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

Authors: Ye-Guo Sun; Qing-Zheng Gao
Journal: Mathematical Problems in Engineering
 (2012)

---

## Abstract

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-to-actuator 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.


---

## Body

## 1. 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 [13]. For different scheduling protocols, the network-induced delay may be constant, or time-varying, but in most cases, it is random [14]. Hence, systems with random time delay attract considerable attention [15–18]. Based on stochastic control theory and a separation property, the effect of random delay is treated as an LQG problem in [15]. However, the network-induced random delay has to be less than one sampling interval. The results in [15] have recently been extended to the case with longer delays in [16]. It is noted that the given controller depends only on sensor-to-controller delay. In [17], a control problem for Bernoulli binary random delay is considered, and a linear matrix inequalities (LMIs) problem for the analysis of stochastic exponential mean square stability is established. The model-based NCSs with random transmission delay is studied in [18]. Sufficient conditions for almost sure stability and stochastic exponential mean square stability are presented.On the other hand, the study of stochastic systems has attracted a great deal of attention [19–38]. Some of these results are applied to networked control systems with random time delays [39–43]. In [39, 40], the network-induced random delays are modeled as Markov chains such that the closed-loop systems are jump linear systems with one mode. It is noticed that in [39], the state feedback gain is mode independent, and in [40], the state feedback gain only depends on the delay from sensor to controller. Recently, stabilization of networked control systems with the sensor-to-controller and controller-to-actuator delays are considered in [41]. In [42, 43], a class of Markovian jump linear systems with time delays both in the system state and in the mode signal is considered. Based on Lyapunov method, a time-delayed, mode-dependent, and state feedback controller such that the closed-loop system is stochastically stable is designed. It is noticed that the time delay in the mode signal is constant in [42, 43], and the time delay in the mode signal is random. It is worth pointing out that in all of the aforementioned papers, the plant is in the discrete-time domain. To the best of the authors' knowledge, the stability and stabilization problems for NCSs with the plant being in the continuous-time domain have not been fully investigated to date. Especially for the case where both sensor-to-controller and controller-to-actuator network-induced delays are random and longer than one sampling interval, very few results related to NCSs have been available in the literature so far, which motivates the present study.The aim of this paper is to consider a class of networked control systems with sensors and actuators connected to a controller via two communication networks in the continuous-time domain. Two Markov processes are introduced to describe sensor-to-controller transmission delay and the controller-to-actuator transmission delay. Based on Lyapunov stability theorem, a method for designing a mode-dependent state feedback controller that stabilizes this class of networked control systems is proposed. The existence of such a controller is given in terms of BMIs, which can be solved by the proposed algorithm.This paper is organized as follows. In Section2, the problem is stated and some useful definitions and lemmas are given, and then the main results of this paper are given in Section 3. Simulation results are presented in Section 4. Finally, the conclusions are provided in Section 5.Notation.
ℝn denotes the n-dimensional Euclidean space, and I is identity matrix. AT 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, [XY*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.


## 2. Problem Formulation

Consider linear systems described by the differential equation(2.1)ẋ=Ax(t)+Bu(t),
where x(t)∈ℝn is the state vector, and u(t)∈ℝm is the control input. Matrices A and B are known matrices of appropriate dimensions.The plant is interconnected by a controller over a communication network, see Figure1. The sensor and controller are periodically sampled with the sampling interval T. We describe the sensor-to-controller transmission delay as τsc(rt) and the controller-to-actuator transmission delay as τca(ηt). The mode switching of τsc(rt) is governed by the continuous-time discrete-state Markov process rt taking the values in the finite set ςr:={1,…,Nr} with generator Λ=(λij),i,j∈ςr given by
(2.2)P[rt+h=j∣rt=i]={λijh+o(h),i≠j,1+λiih+o(h),i=j,
where λij is the transition rate from mode i to j with λij≥0 when i≠j and λii=-∑j=1,j≠iNrλij, and o(h) is such that limh→0o(h)/h=0. The mode switching of τca(ηt) is governed by the continuous-time discrete-state Markov process ηt taking the values in the finite set ςη:={1,…,Nη} with generator Π=(πkl),k,l∈ςη given by (2.3)P[ηt+h=l∣ηt=k]={πklh+o(h),k≠l,1+πkkh+o(h),k=l,
with πkl≥0 and πkk=-∑l=1,l≠kNηπkl.Figure 1
Illustration of NCSs over communication network.Throughout the paper, the following assumption is needed for the considered networked control systems.Assumption 2.1.
The switching difference of consecutive delays is less than one sampling interval, that is,(2.4)P(|τsc(rtk+1)-τsc(rtk)|≥T)=0,P(|τca(ηtk+1)-τca(ηtk)|≥T)=0,
where tk=kT is the kth sampling instant.Remark 2.2.
Although Assumption2.1 restricts that the switching difference of consecutive delays is less than one sampling interval T, this does not imply that the network delay τsc(rtk) and τca(ηtk) are less than T.According to Figure1, for tk≤t<tk+1, the control law has the form:(2.5)u(t)=K(rt,ηt)x(tk-τsc(rt)-τca(ηt)).Define the time delayτ(rt,ηt) as follows:(2.6)τ(rt,ηt)=t-tk+τsc(rt)+τca(ηt),
which can be illustrated by Figure 2.Figure 2
Illustration of the time delay.Then, we have(2.7)u(t)=K(rt,ηt)x(t-τ(rt,ηt)).
The associated upper bounds of τ(rt,ηt) are defined as(2.8)τ¯=T+max⁡i∈ςrτsc(i)+max⁡k∈ςητca(k).Applying controller (2.7) to the open-loop system (2.1) results in the closed-loop networked control system(2.9)ẋ(t)=Ax(t)+BK(rt,ηt)x(t-τ(rt,ηt)),x(θ)=ϕ(θ),θ∈[-τ¯,0],
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(r0,η0,ϕ(·)) such that
(2.10)E[∫0∞‖x(s)‖2ds∣(r0,η0,ϕ(⋅))]≤T(r0,η0,ϕ(⋅)),
for any initial condition x(r0,η0,ϕ(·)).The following lemmas will be essential for the proofs in Section3.Lemma 2.4 (see [44]).
Given any real matricesΣ1,Σ2,Σ3 of appropriate dimensions and a scalar ϵ>0 such that Σ3=Σ3T>0, Then the following inequality holds:
(2.11)∑1T∑2+∑2T∑1≤ϵ∑1T∑3∑1+ϵ-1∑2T∑3-1∑2.For the delay functional differential equation,(2.12)ẋ(t)=f(t,xt),
where (2.13)f:[0,+∞)×C([-τ,0],Rn)⟶Rn,
is completely continuous, f(t,0)=0, and xt(θ) is defined as (2.14)xt(θ)=x(t+θ),θ∈[-τ,0].
Then we have the following Razumikhin lemma.Lemma 2.5 (see [45]).
Suppose thatu,v,w:ℝ+→ℝ+ are continuous, strictly monotonous increasing functions, then u(s),v(s), and w(s) are positive for s>0, and u(0)=v(0)=0. If there is a continuous function V:[-τ,+∞)×ℝn→ℝ+ such that
(2.15)u(‖x‖)≤V(t,x)≤v(‖x‖),t∈[-τ,+∞),x∈R,
and there is a continuous nondecreasing function p(s)>s for s>0, and for any t0∈ℝ+,
(2.16)V̇(t,x)≤-w(‖x‖)
if
(2.17)V(x(t+θ,t+θ))<p(V(t,x)),θ∈[-τ,0],t≥t0,
then the zero solution of (2.12) is uniformly asymptotically stable.


## 3. 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 ϵ1, ϵ2 such that the following matrix inequalities hold for all i∈ςr and k∈ςη,
(3.1)[J(i,k)φ1(i,k)φ2(i,k)*-ψ10**-ψ2]<0,(3.2)[-ε1Q(i,k)AQ(i,k)*-Q(i,k)]<0,(3.3)[-ε2Q(i,k)BY(i,k)*-Q(i,k)]<0,
where
(3.4)J(i,k)=Q(i,k)AT+AQ(i,k)+YT(i,k)BT+BY(i,k)+τ¯(ϵ1+3ϵ2)Q(i,k)+λiiQ(i,k)+πkkQ(i,k),φ1(i,k)=[λi,1Q(i,k),…,λi,i-1Q(i,k),λi,i+1Q(i,k),…,λi,NrQ(i,k)],φ2(i,k)=[πk,1Q(i,k),…,πk,k-1Q(i,k),πk,k+1Q(i,k),…,πk,NηQ(i,k)],ψ1=diag⁡[Q(1,k),…,Q(i-1,k),Q(i+1,k),…,Q(Nr,k)],ψ2=diag⁡[Q(i,1),…,Q(i,k-1),Q(i,k+1),…,Q(i,Nη)],
with P(i,k)=Q-1(i,k), then the system is stochastically stable with the state feedback gain:
(3.5)K(i,k)=Y(i,k)Q-1(i,k).Proof.
Consider the following Lyapunov candidate:(3.6)V(x(t),rt,ηt)=xT(t)P(rt,ηt)x(t),
where P(rt,ηt) is the positive symmetric matrix. From (3.6), it follows that
(3.7)β1‖x(t)‖2≤V(x(t),rt,ηt)≤β2‖x(t)‖2,
where
(3.8)β1=min⁡rt∈ςr,ηt∈ςηλmin⁡(P(rt,ηt)),β2=max⁡rt∈ςr,ηt∈ςηλmax⁡(P(rt,ηt)).
Note that
(3.9)x(t-τ(rr,ηt))=x(t)-∫-τ(rr,ηt)0ẋ(t+θ)dθ=x(t)-∫-τ(rr,ηt)0[Ax(t+θ),+BK(rt,ηt)x(t-τ(rr,ηt)+θ)]dθ.
Thus, the closed-loop system (2.9) can be rewritten as
(3.10)ẋ(t)=[A+BK(rt,ηt)]x(t)-BK(rt,ηt)∫-τ(rr,ηt)0[Ax(t+θ)+BK(rt,ηt)x(t-τ(rr,ηt)+θ)]dθ.
Let ℒ(·) be the weak infinitesimal generator of {x(t),rt,ηt,t≥0}, then for rt=i∈ςr,ηt=k∈ςη, we have(3.11)LV(x(t),i,k)=ẋT(t)P(i,k)x(t)+xT(t)P(i,k)ẋ(t)+∑j=1NrλijxT(t)P(j,k)x(t)+∑l=1NηπklxT(t)P(i,l)x(t)+xT(t)[ATP(i,k)+P(i,k)A+KT(i,k)BTP(i,k)+P(i,k)BK(i,k)+∑j=1NrλijP(j,k)+∑l=1NηπklP(i,l)]x(t)-2∫-τ(i,k)0{xT(t)P(i,k)BK(i,k)×[Ax(t+θ)+BK(i,k)x(t-τ(i,k)+θ)]}dθ.According to Lemma 2.4, we have
(3.12)-2∫-τ(i,k)0{xT(t)P(i,k)BK(i,k)×[Ax(t+θ)+BK(i,k)x(t-τ(i,k)+θ)]}dθ≤τ(i,k)[ϵ1-1xT(t)P(i,k)BK(i,k)AP-1(i,k)×ATKT(i,k)BTP(i,k)x(t)+ϵ2-1xT(t)×P(i,k)BK(i,k)BK(i,k)P-1(i,k)KT(i,k)×BTKT(i,k)BTP(i,k)x(t)]+ϵ1∫-τ(i,k)0xT(t+θ)P(i,k)x(t+θ)dθ+ϵ2∫-τ(i,k)0xT(t-τ(i,k)+θ)P(i,k)x(t-τ(i,k)+θ)dθ.
From (3.2), (3.3), and Lemma 2.5, we can obtain
(3.13)AP-1(i,k)AT<ϵ1P-1(i,k),BK(i,k)P-1(i,k)KT(i,k)BT<ϵ2P-1(i,k),
which yields
(3.14)-2∫-τ(i,k)0{xT(t)P(i,k)BK(i,k)×[Ax(t+θ)+BK(i,k)x(t-τ(i,k)+θ)]}dθ≤2τ(i,k)ϵ2xT(t)P(i,k)x(t)+ϵ1∫-τ(i,k)0xT(t+θ)P(i,k)x(t+θ)dθ+ϵ2∫-τ(i,k)0xT(t-τ(i,k)+θ)P(i,k)×x(t-τ(i,k)+θ)dθ.
Following Lemma 2.5, for -2τ¯≤θ≤0, we assume that for any δ>1, the following inequality holds:
(3.15)V(x(t+θ),rt+θ,ηt+θ)<δV(x(t),rt,ηt),
then we have
(3.16)LV(x(t),i,k)≤xT(t)H(τ(i,k),δ)x(t),
where ℋ(τ(i,k),δ) is given by
(3.17)H(τ(i,k),δ)=ATP(i,k)+P(i,k)A+KT(i,k)BTP(i,k)+P(i,k)BK(i,k)+∑j=1NrλijP(j,k)+∑l=1NηπklP(i,l)+2τ(i,k)ϵ2P(i,k)+τ(i,k)ϵ1δP(i,k)+τ(i,k)ϵ2δP(i,k),
for some positive scalars ϵ1 and ϵ2. before and after multiplying ℋ(τ(i,k),δ) by Q(i,k)=P-1(i,k) and its transpose, it gives
(3.18)H̃(τ(i,k),δ)=Q(i,k)AT+AQ(i,k)+Q(i,k)KT(i,k)BT+BK(i,k)Q(i,k)+Q(i,k)∑j=1NrλijP(j,k)Q(i,k)+Q(i,k)∑l=1NηπklP(i,l)Q(i,k)+2τ(i,k)ϵ2Q(i,k)+τ(i,k)ϵ1δQ(i,k)+τ(i,k)ϵ2δQ(i,k).
Since
(3.19)0≤τ(i,k)≤τ¯,
we have from (3.16) that
(3.20)LV(x(t),i,k)≤xT(t)H(τ¯,δ)x(t).
From (3.1) and Lemma 2.5, it follows that
(3.21)H̃(τ¯,δ=1)<0,
which is equivalent to
(3.22)H(τ¯,δ=1)<0.
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
(3.23)H(τ¯,δ)<0,
which yields
(3.24)LV(x(t),rt,ηt)≤-β‖x(t)‖2,
where
(3.25)β=min⁡i∈ςr,k∈ςη[λmin⁡(-H(τ¯,δ))]>0.
Applying Dynkin's formula, we have
(3.26)E[V(x(t),i,k)]-E[V(x0,r0,η0)]=E{∫0t[LV(x(s),rs,ηs)ds∣x0,r0,η0]}≤-βE{∫0t[‖x(s)‖2ds∣x0,r0,η0]}.
Note that
(3.27)E[V(x(t),i,k)]≥0,
Then we can obtain
(3.28)βE{∫0t[‖x(s)‖2ds∣x0,r0,η0]}≤E[V(x(t),i,k)]+βE{∫0t[‖x(s)‖2ds∣x0,r0,η0]}≤E[V(x0,r0,η0)].
This completes the proof.Remark 3.2.
In case of constant transmission delay, that is,τsc(rt)=τsc,τca(ηt)=τca,λij=0, and πkl=0, Theorem 3.1 can be directly applied to systems with constant delay.
It should be noted that the termsϵ1Q(i,k) and ϵ2Q(i,k) in (3.1)–(3.3) are bilinear. Therefore, we propose the following algorithm to solve these bilinear matrix inequality problems.
Step 1.
SetQ0(i,k)>0, and Y0(i,k) such that the following LMI holds:
(3.29)[J̃(i,k)φ1(i,k)φ2(i,k)*-ψ10**-ψ2]<0,
where
(3.30)J̃(i,k)=Q(i,k)AT+AQ(i,k)+YT(i,k)BT+BY(i,k)+λiiQ(i,k)+πkkQ(i,k).Step 2.
ForQ(i,k)>0 given in the previous step, find ϵ1s,ϵ2s, and Ys(i,k) by solving the following convex optimization problem:
(3.31)max⁡Y(i,k),ϵ1,ϵ2τ¯(Y(i,k),ϵ1,ϵ2),s.t.(3.1)–(3.3)holdforQ(i,k)>0fixed.Step 3.
ForY(i,k), ϵ2, and ϵ1 given in the previous step, find Qs(i,k)>0 by solving the following quasiconvex optimization problem
(3.32)max⁡Q(i,k)>0τ¯(Q(i,k)),s.t.(3.1)–(3.3)holdforY(i,k),ϵ2,andϵ1fixed.Step 4.
Return to step 2 until the convergence ofτ¯ is attained with a desired precision.Remark 3.3.
For a givenQ(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), ϵ1 and ϵ2, 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.


## 4. 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 (4.1)ddt[qq̇]=[011-50][qq̇]+[01]u(t),
and the third is (4.2)ddt[qq̇]=[011-40][qq̇]+[01]u(t).For simplicity, we only discuss the third joint of ViSHaRD3. Suppose that the sampling interval isT=0.01 s, and the Markov process rt that governs the mode switching of the SC delay takes values in ςr={1,2} and has the generator (4.3)Λ=[-332-2],
and the Markov process ηt that governs the mode switching of the CA delay takes values in ηr={1,2} and has the generator (4.4)Π=[-112-2].Associated with modes 1 and 2, let the system have time delayτsc(1)=0.03 s, τca(1)=0.02 s and τsc(2)=0.025 s, τca(2)=0.015 s, respectively. From (2.8), we have τ¯=0.06 s, and the initial condition is ϕ(θ)=[-1,0]T,θ∈[-0.06,0]. By the proposed algorithm and Theorem 3.1, we can obtain the controllers as follows: (4.5)K(1,1)=[-645.0596-15.9109],K(1,2)=[-623.3689-15.4999],K(2,1)=[-575.1361-14.2296],K(2,2)=[-616.8428-15.3049].The simulations of the state response and the control input for the closed-loop system are depicted in Figures3 and 4, respectively, which shows that the system is stochastically stable.Figure 3
State response of closed-loop system.Figure 4
Control input of closed-loop system.


## 5. 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.



---

*Source: 834643-2012-01-31.xml*