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

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.


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][8][9][10][11][12][13][14][15][16][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][11][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][8][9][10][11][12][13][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 discretetime NCSs was designed.Unfortunately, in [16][17][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.

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, the transition probabilities of   and   are both considered to be partly accessible; that is, some elements in matrix Ξ and Π are unknown.For notational clarity, ∀ ∈ Ω, we denote Ω = Ω where where  is a known real scalar.The dynamic observer-based control scheme is given by Observer: Controller: where x ∈   ,   ∈   , and ŷ ∈   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 ũ 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, ũ 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   = (,   ) − (, x ).

By defining 𝜂
, 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  0 and  0 ∈ Ω,  0 ∈ Γ, there exists a finite matrix  > 0 such that In this paper, our objective is to design the dynamic observerbased control scheme ( 3) and ( 4), such that the closedloop system ( 7) is stochastically stable on condition that the transition probabilities of   and   are both partly inaccessible.

Main Results and Proofs
In this section, we will present the main results.To proceed, we will need the following three lemmas.
From (25), we can see that, for any  ≥ 1, According to Definition 2, the closed-loop system (7) is stochastically stable, 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     ,  0 =  0 = 0.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.
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].

Conclusion
The observer-based controller design problem for a class of nonlinear networked control systems with random timedelays 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.