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

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.

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 systems with 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, ℝ^𝑛 and ℝ^𝑛×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 ℝ^𝑛. If 𝐴 is a square matrix, 𝜆_max(𝐴) (resp., 𝜆_min(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

Let 𝑟(𝑘)(𝑘≥0) be a Markov chain taking values in a finite state space 𝒩={1,2,…,𝑛₀} with probability transition matrix Π=(𝜋_𝑖𝑗)_𝑛0×𝑛0 given by Pr𝑟(𝑘+1)=𝑗∣𝑟(𝑘)=𝑖=𝜋_𝑖𝑗,∀𝑖,𝑗∈𝒩, (1) where 𝜋_𝑖𝑗≥0(𝑖,𝑗∈𝒩) is the transition probability from 𝑖 to 𝑗 and ∑^𝑛0_𝑗=1𝜋_𝑖𝑗=1, for all 𝑖∈𝑁.

Consider a discrete-time nonlinear system with 𝑛₀ modes described by the following dynamical equation:𝑥()(𝑟(𝑘𝑥(𝑘)(𝑟(𝑘𝑓(𝑥(𝑘(𝑟(𝑘𝑔𝑥𝑘+1=𝐴))+𝐵))))+𝐶))𝑘−𝜏_1,𝑟(𝑘)(𝑟(𝑘×+𝐷))_{+∞𝑚=𝜏2,𝑟(𝑘)}𝜇_𝑚ℎ(𝑥((𝑟(𝑘𝑢(𝑘),𝑥(𝑚)(𝑚)𝑘−𝑚))+𝐸))=𝜙for𝑚∈ℕ⁻,𝑟(0)=𝑟₀, (2a)(2b)where 𝑥(𝑘)∈ℝ^𝑛 is the state vector; for 𝑟(𝑘)=𝑖∈𝒩, 𝐴(𝑟(𝑘))∈ℝ^𝑛×𝑛, 𝐵(𝑟(𝑘))∈ℝ^𝑛×𝑛, 𝐶(𝑟(𝑘))∈ℝ^𝑛×𝑛, 𝐷(𝑟(𝑘))∈ℝ^𝑛×𝑛∈ℝ^𝑛×𝑛, and 𝐸(𝑟(𝑘))∈ℝ^𝑛×𝑞 are known constant matrices; 𝑓(⋅), 𝑔(⋅) and ℎ(⋅)∶ℝ^𝑛→ℝ^𝑛 are nonlinear vector functions; 𝜇_𝑚(𝑚=1,2,…) are scalar constants; 𝑢(𝑘)∈ℝ^𝑞 is the control input; 𝜏_1,𝑟(𝑘) stands for the mode-dependent discrete time-delay while 𝜏_2,𝑟(𝑘)≥0 describes a mode-dependent lower bound for the distributed time-delay; 𝜙∶ℕ⁻→ℝ^𝑛 is the initial value; and 𝑟(0)=𝑟₀ is the initial mode of the Markov chain.

For nonlinear vector functions 𝑓, 𝑔, ℎ, we assume that 𝑓(𝑥)−𝐹₁𝑥^𝑇𝑓(𝑥)−𝐹₂𝑥≤0,∀𝑥∈ℝ^𝑛,𝑔(𝑥)−𝐺₁𝑥^𝑇𝑔(𝑥)−𝐺₂𝑥≤0,∀𝑥∈ℝ^𝑛,ℎ(𝑥)−𝐻₁𝑥^𝑇ℎ(𝑥)−𝐻₂𝑥≤0,∀𝑥∈ℝ^𝑛, (2) 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 𝐽=𝔼^∞_𝑘=0𝑥^𝑇(𝑘)𝑅₁(𝑟(𝑘))𝑥(𝑘)+𝑢^𝑇(𝑡)𝑅₂(𝑟(𝑘))𝑢(𝑘)∣𝜙(𝑘),𝑟₀, (4) where 𝑅₁(𝑖)>0 and 𝑅₂(𝑖)>0, for all 𝑖∈𝒩.

Now, consider the following state-feedback control law 𝑢(𝑘)=𝐾(𝑟(𝑘))𝑥(𝑘), where 𝐾(𝑖)∈ℝ^𝑞×𝑛(𝑖∈𝒩) are controller gains to be designed. Then, the closed-loop system can be given as follows:𝑥(𝑘+1)=𝐴_𝐾(𝑟(𝑘𝑥(𝑘)(𝑟(𝑘𝑓(𝑥(𝑘(𝑟(𝑘𝑔𝑥))+𝐵))))+𝐶))𝑘−𝜏_1,𝑟(𝑘)+𝐷(𝑟(𝑘))^+∞_{𝑚=𝜏2,𝑟(𝑘)}𝜇_𝑚ℎ(𝑥(,𝑥(𝑚)(𝑚)𝑘−𝑚))=𝜙for𝑚∈ℕ⁻,𝑟(0)=𝑟₀, (5a)(5b)where 𝐴_𝐾(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: lim_𝑘→∞𝔼|𝑥(𝑘)|²=0. (6)

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 𝐽≤𝛾, (7) 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 controller 𝑢(𝑘)=𝐾(𝑟(𝑘))𝑥(𝑘), which achieves as small value of 𝛾 as possible.

Assumption 5. Constant 𝜇_𝑚≥0 satisfies the following convergent conditions: ^+∞_𝑚=1𝜇_𝑚<+∞,^+∞_𝑚=1𝑚𝜇_𝑚<+∞. (8)

Remark 6. Assumption 5 makes sense as they guarantee that the term 𝐷(𝑟(𝑘))∑^+∞_{𝑚=𝜏2,𝑟(𝑘)}𝜇_𝑚ℎ(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]). Let 𝑀∈ℝ^𝑛×𝑛 be a positive semi definite matrix, 𝐱_𝑖∈ℝ^𝑛 and 𝑎_𝑖≥0(𝑖=1,2,…). If the series concerned are convergent, the following inequality holds: ^+∞_𝑖=1𝑎_𝑖𝐱_𝑖^𝑇𝑀^+∞_𝑖=1𝑎_𝑖𝐱_𝑖≤^+∞_𝑖=1𝑎_𝑖^+∞_𝑖=1𝑎_𝑖𝐱^𝑇_𝑖𝑀𝐱_𝑖. (9)

Lemma 8 (see [10]). Assume that nonlinear function ℏ(⋅)∶ℝ^𝑛→ℝ^𝑛 satisfies ℏ(𝑥)−𝑈₁𝑥^𝑇ℏ(𝑥)−𝑈₂𝑥≤0,∀𝑥∈ℝ^𝑛 (10) with 𝑈₁ and 𝑈₂ being constant matrices. Then, the following matrix inequality holds: 𝑥ℏ(𝑥)^𝑇𝑈̆₁−𝑈̆₂−𝑈̆^𝑇₂𝐼𝑥ℏ(𝑥)≤0, (11) or 𝑥^𝑇𝑈̆₁𝑥−2𝑥^𝑇𝑈̆₂ℏ(𝑥)+ℏ^𝑇(𝑥)ℏ(𝑥)≤0, (12) where 𝑈̆₁=(𝑈^𝑇₁𝑈₂+𝑈^𝑇₂𝑈₁)/2 and 𝑈̆₂=(𝑈^𝑇₁+𝑈^𝑇₂)/2.

Proof. It can be verified by simple matrix operations.

Lemma 9 (Schur complement [20]). Given constant matrices Ω₁, Ω₂, Ω₃ where Ω₁=Ω^𝑇₁ and Ω₂>0, then Ω₁+Ω^𝑇₃Ω₂⁻¹Ω₃<0 (13) if and only if Ω₁Ω^𝑇₃Ω₃−Ω₂<0𝑜𝑟−Ω₂Ω₃Ω^𝑇₃Ω₁<0. (14)

Hereafter, one denote 𝜏₁=max_{1≤𝑗≤𝑛0}{𝜏_1,𝑗}, 𝜏₂=max_{1≤𝑗≤𝑛0}{𝜏_2,𝑗}, 𝜏₁=min_{1≤𝑗≤𝑛0}{𝜏_1,𝑗}, 𝜏₂=min_{1≤𝑗≤𝑛0}{𝜏_2,𝑗}, and 𝜋=min_{1≤𝑖≤𝑛0}{|𝜋_𝑖𝑖|}.

One also denotes 𝐹̆₁=𝐹^𝑇₁𝐹₂+𝐹^𝑇₂𝐹₁2,𝐹̆₂=𝐹^𝑇₁+𝐹^𝑇₂2,𝐺̆₁=𝐺^𝑇₁𝐺₂+𝐺^𝑇₂𝐺₁2,𝐺̆₂=𝐺^𝑇₁+𝐺^𝑇₂2,𝐻̆₁=𝐻^𝑇₁𝐻₂+𝐻^𝑇₂𝐻₁2,𝐻̆₂=𝐻^𝑇₁+𝐻^𝑇₂2,𝜎_𝑚=^+∞_𝜄=𝑚𝜇_𝜄,𝜇=max𝜇_𝑚∣𝜏₂≤𝑚≤𝜏₂−1. (15) 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 controller 𝑢(𝑘)=𝐾(𝑟(𝑘))𝑥(𝑘). If there exist a set of positive definite matrices 𝑃_𝑖(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: 𝐽≤𝑥^𝑇(0)𝑃_𝑟0𝑥(0)+⁻¹_{𝑣=−𝜏01,𝑟}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣+))1−𝜋^𝜏1−1_𝜄=𝜏1⁻¹_𝑣=−𝜄𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣+))_{+∞𝜄=𝜏02,𝑟}𝜇_𝜄⁻¹_𝑣=−𝜄ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣+))1−𝜋𝜇^𝜏2−1_𝑠=𝜏2^𝑠−1_𝜄=1⁻¹_𝑣=−𝜄ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣,Φ(𝑖)=))−𝑃_𝑖−Ξ+𝑅₁(𝑖)+𝐾^𝑇(𝑖)𝑅₂𝐾(𝑖)Θ𝐴^𝑇_𝐾(𝑖)𝑃_𝑖Θ^𝑇Υ(𝑖)Σ^𝑇(𝑖)𝑃_𝑖𝑃_𝑖𝐴_𝐾(𝑖)𝑃_𝑖Σ(𝑖)−𝑃_𝑖<0,𝑖∈𝒩, (16)(17) where Ξ=𝐹̆₁+𝐺̆₁+𝐻̆₁,𝑃_𝑖=^𝑛0_𝑗=1𝜋_𝑖𝑗𝑃_𝑗,Θ=𝐹̆₂𝐺̆₂0𝐻̆₂0,Σ(𝑖)=𝐵(𝑖)(𝑖)(𝑖),Υ(𝑖)0𝐶0𝐷=diag−𝐼,𝛼₁𝑄₁−𝐼,−𝑄₁,𝛼₂𝑄₂1−𝐼,−𝜎_𝜏2,𝑖𝑄₂ (18) with 𝛼₁=1−𝜋𝜏₁−𝜏₁𝛼+1,₂=𝜎_𝜏2+12𝜇1−𝜋𝜏₂−𝜏₂𝜏₂+𝜏₂−3. (19)

Proof. For convenience, we denote𝒜_𝐾(𝑖)=𝐴_𝐾(𝑖)𝐵(𝑖)(𝑖)(𝑖),𝜉𝐱0𝐶0𝐷_𝑘,𝑖=𝑥^𝑇(𝑘)𝑓^𝑇(𝑥(𝑘))𝑔^𝑇(𝑥(𝑘))𝑔^𝑇𝑥𝑘−𝜏_1,𝑟(𝑘)ℎ^𝑇(𝑥(𝑘))^+∞_{𝑚=𝜏2,𝑖}𝜇_𝑚ℎ^𝑇(𝑥(𝑘−𝑚))^𝑇. (20)
By Lemma 9, the inequality (17) is equivalent to Φ(𝑖)+𝒜^𝑇_𝐾(𝑖)𝑃_𝑖𝒜_𝐾(𝑖)<0, (21) where Φ(𝑖)=−𝑃_𝑖−Ξ+𝑅₁(𝑖)+𝐾^𝑇(𝑖)𝑅₂(𝑖)𝐾(𝑖)ΘΘ^𝑇Υ(𝑖). (22)
Define 𝐱_𝑘∶ℕ⁻→ℝ^𝑛 by 𝐱_𝑘(𝑚)=𝑥(𝑘+𝑚)for𝑚∈ℕ⁻. To proceed the stability analysis, we construct the following Lyapunov-Krasovskii functional for the system (5a) and (5b): 𝑉𝐱_𝑘,𝑘,𝑟(𝑘)=⁵_𝑖=1𝑉_𝑖𝐱_𝑘,𝑘,𝑟(𝑘), (23) where 𝑉₁𝐱_𝑘,𝑘,𝑟(𝑘)=𝑥^𝑇(𝑘)𝑃_𝑟(𝑘)𝑥(𝑘),𝑉₂𝐱_𝑘,𝑘,𝑟(𝑘)=^𝑘−1_{𝑣=𝑘−𝜏1,𝑟(𝑘)}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣,𝑉))₃𝐱_𝑘,𝑘,𝑟(𝑘)=1−𝜋^𝜏1−1_𝜄=𝜏1^𝑘−1_{𝑣=𝑘−𝜄}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣,𝑉))₄𝐱_𝑘,𝑘,𝑟(𝑘)=^+∞_{𝜄=𝜏2,𝑟(𝑘)}𝜇_𝜄^𝑘−1_{𝑣=𝑘−𝜄}ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣,𝑉))₅𝐱_𝑘,𝑘,𝑟(𝑘)=1−𝜋𝜇^𝜏2−1_𝑠=𝜏2^𝑠−1_𝜄=1^𝑘−1_{𝑣=𝑘−𝜄}ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣)). (24)
For 𝑖∈𝒩, associated with the closed-loop system (5a) and (5b) we can carry out the following computation: 𝔼𝑉₁𝐱_𝑘+1,𝑘+1,𝑟(𝑘+1)∣𝐱_𝑘,𝑟(𝑘)=𝑖−𝑉₁𝐱_𝑘=,𝑘,𝑖_𝑛0𝑗=1𝜋_𝑖𝑗𝑥^𝑇(𝑘+1)𝑃_𝑗𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘)=𝜉^𝑇𝐱_𝐤,𝑖𝒜^𝑇_𝐾(𝑖)𝑃_𝑖𝒜_𝐾(𝑖)𝜉𝐱_𝐤,𝑖−𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘);𝔼𝑉₂𝐱_𝑘+1,𝑘+1,𝑟(𝑘+1)∣𝐱_𝑘,𝑟(𝑘)=𝑖−𝑉₂𝐱_𝑘=,𝑘,𝑖_𝑛0𝑗=1𝜋^𝑘_{𝑖𝑗𝑣=𝑘−𝜏1,𝑗+1}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣−))_{𝑘−1𝑣=𝑘−𝜏1,𝑖}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣=))_𝑛0𝑗=1𝜋_𝑖𝑗𝑔^𝑇(𝑥(𝑘))𝑄₁𝑔(𝑥(𝑘))−𝑔^𝑇𝑥𝑘−𝜏_1,𝑖𝑄₁𝑔𝑥𝑘−𝜏_1,𝑖+^𝑛0_𝑗=1𝜋_𝑖𝑗^𝑘−1_{𝑣=𝑘−𝜏1,𝑗+1}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣−))_{𝑘−1𝑣=𝑘−𝜏1,𝑖+1}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣))=𝑔^𝑇(𝑥(𝑘))𝑄₁𝑔(𝑥(𝑘))−𝑔^𝑇𝑥𝑘−𝜏_1,𝑖×𝑄₁𝑔𝑥𝑘−𝜏_1,𝑖+_𝑗≠𝑖𝜋_𝑖𝑗^𝑘−1_{𝑣=𝑘−𝜏1,𝑗+1}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣−))_{𝑘−1𝑣=𝑘−𝜏1,𝑖+1}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣))≤𝑔^𝑇(𝑥(𝑘))𝑄₁𝑔(𝑥(𝑘))−𝑔^𝑇𝑥𝑘−𝜏_1,𝑖𝑄₁𝑔𝑥𝑘−𝜏_1,𝑖+_𝑗≠𝑖𝜋_𝑖𝑗^𝑘−𝜏1_{𝑣=𝑘−𝜏1+1}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣))≤𝑔^𝑇(𝑥(𝑘))𝑄₁𝑔(𝑥(𝑘))−𝑔^𝑇𝑥𝑘−𝜏_1,𝑖𝑄₁𝑔𝑥𝑘−𝜏_1,𝑖+1−𝜋^𝑘−𝜏1_{𝑣=𝑘−𝜏1+1}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣;𝔼𝑉))₃𝐱_𝑘+1,𝑘+1,𝑟(𝑘+1)∣𝐱_𝑘,𝑟(𝑘)=𝑖−𝑉₃𝐱_𝑘=,𝑘,𝑖1−𝜋^𝜏1−1_𝜄=𝜏1^𝑘_{𝑣=𝑘−𝜄+1}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣−))_{𝜏1−1𝜄=𝜏1𝑘−1𝑣=𝑘−𝜄}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣=))1−𝜋^𝜏1−1_𝜄=𝜏1𝑔^𝑇(𝑥(𝑘))𝑄₁𝑔(𝑥(𝑘))−𝑔^𝑇(𝑥(𝑘−𝜄))𝑄₁𝑔(𝑥(=𝑘−𝜄))1−𝜋𝜏₁−𝜏₁𝑔^𝑇(𝑥(𝑘))𝑄₁𝑔(𝑥(𝑘−))1−𝜋^𝑘−𝜏1_{𝑣=𝑘−𝜏1+1}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣,𝔼𝑉))₄𝐱_𝑘+1,𝑘+1,𝑟(𝑘+1)∣𝐱_𝑘,𝑟(𝑘)=𝑖−𝑉₄𝐱_𝑘=,𝑘,𝑖_𝑛0𝑗=1𝜋_𝑖,𝑗^+∞_{𝜄=𝜏2,𝑗}𝜇_𝜄^𝑘_{𝑣=𝑘−𝜄+1}ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣−))_{+∞𝜄=𝜏2,𝑖}𝜇_𝜄^𝑘−1_{𝑣=𝑘−𝜄}ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣))≤_𝑗≠𝑖𝜋_𝑖,𝑗^+∞_{𝜄=𝜏2,𝑗}𝜇_𝜄^𝑘−1_{𝑣=𝑘−𝜄+1}ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣−))_{+∞𝜄=𝜏2,𝑖}𝜇_𝜄^𝑘−1_{𝑣=𝑘−𝜄+1}ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣))+𝜎_𝜏2ℎ^𝑇(𝑥(𝑘))𝑄₂ℎ(𝑥(𝑘−))_{+∞𝜄=𝜏2,𝑖}𝜇_𝜄ℎ^𝑇(𝑥(𝑘−𝜄))𝑄₂ℎ(𝑥(𝑘−𝜄))≤_𝑗≠𝑖𝜋_𝑖,𝑗^+∞_𝜄=𝜏2𝜇_𝜄^𝑘−1_{𝑣=𝑘−𝜄+1}ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣−))_{+∞𝜄=𝜏2}𝜇_𝜄^𝑘−1_{𝑣=𝑘−𝜄+1}ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣))+𝜎_𝜏2ℎ^𝑇(𝑥(𝑘))𝑄₂ℎ(𝑥(𝑘−))_{+∞𝜄=𝜏2,𝑖}𝜇_𝜄ℎ^𝑇(𝑥(𝑘−𝜄))𝑄₂ℎ(𝑥(≤𝑘−𝜄))1−𝜋^𝜏2−1_𝜄=𝜏2𝜇_𝜄^𝑘−1_{𝑣=𝑘−𝜄+1}ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣))+𝜎_𝜏2ℎ^𝑇(𝑥(𝑘))𝑄₂ℎ(𝑥(𝑘−))_{+∞𝜄=𝜏2,𝑖}𝜇_𝜄ℎ^𝑇(𝑥(𝑘−𝜄))𝑄₂ℎ(𝑥(≤𝑘−𝜄))1−𝜋𝜇^𝜏2−1_𝜄=𝜏2^𝑘−1_{𝑣=𝑘−𝜄+1}ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣))+𝜎_𝜏2ℎ^𝑇(𝑥(𝑘))𝑄₂ℎ(𝑥(𝑘−))_{+∞𝑚=𝜏2,𝑖}𝜇_𝑚ℎ^𝑇(𝑥(𝑘−𝑚))𝑄₂ℎ(𝑥(;𝔼𝑉𝑘−𝑚))₅𝐱_𝑘+1,𝑘+1,𝑟(𝑘+1)∣𝐱_𝑘,𝑟(𝑘)=𝑖−𝑉₅𝐱_𝑘=,𝑘,𝑖1−𝜋𝜇^𝜏2−1_𝑠=𝜏2^𝑠−1𝑘_{𝜄=1𝑣=𝑘−𝜄+1}ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣−))_{𝜏2−1𝑠=𝜏2𝑠−1𝜄=1𝑘−1𝑣=𝑘−𝜄}ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣=))1−𝜋𝜇^𝜏2−1_𝑠=𝜏2^𝑠−1_𝜄=1ℎ^𝑇(𝑥(𝑘))𝑄₂ℎ(𝑥(𝑘))−ℎ^𝑇(𝑥(𝑘−𝜄))𝑄₂ℎ(𝑥(=𝑘−𝜄))1−𝜋𝜇12𝜏₂−𝜏₂𝜏₂+𝜏₂−3ℎ^𝑇(𝑥(𝑘))𝑄₂ℎ×(𝑥(𝑘))−^𝜏2−1_𝜄=𝜏2^𝑘−1_{𝑣=𝑘−𝜄+1}ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣)). (25)
Therefore, we have 𝔼𝑉𝐱_𝑘+1,𝑘+1,𝑟(𝑘+1)∣𝐱_𝑘,𝑟(𝑘)=𝑖−𝑉𝐱_𝑘=,𝑘,𝑖_5𝑗=1𝔼𝑉_𝑗𝐱_𝑘+1,𝑘+1,𝑟(𝑘+1)∣𝐱_𝑘,𝑟(𝑘)=𝑖−𝑉_𝑗𝐱_𝑘,𝑘,𝑖≤𝜉^𝑇𝐱_𝐤,𝑖𝒜^𝑇_𝐾(𝑖)𝑃_𝑖𝒜_𝐾(𝑖)𝜉𝐱_𝐤,𝑖−𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘)+𝛼₁𝑔^𝑇(𝑥(𝑘))𝑄₁𝑔(𝑥(𝑘))−𝑔^𝑇𝑥𝑘−𝜏_1,𝑖𝑄₁𝑔𝑥𝑘−𝜏_1,𝑖+𝛼₂ℎ^𝑇(𝑥(𝑘))𝑄₂ℎ(𝑥(𝑘−))_{+∞𝑚=𝜏2,𝑖}𝜇_𝑚ℎ^𝑇(𝑥(𝑘−𝑚))𝑄₂ℎ(𝑥(𝑘−𝑚)), (26) where 𝛼₁=(1−𝜋)(𝜏₁−𝜏₁)+1, 𝛼₂=𝜎_𝜏2+(1/2)𝜇(1−𝜋)(𝜏₂−𝜏₂)(𝜏₂+𝜏₂−3).
By Lemma 7, it is clear that −^+∞_{𝑚=𝜏2,𝑖}𝜇_𝑚ℎ^𝑇(𝑥(𝑘−𝑚))𝑄₂ℎ(𝑥(1𝑘−𝑚))≤−𝜎_𝜏2,𝑖^+∞_{𝑚=𝜏2,𝑖}𝜇_𝑚ℎ^𝑇(𝑥(𝑘−𝑚))𝑄₂^+∞_{𝑚=𝜏2,𝑖}𝜇_𝑚ℎ(𝑥(𝑘−𝑚)). (27)
Also, from the conditions (2) and Lemma 8, it follows that 𝑥^𝑇(𝑘)𝐹̆₁𝑥(𝑘)−2𝑥^𝑇(𝑘)𝐹̆₂𝑓(𝑥(𝑘))+𝑓^𝑇(𝑥(𝑘𝑓(𝑥(𝑘𝑥))))≤0,_𝑇(𝑘)𝐺̆₁𝑥(𝑘)−2𝑥^𝑇(𝑘)𝐺̆₂𝑔(𝑥(𝑘))+𝑔^𝑇(𝑥(𝑘𝑔(𝑥(𝑘𝑥))))≤0,_𝑇(𝑘)𝐻̆₁𝑥(𝑘)−2𝑥^𝑇(𝑘)𝐻̆₂ℎ(𝑥(𝑘))+ℎ^𝑇(𝑥(𝑘))ℎ(𝑥(𝑘))≤0. (28) From (26)–(28), it follows readily that 𝔼𝑉𝐱_𝑘+1,𝑘+1,𝑟(𝑘+1)∣𝐱_𝑘,𝑟(𝑘)=𝑖−𝑉𝐱_𝑘,𝑘,𝑖≤𝜉^𝑇𝐱_𝐤,𝑖𝒜^𝑇_𝐾(𝑖)𝑃_𝑖𝒜_𝐾(𝑖)𝜉𝐱_𝐤,𝑖−𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘)+𝛼₁𝑔^𝑇(𝑥(𝑘))𝑄₁𝑔(𝑥(𝑘))−𝑔^𝑇𝑥𝑘−𝜏_1,𝑖𝑄₁𝑔𝑥𝑘−𝜏_1,𝑖+𝛼₂ℎ^𝑇(𝑥(𝑘))𝑄₂ℎ(𝑥(𝑘−1))𝜎_𝜏2,𝑖^+∞_{𝑚=𝜏2,𝑖}𝜇_𝑚ℎ^𝑇(𝑥(𝑘−𝑚))𝑄₂^+∞_{𝑚=𝜏2,𝑖}𝜇_𝑚ℎ^𝑇(𝑥(−𝑥𝑘−𝑚))_𝑇(𝑘)𝐹̆₁𝑥(𝑘)−2𝑥^𝑇(𝑘)𝐹̆₂𝑓(𝑥(𝑘))+𝑓^𝑇(𝑥(𝑘𝑓(𝑥(𝑘−𝑥))))_𝑇(𝑘)𝐺̆₁𝑥(𝑘)−2𝑥^𝑇(𝑘)𝐺̆₂𝑔(𝑥(𝑘))+𝑔^𝑇(𝑥(𝑘𝑔(𝑥(𝑘−𝑥))))_𝑇(𝑘)𝐻̆₁𝑥(𝑘)−2𝑥^𝑇(𝑘)𝐻̆₂ℎ(𝑥(𝑘))+ℎ^𝑇(𝑥(𝑘))ℎ(𝑥(𝑘))=𝜉^𝑇𝐱_𝐤,𝑖Φ(𝑖)+𝒜^𝑇_𝐾(𝑖)𝑃_𝑖𝒜_𝐾(𝑖)𝜉𝐱_𝐤,𝑖−𝑥^𝑇(𝑘)𝑅₁(𝑖)+𝐾^𝑇(𝑖)𝑅₂𝐾(𝑖)𝑥(𝑘), (29) which, together with (21), implies 𝔼𝑉𝐱_𝑘+1,𝑘+1,𝑟(𝑘+1)−𝔼𝑉𝐱_𝑘(𝑘)𝑥,𝑘,𝑟≤−𝔼^𝑇(𝑘)𝑅₁(𝑟(𝑘))+𝐾^𝑇(𝑟(𝑘))𝑅₂𝐾(𝑟(𝑘))𝑥(𝑘). (30)
Therefore, 𝔼𝑉𝐱_𝑘+1,𝑘+1,𝑟(𝑘+1)−𝔼𝑉𝐱_𝑘,𝑘,𝑟(𝑘)≤−𝜆₀𝔼|𝑥(𝑘)|², (31) where 𝜆₀=min_𝑖∈𝒩{𝜆_min(𝑅₁(𝑖))}.
Let 𝑠 be an arbitrary positive integer; then it can be inferred from (31) that 𝔼𝑉𝐱_𝑠+1,𝑠+1,𝑟(𝑠+1)−𝔼𝑉𝐱₀,0,𝑟(0)≤−𝜆₀^𝑠_𝑘=0𝔼|𝑥(𝑘)|², (32) or 𝔼𝑉𝐱_𝑠+1,𝑠+1,𝑟(𝑠+1)−𝑉𝐱₀,0,𝑟(0)≤−𝜆₀^𝑠_𝑘=0𝔼|𝑥(𝑘)|², (33) which results in ^𝑠_𝑘=0𝔼|𝑥(𝑘)|²≤1𝜆₀𝑉𝐱₀,0,𝑟(0). (34) It can now be concluded that the series ∑^+∞_𝑘=0𝔼[|𝑥(𝑘)|²] is convergent, and therefore lim_𝑘→+∞𝔼|𝑥(𝑘)|²=0. (35)
Therefore, the closed-loop system (5a) and (5b) is asymptotically stable in mean square.
On the other hand, for any positive integer 𝑠, from (30) we have 𝔼^𝑠_𝑘=0𝑥^𝑇(𝑘)𝑅₁(𝑟(𝑘))𝑥(𝑘)+𝑢^𝑇(𝑘)𝑅₂(𝑟(𝑘))𝑢(𝑘)=𝔼^𝑠_𝑘=0𝑥^𝑇(𝑘)𝑅₁(𝑟(𝑘))+𝐾^𝑇(𝑟(𝑘))𝑅₂(𝑟(𝑘𝐾(𝑟(𝑘(𝑘)𝐱))))×𝑥+𝑉_𝑘+1,𝑘+1,𝑟(𝑘+1)−𝑉𝐱_𝑘(𝑘𝑉𝐱,𝑘,𝑟)−𝔼_𝑠+1,𝑠+1,𝑟(𝑠+1)+𝑉𝐱₀(0)𝑥,0,𝑟≤𝑉₀,0,𝑟(0). (36) Letting 𝑠→+∞, we have 𝐽≤𝑉𝑥₀,0,𝑟₀; (37) namely, (16) holds. This completes the proof of the theorem.

Theorem 10 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 𝑋_𝑖(𝑖∈𝒩), a set of matrices 𝑌_𝑖(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)=𝑌_𝑖𝑋_𝑖⁻¹ is a guaranteed cost controller for the system (2a) and (2b), and the cost function satisfies the following bound:𝐽≤𝛾=𝑥^𝑇(0)𝑋_𝑟⁻¹₀𝑥(0)+⁻¹_{𝑣=−𝜏01,𝑟}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣))+1−𝜋^𝜏1−1_𝜄=𝜏1⁻¹_𝑣=−𝜄𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣+))_{+∞𝜄=𝜏02,𝑟}𝜇_𝜄⁻¹_𝑣=−𝜄ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣))+1−𝜋𝜇^𝜏2−1_𝑠=𝜏2^𝑠−1_𝜄=1⁻¹_𝑣=−𝜄ℎ^𝑇(𝑥(𝑣))𝑄₂ℎ(𝑥(𝑣));Φ(𝑖)=^Δ−𝑋_𝑖𝑋_𝑖Θ𝑋_𝑖𝑋_𝑖𝑌^𝑇_𝑖𝐴(𝑖)𝑋_𝑖+𝐸(𝑖)𝑌_𝑖^𝑇𝑊_𝑖Θ^𝑇𝑋_𝑖Υ(𝑖)000Σ^𝑇(𝑖)𝑊_𝑖𝑋_𝑖0Ξ⁻¹𝑋000_𝑖00−𝑅₁⁻¹(𝑖)𝑌00_𝑖000−𝑅₂⁻¹(𝑖)0𝑊^𝑇_𝑖𝐴(𝑖)𝑋_𝑖+𝐸(𝑖)𝑌_𝑖𝑊^𝑇_𝑖Σ(𝑖)000−𝒳<0, (38)(39)where 𝑊_𝑖=[√𝜋_𝑖1𝐼,√𝜋_𝑖2𝐼,…,𝜋_𝑖𝑛0𝐼], 𝒳=diag{𝑋₁,𝑋₂,…,𝑋_𝑛0}, and Θ, Υ(𝑖), Ξ and Σ(𝑖) are defined as in Theorem 10.

Proof. Let 𝑃_𝑖=𝑋_𝑖⁻¹, 𝑌_𝑖=𝐾(𝑖)𝑃_𝑖⁻¹, and 𝜗(𝑖)=diag{𝑃_𝑖,𝐼,…⁶,𝐼}. Then, inequality (39) is equivalent to 𝜗(𝑖)Φ(𝑖)𝜗(𝑖)<0; (40) namely, −𝑃_𝑖Θ𝐼𝐼𝐾^𝑇(𝑖)𝐴^𝑇_𝐾(𝑖)𝑊_𝑖Θ^𝑇Υ(𝑖)000Σ^𝑇(𝑖)𝑊_𝑖𝐼0Ξ⁻¹000𝐼00−𝑅₁⁻¹(𝑖)𝐾(𝑖)00000−𝑅₂⁻¹(𝑖)0𝑊^𝑇_𝑖𝐴_𝐾(𝑖)𝑊^𝑇_𝑖Σ(𝑖)000−𝒫⁻¹<0, (41) where 𝒫=diag{𝑃₁,𝑃₂,...,𝑃_𝑛0}.
From Lemma 9, it follows readily that (41) is equivalent to (21) and is therefore equivalent to (17).
By Theorem 10, 𝑢(𝑘)=𝐾(𝑟(𝑘))𝑥(𝑘) with 𝐾(𝑖)=𝑌_𝑖𝑋_𝑖⁻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 Theorem 11, the bound 𝛾 of the cost function depends on the parameters 𝑋_𝑖,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 min_{𝛽0,𝑋𝑖,𝑌𝑖𝑄1,𝑄2,𝑀𝑖}𝛽₀+Tr𝑀₁+𝑀₂+𝑀₃+𝑀₄ (42) subject to LMI constraints (i)()()𝐿𝑀𝐼39ii−𝛽₀𝑥^𝑇(0)𝑥(0)−𝑋_𝑟0()Ω<0,iii_𝑘<0,(𝑘=1,2,3,4) (43) has a set of solutions 𝛽₀,𝑋_𝑖(𝑖∈𝒩), 𝑌_𝑖(𝑖∈𝒩), 𝑄₁, 𝑄₂, 𝑀_𝑘(𝑘=1,2,3,4), then 𝑢(𝑘)=𝐾(𝑟(𝑘))𝑥(𝑘) with 𝐾(𝑖)=𝑌_𝑖𝑋_𝑖⁻¹ is an optimal guaranteed cost controller for the system (2a) and (2b), which minimizes the guaranteed cost (38).

Here, Ω₁=−𝑀₁𝑁^𝑇₁𝑄₁𝑄₁𝑁₁−𝑄₁,Ω₂=−𝑀₂𝑁^𝑇₂𝑄₁𝑄₁𝑁₂−𝑄₁,Ω₃=−𝑀₃𝑁^𝑇₃𝑄₂𝑄₂𝑁₃−𝑄₂,Ω₄=−𝑀₄𝑁^𝑇₄𝑄₂𝑄₂𝑁₄−𝑄₂, (44) where 𝑁_𝑘(𝑘=1,2,3,4) satisfy 𝑁₁𝑁^𝑇₁=⁻¹_{𝑣=−𝜏01,𝑟}𝑔(𝑥(𝑣))𝑔^𝑇(𝑥(𝑣,𝑁))₂𝑁^𝑇₂=1−𝜋^𝜏1−1_𝜄=𝜏1⁻¹_𝑣=−𝜄𝑔(𝑥(𝑣))𝑔^𝑇(𝑥(𝑣,𝑁))₃𝑁^𝑇₃=^+∞_{𝜄=𝜏02,𝑟}𝜇_𝜄⁻¹_𝑣=−𝜄ℎ(𝑥(𝑣))ℎ^𝑇(𝑥(𝑣,𝑁))₄𝑁^𝑇₄=1−𝜋𝜇^𝜏2−1_𝑠=𝜏2^𝑠−1_𝜄=1⁻¹_𝑣=−𝜄ℎ(𝑥(𝑣))ℎ^𝑇(𝑥(𝑣)). (45)

Proof. According to Theorem 11, 𝑢(𝑘)=𝐾(𝑟(𝑘))𝑥(𝑘) with 𝐾(𝑖)=𝑌_𝑖𝑋_𝑖⁻¹ is a guaranteed cost controller for the system (2a) and (2b) if LMI (39) has a set of solutions 𝑋_𝑖, 𝑌_𝑖, 𝑄₁, and 𝑄₂.
On the other hand, by Lemma 9, the inequality in (ii) is equivalent to 𝑥^𝑇(0)𝑋_𝑟⁻¹₀𝑥(0)<𝛽₀. Notice that Ω₁<0 is equivalent to 𝑁^𝑇₁𝑄₁𝑁₁<𝑀₁,_{−1𝑣=−𝜏01,𝑟}𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣=))_{−1𝑣=−𝜏01,𝑟}Tr𝑔^𝑇(𝑥(𝑣))𝑄₁𝑔(𝑥(𝑣=))_{−1𝑣=−𝜏01,𝑟}Tr𝑔(𝑥(𝑣))𝑔^𝑇(𝑥(𝑣))𝑄₁=Tr⁻¹_{𝑣=−𝜏01,𝑟}𝑔(𝑥(𝑣))𝑔^𝑇(𝑥(𝑣))𝑄₁=Tr𝑁₁𝑁^𝑇₁𝑄₁=Tr𝑁^𝑇₁𝑄₁𝑁₁((𝑀Thanksto46))<Tr₁. (46)(47) Similarly, from Ω_𝑘<0(𝑘=2,3,4) it follows that 1−𝜋^𝜏1−1_𝜄=𝜏1⁻¹_𝑣=−𝜄𝑔(𝑥(𝑣))𝑔^𝑇(𝑥(𝑣))<Tr𝑀₂,_{+∞𝜄=𝜏02,𝑟}𝜇_𝜄⁻¹_𝑣=−𝜄ℎ(𝑥(𝑣))ℎ^𝑇(𝑥(𝑣))≤Tr𝑀₃,1−𝜋𝜇^𝜏2−1_𝑠=𝜏2^𝑠−1_𝜄=1⁻¹_𝑣=−𝜄ℎ(𝑥(𝑣))ℎ^𝑇(𝑥(𝑣))≤Tr𝑀₄. (3) 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=^{0.60.30.30.10.60.3}_0.10.10.8, and the following matrix parameters: 𝐴(1)=(1)=,𝐶(1)=(1)=,𝐸(1)=10,𝐴(2)=(2)=,𝐶(2)=(2)=,𝐸(2)=0,𝐴(3)=(3)=,𝐶(3)=(3)=,𝐸(3)=0.1.20.20.20.2,𝐵0.30.30.2−0.2−0.20.30.20.3,𝐷0.2−0.20.20.21.50.30.10.2,𝐵0.30.20.30.20.40.30.20.2,𝐷0.30.20.20.31.520.10.20.3,𝐵0.20.20.30.20.20.300.2,𝐷0.200.20.31.2 (50)

In addition, the parameters for time delays are listed as 𝜏_1,1=6, 𝜏_1,2=8, 𝜏_1,3=9, 𝜏_2,1=9, 𝜏_2,2=8, 𝜏_2,3=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 𝑓(𝑥)(𝑥)(𝑥)==𝑔=ℎ0.1𝑥₁+0.2𝑥₂+0.4𝑥₁sin𝑥₂,0.3𝑥₁+0.1𝑥₂+0.4𝑥₂cos𝑥₁^𝑇. (51)

It can also be seen from (51) that 𝐹̆₁=,𝐹̆−0.06000.05000.0500−0.1100²=0.10000.30000.20000.1000. (52) 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 𝑋_𝑖, 𝑌₁, 𝑄₁, 𝑄₂, 𝛽₀, and 𝑀_𝑖 (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.

(a) The state evolution of the unforced system

(b) The state evolution of the closed-loop system

Figure 1 

The comparison of state trajectories of the unforced and the controlled systems.

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.

Acknowledgments

This work was supported in part by the International Travel Grant 2011/R3 sponsored by the Royal Society of the U.K., the National Natural Science Foundation of China under Grant 61074129, and the Natural Science Foundation of Jiangsu Province of China under Grant BK2012682.