# Finite-Time Control for Markovian Jump Systems with Polytopic Uncertain Transition Description and Actuator Saturation

Authors: Zhongyi Tang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The problem of finite-timeL2-L∞  control for Markovian jump systems (MJS) is investigated. The systems considered time-varying delays, actuator saturation, and polytopic uncertain transition description. The purpose of this paper is to design a state feedback controller such that the system is finite-time bounded (FTB) and a prescribed L2-L∞ disturbance attenuation level during a specified time interval is guaranteed. Based on the Lyapunov method, a linear matrix inequality (LMI) optimization problem is formulated to design the delayed feedback controller which satisfies the given attenuation level. Finally, illustrative examples show that the proposed conditions are effective for the design of robust state feedback controller.


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## Body

## 1. Introduction

In the aspect of modeling practical systems with abrupt random changes, such as manufacturing system, telecommunication, and economic systems, MJS have powerful ability. MJS have been extensively studied during the past decades and many systematic results have been obtained [1–3]. The peak-to-peak filtering problem was studied for a class of Markov jump systems with uncertain parameters in [4]. A robust H2 state feedback controller for continuous-time Markov jump linear systems subject to polytopic-type parameter uncertainty was designed in [5]. In [6], the authors address the stabilization problem for single-input Markov jump linear systems via mode-dependent quantized state feedback for control.Actuator saturation which can lead to poor performance of the closed-loop system is another active research area. In practical situations, it may be encountered sometimes. How to preserve the closed-loop system performance in the case of actuator saturation would be more meaningful. In [7], the H∞ control problem for discrete-time singular Markov jump systems with actuator saturation was considered. In [8] the stochastic stabilization problem for a class of Markov jump linear systems subject to actuator saturation was considered.In some practical applications, the behavior of the system over a finite-time interval is mainly considered. Finite-time stable (FTS) and Lyapunov asymptotic stability are independent concepts. The concept of FTS was first introduced in [9]. A system is said to be finite-time stable if, given a bound on the initial condition, its state does not exceed a certain threshold during a specified time interval. FTS of linear time-varying systems was considered in [10]. Sufficient conditions for the solvability of both the state and the output feedback problems are stated. Amato [11] provided a necessary and sufficient condition for the FTS of linear-varying systems with jumps. Recently, robust finite-time H∞ control of jump systems was dealt with in [12–14]. In [15], the problems of finite-time stability analysis were investigated for a class of Markovian switching stochastic systems. To the best of authors’ knowledge, however, the problem of finite-time L2-L∞ performance for discrete-time MJS with imprecise transition probabilities and time-varying delays has not been well addressed, which motivates our work.This paper deals with this problem. More specifically, the actuator is saturation. By using the Lyapunov-Krasovskii functional, a new sufficient condition for stochastic asymptotic stability with finite-timeL2-L∞ performance is derived in terms of LMI. Based on this, the existence condition of the desired performance which guarantees finite-time stability and an L2-L∞ performance of the MJS is presented. A numerical example is provided to show the effectiveness of the proposed results.Throughout the paper, if not explicitly stated, matrices are assumed to have compatible dimensions. The notationW>(≥,<,≤)0 is used to denote a symmetric positive definite (positive semidefinite, negative, negative semidefinite) matrix. λmin⁡(·) and λmax⁡(·) represent the minimum and maximum eigenvalues of the corresponding matrix, respectively. I is the identity matrix with compatible dimensions. ∥·∥ refers to the Euclidean norm of vectors and E[·] stands for the mathematical expectation. For a symmetric block matrix, “*” is used as an ellipsis for the terms that are obtained by symmetry.


## 2. Problem Statement and Preliminaries

Consider a discrete-time MJS with actuator saturation and delay in the state. Let the system dynamics be described by the following:(1)x(k+1)=Aθ1(rk)x(k)+Aθ2(rk)x(k-d)+Bθ1(rk)σ(uk)+Bθ2(rk)wk,z(k)=Cθ1(rk)x(k)+Cθ2(rk)x(k-d)+Dθ1(rk)wk,
where xk∈Rn is the system state, zk∈Rn is the system output, uk∈Rm is the control input, wk∈Rq is the disturbance input which belongs to L2[0,∞) and ∑k=0∞wkTwk<κ2, andκ is a given positive scalar. Aθ1(rk),Aθ2(rk),Bθ1(rk),Bθ2(rk),Cθ1(rk),Dθ1(rk), and Dθ2(rk) are appropriately dimensioned real-valued matrices, which belong to the part of convex polyhedron Φ(rk):
(2)Φ(rk)={∑l=1Lθl[Al1(rk),Al2(rk),Bl1(rk),hhhhhhhBl2(rk),Cl1(rk),Cl2(rk),hhhhhhhDl1(rk),Dl2(rk)],∑l=1Lθl=1,θl≥0},
where Al1(rk),Al2(rk),Bl1(rk),Bl2(rk),Cl1(rk),Cl2(rk), and Dl1(rk) are matrix functions of the random jumping process {rk} (Figure 1), which is a discrete-time Markov chain taking values in a finite set Ω={1,2,…,S} with transition probabilities:
(3)P{rk+1=j∣rk=i}=πij.
Here πij≥0 and for any i,j∈Ω,∑j=1sπij=1. Assuming that the transition probability πij is not exactly known, a certain range can only be given
(4)[πv(i,1),πv(i,2),…,πv(i,S)]=∑m=1Mvm[πm(i,1),πm(i,2),…,πm(i,S)],
where v=[v1⋯vM]T∈RM and ∑m=1Mvm=1, and the transition probability belongs to the following convex polyhedron:
(5)ℵ(rk=i)=Co{[π1(i,1),π1(i,2),…,π1(i,N)][πM(i,1),πM(i,2),…,πM(i,N)]}.
When the system operates in the ith mode (rk=i), for simplicity, the matrices Aθ1(rk),Aθ2(rk),Bθ1(rk),Bθ2(rk),Cθ1(rk), and Dθ1(rk) are denoted as Aθ1i,Aθ2i,Bθ1i, Bθ2i, Cθ1i, and Dθ1i, respectively. d is a positive integer denoting the constant delay of the system state (Figures 2 and 3).Figure 1
Jumping mode.Figure 2
Response of the system statex1.Figure 3
Response of the system statex2.In system (1), σ(·):Rm→Rm is the vector-valued standard saturation function defined as follows:
(6)σ(u)=[σ(u1),σ(u2),…,σ(um)]T,
where σ(uθ)=sign⁡(uθ)min⁡{1,|uθ|}. It is assumed that system (1) is completely controllable. A mode-dependent controller is considered here with the following form:
(7)σ(u(k))=σ(Kix(k)),
where Ki∈Rm×n(∀rk=i∈Ω) is the controller gain to be determined.LetM be the set of m×m diagonal matrices whose diagonal elements are either 1 or 0. Suppose each element of M is Mj, j=1,…,2m, and denote Mj-=I-Mj. Note that Mj- is also an element of M if Mj∈M. Let hij be the jth row of the matrix Hi, and define the symmetric polyhedron by φ(Hi)={x(t)∈Rn:|fijx(t)|≤1,i=1,2,…,m}.Lemma 1 (see [8]).
LetKi,Hi∈Rm×n be given matrix. For x(t)∈Rn, if x(t)∈φ(Hi), then
(8)σ(Kix(t))=∑r=12mζr(MrKi+Mr-Hi)x(t),
where 0≤ζr≤1, ∑r=12mζr=1.
By the connection of (6), (7) and (8), the following closed-loop MJS are obtained:(9)x(k+1)=(Aθ1(rk)+Bθ1(rk)∑r=12mζr(MrKi+Mr-Hi)x(k)hh×∑r=12mζr(MrKi+Mr-Hi))x(k)+Aθ2(rk)x(k-d)+Bθ2(rk)wk.To describe the main objective of this note more precisely, let us now introduce the following definition for the underlying system.Definition 2 (see [13]).
Given a time constantT>0, the MJS
(10)x(k+1)=Aθ1(rk)x(k)+Aθ2(rk)x(k-d)
are said to be FTS with respect to (ℏ1ℏ2TRi), if there exist positive matrix Ri>0, scalars ℏ1>0 and ℏ2>0, and
(11)E{xT(k1)Rix(k1)}≤ℏ1⟹E{xT(k2)Rix(k2)}≤ℏ2,k1∈{-h,…,0},k2∈{1,2,…,T}.Definition 3 (see [13]).
Given a time constantT>0, the MJS
(12)x(k+1)=Aθ1(rk)x(k)+Aθ2(rk)x(k-d)+Bθ2(rk)wk
are said to be finite-time bounded (FTB) with respect to (ℏ1ℏ2TRi), if there exist positive matrix Ri>0 and scalars ℏ1>0 and ℏ2>0, and satisfied (11).In general, FTB and FTS are different. If there is external disturbance in systems, the concept of FTB is used. Conversely, FTS is addressed.The objective of this paper is to design a delayed feedback controller which satisfies the given attenuation level of system (1). The design procedure is given in the next section.Definition 4.
The time-delay MJS (1) is said to be finite-time L2-L∞ control with respect to (ℏ1ℏ2TRi) and performance γ, where Ri>0, γ>0,ℏ1>0, and ℏ2>0, if the time-delay MJS (1) is stochastically FTB and under the zero-initial condition the output z(k) satisfies
(13)∥z(k)∥∞<γ∥w(k)∥2
for all nonzero w(k)∈L2[0,∞) subject to the zero-initial condition.


## 3. Main Results

In this section, firstly stochastic FTB analysis of nominal time-delay MJS (1) is provided. Then, these results will be extended to the MJS (1) with actuator saturation and uncertain transition probability. LMI conditions are established.Lemma 5.
System (1) with σ(uk)≡0 is stochastic FTB with respect to (ℏ1ℏ2dRiN); if for scalars ς≥1, ℏ1>0, and ℏ2>0, there exist symmetric matrices Ri>0(i∈Ω) and Qi>0(i∈Ω), such that the following matrix inequalities hold:
(14)Λ=[Al1iTP-iAl1i-Pi+Q**Al2iTP-iAl1i-Q+Al2iTP-iAl2i*Bl2iTP-iAl1iBl2iTP-iAl2iBl2iTP-iBl2i-I]<0(15)ςk[λmax⁡(P~r(0))+λmax⁡(Q)·d]c1≤c2·λmin⁡(P~i),
where P-i=∑j=1SπvijPj.Proof.
Choose the following Lyapunov functional:(16)V(k)=xT(k)Pix(k)+∑n=k-dk-1xT(n)Qx(n).
The proof of Lemma 5 is divided into two parts. In the first part, the following inequality is obtained:
(17)E{V(k)}<ςkE{V(0)}+ςkwT(k)w(k).
Then, we compute(18)ΔV(k)=E{V(k+1)}-V(k)=∑j=1SπvijxT(k+1)Pjx(k+1)-xT(k)Pix(k)+∑j=1SπvijxT(k)Qx(k)-xT(k-d)Qx(k-d)=ϑT(k)[Aθ1iTP-iAθ1i-Pi+Q**Aθ2iTP-iAθ1i-Q+Aθ2iTP-iAθ2i*Bθ2iTP-iAθ1iBθ2iTP-iAθ2iBθ2iTP-iBθ2i]ϑ(k)=ϑT(k)[(∑l=1Lθl)(∑l=1Lθl)[Al1iTP-iAl1i-Pi+Q**Al2iTP-iAl1i-Q+Al2iTP-iAl2i*Bl2iTP-iAl1iBl2iTP-iAl2iBl2iTP-iBl2i]]ϑ(k),where ϑ(k)=[x(k)x(k-d)w(k)].
Note condition (14); it follows that
(19)E{V(k+1)}-V(k)<(ς-1)V(k)+wT(k)w(k),ς≥1.
Therefore, we obtain that
(20)E{V(k+1)}<ςV(k)+wT(k)w(k).
That is,
(21)E{V(x(1),r(1))}<ςV(x(0),r(0))+wT(k)w(k)⋮⋮⋮E{V(x(k+1),r(k+1))}<ςE{V(x(k),r(k))}+wT(k)w(k).
By recursive,
(22)E{V(k)}<ςkE{V(0)}+E{∑τ=0k-1ςk-τ-1wT(τ)w(τ)}.
Then the inequality in (17) is obtained.
In the second part, stochastic FTB is established:(23)E{V(k)}=E{xT(k)Pix(k)+∑n=k-dk-1xT(n)Qx(n)}≥λmin⁡(P~i)E{xT(k)Rx(k)}.
On the other hand,
(24)ςkE{V(0)}≤ςk[λmax⁡(P~r(0))E{xT(0)Rx(0)}{∑n=-d-1xT(n)Rx(n)}hhh+λmax⁡(Q)E{∑n=-d-1xT(n)Rx(n)}].
From Definition 2, we have
(25)ςkE{V(0)}≤ςk{[λmax⁡(P~r(0))+λmax⁡(Q)·d]}ℏ1+ςk.
By (23) and (25), we know
(26)E{xT(k)Rx(k)}≤ςk{[λmax⁡(P~r(0))+λmax⁡(Q)·d]+1}ℏ1λmin⁡(P~i)≤ℏ2.
This completes the proof.Theorem 6.
System (1) is finite-time L2-L∞ control and satisfies the given lever γ with respect to (ℏ1ℏ2dRiN); if for scalars ς≥1, ℏ1>0, and ℏ2>0, there exist symmetric matrices Ri>0(i∈Ω) and Q>0, such that the following matrix inequalities hold:
(27)Θ1=[Al1iTP-iAl1i-Pi+Q**Al2iTP-iAl1i-Q+Al2iTP-iAl2i*Bl2iTP-iAl1iBl2iTP-iAl2iBl2iTP-iBl2i-I]<0,(28)Θ2=[-Pi***0-Q**00-I*Cl1kCl2kDl2k-γ2I]ςk[λmax⁡(P~r(0))+λmax⁡(Q)·d]c1≤c2·λmin⁡(P~i).Proof.
System (1) with σ(uk)≡0 is FTB according to Lemma 5 and inequality (27).
Subsequently, to establish the energy-to-peak performance for the system (1), assume that the initial values for the plant are zeros and consider the following function:
(29)ℵ:=E{V(k)}-∑i=0k-1wiTwi.
For any nonzero wk∈l2[0,∞) and k>0, it follows from (18) that
(30)ℵ≔E{∑i=0k-1ΔV(i)-∑i=0k-1wiTwi}=ϑT(k)Θ1ϑ(k).
It follows from (27) that E{V(k)}<∑i=0k-1wiTwi.
For all the time instantsk>0, the expectation of the output can be evaluated as
(31)E{zkTzk}=E{ϑT(k)[Cθ1kCθ2kDθ1k]Thhl[Cθ1kCθ2kDθ1k]T×[Cθ1kCθ2kDθ1k]ϑ(K)}<E{γ2ϑT(k)[Pi00*Q-i0**I]ϑ(k)}<γ2E{∑i=0kwiTwi}<γ2E{∑i=0∞wiTwi}=γ2∥w∥22.
Applying Definition 4, the statement of Theorem 6 is true.Theorem 7.
Consider the uncertain time-delay system (1); there exists a state feedback controller σ(Kix(t)) such that the uncertain time-delay system (1) is finite-time L2-L∞ control with respect to (ℏ1ℏ2dRiN), if the following LMIs hold:
(32)Λ1=[-Xi00ε14⋯ε16Xi*-R0ε24⋯ε260**-Iε34⋯ε360***ε44⋯00****⋱00*****ε660******-R]<0,(33)Λ2=[-Pi***0-Q**00-I*Cl1kCl2kDl2k-γ2I]<0,ςk[λmax⁡(P~r(0))+λmax⁡(Q)·d]c1≤c2·λmin⁡(P~i),
where ε14=πi1(Al1i+Bl1i(MrYi+Mr-Zi), ε24=πi1RAl2i, ε34=πi1RBl2i, ε44=-X1, ε16=πiS(Al1i+Bl1i(MrYi+Mr-Zi)i), ε26=πiSRAl2i, ε36=πiSRBl2i, and ε66=-XS.The state feedback controller is designed asσ(Kix(t))=∑r=12mζr(MrKi+Mr-Hi)x(t).Proof.
Noting condition (27) and P-i=ΓiκΓiT, where κ=diag⁡{P1,…,PS}, Γi=[pi1I,…,piSI] thus Θ1 can be rewritten as
(34)Θ1=[-Pi+Q**0-Q*00-I]+[Al1iTAl2iTBl2iT]κ[Al1iAl2iBl2i]<0.
Using Schur complement, it can be obtained
(35)[-P+Q00Al1iTΓi*-Q0Al2iTΓi**-IBl2iTΓi***-κ-1]<0.
Let Xi=Pi-1,R=Q-1, Yi=KiXi, and Zi=HiXi. Pre- and postmultiplying (35) by diag⁡{Xi,R,I,I} and then using Schur complement, then inequality (32) is obtained. Implying Theorem 6, we can conclude that the corresponding closed-loop system is finite-time L2-L∞ control. This completes the proof.


## 4. Numeral Example

To illustrate the proposed results, a numerical example is considered for finite-timeL2-L∞ control. The system is described by (1) and assumed to have two modes; Ω={1,2}. The mode switching is governed by a Markov chain that has the following transition probability matrix:
(36)P=[0.20.80.40.6].
The system matrices are as follows:
(37)A111=A211=[0.30.102-0.6630.3],A112=A212=[0.80.0539-0.86550.8],A121=A221=[0.50.06-0.8430.5],A122=A222=[0.90.0766-0.76610.9],B111=B211=[0.00050.0539],B212=B112=[0.0050.1078],B121=B221=[0.00450.0539],B122=B222=[0.00450.1078],C111=C211=C112=C212=[00.2],C121=C221=C122=C222=[0.30],D111=D211=D112=D212=0.3.AssumeL2-L∞ performance of level γ=0.3; by applying Theorem 7, we can explicitly compute the optimally achievable closed-loop L2-L∞ performance γ from Theorem 7 as γ=0.2056. Response of the system state is depicted in Figures 2 and 3.


## 5. Conclusion

The problem of finite-timeL2-L∞ control for MJS has been studied. By using the Lyapunov functional approach, a sufficient condition is derived such that the closed-loop MJS are stochastic FTB and satisfy the given level. The controller can be obtained by using the exiting LMI optimization techniques. Finally, numerical and simulation results demonstrate the effectiveness of the results of the paper.



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