Stability and Stabilization of Networked Control System with Forward and Backward Random Time Delays

This paper deals with the problem of stabilization for a class of networked control systems NCSs with random time delay via the state feedback control. Both sensor-to-controller and controller-toactuator delays are modeled as Markov processes, and the resulting closed-loop system is modeled as a Markovian jump linear system MJLS . Based on Lyapunov stability theorem combined with Razumikhin-based technique, a new delay-dependent stochastic stability criterion in terms of bilinear matrix inequalities BMIs for the system is derived. A state feedback controller that makes the closed-loop system stochastically stable is designed, which can be solved by the proposed algorithm. Simulations are included to demonstrate the theoretical result.


Introduction
Feedback control systems in which the control loops are closed through a real-time network are called networked control systems NCSs 1 .Recently, much attention has been paid to the study of stability analysis and controller design of NCSs 2, 3 due to their low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability.Consequently, NCSs have been applied to various areas such as mobile sensor networks 4 , remote surgery 5 , haptics collaboration over the Internet 6-8 , and automated highway systems and unmanned aerial vehicles 9, 10 .However, the sampling data and controller signals are transmitted through a network, so network-induced delays in NCSs are always inevitable 11, 12 .One of the main issues in NCSs is network-induced delays, which are usually the major causes for the deterioration of system performance and potential system instability Notation.R n denotes the n-dimensional Euclidean space, and I is identity matrix.A T stands for the transpose of the corresponding matrix A. The notation A ≥ 0 A > 0 means that the matrix A is a positive semidefinite positive definite matrix.For an arbitrary matrix Y and two symmetric matrices X and Z, X Y * Z denotes a symmetric matrix, where * denotes a block matrix entry implied by symmetry, and • refers to the Euclidean norm for vectors and induced 2-norm for matrices.E • stands for the mathematical expectation operator, and P • for probability operator.The plant is interconnected by a controller over a communication network, see Figure 1.The sensor and controller are periodically sampled with the sampling interval T .We describe the sensor-to-controller transmission delay as τ sc r t and the controller-to-actuator transmission delay as τ ca η t .The mode switching of τ sc r t is governed by the continuoustime discrete-state Markov process r t taking the values in the finite set ς r : {1, . . ., N r } with generator Λ λ ij , i, j ∈ ς r given by

Problem Formulation
where λ ij is the transition rate from mode i to j with λ ij ≥ 0 when i / j and λ ii − N r j 1,j / i λ ij , and o h is such that lim h → 0 o h /h 0. The mode switching of τ ca η t is governed by the continuous-time discrete-state Markov process η t taking the values in the finite set ς η : {1, . . ., N η } with generator Π π kl , k, l ∈ ς η given by  Assumption 2.1.The switching difference of consecutive delays is less than one sampling interval, that is, where t k kT is the kth sampling instant.
Remark 2.2.Although Assumption 2.1 restricts that the switching difference of consecutive delays is less than one sampling interval T , this does not imply that the network delay τ sc r t k and τ ca η t k are less than T .
According to Figure 1, for t k ≤ t < t k 1 , the control law has the form: Define the time delay τ r t , η t as follows: which can be illustrated by Figure 2.Then, we have The associated upper bounds of τ r t , η t are defined as Applying controller 2.7 to the open-loop system 2.1 results in the closed-loop networked control system ẋ t Ax t BK r t , η t x t − τ r t , η t , where φ θ , θ ∈ −τ, 0 is the initial function.
We have the following stochastic stability concept for system 2.9 .
Definition 2.3.The system 2.9 is said to be stochastically stable if there exists a constant T r 0 , η 0 , φ • such that for any initial condition x r 0 , η 0 , φ • .
The following lemmas will be essential for the proofs in Section 3.

2.11
For the delay functional differential equation, ẋ t f t, x t , 2.12 where is completely continuous, f t, 0 0, and x t θ is defined as Then we have the following Razumikhin lemma.
Lemma 2.5 see 45 .Suppose that u, v, w : R → R are continuous, strictly monotonous increasing functions, then u s , v s , and w s are positive for s > 0, and

Mathematical Problems in Engineering
and there is a continuous nondecreasing function p s > s for s > 0, and for any then the zero solution of 2.12 is uniformly asymptotically stable.

Main Results
The following theorem provides sufficient conditions for existence of a mode-dependent state feedback controller for the system 2.9 .
Theorem 3.1.Consider the closed-loop system 2.9 satisfying Assumption 2.1.If there exist symmetric matrix Q i, k > 0, matrix Y i, k , and positive scalar 1 , 2 such that the following matrix inequalities hold for all i ∈ ς r and k ∈ ς η , where with P i, k Q −1 i, k , then the system is stochastically stable with the state feedback gain: Proof.Consider the following Lyapunov candidate: V x t , r t , η t x T t P r t , η t x t , 3.6 where P r t , η t is the positive symmetric matrix.From 3.6 , it follows that

3.9
Thus, the closed-loop system 2.9 can be rewritten as ẋ t A BK r t , η t x t −BK r t , η t 0 −τ r r ,η t Ax t θ BK r t , η t x t − τ r r , η t θ dθ.

3.10
Let L • be the weak infinitesimal generator of {x t , r t , η t , t ≥ 0}, then for r t i ∈ ς r , η t k ∈ ς η , we have

Mathematical Problems in Engineering
According to Lemma 2.4, we have

3.14
Following Lemma 2.5, for −2τ ≤ θ ≤ 0, we assume that for any δ > 1, the following inequality holds: V x t θ , r t θ , η t θ < δV x t , r t , η t , 3.15 then we have where H τ i, k , δ is given by 3.17 for some positive scalars 1 and 2 .before and after multiplying H τ i, k , δ by Q i, k P −1 i, k and its transpose, it gives

3.18
Since we have from 3.16 that

3.20
From 3.1 and Lemma 2.5, it follows that which is equivalent to Using the continuity properties of the eigenvalues of H with respect to δ, then there exists a δ > 1 sufficiently small such that 3.21 still holds.Thus, for such a δ, we have where Mathematical Problems in Engineering Applying Dynkin's formula, we have x s 2 ds | x 0 , r 0 , η 0 .

3.26
Note that Then we can obtain

3.28
This completes the proof.
Remark 3.2.In case of constant transmission delay, that is, τ sc r t τ sc , τ ca η t τ ca , λ ij 0, and π kl 0, Theorem 3.1 can be directly applied to systems with constant delay.
It should be noted that the terms 1 Q i, k and 2 Q i, k in 3.1 -3.3 are bilinear.Therefore, we propose the following algorithm to solve these bilinear matrix inequality problems.
Step 1. Set Q 0 i, k > 0, and Y 0 i, k such that the following LMI holds: where Step 2. For Q i, k > 0 given in the previous step, find 1s , 2s , and Y s i, k by solving the following convex optimization problem:

3.31
Step 3.For Y i, k , 2 , and 1 given in the previous step, find Q s i, k > 0 by solving the following quasiconvex optimization problem max 3.1 -3.3 hold for Y i, k , 2 , and 1 fixed.

3.32
Step 4. Return to step 2 until the convergence of τ is attained with a desired precision.
Remark 3.3.For a given Q i, k , the considered optimization problem consists of minimizing an eigenvalue problem which is a convex one.On the other hand, for given Y i, k , 1 and 2 , the considered optimization problem consists of minimizing a generalized eigenvalue problem which is a quasiconvex optimization problem.Therefore, the proposed algorithm gives a suboptimal solution.

Simulations
In this section, simulations of the position control for robotic manipulator ViSHaRD3 46 are included to illustrate the effectiveness of the proposed method.Combining computed torque feedback approach 47 with friction compensation, the system is decoupled into three systems.The first and second joints of the ViSHaRD3 are and the third is For simplicity, we only discuss the third joint of ViSHaRD3.Suppose that the sampling interval is T 0.01 s, and the Markov process r t that governs the mode switching of the SC delay takes values in ς r {1, 2} and has the generator

4.5
The simulations of the state response and the control input for the closed-loop system are depicted in Figures 3 and 4, respectively, which shows that the system is stochastically stable.

Conclusions
In this paper, a technique of designing a mode-dependent state feedback controller for networked control systems with random time delays has been proposed.The main contribution of this paper is that both the sensor-to-controller and controller-to-actuator delays have been taken into account.Two Markov processes have been used to model these two time delays.Based on Lyapunov stability theorem combined with Razumikhin-based technique, some new delay-dependent stability criteria in terms of BMIs for the system are derived.A state feedback controller that makes the closed-loop system stochastically stable is designed, which can be solved by the proposed algorithm.Simulations results are presented to illustrate the validity of the design methodology.

Figure 2 :
Figure 2: Illustration of the time delay.

Figure 3 :
Figure 3: State response of closed-loop system.

Figure 4 :
Figure 4: Control input of closed-loop system.
3and the Markov process η t that governs the mode switching of the CA delay takes values in η r {1, 2} and has the generator