#### Abstract

This paper investigates the observer-based controller design problem for a class of nonlinear networked control systems with random time-delays. The nonlinearity is assumed to satisfy a global Lipschitz condition and two dependent Markov chains are employed to describe the time-delay from sensor to controller (S-C delay) and the time-delay from controller to actuator (C-A delay), respectively. The transition probabilities of S-C delay and C-A delay are both assumed to be partly inaccessible. Sufficient conditions on the stochastic stability for the closed-loop systems are obtained by constructing proper Lyapunov functional. The methods of calculating the controller and the observer gain matrix are also given. Two numerical examples are used to illustrate the effectiveness of the proposed method.

#### 1. Introduction

Networked control systems (NCSs) are spatially distributed systems where the communication between sensor, controller, and actuator is carried out by a shared band limited digital network [1, 2]. NCSs are used in a wide range of areas such as robots, industrial manufacturing plants, and remote surgery due to their advantages in practical applications, for example, flexible architectures, the reduced weight, simple installation, and maintenance as well as high flexibility and reliability [3, 4]. However, the communication networks also present some constraints such as time-delays and packet dropouts result from the limited bandwidth. It is generally known that the time-delay maybe degrades the performance or even causes instability [5, 6].

The Markov chain which is a discrete-time stochastic process with the Markov property can be effectively used to model the time-delay in NCSs. The random time-delays in NCSs modeled as Markov chains have been researched in the past several years, and many results have been reported [7–17]. In [7], the time-delay of NCSs was modeled as a Markov chain, and further a LQG optimal controller design method was proposed. In [8], the NCSs were molded as Markov jump linear systems (MJLSs) where the S-C delay was molded as a finite state Markov chain, and a V-K iteration method was proposed to get a stabilizing controller. In [9], a buffer was added ahead of the actuator, and the time-delays from sensor to actuator were lumped together which was molded as a Markov chain, and then the mean-square stability of the closed-loop system was derived. In [10–12], for the NCSs with S-C delay, the problem of control was investigated using the Bounded Real Lemma and the Markov jump theory. In [13, 14], the S-C delay and C-A delay were modeled as two independent Markov chains. The resulting closed-loop systems were transformed to control systems which contain two Markov chains. The sufficient and necessary conditions for the stochastic stability of the resulting closed-loop systems were established, and the mode-dependent state feedback and output feedback controller were designed, respectively.

The transition probabilities of time-delays in [7–14] were assumed to be completely accessible. However, in practical applications, this assumption is too ideal and hence will limit the application of the derived results due to the difficulty in obtaining all the transition probabilities of time-delay precisely. Some results have been obtained when the transition probabilities of the time-delays (data packet dropout) are partly inaccessible. In [15], the control problem was investigated for the NCSs with random data packet dropouts. The closed-loop systems were modeled as MJLSs with four modes and partly inaccessible transition probabilities. In [16, 17], the closed-loop systems were modeled as MJLSs with partly inaccessible transition probabilities of the S-C delay, and the stabilization controller was designed though the linear matrix inequality (LMI) method. In [18], the transition probabilities of the time-delay were assumed to be partly inaccessible, and the fault-tolerant controller for the discrete-time NCSs was designed. Unfortunately, in [16–18], only the S-C delay was considered and an improved controller should take both S-C delay and C-A delay into consideration.

It is well known that nonlinearities usually exist in practical systems. Hence, research about nonlinear NCSs is important in both application and theory. To the best of the authors’ knowledge, up to now, involving both S-C delay and C-A delay to design the controller for nonlinear NCSs when transition probabilities of S-C delay and C-A delay are both partly inaccessible has not been investigated, which motivates our investigation.

In this paper, we propose two controller design methods for a kind of nonlinear NCSs with both S-C delay and C-A delay based on observer. Compared to the previous relevant works, the main contribution of this paper is that the proposed methods can deal with the situations of both complete accessible transition probabilities and partly inaccessible transition probabilities. The rest of this paper is organized as follows. The closed-loop system model with Markov delays is obtained in Section 2. The main results and proofs are given in Section 3. Section 4 presents the simulation results, and the conclusions are provided in Section 5.

#### 2. Problem Formulation

The configuration of the NCSs considering time-delays is depicted in Figure 1 where and denote the S-C delay and C-A delay, respectively.

In this paper, and are modeled as two homogeneous Markov chains which take value in the set and , and their transition probability matrices are and , respectively. That is, and jump from mode to and from to , respectively, with probabilities and , which are defined by , , where , , and , , for all and .

In this paper, the transition probabilities of and are both considered to be partly accessible; that is, some elements in matrix and are unknown. For notational clarity, , we denote with , . Moreover, if , it is further described as (), and is described as , where is the number of elements in the set .

Similarly, , we denote with , . If , it is further described as () and is described as , where is the number of elements in the set .

Considering the following controlled plant after sampling,where is the state vector, is the control input, and is the measured output. , , and are known real constant matrices with appropriate dimensions. is a nonlinear vector function which satisfies the following global Lipschitz conditions:where is a known real scalar.

The dynamic observer-based control scheme is given bywhere , , and are the state vector, control input, and output vector of the observer, respectively. is the controller gain and is the observer gain.

*Remark 1. *It should be pointed out that the control input of the controlled plant (1) is different from the control input of the observer (3) due to the existence of the C-A delay , while, in the most of the observer-based controller design problem, and were assumed to be identical.

Define the state estimate errors as Substituting (3) and (4) into (1) and (5), the closed-loop system can be obtained as follows:where .

By defining , the closed-loop system (6) can be written in a compact form as follows:where

*Definition 2 (see [13]). *System (7) is stochastically stable if, for every finite and , , there exists a finite matrix such thatIn this paper, our objective is to design the dynamic observer-based control scheme (3) and (4), such that the closed-loop system (7) is stochastically stable on condition that the transition probabilities of and are both partly inaccessible.

#### 3. Main Results and Proofs

In this section, we will present the main results. To proceed, we will need the following three lemmas.

Lemma 3 (see [19]). *For any positive-definite matrix , scalars satisfying and vector function , one has .*

Lemma 4 (see [20], S-procedure). *Let be symmetric matrices. The conditions on , s.t. hold if there exist scalars such that .*

Lemma 5 (see [21]). *For given scalars , and matrix , the following inequality always holds:*

In the following, a sufficient condition such that the closed-loop system (7) is stochastically stable will be derived.

Theorem 6. *Taking the controller gain matrix and observer gain matrix , if there exist positive-definite matrices , , , , , , , , , , , and scalar , such that the following inequalitywhere holds for all and , the closed-loop system (7) is stochastically stable.*

*Proof. *Choose the Lyapunov function candidate as , whereIn the following, we denote as when and . Then along the solution of system (7), we have It is noticed thatTherefore, we obtain Note that We can obtain the following inequality by Lemma 3:Similarly, we have From (14)–(19), we havewhereIt follows from (2) that which imply thatBy the well-known S-procedure, that is, Lemma 4, we can get with constraints (23) holding if there exist nonnegative real scalars , such thatFrom (24), we havewhere

From (25), we can see that, for any ,According to Definition 2, the closed-loop system (7) is stochastically stable, which completes the proof.

Theorem 7. *Taking the controller gain matrix and observer gain matrix , the closed-loop system (7) is stochastically stable if there exist positive-definite matrices , , , , , , , , , , , , , , , , and scalar , such thatwherehold for all and .*

*Proof. *Making use of Lemma 5, we haveLetting , , , , and by Schur complement Lemma, (27) can be obtained from (11) and (30)–(32), which complete the proof.

In Theorem 7, in order to get a desired controller (3)-(4) for the closed-loop NCS (7) with partly inaccessible transition probabilities, Lemma 5 is used to separate from . However, this could lead to conservativeness to some extents as shown in the Numerical Example. In the following, a less conservative result will be given in the following theorem.

Theorem 8. *There exists dynamic observer-based control scheme (3)-(4) such that the resulting closed-loop system (7) is stochastically stable if, for all and , there exist matrices , , , , , , , , , , , , , , , , and scalar , such thatwhere*

*Proof. *Equation (11) is equivalent to that is The above inequality can be written as By Schur complement Lemma,is equivalent to the following inequality