# Output-Feedback Controller Design of a Wireless Networked Control System with Packet Loss and Time Delay

Authors: Zhen Hong; JinFeng Gao; Ning Wang
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper investigates the problem of modeling
and stabilization of a wireless network control system (NCS)
with both time-varying delay and packet-dropout. And the time-varying
delay can be more or less than one sampling period. The wireless NCS is modeled as an asynchronous dynamic
system (ADS) with three subsystems. Sufficient condition of the
closed-loop NCS to be stable is obtained by using the ADS
approach. A numerical example is presented to demonstrate
the effectiveness of the proposed result.


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## Body

## 1. Introduction

In modern control systems, the sensors, controllers, and controlled process are often connected by a real-time network medium. Such systems are called network control systems (NCSs) [1]. In the last decade, the general theory for NCSs has been widely investigated. It is widely used at almost all levels of operation and information processing in various areas, including manufacturing plants, automobiles, aircraft, remote operation, and teleautonomy [2–8]. As an alternative item for the wired network, wireless NCSs are becoming fundamental components of modern control systems due to their flexibility, ease of deployment, and low cost [9–12]. Thus, wireless NCS is considered in this paper.One of the main issues in the NCS is the effect of network-induced delay that occurs when sensors, actuators, and controllers exchange data across the shared network. Without considering the delay, it not only degrades the performance of the control system but also even destabilizes the system. On the other hand, packet-dropout results from the network traffic congestion and the limited network reliability. When a data packet is dropped, the complete information of the NCS becomes unavailable. In this case, the controller or actuator has to decide what control signal is output with incomplete information.Recently, many researchers have tried to solve the above problems with network-induced delay and packet-dropout in wireless NCS. For the problem with delay, for example, Hu and Yuan [2] introduce a finite sum equality based on quadratic terms to H∞ output-feedback control for switched linear discrete-time systems. Together with a Lyapunov sequence, a novel delay-dependent condition is implemented for H∞ without ignoring the useful terms. It can obtain the suboptimal H∞ static and dynamic output-feedback controllers at last, when a procedure involving a modified iterative algorithm is performed. For the problems with packet-dropout, for instance, the sufficient conditions for the exponential stability of the closed-loop NCSs using the average dwell time method are proposed in [5]. Furthermore, the relation between the packet-dropout rate and the stability of the closed-loop NCSs is also explicitly established in order to prove the effectiveness.Unlike separately considered these two issues of the delay and the packet-dropout, this paper intends to deal with the modelling, analysis, and synthesis for the wireless NCS with both delay and packet-dropout as shown in Figure1. An asynchronous dynamic system (ADS) approach is presented to stabilize the wireless NCS. Firstly, a switched system with time-varying delay model is presented to describe the wireless NCS. In [4], a new switched linear system model is proposed to describe NCS while the delay is assumed to be less than one sampling period with the state feedback controller. And stochastic optimal control method is used by an adaptive estimator (AE) and ideas from Q-learning to solve the infinite horizon optimal regulation of unknown wireless NCS with time-varying system matrices in [13]. Nevertheless, the computational complexity of the controller will increase when the delay bound is increased in NCS in [13]. Furthermore, the network system identification problem and estimation problem are also studied in [14, 15] based on the Matlab/Simulink simulator TrueTime and orthogonal projection principle, respectively, which aim at identifying mathematical models required in network control/estimation/filtering systems while stabilization is not considered. Similarly, we can also get the new ideas and future applications from the related papers [16, 17], respectively.Figure 1
Diagram of network control system with network-induced delay and packet-dropout.Summarizing the aforementioned discussion, in this paper, we aim to investigate the problem of modeling and stabilization of a wireless NCS with both time-varying delay and packet-dropout, where the wireless NCS is modeled as an ADS with three subsystems, and sufficient condition of the closed-loop NCS is obtained to be stable using this ADS method. The main contribution of this paper is highlighted as follows: to obtain the observer-based output-feedback controller, particularly, when the output of the sensor is dropped and delay appeared to the controller input, the ADS approach is presented to stabilize the considered wireless NCS with delay and packet-dropout.Notations. Throughout this paper, R denotes the set of real numbers, Rn denotes the n-dimensional Euclidean space, and Rn×m refers to the set of all n×m real matrices. AT represents the transpose of the matrix A, while A-1 denotes the inverse of A. For real symmetric matrices, X and Y, the notation X≥Y (resp., X>Y) means that the matrix X-Y is positive semidefinite (resp., positive-definite). I is the identity matrix with appropriate dimensions. ∥x∥ refers to the Euclidean norm of the vector x; that is, ∥x∥=xTx. For a symmetric matrix, * denotes the matrix entries implied by symmetry.


## 2. Problem Formulation and NCS Modeling

The NCS with packet-dropout and possible delay is illustrated in Figure1, where the plant is described by the following model denoted by Ω:
(1)x(k+1)=Ax(k)+Bu(k),y(k)=Cx(k),
where x(k)∈Rn is the system state, u(k)∈Rm is the control input, and y(k)∈Rr is the measured output. And the model of an observer-based output-feedback controller is described as follows:
(2)x^(k+1)=Ax^(k)+Bu(k)+L[ω(k)-y^(k)],y^(k)=Cx^(k),u(k)=Kx^(k),
where x^(k)∈Rn is the estimated state of the system (1) and y^(k)∈Rr is the estimated output. L∈Rn×r and K∈Rm×n are the observer and controller gain, respectively. It is also assumed that the pairs (A,B) are controllable and (C,A) are observable. In Figure 1, we can use a switch to denote the packet loss of the states in the network channel. If the switch is closed, the data packet is successfully transmitted. And we have ω(k)=y(k) without network delay or ω(k)=y(k-dk) with network delay. When the switch is open, the previous value of the switch output will be used in the controller (2) and a packet is dropped. Then we have ω(k)=ω(k-1) in this case.Under consideration for the NCS, without loss of generality, we give the following assumptions, which will be useful in our main results.Assumption 1.
(
1
) The sensors and controllers are all time-driven and synchronized.
(
2
) Time-stamping of measurements is necessary to reorder data packet at the observer side since they can arrive out of order. And the controller can get the delay of each data packet.
(
3
) The maximum delay in the network is dM that is a known integer.Define the estimation error bye(k)=x(k)-x^(k) and let
(3)z(k)=[xT(k)eT(k)wT(k-1)]T.
Then the dynamics of the closed-loop system can be described by the following three subsystems.(S1)
There is packet loss and the corresponding controller gain isK1. Then the closed-loop NCS is described as
(4)Ω1:z(k+1)=A1z(k),A1=[A+BK1-BK10LCA-LC-L00I].(S2)
The data packet is transmitted successfully without network delay and the corresponding controller gain isK2 in this case. Then the closed-loop NCS can be described as
(5)Ω2:z(k+1)=A2z(k),A2=[A+BK2-BK200A-LC0C00].(S3)
The data packet is transmitted successfully with network delaydk and the corresponding controller gain becomes K3,dk here. Then the closed-loop NCS is as follows:
(6)Ω3:z(k+1)=A3z(k)+Ad3z(k-dk),A3=[A+BK3,dk-BK3,dk0LCA-LC0000],Ad3=[000-LC00C00].From the above analysis, we can conclude that there are three different cases which may appear during every sampling period. So the closed-loop NCS can be described as a discrete-time switched system within three subsystemsΩ1 to Ω3. In subsystem Ω3, when dk=0, Ω3 turns to Ω2. And the system matrices Ω1 and Ω2 are similar. So subsystem Ω3 in case 3 includes cases 1 and 2 by appropriately choosing the value of the matrices. Then the wireless NCS can be represented by the following switched system with time-varying delay:
(7)z(k+1)=Aiz(k)+Adiz(k-dk),i=1,2,3,
where Ad1=0, Ad2=0, and dk=1,2,…,dM.To end this section, the following definition and lemma are introduced to obtain our main results.Definition 2.
For any given initial conditions(k0,ϕ)∈ℜ+×Cn, (7) is globally exponentially stable if the solutions of (10) satisfy
(8)∥x(k)∥≤aλ-(k-k0)∥x(k0)∥,∀k≥k0,
where a>0 is a constant and λ>1 is the decay rate.Lemma 3 (see [2]).
For any appropriately dimensional matricesR=RT>0, N, X, η(l)≜x(l+1)-x(l) and two positive integer time-varying d(k1), d(k2) satisfying d(k1)+1≤d(k2)≤dM, the following equality holds:
(9)-∑l=k-d(k2)k-d(k1)-1‍ηT(l)Rη(l)=2ξT(k)N[x(k-d(k1))-x(k-d(k2))]+(d(k2)-d(k1))ξT(k)Xξ(k)-∑l=k-d(k2)k-d(k1)-1‍[ξ(k)η(l)]T[XN*R][ξ(k)η(l)].


## 3. Stability Analysis of the Wireless NCS

More generally, we consider the following discrete-time switched system with time-varying delay:(10)z(k+1)=Aiz(k)+Adiz(k-dk),i=1,2,…,N,iiiiiiiiiiiiiiiiiiiiiiiiiidk=1,2,…,dM,
where N is the number of the subsystems. Suppose that the event rates of the described subsystems Si are defined as r1,r2,…,rN. The time interval [0,kT] will be simplified [0,k] in the following text. Let ni, i=1,2,…,N denote the times in which the subsystems Si are activated on the interval [0,k]. Then we can obtain
(11)k=∑i=1N‍ni;ri=nik,i=1,2,…,N;∑i=1N‍ri=1.The following theorem gives a criterion to guarantee that the Lyapunov functionV(k) exponentially decays along state trajectory of system (10).Theorem 4.
Given scalarλ>0 and any delay satisfying 0≤dk≤dM, if there exist appropriate dimensional symmetric positive-definite matrices P, Q1, Q2, R, symmetric matrix X=[X1X2*X3]≥0, and matrices G,N=[N1TN2T]T and M=[M1TM2T]T, such that the following matrix inequalities hold:
(12)Φ(dk)=[Φ11Φ12-N1P+(A^iT-I)GT-G*Φ22-N2A^diTGT**-Q20***P+dMR-G-GT]<0,(13)[XN*R]>0,[XM*R]>0,
where dk=0,…,dM, A^i=λAi, A^di=λ1+dkAdi, and
(14)Φ11=dMQ1+Q2+dMX1+Sym(M1+G(A^i-I)),Φ12=dMX2+N1-M1+M2T+GA^di,Φ22=dMX3-Q1+Sym(N2-M2),
then
(15)V(k)<λ-2(k-k0)V(k0).Proof.
The following expression ofV(k) is the Lyapunov function of system (10):
(16)V1(k)=zT(k)Pz(k),V2(k)=∑l=k-dkk-1‍λ2(l-k)zT(l)Q1z(l)V2(k)=+∑m=-dM+20‍∑l=k+m-1k-1‍λ2(l-k)zT(l)Q1z(l),V3(k)=∑l=k-dMk-1‍λ2(l-k)zT(l)Q2z(l),V4(k)=∑m=-dM-1‍∑l=k+mk-1‍λ2(l-k)(λz(l+1)-z(l))TV4(k)=×R(λz(l+1)-z(l)).
And defining ξ(k)=λk-k0z(k), λ>0, A^i=λAi, A^di=λ1+dkAdi, combined with (10), we have
(17)ξ(k+1)=A^iξ(k)+A^diξ(k-dk),i=1,2,…,N,dk=1,2,…,dM.
Choose the following Lyapunov function of system (17):
(18)W(k)=ξT(k)Pξ(k)+∑l=k-dkk-1‍ξT(l)Q1ξ(l)+∑m=-dM+20‍∑l=k+m-1k-1‍ξT(l)Q1ξ(l)+∑l=k-dMk-1‍ξT(l)Q2ξ(l)+∑m=-dM-1‍∑l=k+mk-1‍δT(l)Rδ(l),
where δ(l)=ξ(l+1)-ξ(l). Then the forward difference for W(k) along any trajectory of system (17) is given by
(19)ΔW(k)=W(k+1)-W(k)=ξT(k+1)Pξ(k+1)-ξT(k)Pξ(k)+dMξT(k)Q1ξ(k)+∑l=k+1-dk+1k-1‍ξT(l)Q1ξ(l)-∑l=k-dM+1k-1‍ξT(l)Q1ξ(l)-∑l=k-dkk-1‍ξT(l)Q1ξ(l)+ξT(k)Q2ξ(k)-ξT(k-dM)Q2ξ(k-dM)+dMδT(k)Rδ(k)+dM[ξ(k)ξ(k-dk)]TX[ξ(k)ξ(k-dk)]+2[ξ(k)ξ(k-dk)]T[N1N2]×(ξ(k-dk)-ξ(k-dM))+2[ξ(k)ξ(k-dk)]T[M1M2](ξ(k)-ξ(k-dk))-∑l=k-dMk-dk-1‍[ξ(k)ξ(k-dk)δ(l)]T[XN*R][ξ(k)ξ(k-dk)δ(l)]-∑l=k-dkk-1‍[ξ(k)ξ(k-dk)δ(l)]T[XM*R][ξ(k)ξ(k-dk)δ(l)]+2(ξT(k)+δT(k))G×[(Ai-I)ξT(k)+Adiξ(k-dk)-δ(k)]<θT(k)Φθ(k)<0,
where θT(k)=[ξT(k)ξT(k-dk)ξT(k-dM)δT(k)], which means that W(k)<W(k0) for any k≥k0. Furthermore,
(20)V1(k)=zT(k)Pz(k)=λ-2(k-k0)ξT(k)Pξ(k)V2(k)=∑l=k-dkk-1‍λ2(l-k)λ-2(l-k0)ξT(l)Q1ξ(l)+∑m=-dM+20∑l=k+m-1k-1‍λ2(l-k)λ-2(l-k0)ξT(l)Q1ξ(l)V3(k)=∑l=k-dMk-1‍λ2(l-k)λ-2(l-k0)ξT(l)Q2ξ(l)V4(k)=∑m=-dM-1‍∑l=k+mk-1‍λ2(l-k)λ-2(l-k0)δT(l)Rδ(l)V(k)=V1(k)+V2(k)+V3(k)+V4(k)=λ-2(k-k0)W(k).
As W(k0)=V(k0), it is easy to verify that V(k)<λ-2(k-k0)V(k0). This completes the proof.The following theorem gives a sufficient condition for the closed-loop NCS (10) to be exponentially stable.Theorem 5.
The discrete-time switched system (10) is stable if there exist Lyapunov function V(k), ri defined in (11), and some positive scalars μi, i=1,2,…,N which correspond to each subsystem such that the following inequalities hold:
(21)V(k)<μi-2(k-k0)V(k0),∏i=1N‍μiri<1.Proof.
Defining the transition time of the subsystems to bet1=0, t2,…,tk=k, then
(22)V(k)<μtktk-tk-1V(tk-1)<μtktk-tk-1μtk-1tk-1-tk-2V(tk-2)<⋯<∏i=1N‍μiniV(0)=(∏i=1N‍μiri)kV(0).
So the system is stable if ∏i=1N‍μiri<1. This completes the proof.


## 4. Output-Feedback Controller Design

An algorithm to design the observer-based output-feedback controller of the wireless NCS is presented in this section. Since the data packets are time-stamped, the packet loss rate and time delay are known to the controller, which is designed to depend on both the packet loss rate and time delay. Firstly,Ai can be rewritten as Ai=A0i+B00DiC0i, i=1,2,3, where
(23)A01=[A000A000I],B00=[B00I00],Di=[Ki00L],C01=[I-I0C-C-I]A02=[A000A0C00],C02=[I-I00C0]A03=[A000A0000],C03=[I-I0C-C0]Theorem 6.
The system (7) is stable if there exist positive scalars μi, i=1,2,3 satisfying ∏i=13‍μiri<1 and appropriate dimensional matrices P, Q1, Q2, R, X1, X3, matrices M1, M2, N1, N2, X2 and matrices G1, G2, G3, and matrices K1, K2, K3,dk such that the following matrix inequality holds:
(24)Φ0+Φ1<0,Φ0+Φ2<0,Φ0+Φ3(dk)<0,dk=1,2,…,dM,[XN*R]>0,[XM*R]>0,
where
(6)Φ0=[Σ11Σ12-N1P-GT-G*Σ22-N20**-Q20***P+dMR-G-GT],Σ11=dMQ1+Q2+dMX1+Sym(M1-G),Σ12=dMX2+N1-M1+M2T,Σ22=dMX3-Q1+Sym(N2-M2),G=[G1000G2000G3],Φ1=[ϕ¯1100ϕ¯14*000**00***0],ϕ¯11=[ψ¯11ψ¯120*ψ¯22-μ1G2L**μ1G3+μ1G3T],ψ¯11=Sym(μ1G1A+μ1G1BK1),ψ¯12=-μ1G1BK1+μ1CTLTG2T,ψ¯22=Sym(μ1G2A-μ1G2LC),ϕ¯14=[φ¯11-μ1CTLTG2T0-μ1K1TBTG1Tφ¯2200-μ1LTG2Tμ1G3T],φ¯11=μ1ATG1T+μ1K1TBTG1T,φ¯22=μ1ATG2T-μ1CTLTG2T,Φ2=[ϕ^1100ϕ^14*000**00***0],ϕ^11=[ψ^11ψ^120*ψ^220**0],ψ^11=Sym(μ2G1A+μ2G1BK2),ψ^12=-μ2G1BK2,ψ^22=Sym(μ2G2A-μ2G2LC),ϕ^14=[φ^110μ2CTG3T-μ2K2TBTG1Tφ^220000],φ^11=μ2ATG1T+μ2K2TBTG1T,φ^22=μ2ATG2T-μ2CTLTG2T,Φ3=[ϕ~11ϕ~120ϕ~14*00ϕ~24**00***0],ϕ~11=[ψ~11ψ~120*ψ~220**0],ψ~11=Sym(μ3G1A+μ3G1BK3,dK),ψ~12=-μ3(G1BK3,dK+CTLTG2T),ψ~22=Sym(μ3G2A-μ3G2LC),ϕ~12=[000-μ1+dkG2LC00000],ϕ~24=ϕ~12T,ϕ~14=[φ~11μ3CTLTG2T0-μ3K3,dkTBTG1Tφ~220000],φ~11=μ3ATG1T+μ3K3,dkTBTG1T,φ~22=μ3ATG2T-μ3CTLTG2T.Replacing the system matrices of (12) by (7), it is easy to obtain the above results according to Theorems 4 and 5. Furthermore, let G2L=L¯, G1BKi=K¯i, i=1,2, G1BK3=K¯3,dk, and the controllers gain matrices Ki, L can be gained by solving the corresponding linear matrix inequalities.


## 5. Simulation Results

A numerical example of output-feedback stabilization of wireless NCS is evaluated in this section.Consider the following discrete-time system:(26)A=[0.68520.33580.58320.80980.78960.43890.20650.66330.8528],B=[011],C=[110]T.
The eigenvalues of A are 1.7183 and 0.3142±0.2595i and the system is unstable without control. Let the event rates of the packet loss and time delay be r1=0.5 and r3=0.2, respectively, and let the maximum delay be dM=3. Choose μ1=1.8, μ2=0.5, and μ3=0.6. So
(27)μ1r1μ2r2μ3r3=1.80.5×0.50.3×0.60.2=0.9839<1.By using Theorem6, a suitable controller gain matrix can be obtained. The simulation result is shown in Figure 2. The above subgraph depicts the packet loss and the time delay of the wireless NCS in Figure 2. When the delay value is -1, it means that this packet is lost. And the below graph is to describe the state trajectories of closed-loop wireless NCS. From Figure 2, the states of the system diverge at the case when the data packets are dropped, but they converge to zero finally. Therefore, the example illustrates the effectiveness of the proposed method.Figure 2
State trajectories of NCS with delay and packet loss.


## 6. Conclusions

The problem of modeling and stabilization of wireless NCS with both packet loss and time-varying delay is discussed in this paper. The output-feedback controller based on state observer is designed to stabilize the closed-loop wireless NCS by using ADS approach. And the numerical example is presented to demonstrate the effectiveness of the proposed result.



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*Source: 206475-2014-03-25.xml*