Stabilization for Networked Control System With Time-Delay and Packet Loss in Both S-C Side and C-A Side

The stabilization problem for a class of discrete network control system with time-delay and packet loss in both S-C side and C-A side is researched in this paper. Firstly, two independent discrete Markov chains are used to describe the network time-delay from sensor to controller and the network time-delay from controller to actuator. Two random variables obeying the Bernoulli distribution are employed to describe the packet loss between the sensor and the controller and the packet loss between the controller and the actuator. Secondly, a mathematical model for closed-loop system is established. By constructing the appropriate Lyapunov-Krasovskii functional, the sufficient conditions for the existence of the controller and observer gain matrix are obtained under the condition that the transition probabilities of S-C time-delay and C-A time-delay are both partly unknown. Finally, two examples are exploited to illustrate the effectiveness of the proposed method.

considered the packet loss of two networks, and some literatures only consider the time-delay and packet loss of S-C side or C-A side. The existing literatures on controller design for NCS can be divided into the following three types: The first type of literatures only considered time-delay. The S-C time-delay was described by a finite-state discrete Markov chain, and the closed-loop NCS was modelled as a Markov jump linear system [12]. Two independent discrete Markov chains were employed to describe the S-C timedelay σ k and C-A time-delay φ k respectively, and the mathematical model of the closed-loop system was established by the method of state augmentation. The necessary and sufficient conditions for the stochastic stability of the closedloop system were obtained, and the solution method of the state feedback controller was proposed [13]. Considering S-C time-delay σ k and C-A time-delay φ k , the H 2 /H ∞ control problem for a class of discrete-time NCS was investigated. Two independent Markov chains were exploited to model the time-delay in S-C side and C-A side. The resulting closed-loop system was a jump linear time-delay induced by two Markov chains. Sufficient conditions for existence of H 2 /H ∞ controller were established based on the free weight matrix method [14]. The robust H ∞ fault detection problem was investigated for the discrete NCS with time-delay on condition that the transition probabilities of time-delay were partly unknown. The closed-loop NCS was molded as a control system which contained two Markov chains, and the relationship between transition probabilities and the minimum H ∞ attenuation level was also obtained [15].
The second type of literatures only considered packet loss. Considering the S-C packet loss and C-A packet loss, the observer-based stabilization controller design problem was researched for a class of nonlinear NCS. The S-C packet loss and C-A packet loss were described by two random variables obeying the Bernoulli distribution. The controller that made the closed-loop system stochastically mean square stable and meet certain H ∞ performance was designed [16]. For a class of nonlinear NCS with S-C packet loss and C-A packet loss, the H ∞ controller was designed as an observerbased dynamic, such that the closed-loop system was exponentially mean square stable and the effect of the disturbance input on the controlled output was less than a minimum level γ for all admissible uncertainties [17].
The third type of literatures only considered time-delay and packet loss in S-C side or time-delay and packet loss in C-A side. The dynamic output feedback controller was designed for nonlinear NCS with time-delay and packet loss in S-C side. The time-delay and packet loss were modeled as two independent random variables. An observer-based dynamic output feedback controller was designed based upon the Lyapunov theory.The quantitative relationship between the packet loss rate and nonlinear level was derived by solving a set of linear matrix inequalities (LMIs) [18]. For the NCS with time-delay and packet loss, the sufficient conditions for the existence of the fault detection filter which made the closed-loop system stable and achieve given H ∞ attenuation performance were established. Although the time-delay in S-C side and C-A side were considered, but the packet loss in C-A side was ignored [19].
Due to the limitation of environmental or economic conditions, it is usually difficult to measure the entire states of the controlled plant, which makes state feedback difficult to achieve. Hence, the state observer needs to be designed, and the state of the controlled plant can be reconstructed through the observer to achieve the required feedback. Therefore, it is of great practical significance to research the observer-based stabilization for NCS [20].
In summary, the current research on the controller design of NCS is not sufficient. To the best of our knowledge, for NCS with time-delay and packet loss in both S-C side and C-A side, the stabilization problem under the condition that the transition probabilities of S-C time-delay and C-A time-delay are both partly unknown has not been researched, which motivates our investigation. Compared to the previous relevant literatures, the main contribution of this paper is that a mathematical model of NCS with time-delay and packet loss in both S-C side and C-A side has been proposed. By constructing proper Lyapunov-Krasovskii functional, and separating unknown probabilities from the known ones, the proposed controller design method is applicable not only to the case that the transition probabilities of the time-delay are partially unknown, but also to the case where the transition probabilities of the time-delay are known, which is less conservative than the existing literatures.
The rest of this paper is organized as follows. The mathematical model of NCS with time-delay and packet loss in both S-C side and C-A side is obtained in Section II. The main results are provided in Section III. Section IV presents a simulation example, and the conclusions are given in Section V.
Notations: Throughout the paper, Pr{·} means mathematical probability, E{·} stands for mathematical expectation and Var{·} denotes variance. The superscript ''T'' and ''-1'' stands for the transpose and inverse of a matrix, respectively. Diag{· · · } stands for a block-diagonal matrix. The symbol ''*'' denotes the symmetric part in a symmetric matrix. P > 0 denotes a positive definite matrix.

II. PROBLEM FORMULATION AND PRELIMINARIES
The structure of the NCS considered in this paper is shown in Figure 1, where the switch closure indicates that the packet transmission is successful, and the switch open indicates that a packet loss has occurred. σ k and φ k denotes the time-delay in S-C side and C-A side and takes value from = {0, · · · , σ M } and = {0, · · · , φ M }, respectively. The transition probability matrix of σ k and φ k is = [µ ab ], = [ν mn ], respectively, where µ ab and ν mn is defined as µ ab = Pr{µ k+1 = b|µ k = a}, ν mn = Pr{ν k+1 = n|ν k = m}, respectively, where µ ab ≥ 0, ν mn ≥ 0,  It is usually difficult to obtain the all transition probabilities of the time-delay, so it is assumed that there are some unknown elements in the transition probability matrix of the time-delay. For notational clarity, ∀b ∈ , let = a k + a uk with a k = {b : µ ab is known}, a uk = {b : µ ab is unknown}. If a k is not an empty set, it is further described as a k ={ k a 1 , k a 2 , · · · , k a p }, where k a p represents the pth known element in the ath row of matrix with the index The random variable α k , β k which obeys Bernoulli distribution is used to describe the packet loss in S-C side and C-A side, respectively. When the random variable takes the value of 1, it indicates that the data packet was successfully transmitted. Otherwise, it indicates that the data packet transmission failed. Random variables α k , β k satisfy the following characteristics: The discrete NCS equation considered in this paper are as follows: where x k is the system state vector, u k is the control input vector, y k is the system measurement output vector, A, B, C are known real constant matrices with appropriate dimensions.
The state equation of the observer is as follows: wherex k is the state vector of the observer,ŷ k is the out vector of the observer, L is the gain matrix to be determined,ȳ k is the system output received by the observer andū k is the control input of the observer which expressed as Considering the time-delay and packet loss, the system output y k received by the observer and the control input u k received by the actuator can be expressed as: Define the following state estimation error and augmentation vector: The state equation of the closed-loop system can be obtained from (1)- (7): In order to deal with the stochastic parameter in closedloop system (8), it is necessary to introduce the following definition.
Definition 1 [13]: For any initial system state ζ 0 and initial time-delay mode σ 0 ∈ , φ 0 ∈ , if there exists a positive definite matrix W , such that the closed-loop system (8) is said to be stochastically stable. Remark 1: Because of the existence of the time-delay and the packet loss in C-A side, the control input of the observer u k in (3) is different from the control input of the controlled system u k in (1), which brings difficulties in the controller design.

III. MAIN RESULTS
In this section, the main resuls of this paper are presented. To proceed, the following lemma is needed.
Lemma 1 [21]: For any positive definite matrix H and two scalar θ, θ 0 satisfying θ ≥ θ 0 ≥ 1, the following formula always holds: The following theorem presents a sufficient condition on the stochastic stability of the system (8).
Remark 2: In dealing with the unknown time-delay transition probabilities, another method is to separate the unknown probabilities from the correlation matrices and estimate the unknown probabilities with the known ones by the related lemma [15], for example, certain conservativeness. In this paper, the unknown probabilities are separated from the known ones, and the obtained result is less conservative, as shown in Example 2. Remark 3: This paper deals with time-delay by constructing a proper Lyapnov-Krasovskii functional. Another method is to convert the time-delay into the parameter matrix of the closed-loop system by the state augmentation technique [13]. However, as the time-delay mode increases, the dimension of VOLUME 8, 2020 the closed-loop system will become high, which increases the controller solution time. The method in this paper reduces the dimension of the matrix for the closed-loop system.
The conditions in Theorem 2 are a set of LMIs with nonconvex constraints which can be solved by several existing iterative algorithms. The cone complementarity linearization (CCL) method [22] is used to transform the conditions in Theorem 2 into the following nonlinear minimization problem with LMI constraints. (26) and (27):

Min tr
The procedure for solving the controller and observer gain matrix is presented in Algorithm 1.
Remark 4: The method proposed in this paper can also be applied to the H ∞ control and guaranteed performance control where the relationship among the system performance, packet loss probability and the information amount of timedelay transition probability can be further researched.

IV. NUMERICAL EXAMPLE
In this section, two examples are presented to illustrated the effectiveness of the proposed method. By the method in [15], a set of feasible solutions can also be gotten as follows: The initial state of the system is x T 0 = 1 −0.5 . Figure 2 and Figure 3 illustrate the response of system state x 1 and x 2 using   the proposed method and the method in [15]. It is observed that the proposed method outperforms the method in [15].
Example 2: Considering the angular position tracking system [23] shown in Figure 4, where ϕ r is the angular position of the moving object, and ϕ is the angular position of the antenna. The function of this system is that the antenna can rotate with the movement of the target object by applying a voltage to the motor and satisfy ϕ ∼ = ϕ r . The state space model parameters of the angular position tracking system are as follows: Obviously this system is unstable. Assume S-C time-delay σ k ∈ = {0, 1}, and C-A time-delay φ k ∈ = {0, 1} , the transition probability matrix of σ k and φ k is as follows,   Assume that the initial state of the system x 0 = 2 −1 T , The time-delay σ k and φ k is shown in Figure 5 and Figure 6, respectively. The closed-loop system  state response curve under the controller designed in this paper is shown in Figure 7 and Figure 8. Due to the introduced conservativeness, one cannot use the method in [15] to obtain a set of feasible solutions for K and L. Therefore, the method proposed in this paper is less conservative than the method in [15].

V. CONCLUSION
In this paper, the observer-based stabilization problem is studied for NCS with time-delay and packet loss in both S-C side and C-A side. Under the condition that the S-C and C-A time-delay transition probability are partially unknown, the sufficient conditions for the stability of the closed-loop system are obtained. The method of solving the controller and observer gain matrix of the NCS is also proposed. Her research interests include networked control system fault detection and fault tolerant control.
WEI WEI received the M.S. and Ph.D. degrees from Xi'an Jiaotong University, Xi'an, China, in 2005 and 2011, respectively. He is currently an Associate Professor with the School of Computer Science and Engineering, Xi'an University of Technology, Xi'an. He was supported by many funded research projects as a principal investigator and a technical member. He has published around 100 research articles in international conferences and journals. His current research interests include wireless networks, wireless sensor networks, image processing, mobile computing, distributed computing, and pervasive computing, the Internet of Things, and sensor data clouds. He is also a member of the Institute of Electronics, Information, and Communication Engineers. He is currently a Full Professor with the China University of Geosciences, Wuhan. He has been supported by several National Natural Science Funds. He has published more than 50 relevant academic articles in SCI-indexed journals. His current research interests include the qualitative theory of differential equations, and chaos and bifurcation theory.