi0L-iTL-1iTC-ziT*-γ2μ-NIKwiTL2iTDzw,iT**-2Si-He(K-ϕ,i)L-3iTD-zq,iT***-W0****-I]<0,(60)[μ-N(d2γ2-ϵ2)*δx-σ1]<0,(61)[Xi*K-i+L-iu0(k)2]>0,k=1,…,m,(62)σ1Ri-1<Xi<Ri-1,
with
(63)C-zi=[(Cpz,i+Dp,zu,iDciCpy,i)XiDp,zu,iXiI2C-aw,i],D-zq,iT=I2D-aw,i+Dp,zu,i.Proof.
Choose the similar Lyapunov function as Theorem7 and denote
(64)Π(x(k),w(k),r(k)=i)=E{V(k+1)}-μV(k)+z(k)Tz(k)-γ2μ-Nw(k)Tw(k).
Thus, in light of Lemma 5, we have
(65)Π(x(k),w(k),rk=i)≤ξ(k)T[L1iL2iL3i]TW-[L1iL2iL3i]ξ(k)+ξ(k)T[CziDzw,iDzq,i]T[CziDzw,iDzq,i]ξ(k)+ξT(k)[-μXi0LiTTi*-γ2μ-NIKwiT**-2Ti-He{TiKϕ,i}]ξ(k).
Then pre- and postmultiply (59) by diag⁡{Pi,I,Ti,I}, and considering Schur complement lemma and (65), we derive that
(66)Π(x(k),w(k),r(k)=i)<0
holds for all rk=i∈S. According to (66), one can obtain that
(67)E{V(k+1)}<μE{V(k)}-E{z(k)Tz(k)}+γ2μ-NE{w(k)Tw(k)}.
Then, we have
(68)E{V(k)}<μkE{V(0)}-∑j=0k-1μk-j-1E{z(j)Tz(j)}+γ2μ-NE{∑j=0k-1μk-j-1w(j)Tw(j)}.
Under the zero-value initial condition and noting that V(k)≥0, for all K∈Zk≥0, it is shown that
(69)∑j=0k-1μk-j-1E{z(j)Tz(j)}<γ2μ-NE{∑j=0k-1μk-j-1w(j)Tw(j)}.
Since μ≥1 and from (69), we have
(70)E{∑j=0Nz(j)Tz(j)}=∑j=0NE{z(j)Tz(j)}≤∑j=0NE{μN-jz(j)Tz(j)}≤γ2μ-NE{∑j=0NμN-jw(j)Tw(j)}≤γ2E{∑j=0Nw(j)Tw(j)}.
The following proof is similar to the process of Zhang and Liu [17].
Sinceε(Pi,1)⊂D(u0), it follows that
(71)[Pi*Ki+Liu0(k)2]>0,k=1,…,m,
and then pre- and post-multiply (71) by diag⁡(Xi,I) and its transpose, respectively; we derive condition (61). This completes the proof.


## 4. Illustrative Examples

In this section, a numerical example is provided to demonstrate the effectiveness of the proposed method. Consider the following systems with four operation modes.Mode 1 is(72)Ap1=[0.75-0.751.5-1.5],Bpu,1=[10],Bpw,1=[10],Cpy,1=[-0.1-0.2],Cpz,1=[10],Dpyw,1=1,Dpzu,1=1,Dpzw,1=0.8.
Mode 2 is
(73)Ap2=[0.154.52.10-0.4],Bpu,2=[10],Bpw,2=[10],Cpy,2=[-0.1-0.1],Cpz,2=[10],Dpyw,2=1,Dpzu,2=0.9,Dpzw,2=0.8.
Mode 3 is
(74)Ap3=[0.242.501.2-2.1],Bpu,3=[0.90],Bpw,3=[10],Cpy,3=[-0.10],Cpz,3=[10],Dpyw,3=0.8,Dpzu,3=1,Dpzw,3=1.2.
Mode 4 is
(75)Ap4=[1-0.251.5-1.5],Bpu,4=[10],Bpw,4=[10],Cpy,4=[10],Cpz,4=[10],Dpyw,4=1,Dpzu,4=0.5,Dpzw,4=1.
With the given designed controllers,
(76)Ac1=-5.5,Bcy,1=-1,Bcw,1=1,Cc1=-1,Dcy,1=-0.1,Dcw,1=0.5,Ac2=-5,Bcy,2=-0.9,Bcw,2=1,Cc2=-1,Dcy,2=5.9,Dcw,2=1,Ac3=-4.5,Bcy,3=-1,Bcw,3=1,Cc3=-1.5,Dcy,3=5.1,Dcw,3=1,Ac4=-7,Bcy,4=-1,Bcw,4=1,Cc4=-1.5,Dcy,4=-2,Dcw,4=1.
The transition rate matrix is given by the following:
(77)[0.30.30.20.20.40.30.20.10.20.10.40.30.20.30.10.4].
In this case, we choose the initial values for Ri=I2,i=1,2,3,4, δx=1,N=5,α=10-10,μ=2.5,d=1, and w(k)=0.5(1+cos⁡x(k)); Theorem 7 yields to ϵ=36.2671,σ1=0.4906,σ2=13.7421, and the bounds of the input saturation u0=0.08.Based on Theorem9, we derive
(78)Aaw,1=-2.67,Aaw,2=-1.86,Aaw,3=-1.88,Aaw,4=-2.59,Baw,1=-0.02,Baw,2=-0.01,Baw,3=-0.01,Baw,4=-0.02,Caw,1=[17.270.68],Caw,2=[68.44-48.31],Caw,3=[68.74-49.29],Caw,4=[17.270.66],Daw,1=[0.210.1],Daw,2=[0.18-0.1],Daw,3=[0.190],Daw,4=[0.20.1].Remark 10.
Figures2, 3, and 4 are given on the last page. Figure 1 is rk of the jump rates, Figure 2 and Figure 3 are state response of open and closed-loop system. Based on the figures provided, the controller and the compensator we designed guarantee that the resulting closed-loop systems are mean-square locally asymptotically finite-time stabilizable.Figure 2
r
k of jump rates.Figure 3
x
(
k
) of the system (1)–(5).Figure 4
x
(
k
) of the closed-loop system (1)–(7).


## 5. Conclusions and Future Work

In this paper, the finite-timeH∞ stabilization problem for a class of discrete-time Markov jump systems with input saturation has been investigated. Based on stochastic finite-time stability analysis, a controller designed for the unconstrained system with a dynamic antiwindup compensator subject to actuator saturation is given to guarantee the stochastic finite-time boundedness and stochastic finite-time stabilization of the considered closed-loop system for all admissible disturbances. Finally, the effectiveness of the proposed approach has been illustrated by simulation results. The finite-time stabilization problem of Markov jump systems with constrained input and time-delay will be considered in the future work.



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*Source: 906902-2014-03-17.xml*