# Optimal Guaranteed Cost Control of a Class of Discrete-Time Nonlinear Systems with Markovian Switching and Mode-Dependent Mixed Time Delays

Authors: Yurong Liu; Hamid Reza Karimi; Xiaohui Liu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The guaranteed cost control problem is investigated
for a class of nonlinear discrete-time  systems with  Markovian
jumping parameters and mixed time delays. The mixed time delays
involved consist of both the mode-dependent discrete delay and the
distributed delay with mode-dependent lower bound. The associated
cost function is of a quadratic summation form over the infinite
horizon. The nonlinear functions are assumed to satisfy
sector-bounded conditions. By introducing new Lyapunov-Krasovskii
functionals and developing some new analysis techniques,
sufficient conditions for the existence of guaranteed cost
controllers are derived with respect to the given cost function.
Moreover, a convex optimization approach is applied to search for
the optimal guaranteed cost controller by minimizing the
guaranteed cost of the closed-loop system. Numerical simulation is
further carried out to demonstrate the effectiveness of the
proposed methods.


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## Body

## 1. Introduction

In the past decades, the control problems for the linear or nonlinear systems have attracted considerable research interest and significant advances on this topic have been made; see, for example, [1–15] and the references therein. It is well known that the time delay in feedback control can be caused by physical properties of control equipments, measurements of system responses, and data processing, calculating and executing control forces, and so forth. The time delay in feedback control may not only deteriorate the performance of controlled systems but also destabilize the controlled systems. There have been a lot of reports on the dynamics analysis of time delay feedback controlled systems. Various sufficient conditions, either delay dependent or delay independent, have been proposed to guarantee the stability for the delayed systems; see, for example, [2, 9, 12, 13] for some recent publications.On the other hand, a great deal of attention has recently been devoted to the study of Markovian jump systems. This class of systems can be modeled with variable structure subject to random abrupt changes resulting from the occurrence of some inner discrete events in the system such as failures and repairs of machine in manufacturing systems, random failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, and so on. Recently, stability and control problems for Markovian jump systems have been extensively investigated; see, for example, [16, 17] and the references therein.It is also noted that in practical applications, the choice of control policy depends upon the optimization of some preassigned performance criterion. When designing a controller for a real system, it is often desirable to make the controlled system not only stable but also guarantee an adequate level of performance. To deal with such control problems, the so-called guaranteed cost control approach was first introduced by Chang and Peng [2]. The objective of this approach is to establish an upper bound on a given performance index so that the system performance degradation incurred by the uncertainties is guaranteed to be less than this bound. For guaranteed cost control, a great number of results on this topic have been reported in the literature and various approaches have been proposed. For example, in [18], notion of the quadratic guaranteed cost control was introduced to allow for a quadratic performance index and a Riccati equation approach was presented for designing quadratic guaranteed cost controllers, where the system was delay-free. The authors in [19] extended the Riccati equation approach given in [18] to uncertain delayed systems and proposed a guaranteed cost controller design method by solving a certain parameter-dependent Riccati equation. In [15], an LMI approach [20] was proposed to deal with the guaranteed cost control problem for a class of linear time delay systems with time-varying norm-bounded parameter uncertainty, and a sufficient condition for the existence of memoryless state-feedback guaranteed cost controllers was derived. In [21], the solutions to the guaranteed cost control problem via state-feedback are presented for a class of uncertain Markovian jump systems with mode-dependent delays in LMI framework, and the delay-dependent/independent sufficient conditions for the existence of guaranteed cost state-feedback controllers have been derived.Based on LMI approach, [22] considered the robust guaranteed cost control problem for uncertain linear discrete-time systems subject to actuator saturation, where the saturation nonlinearity was transformed into a convex polytope of linear systems, and then this problem was formulated into a convex optimization problem with constraints given by a set of LMIs. Very recently, the filtering problems have been investigated for discrete-time nonlinear stochastic systems with network-induced phenomena in [7, 8, 23, 24]. As far as we know, however, little research has been focused on the guaranteed cost control problem for discrete-time systems with distributed time delay and Markovian jumping parameters.In this paper, we consider the guaranteed cost control problem for a class of nonlinear discrete-time systemswith Markovian jumping parameters and mixed time delays. The mixed time delays involved consist of both the mode-dependent discrete delay and the infinite distributed delay with mode-dependent lower bound. The relevant cost function is chosen as a quadratic summation form over the infinite horizon. The nonlinear functions are assumed to satisfy sector-bounded conditions. By constructing novel Lyapunov-Krasovskii functionals and employing some new analysis techniques, sufficient conditions for the existence of guaranteed cost controllers are derived with respect to the given cost function. In addition, a convex optimization approach is applied to search for the optimal guaranteed cost controller by minimizing the guaranteed cost of the closed-loop system. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed methods.Notations. Throughout this paper, ℝn and ℝn×m denote, respectively, the n-dimensional Euclidean space and the set of all n×m real matrices; ℕ- stands for the set of all the negative integers and zero. The superscript “T” denotes matrix transposition. The notation X≥Y (resp., X>Y), where X and Y are symmetric matrices, means that X-Y is positive semidefinite (resp., positive definite). diag{⋯} stands for a block-diagonal matrix, I is the identity matrix with compatible dimension, and |·| denotes the Euclidean norm in ℝn. If A is a square matrix, λmax(A) (resp., λmin(A)) denotes the largest (resp., smallest) eigenvalue of A, and Tr(A) denotes the trace of A. In symmetric block matrices, an asterisk “*” is used to represent a term that is induced by symmetry. 𝔼[x] and 𝔼[x∣y] will, respectively, mean the expectation of x and the expectation of x conditional on y. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.


## 2. Problem Formulation

Letr(k)(k≥0) be a Markov chain taking values in a finite state space 𝒩={1,2,…,n0} with probability transition matrix Π=(πij)n0×n0 given by
(1)Pr{r(k+1)=j∣r(k)=i}=πij,∀i,j∈𝒩,
where πij≥0(i,j∈𝒩) is the transition probability from i to j and ∑j=1n0‍πij=1, for all i∈N.Consider a discrete-time nonlinear system withn0 modes described by the following dynamical equation:(2a)x(k+1)=A(r(k))x(k)+B(r(k))f(x(k))+C(r(k))g(x(k-τ1,r(k)))+D(r(k))×∑m=τ2,r(k)+∞‍μmh(x(k-m))+E(r(k))u(k),(2b)x(m)=ϕ(m)form∈ℕ-,r(0)=r0,where x(k)∈ℝn is the state vector; for r(k)=i∈𝒩, A(r(k))∈ℝn×n, B(r(k))∈ℝn×n, C(r(k))∈ℝn×n, D(r(k))∈ℝn×n∈ℝn×n, and E(r(k))∈ℝn×q are known constant matrices; f(·), g(·) and h(·):ℝn→ℝn are nonlinear vector functions; μm(m=1,2,…) are scalar constants; u(k)∈ℝq is the control input; τ1,r(k) stands for the mode-dependent discrete time-delay while τ2,r(k)≥0 describes a mode-dependent lower bound for the distributed time-delay; ϕ:ℕ-→ℝn is the initial value; and r(0)=r0 is the initial mode of the Markov chain.For nonlinear vector functionsf, g, h, we assume that
(2)(f(x)-F1x)T(f(x)-F2x)≤0,∀x∈ℝn,(g(x)-G1x)T(g(x)-G2x)≤0,∀x∈ℝn,(h(x)-H1x)T(h(x)-H2x)≤0,∀x∈ℝn,
where F1,F2,G1,G2,H1,andH2∈ℝ(n×n) are known constant matrices.Remark 1.
The conditions (2) are quite general, and such a description, compared with the usual Lipschitz condition, is very helpful for using LMI-based approach to reduce the possible conservatism. The similar form of the conditions has been used, for example, by the authors in [10].Remark 2.
It is not difficult to verify that the conditions (2) imply that f(0)=g(0)=h(0)=0, and x=0 is therefore an equilibrium point.The cost function associated with system (2a) and (2b) is
(4)J=𝔼[∑k=0∞(xT(k)R1(r(k))x(k)+uT(t)R2(r(k))u(k))∣ϕ(k),r0∑m=1+∞‍],
where R1(i)>0 and R2(i)>0, for all i∈𝒩.Now, consider the following state-feedback control lawu(k)=K(r(k))x(k), where K(i)∈ℝq×n(i∈𝒩) are controller gains to be designed. Then, the closed-loop system can be given as follows:(5a)x(k+1)=AK(r(k))x(k)+B(r(k))f(x(k))+C(r(k))g(x(k-τ1,r(k)))+D(r(k))∑m=τ2,r(k)+∞‍μmh(x(k-m)),(5b)x(m)=ϕ(m)form∈ℕ-,r(0)=r0,where AK(r(k))=A(r(t))+E(r(k))K(r(k)).Definition 3.
System (2a) and (2b) with u(k)≡0 is said to be asymptotically stable in mean square if, for any solution x(k) of system (2a) and (2b), the following holds:
(6)limk→∞𝔼[|x(k)|2]=0.Definition 4.
Consider the system (2a) and (2b). If there exists a state-feedback control law u(k) and a positive number γ such that the closed-loop system (5a) and (5b) is asymptotically stable in mean square and the resulting cost function satisfies
(7)J≤γ,
then γ is said to be a guaranteed cost and u(k) is said to be a guaranteed cost controller for the system (2a) and (2b).The objective of this paper is to develop a procedure to design a memoryless state-feedback guaranteed cost controlleru(k)=K(r(k))x(k), which achieves as small value of γ as possible.Assumption 5.
Constantμm≥0 satisfies the following convergent conditions:
(8)∑m=1+∞‍μm<+∞,∑m=1+∞‍mμm<+∞.Remark 6.
Assumption5 makes sense as they guarantee that the term D(r(k))∑m=τ2,r(k)+∞μmh(x(k-m)) in (2a) is convergent, which is necessary for the subsequent analysis.


## 3. Main Results and Proofs

We first introduce some lemmas to be used in deriving our results.Lemma 7 (see [25]).
LetM∈ℝn×n be a positive semi definite matrix, xi∈ℝn and ai≥0(i=1,2,…). If the series concerned are convergent, the following inequality holds:
(9)(∑i=1+∞‍aixi)TM(∑i=1+∞‍aixi)≤(∑i=1+∞‍ai)∑i=1+∞‍aixiTMxi.Lemma 8 (see [10]).
Assume that nonlinear functionℏ(·):ℝn→ℝn satisfies
(10)(ℏ(x)-U1x)T(ℏ(x)-U2x)≤0,∀x∈ℝn
with U1 and U2 being constant matrices. Then, the following matrix inequality holds:
(11)[xℏ(x)]T[U˘1-U˘2-U˘2TI][xℏ(x)]≤0,
or
(12)xTU˘1x-2xTU˘2ℏ(x)+ℏT(x)ℏ(x)≤0,
where U˘1=(U1TU2+U2TU1)/2 and U˘2=(U1T+U2T)/2.Proof.
It can be verified by simple matrix operations.Lemma 9 (Schur complement [20]).
Given constant matricesΩ1, Ω2, Ω3 where Ω1=Ω1T and Ω2>0, then
(13)Ω1+Ω3TΩ2-1Ω3<0
if and only if
(14)[Ω1Ω3TΩ3-Ω2]<0or[-Ω2Ω3Ω3TΩ1]<0.Hereafter, one denoteτ¯1=max1≤j≤n0{τ1,j}, τ¯2=max1≤j≤n0{τ2,j}, τ_1=min1≤j≤n0{τ1,j}, τ_2=min1≤j≤n0{τ2,j}, and π_=min1≤i≤n0{|πii|}.One also denotes(15)F˘1=(F1TF2+F2TF1)2,F˘2=(F1T+F2T)2,G˘1=(G1TG2+G2TG1)2,G˘2=(G1T+G2T)2,H˘1=(H1TH2+H2TH1)2,H˘2=(H1T+H2T)2,σm=∑ι=m+∞‍μι,μ^=max{μm∣τ_2≤m≤τ¯2-1}.
The following is a sufficient condition for the existence of state-feedback guaranteed cost control laws for the system (2a) and (2b).Theorem 10.
Given a state-feedback controlleru(k)=K(r(k))x(k). If there exist a set of positive definite matrices Pi(i∈𝒩), and two positive definite matrices Q1 and Q2 such that the LMIs (17) hold, then u(k)=K(r(k))x(k) is a guaranteed cost controller for the system (2a) and (2b), and the cost function satisfies the following bound:
(16)J≤xT(0)Pr0x(0)+∑v=-τ1,r0-1‍gT(x(v))Q1g(x(v))+(1-π_)∑ι=τ_1τ¯1-1‍∑v=-ι-1‍gT(x(v))Q1g(x(v))+∑ι=τ2,r0+∞‍μι∑v=-ι-1‍hT(x(v))Q2h(x(v))+(1-π_)μ^∑s=τ_2τ¯2-1‍∑ι=1s-1‍∑v=-ι-1‍hT(x(v))Q2h(x(v)),(17)Φ(i)=[-Pi-Ξ+R1(i)+KT(i)R2K(i)ΘAKT(i)P-iΘTΥ(i)ΣT(i)P-iP-iAK(i)P-iΣ(i)-P-i]<0,i∈𝒩,
where
(18)Ξ=F˘1+G˘1+H˘1,P-i=∑j=1n0‍πijPj,Θ=[F˘2G˘20H˘20],Σ(i)=[B(i)0C(i)0D(i)],Υ(i)=diag{-I,α1Q1-I,1στ2,i=diag-Q1,α2Q2-I,-1στ2,iQ2}
with
(19)α1=(1-π_)(τ¯1-τ_1)+1,α2=στ_2+12μ^(1-π_)(τ¯2-τ_2)(τ¯2+τ_2-3).Proof.
For convenience, we denote(20)𝒜K(i)=[AK(i)B(i)0C(i)0D(i)],ξ(xk,i)=[xT(k)fT(x(k))gT(x(k))gT(x(k-τ1,r(k)))hT(x(k))∑m=τ2,i+∞‍μmhT(x(k-m))]T.
By Lemma9, the inequality (17) is equivalent to
(21)Φ-(i)+𝒜KT(i)P-i𝒜K(i)<0,
where
(22)Φ-(i)=[-Pi-Ξ+R1(i)+KT(i)R2(i)K(i)ΘΘTΥ(i)].
Definexk:ℕ-→ℝn by xk(m)=x(k+m)form∈ℕ-. To proceed the stability analysis, we construct the following Lyapunov-Krasovskii functional for the system (5a) and (5b):
(23)V(xk,k,r(k))=∑i=15‍Vi(xk,k,r(k)),
where
(24)V1(xk,k,r(k))=xT(k)Pr(k)x(k),V2(xk,k,r(k))=∑v=k-τ1,r(k)k-1‍gT(x(v))Q1g(x(v)),V3(xk,k,r(k))=(1-π_)∑ι=τ_1τ¯1-1‍∑v=k-ιk-1‍gT(x(v))Q1g(x(v)),V4(xk,k,r(k))=∑ι=τ2,r(k)+∞‍μι∑v=k-ιk-1‍hT(x(v))Q2h(x(v)),V5(xk,k,r(k))=(1-π_)μ^∑s=τ_2τ¯2-1‍∑ι=1s-1‍∑v=k-ιk-1‍hT(x(v))Q2h(x(v)).
Fori∈𝒩, associated with the closed-loop system (5a) and (5b) we can carry out the following computation:
(25)𝔼[V1(xk+1,k+1,r(k+1))∣xk,r(k)=i]-V1(xk,k,i)=∑j=1n0‍πijxT(k+1)Pjx(k+1)-xT(k)Pix(k)=ξT(xk,i)𝒜KT(i)P-i𝒜K(i)ξ(xk,i)-xT(k)Pix(k);𝔼[V2(xk+1,k+1,r(k+1))∣xk,r(k)=i]-V2(xk,k,i)=∑j=1n0‍πij∑v=k-τ1,j+1k‍gT(x(v))Q1g(x(v))-∑v=k-τ1,ik-1‍gT(x(v))Q1g(x(v))=∑j=1n0‍πijgT(x(k))Q1g(x(k))-gT(x(k-τ1,i))Q1g(x(k-τ1,i))+∑j=1n0‍πij∑v=k-τ1,j+1k-1‍gT(x(v))Q1g(x(v))-∑v=k-τ1,i+1k-1‍gT(x(v))Q1g(x(v))=gT(x(k))Q1g(x(k))-gT(x(k-τ1,i))×Q1g(x(k-τ1,i))+∑j≠i‍πij(∑v=k-τ1,j+1k-1‍gT(x(v))Q1g(x(v))+∑j≠i‍πij-∑v=k-τ1,i+1k-1‍gT(x(v))Q1g(x(v)))≤gT(x(k))Q1g(x(k))-gT(x(k-τ1,i))Q1g(x(k-τ1,i))+∑j≠i‍πij∑v=k-τ¯1+1k-τ_1‍gT(x(v))Q1g(x(v))≤gT(x(k))Q1g(x(k))-gT(x(k-τ1,i))Q1g(x(k-τ1,i))+(1-π_)∑v=k-τ¯1+1k-τ_1‍gT(x(v))Q1g(x(v));𝔼[V3(xk+1,k+1,r(k+1))∣xk,r(k)=i]-V3(xk,k,i)=(1-π_)(∑ι=τ_1τ¯1-1‍∑v=k-ι+1k‍gT(x(v))Q1g(x(v))=(1-π_)-∑ι=τ_1τ¯1-1‍∑v=k-ιk-1‍gT(x(v))Q1g(x(v)))=(1-π_)∑ι=τ_1τ¯1-1‍(gT(x(k))Q1g(x(k))=(1-π_)-gT(x(k-ι))Q1g(x(k-ι)))=(1-π_)(τ¯1-τ_1)gT(x(k))Q1g(x(k))-(1-π_)∑v=k-τ¯1+1k-τ_1‍gT(x(v))Q1g(x(v)),𝔼[V4(xk+1,k+1,r(k+1))∣xk,r(k)=i]-V4(xk,k,i)=∑j=1n0‍πi,j∑ι=τ2,j+∞‍μι∑v=k-ι+1k‍hT(x(v))Q2h(x(v))-∑ι=τ2,i+∞‍μι∑v=k-ιk-1‍hT(x(v))Q2h(x(v))≤∑j≠i‍πi,j(∑ι=τ2,j+∞‍μι∑v=k-ι+1k-1‍hT(x(v))Q2h(x(v))≤∑j≠i‍πi,j-∑ι=τ2,i+∞‍μι∑v=k-ι+1k-1‍hT(x(v))Q2h(x(v)))+στ_2hT(x(k))Q2h(x(k))-∑ι=τ2,i+∞‍μιhT(x(k-ι))Q2h(x(k-ι))≤∑j≠i‍πi,j(∑ι=τ_2+∞‍μι∑v=k-ι+1k-1‍hT(x(v))Q2h(x(v))≤∑j≠i‍πi,j-∑ι=τ¯2+∞‍μι∑v=k-ι+1k-1‍hT(x(v))Q2h(x(v)))+στ_2hT(x(k))Q2h(x(k))-∑ι=τ2,i+∞‍μιhT(x(k-ι))Q2h(x(k-ι))≤(1-π_)∑ι=τ_2τ¯2-1‍μι∑v=k-ι+1k-1‍hT(x(v))Q2h(x(v))+στ_2hT(x(k))Q2h(x(k))-∑ι=τ2,i+∞‍μιhT(x(k-ι))Q2h(x(k-ι))≤(1-π_)μ^∑ι=τ_2τ¯2-1‍∑v=k-ι+1k-1‍hT(x(v))Q2h(x(v))+στ_2hT(x(k))Q2h(x(k))-∑m=τ2,i+∞‍μmhT(x(k-m))Q2h(x(k-m));𝔼[V5(xk+1,k+1,r(k+1))∣xk,r(k)=i]-V5(xk,k,i)=(1-π_)μ^[∑s=τ_2τ¯2-1‍∑ι=1s-1‍∑v=k-ι+1k‍hT(x(v))Q2h(x(v))=(1-π_)μ^-∑s=τ_2τ¯2-1‍∑ι=1s-1‍∑v=k-ιk-1‍hT(x(v))Q2h(x(v))]=(1-π_)μ^[∑s=τ_2τ¯2-1‍∑ι=1s-1‍(hT(x(k))Q2h(x(k))=(1-π_)μ^-hT(x(k-ι))Q2h(x(k-ι)))∑s=τ_2τ¯2-1‍]=(1-π_)μ^[12(τ¯2-τ_2)(τ¯2+τ_2-3)hT(x(k))Q2h∑ι=τ_2τ¯2-1‍=(1-π_)μ^×(x(k))-∑ι=τ_2τ¯2-1‍∑v=k-ι+1k-1‍hT(x(v))Q2h(x(v))].
Therefore, we have(26)𝔼[V(xk+1,k+1,r(k+1))∣xk,r(k)=i]-V(xk,k,i)=∑j=15‍[𝔼[Vj(xk+1,k+1,r(k+1))∣xk,r(k)=i]-Vj(xk,k,i)]≤ξT(xk,i)𝒜KT(i)P-i𝒜K(i)ξ(xk,i)-xT(k)Pix(k)+α1gT(x(k))Q1g(x(k))-gT(x(k-τ1,i))Q1g(x(k-τ1,i))+α2hT(x(k))Q2h(x(k))-∑m=τ2,i+∞‍μmhT(x(k-m))Q2h(x(k-m)),
where α1=(1-π_)(τ¯1-τ_1)+1, α2=στ_2+(1/2)μ^(1-π_)(τ¯2-τ_2)(τ¯2+τ_2-3).
By Lemma7, it is clear that
(27)-∑m=τ2,i+∞‍μmhT(x(k-m))Q2h(x(k-m))≤-1στ2,i∑m=τ2,i+∞‍μmhT(x(k-m))Q2∑m=τ2,i+∞‍μmh(x(k-m)).
Also, from the conditions (2) and Lemma 8, it follows that
(28)xT(k)F˘1x(k)-2xT(k)F˘2f(x(k))+fT(x(k))f(x(k))≤0,xT(k)G˘1x(k)-2xT(k)G˘2g(x(k))+gT(x(k))g(x(k))≤0,xT(k)H˘1x(k)-2xT(k)H˘2h(x(k))+hT(x(k))h(x(k))≤0.
From (26)–(28), it follows readily that
(29)𝔼[V(xk+1,k+1,r(k+1))∣xk,r(k)=i]-V(xk,k,i)≤ξT(xk,i)𝒜KT(i)P-i𝒜K(i)ξ(xk,i)-xT(k)Pix(k)+α1gT(x(k))Q1g(x(k))-gT(x(k-τ1,i))Q1g(x(k-τ1,i))+α2hT(x(k))Q2h(x(k))-1στ2,i∑m=τ2,i+∞‍μmhT(x(k-m))Q2∑m=τ2,i+∞‍μmhT(x(k-m))-(xT(k)F˘1x(k)-2xT(k)F˘2f(x(k))+fT(x(k))f(x(k)))-(xT(k)G˘1x(k)-2xT(k)G˘2g(x(k))+gT(x(k))g(x(k)))-(xT(k)H˘1x(k)-2xT(k)H˘2h(x(k))+hT(x(k))h(x(k)))=ξT(xk,i)[Φ-(i)+𝒜KT(i)P-i𝒜K(i)]ξ(xk,i)-xT(k)[R1(i)+KT(i)R2K(i)]x(k),
which, together with (21), implies
(30)𝔼[V(xk+1,k+1,r(k+1))]-𝔼[V(xk,k,r(k))]≤-𝔼[xT(k)(R1(r(k))+KT(r(k))R2K(r(k)))x(k)].
Therefore,(31)𝔼[V(xk+1,k+1,r(k+1))]-𝔼[V(xk,k,r(k))]≤-λ0𝔼|x(k)|2,
where λ0=mini∈𝒩{λmin(R1(i))}.
Lets be an arbitrary positive integer; then it can be inferred from (31) that
(32)𝔼[V(xs+1,s+1,r(s+1))]-𝔼[V(x0,0,r(0))]≤-λ0∑k=0s‍𝔼[|x(k)|2],
or
(33)𝔼[V(xs+1,s+1,r(s+1))]-V(x0,0,r(0))≤-λ0∑k=0s‍𝔼[|x(k)|2],
which results in
(34)∑k=0s‍𝔼[|x(k)|2]≤1λ0V(x0,0,r(0)).
It can now be concluded that the series ∑k=0+∞𝔼[|x(k)|2] is convergent, and therefore
(35)limk→+∞𝔼[|x(k)|2]=0.
Therefore, the closed-loop system (5a) and (5b) is asymptotically stable in mean square.
On the other hand, for any positive integers, from (30) we have
(36)𝔼[∑k=0s‍(xT(k)R1(r(k))x(k)+uT(k)R2(r(k))u(k))]=𝔼[∑k=0s‍xT(k)(R1(r(k))+KT(r(k))R2(r(k))K(r(k)))𝔼×x(k)+V(xk+1,k+1,r(k+1))-V(xk,k,r(k))∑k=0s‍]-𝔼[V(xs+1,s+1,r(s+1))]+V(x0,0,r(0))≤V(x0,0,r(0)).
Letting s→+∞, we have
(37)J≤V(x0,0,r0);
namely, (16) holds. This completes the proof of the theorem.Theorem10 provides a sufficient condition to determine if a given controller is a guaranteed cost controller. Next, we turn to the design problem of guaranteed cost controller for the system (2a) and (2b). For this, we have the following results.Theorem 11.
Consider the system (2a) and (2b). If there exist a set of positive definite matrices Xi(i∈𝒩), a set of matrices Yi(i∈𝒩), and two positive definite matrices Q1 and Q2 such that the LMIs (39) hold, then u(k)=K(r(k))x(k) with K(i)=YiXi-1 is a guaranteed cost controller for the system (2a) and (2b), and the cost function satisfies the following bound:(38)J≤γ=xT(0)Xr0-1x(0)+∑v=-τ1,r0-1‍gT(x(v))Q1g(x(v))+(1-π_)∑ι=τ_1τ¯1-1‍∑v=-ι-1‍gT(x(v))Q1g(x(v))+∑ι=τ2,r0+∞‍μι∑v=-ι-1‍hT(x(v))Q2h(x(v))+(1-π_)μ^∑s=τ_2τ¯2-1‍∑ι=1s-1‍∑v=-ι-1‍hT(x(v))Q2h(x(v));(39)Φ-(i)=Δ[-XiXiΘXiXiYiT(A(i)Xi+E(i)Yi)TWiΘTXiΥ(i)000ΣT(i)WiXi0Ξ-1000Xi00-R1-1(i)00Yi000-R2-1(i)0WiT(A(i)Xi+E(i)Yi)WiTΣ(i)000-𝒳]<0,where Wi=[πi1I,πi2I,…,πin0I], 𝒳=diag{X1,X2,…,Xn0}, and Θ, Υ(i), Ξ and Σ(i) are defined as in Theorem 10.Proof.
LetPi=Xi-1, Yi=K(i)Pi-1, and ϑ(i)=diag{Pi,I,…,I︷6}. Then, inequality (39) is equivalent to
(40)ϑ(i)Φ-(i)ϑ(i)<0;
namely,
(41)[-PiΘIIKT(i)AKT(i)WiΘTΥ(i)000ΣT(i)WiI0Ξ-1000I00-R1-1(i)00K(i)000-R2-1(i)0WiTAK(i)WiTΣ(i)000-𝒫-1]<0,
where 𝒫=diag{P1,P2,...,Pn0}.
From Lemma9, it follows readily that (41) is equivalent to (21) and is therefore equivalent to (17).
By Theorem10, u(k)=K(r(k))x(k) with K(i)=YiXi-1 is a guaranteed cost controller for the system (2a) and (2b), and the cost function satisfies the bound as shown in (38).Remark 12.
In Theorem11, the bound γ of the cost function depends on the parameters Xi,Q1, and Q2 in addition to the initial value of the state and mode of the system. Next, we will design an optimal state-feedback guaranteed cost controller u(k)=K(r(k))x(k), which minimizes the bound of the guaranteed cost function.Theorem 13.
Consider the system (2a) and (2b) with cost function (4). If the following optimal problem of a linear objective
(42)minβ0,Xi,YiQ1,Q2,Mi(β0+Tr(M1+M2+M3+M4))
subject to LMI constraints
(43)(i)LMI(39)(ii)[-β0xT(0)x(0)-Xr0]<0,(iii)Ωk<0,(k=1,2,3,4)
has a set of solutions β0,Xi(i∈𝒩), Yi(i∈𝒩), Q1, Q2, Mk(k=1,2,3,4), then u(k)=K(r(k))x(k) with K(i)=YiXi-1 is an optimal guaranteed cost controller for the system (2a) and (2b), which minimizes the guaranteed cost (38).Here,(44)Ω1=[-M1N1TQ1Q1N1-Q1],Ω2=[-M2N2TQ1Q1N2-Q1],Ω3=[-M3N3TQ2Q2N3-Q2],Ω4=[-M4N4TQ2Q2N4-Q2],
where Nk(k=1,2,3,4) satisfy
(45)N1N1T=∑v=-τ1,r0-1‍g(x(v))gT(x(v)),N2N2T=(1-π_)∑ι=τ_1τ¯1-1‍∑v=-ι-1‍g(x(v))gT(x(v)),N3N3T=∑ι=τ2,r0+∞‍μι∑v=-ι-1‍h(x(v))hT(x(v)),N4N4T=(1-π_)μ^∑s=τ_2τ¯2-1‍∑ι=1s-1‍∑v=-ι-1‍h(x(v))hT(x(v)).Proof.
According to Theorem11, u(k)=K(r(k))x(k) with K(i)=YiXi-1 is a guaranteed cost controller for the system (2a) and (2b) if LMI (39) has a set of solutions Xi, Yi, Q1, and Q2.
On the other hand, by Lemma9, the inequality in (ii) is equivalent to xT(0)Xr0-1x(0)<β0. Notice that Ω1<0 is equivalent to
(46)N1TQ1N1<M1,(47)∑v=-τ1,r0-1‍gT(x(v))Q1g(x(v))=∑v=-τ1,r0-1‍Tr(gT(x(v))Q1g(x(v)))=∑v=-τ1,r0-1‍Tr(g(x(v))gT(x(v))Q1)=Tr(∑v=-τ1,r0-1‍g(x(v))gT(x(v))Q1)=Tr(N1N1TQ1)=Tr(N1TQ1N1)(Thanksto(46))<Tr(M1).
Similarly, from Ωk<0(k=2,3,4) it follows that
(3)(1-π_)∑ι=τ_1τ¯1-1‍∑v=-ι-1‍g(x(v))gT(x(v))<Tr(M2),∑ι=τ2,r0+∞‍μι∑v=-ι-1‍h(x(v))hT(x(v))≤Tr(M3),(1-π_)μ^∑s=τ_2τ¯2-1‍∑ι=1s-1‍∑v=-ι-1‍h(x(v))hT(x(v))≤Tr(M4).
Accordingly, it follows that γ<λ0+Tr(M1+M2+M3+M4), where γ is defined in (38). Since the optimal problem (42) has a set of solutions, the minimization of the guaranteed cost for the system (2a) and (2b) follows from the minimization of λ0+Tr(M1+M2+M3+M4). The proof of this theorem is completed.


## 4. A Numerical Example

In this section, an example is presented to demonstrate the effectiveness of our main results.Example 1.
For simplicity, consider a two-dimensional system (2a) and (2b) with probability transition matrix Π=[0.60.30.30.10.60.30.10.10.8], and the following matrix parameters:
(50)A(1)=[1.20.20.20.2],B(1)=[0.30.30.2-0.2],C(1)=[-0.20.30.20.3],D(1)=[0.2-0.20.20.2],E(1)=[10],A(2)=[1.50.30.10.2],B(2)=[0.30.20.30.2],C(2)=[0.40.30.20.2],D(2)=[0.30.20.20.3],E(2)=[1.50],A(3)=[20.10.20.3],B(3)=[0.20.20.30.2],C(3)=[0.20.300.2],D(3)=[0.200.20.3],E(3)=[1.20].In addition, the parameters for time delays are listed asτ1,1=6, τ1,2=8, τ1,3=9, τ2,1=9, τ2,2=8, τ2,3=10, and the initial mode of Markov chain is r0=1, and the initial value of the system is x(m)=(1.6969,-0.4770)T for m∈(-∞,0], which is stochastically produced by Matlab.Also, the nonlinear functions are taken as(51)f(x)=g(x)=h(x)=(0.1x1+0.2x2+0.4x1sinx2,0.3x1+0.1x2+0.4x2cosx1)T.It can also be seen from (51) that
(52)F˘1=[-0.06000.05000.0500-0.1100],F˘2=[0.10000.30000.20000.1000].
With the previous parameters, based on Theorem 13 and by using Matlab LMI Toolbox, we solve the linear objective minimization problem (42) and obtain the feasible solutions for Xi, Y1, Q1, Q2, β0, and Mi (the values are omitted for space saving). Here, we just give the corresponding optimal control gain matrices K(1)=-[1.2941,0.3499], K(2)=-[1.1552,0.3687], K(3)=-[1.7817,0.2616], and the minimal upper bound γ=5.3853 of the guaranteed cost. Moreover, the dynamical comparison between the unforced system and closed-loop system is shown in Figure 1.The comparison of state trajectories of the unforced and the controlled systems.
(a)
The state evolution of the unforced system
(b)
The state evolution of the closed-loop system


## 5. Conclusions

In this paper, we have dealt with the guaranteed cost control problem for a class of nonlinear discrete-time systems with Markovian jumping parameters and mode-dependent mixed time delays. The sufficient conditions for the existence of guaranteed cost controllers are established for the system under consideration and related cost function. Furthermore, an LMI-based approach to design the optimal guaranteed cost controller has been formulated to minimize the guaranteed cost of the closed-loop system. A numerical example is also given to illustrate the effectiveness of the proposed methods. It will be interesting to extend the present results to more general cases, for example, the case where the quantized state-feedback is used for stabilization of the system concerned, and the case when the nonlinear stochastic systems are considered with missing measurements. And it would be one of the future research topics.



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