Nonfragile H ∞ Control for Stochastic Systems with Markovian Jumping Parameters and Random Packet Losses

and Applied Analysis 3 Therefore, in this paper, the packet losses phenomena are considered in communication links. As a result, one has u(k) ̸ = u oc and x(k) ̸ = x ic (k), and the relations between them are modeled by using a stochastic method as follows: x ic (k) = ρ k x (k) , u (k) = σ k u oc . (8) Here, {ρ k } and {σ k } are two independent Bernoulli processes. As shown in (8), {ρ k } models the unreliable communication link from the sensor to the controller and {σ k } models the unreliable communication link from the controller to the actuator. Inspired by [27], a natural assumption on {ρ k } and {σ k } can be made as follows: Prob {ρ k = 1} = E {ρ k } = ρ, Prob {ρ k = 0} = 1 − ρ, Prob {σ k = 1} = E {σ k } = σ, Prob {σ k = 0} = 1 − σ, (9) where either ρ or σ is a known constant satisfying ρ ∈ [0, 1] and σ ∈ [0, 1]. Clearly, for {ρ k }, when ρ = 0 (resp., ρ = 1), it means that the communication link from the sensor to the controller fails (resp., successful transmission), and {σ k } also has a similar inference. Throughout this paper, we also assume that the sequences ω(k), {δ k }, {ρ k }, and {σ k } are mutually independent. Clearly, one can get u (k) = σ k ρ k (K α +M α H α (k)N α ) x (k) . (10) In order to address the considered problem, before presenting further results, let us introduce a new Bernoulli process {󰜚 k } satisfying 󰜚 k ≡ σ k ρ k . Then, simple computation yields Prob {󰜚 k = 1} = E {󰜚 k } = 󰜚 = ρσ, Prob {󰜚 k = 0} = 1 − ρσ, (11) which implies that u (k) = σ k ρ k (K α +M α H α (k)N α ) x (k) = 󰜚 k (K α +M α H α (k)N α ) x (k) = 󰜚 (K α +M α H α (k)N α ) x (k) + (󰜚 k − 󰜚) (K α +M α H α (k)N α ) x (k) . (12) Under the control law (12), the resulting closed-loop system can be obtained as (Σ) : x (k + 1) = Ω 1α (k) x (k) + (󰜚 k − 󰜚)Ω 2α (k) x (k) + C α υ (k) + [E α x (k) + F α υ (k)] ω (k) , z (k) = Ω 3α (k) x (k) + (󰜚 k − 󰜚)Ω 4α (k) x (k) + G α υ (k) , (13) where Ω 1α (k) = A α + 󰜚B 1α (K α +M α H α (k)N α ) , Ω 2α (k) = B 1α (K α +M α H α (k)N α ) , Ω 3α (k) = D α + 󰜚B 2α (K α +M α H α (k)N α ) , Ω 4α (k) = B 2α (K α +M α H α (k)N α ) . (14) Before formulating the problem to be investigated, we first introduce the following definition for system (Σ). Definition 2. The closed-loop system in (13) with υ(k) ≡ 0 is said to be stochastically mean square stable (SMSS) if there exists a κ > 0 such that E {‖x (k)‖} 󳨀→ 0 as k 󳨀→ ∞, (15) for any initial condition ‖x(0)‖ < κ. Definition 3. System (Σ) is said to be SMSS with a guaranteed H ∞ performance level γ, if system (Σ) is SMSS according to Definition 2, and the prescribed disturbance attenuation level γ is made small in the feasibility of ‖z (k)‖ E ≤ γ‖υ (k)‖ 2 , (16) for all nonzero ω(k) ∈ l 2 [0,∞) under zero initial conditions, where


Introduction
During the past several decades, stochastic systems have been the main focus of research receiving much attention since realistic models of most engineering systems involve random exogenous disturbances [1,2].As a simple yet significant mathematical model, stochastic systems have come to play a key role in many branches of science and engineering [3].For this reason, many fundamental issues have been extensively addressed for stochastic systems, and consequently fruitful results have been presented in the literature; see, for example, [2,4,5] and the references therein.
In addition to stochastic systems, there have been great efforts in the research of the modeling of dynamic systems subject to random abrupt changes in their parameters [6][7][8].Such random abrupt changes may be caused by various factors, including the switching between economic scenarios, abrupt changes in the operation point for nonlinear plant, and actuator/sensor failure or repairs, to name just a few.Fortunately, Markov jump systems provide a natural framework for modeling these practical systems subject to random abrupt changes.Since the pioneering work on Markov jump systems was introduced in [9], considerable research results related to Markov jump systems or system with Markovian jumping parameters have been presented in terms of a variety of methods.For more details, we refer to the literature [10][11][12][13][14][15][16][17].When Markovian jumping parameters appear in stochastic systems, many control issues have been studied recently by researchers.For example, robust stability and stabilization problems were investigated in [18], passivity-based control problem was addressed in [19], optimal control problems were studied in [12,[20][21][22], and the sliding-mode control problem was solved in [23].
It is worth noting that the controller design methods proposed in the previous literature require two critical assumptions.One is that the controller can be implemented exactly, and the other is that the communication between the physical plant and controller is always perfect.Such two assumptions, however, may not be unreasonable in practice.Firstly, in the implementation of a design controller, uncertainties or inaccuracies do occur because of roundoff errors in numerical computation.Some existing control synthesis methods have proven to be sensitive, or fragile, with respect to small perturbations in controller parameters.Therefore, it is an important question to design a controller, which guarantees that the controller is insensitive to some amount of errors with its gain, that is, the nonfragile or resilient control problem [24].Secondly, a modern control system can hardly work without the help of the networks and the computers and their intercommunication.They bring a lot of advantages, but the existence of network-induced phenomena is unavoidable [25].For instance, packet losses may occur due to the unreliability of the network links.These may limit the scope of the applications of the existing results related to stochastic systems with Markovian jumping parameters.The main purpose of this paper, therefore, is to shorten such a gap.
In this paper, we make the first attempt to deal with the nonfragile  ∞ control for stochastic systems with Markovian jumping parameters and random packet losses.The packet losses phenomena are assumed to exist in communication links between the physical plant and controller.Attention is focused on the design of a nonfragile controller such that the resulting closed-loop system is stochastically mean square stable, and meanwhile a prescribed  ∞ performance is satisfied.Sufficient conditions for the existence of such a controller are given in terms of LMIs.By solving a convex optimization problem, a desired nonfragile controller can be constructed based on the use of standard numerical algorithms [26].
Notation.Throughout this paper, for symmetric matrices , the notation  ≥ 0 (resp.,  > 0) means that the matrix  is positive semidefinite (resp., positive definite);  and 0 represent the identity matrix and zero matrix with appropriate dimension.The notation   represents the transpose of the matrix ; diag{⋅ ⋅ ⋅ } stands for a block-diagonal matrix.In symmetric block matrices or complex matrix expressions, we employ an asterisk ( * ) to represent a term that is induced by symmetry; E{⋅} denotes the expectation operator with respect to some probability measure P; (Ω, F, P) is a probability space; Ω is the sample space, F is the -algebra of subsets of the sample space, and P is the probability measure on F;  2 [0, ∞) is the space of square-summable infinite vector sequences over [0, ∞); | ⋅ | refers to the Euclidean vector norm; ‖ ⋅ ‖ 2 stands for the usual  2 [0, ∞) norm.Matrices, if not explicitly stated, are assumed to have compatible dimensions. + represents {0, 1, 2, . ..}.

Problem Formulations
Consider the following discrete-time stochastic Markov jump system over a probability space (Ω, F, P): (Σ) :  ( + 1) =  (  )  () +  1 (  )  () +  (  )  () + [ (  )  () +  (  )  ()]  () ,  () =  (  )  () +  2 (  )  () +  (  )  () , where () ∈ R  is the system state vector; () ∈ R  is the controlled input; () ∈ R  is the controlled output; () ∈ R  is the exogenous disturbance input that belongs to  2 [0, ∞).For each   , (  ),  1 (  ), (  ), (  ), (  ), (  ),  2 (  ), and (  ) are real constant matrices with appropriate dimensions.() is a one-dimensional zero mean Gaussian white noise sequence on a probability space (Ω, F, P) with In system (Σ), the system mode switching is governed by a discrete-time homogeneous Markov chain {  } ( ∈  + ), which takes values in a finite state space S = {1, 2, . . ., N} with transition probability matrix Π ≜ {  } given by with 0 ≤   ≤ 1, for any ,  ∈ S, and In practice, it is usually of importance to require very accurate controllers to achieve given engineering specifications.However, the resulting closed-loop systems are sensitive to changes in controller gain.In this case, once there are some small perturbations in the controller parameters, the existence of these perturbations may cause a serious deterioration of system performance.Hence, it is imperative to consider the design of nonfragile controllers [24].Consequently, in this paper, we are interested in designing the controller in the following form: where   () is the input of the controller;   is the output of the controller;   are the gain matrices of the controller, which will be determined; Δ  () are real-valued unknown matrices denoting the additive gain variations as follows: where   and   ,  ∈ S, are the known real constant matrices of appropriate dimensions and   () are unknown time-varying matrix functions, which satisfy the following constraint: Remark 1. Normally, under the implicit assumption that the communication between the plant and the controller is perfect, one can readily get that the controlled input () is equivalent to the the output of controller   and the measurement state of the plant () is also equivalent to the input of the controller   ().As noted in the previous section, such an assumption is sometimes unpractical especially under networked environments because of the existence of the packet losses.
Therefore, in this paper, the packet losses phenomena are considered in communication links.As a result, one has () ̸ =   and () ̸ =   (), and the relations between them are modeled by using a stochastic method as follows: Here, {  } and {  } are two independent Bernoulli processes.
As shown in (8), {  } models the unreliable communication link from the sensor to the controller and {  } models the unreliable communication link from the controller to the actuator.Inspired by [27], a natural assumption on {  } and {  } can be made as follows: Prob where either  or  is a known constant satisfying  ∈ [0, 1] and  ∈ [0, 1].Clearly, for {  }, when  = 0 (resp.,  = 1), it means that the communication link from the sensor to the controller fails (resp., successful transmission), and {  } also has a similar inference.Throughout this paper, we also assume that the sequences (), {  }, {  }, and {  } are mutually independent.Clearly, one can get In order to address the considered problem, before presenting further results, let us introduce a new Bernoulli process {  } satisfying   ≡     .Then, simple computation yields which implies that Under the control law (12), the resulting closed-loop system can be obtained as where Before formulating the problem to be investigated, we first introduce the following definition for system (Σ).
Definition 2. The closed-loop system in (13) with () ≡ 0 is said to be stochastically mean square stable (SMSS) if there exists a  > 0 such that for any initial condition ‖(0)‖ < .
Definition 3. System (Σ) is said to be SMSS with a guaranteed  ∞ performance level , if system (Σ) is SMSS according to Definition 2, and the prescribed disturbance attenuation level  is made small in the feasibility of for all nonzero () ∈  2 [0, ∞) under zero initial conditions, where Now, let us state the problems concerned in this paper, which are listed as follows.
Problem I. Consider the stochastic system (Σ), suppose that the controller gain matrices   and the additive gain variations Δ  () are given, and determine under what condition the system (Σ) is SMSS with a guaranteed  ∞ performance level .
Problem II.Consider the system (Σ), and design a nonfragile controller in the form of (5) such that the resulting closedloop system (Σ) is SMSS with a guaranteed  ∞ performance level , in spite of the presence of packet losses phenomena.

Main Results
In this section, we will give an LMI approach to solving the nonfragile  ∞ control problem formulated in the previous section.Before proceeding further, we shall introduce the following lemmas, which will be used in the proof of the main results.
Next, we will show the  ∞ performance analysis of system (Σ).To this end, we also obtain that each   =  ∈ S and Note that It can be verified that Similar to the derivation of ( 27), we apply the Schur complement to (23) and get + ( ( + 1) ,  + 1) −  ( () , ) } < 0.
In the following, we will present a solution to the nonfragile  ∞ controller design problem for system (Σ) based on Theorem 6.The following theorem proposes a sufficient condition for the existence of such a controller for system (Σ).Theorem 7. Consider system (Σ), let scalars  > 0,  > 0 be given, and let matrices  1 ,  2 , and  3 be fixed.Then, there exists an admissible controller in the form of (5) such that the resulting closed-loop system (Σ) is SMSS with a guaranteed  ∞ performance level , in spite of the presence of packet losses phenomena if there exist matrices   > 0,  such that the following LMIs hold for each  ∈ S: where In this case, a suitable nonfragile  ∞ controller in the form of (5) is given by Proof.Introduce the new variables   =     ; then one can find that which implies that Note that It follows from (37) that In view of Lemma 5, one can get that where Using Lemma 4 and combining (33), (39), and (40) result in Then, by pre-and postmultiplying (42) by diag{ − , , , , , , } and its transpose, one has that inequality (23) holds.Therefore, in light of Theorem 6, we can conclude that the resulting closed-loop system is SMSS with a guaranteed  ∞ performance level .This completes the proof.

An Illustrative Example
In this section, an example is used to illustrate the effectiveness of the presented nonfragile controller design method.Consider the discrete-time stochastic Markov jump system (Σ) over a probability space (Ω, F, P) with two modes ( = 1, 2) and the following parameters: Here, our aim is to design a nonfragile controller in the form of ( 5) such that the resulting closed-loop system is SMSS with a guaranteed  ∞ performance level , in spite of the presence of packet losses phenomena.To this end, we suppose that  = 0. (46)

Conclusions
In this paper, we have studied the problem of nonfragile  ∞ control for stochastic systems with Markovian jumping parameters and random packet losses.An LMI approach has been developed to design a nonfragile controller which ensures both the stochastic mean square stability and a prescribed  ∞ performance level for the resulting closedloop systems in the presence of random packet losses.The proposed approach has been illustrated to be effective by an example.It should be pointed out that the states of the system are assumed to be precisely known, but this is difficult to achieve in practice [30][31][32].Therefore, one of our further research topics is to develop nonfragile output feedback controller design methods for stochastic Markov jump systems.