Journal of Control Science and Engineering
Volume 2017 (2017), Article ID 1523825, 13 pages
https://doi.org/10.1155/2017/1523825

Research Article

Observer-Based Controller Design for a Class of Nonlinear Networked Control Systems with Random Time-Delays Modeled by Markov Chains

Yanfeng Wang, Peiliang Wang, Zuxin Li, and Huiying Chen

School of Engineering, Huzhou University, Huzhou, Zhejiang 313000, China

Correspondence should be addressed to Yanfeng Wang; neu2009wyf@163.com

Received 3 January 2017; Revised 14 February 2017; Accepted 1 March 2017; Published 30 March 2017

Academic Editor: Lei Zou

Copyright © 2017 Yanfeng Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Figure 1: Diagram of the NCSs.

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 by

where , , 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 that

In 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 inequality

where

holds for all and , the closed-loop system (7) is stochastically stable.

Proof. Choose the Lyapunov function candidate as , where

In the following, we denote as when and . Then along the solution of system (7), we have

It is noticed that

Therefore, we obtain

Note that

We can obtain the following inequality by Lemma 3:

Similarly, we have

From (14)–(19), we have

where

It follows from (2) that

which imply that

By the well-known S-procedure, that is, Lemma 4, we can get with constraints (23) holding if there exist nonnegative real scalars , such that

From (24), we have

where
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 that

where

hold for all and .

Proof. Making use of Lemma 5, we have

Letting , , , , 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 that

where

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

where . Let , , , and , ; we can obtain (33) and (37). Hence, (42) holds if (33) holds. Because , , it can be concluded that if (33)–(36) holds, (39) holds, which completes the proof.

The conditions in Theorem 7 are in fact a set of linear matrix inequalities (LMIs) with some inversion constraints. Though they are nonconvex which brings difficulties in solving them using the existing convex optimization tool, we can use the cone complementarity linearization (CCL) algorithm [22] to transform this problem into the nonlinear minimization problem with LMIs constraints as follows:

Furthermore, the iterative algorithm which can be used to calculate the controller gain matrix and observer gain matrix is given as follows.

Algorithm 9.  
Step  1. Find a feasible solution satisfying (27), (45), and (46) and set as ; Let .
Step  2. Solve the following LMI optimization problem for variables . + , subject to (27), (45), and (46); set (, , , , , , , , , , , ).
Step  3. If (27) and (28) are satisfied, then exit the iteration. Otherwise, let , and return to Step  2.

The conditions in Theorem 8 can be solved by the similar procedures.

Remark 10. Making use of Matlab LMI tool box, a feasible solution satisfying (27), (45), and (46) can be obtained by the functions feasp() and dec2mat(). Once the feasible solution is obtained, then the LMI optimization problem , subject to (27), (45), and (46), can be solved by the functions defcx() and mincx().

4. Numerical Example

In this section, we present two examples to illustrate the effectiveness of the proposed method.

Example 1. Consider the nonlinear controlled plant with the following parameter:

We can see that the linear part of the control plant is stable itself. The random S-C delay and C-A delay . Their transition probability matrices are as follows, respectively:

By Theorem 7, we can obtain the controller gain matrix and observer gain matrix as follows:

The initial value , , . The trajectories of the closed-loop system’s states and the corresponding estimated value are shown in Figures 2 and 3 which indicate that the closed-loop system is stochastically stable.
By Theorem 8, the controller gain matrix and observer gain matrix can also be obtained as follows:

The trajectories of the closed-loop system’s states and the corresponding estimated value are omitted.

Figure 2: State and its estimated value .

Figure 3: State and its estimated value .

Example 2. Consider the classical angular positioning system shown in Figure 4, which consists of a rotating antenna at the origin of the plane driven by a motor [23].
Assume that the angular position of the antenna , the angular position of the moving object , and the angular velocity of the antenna are measurable. The state variables are chosen as ; then state equation of the angular positioning system can be obtained with the following parameter, where can be treated as the external disturbance:

Obviously, it can be seen that the linear part of the control plant is unstable. The random S-C delay and C-A delay . Their transition probability matrices are as follows, respectively:

For this unstable plant, we cannot get the controller gain matrix and observer gain matrix by Theorem 7 due to its conservativeness. However, by Theorem 8, we can obtain and as follows:

The initial value , , . The trajectories of the closed-loop system’s states and the corresponding estimated value are shown in Figures 5 and 6 which indicate that the closed-loop system is stochastically stable.

Figure 4: The angular positioning system.

Figure 5: State and its estimated value .

Figure 6: State and its estimated value .

Remark 11. The results in this paper only require a part of transition probabilities while the results in [13, 14] need the full information of the transition probabilities. In view of these, the proposed controller design method is more general than that of [13, 14].

5. Conclusion

The observer-based controller design problem for a class of nonlinear networked control systems with random time-delays is investigated in this paper. Two dependent Markov chains are employed to describe S-C delay and C-A delay, respectively. The transition probabilities of S-C delay and C-A delay are both assumed to be partly accessible. Sufficient conditions on the stochastic stability for the closed-loop systems are obtained by the Lyapunov stability theory. The CCL algorithm is employed to calculate the controller and observer gain matrices. Finally, two examples are used to illustrate the effectiveness of the proposed method.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research is supported by the National Natural Science Foundation of China (Grant nos. 61503136, 61573137, 61573136, and 61603133).

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