# Impulsive Controller Design for Complex Nonlinear Singular Networked Systems with Packet Dropouts

Authors: Xian-Lin Zhao; Ai-Min Wang; Cun-Li Dai; Xiao-Lin Li; Wei Lu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Globally exponential stability of Complex (with coupling) Nonlinear Singular Impulsive Networked Control Systems (CNSINCS) with packet dropouts and time-delay is investigated. Firstly, the mathematic model of CNSINCS is established. Then, by employing the method of Lyapunov functional, exponential stability criteria are obtained and the impulsive controller design method is given. Finally, some simulation results are provided to demonstrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

At present, singular system is widely used in the control of spacecraft, flexible robot, complex power, large chemical and wireless transmission system [1–5]. Many results had been achieved for discrete singular system and time-delay singular system. Such as in [6], the nonlinear discrete singular perturbation model was established and the system condition was given. In [7], chattering free sliding mode control for uncertain discrete time-delay singular system was investigated. The asymptotically stable was established, and the chattering problem that appears in traditional variable structure system was eliminated. As for time-delay singular system, the stability of uncertain time-delay singular systems was researched and the asymptotic stability condition was achieved in [8] by using Jensen integral inequality and feedback control method.On the other hand, singular system has impulsive behavior in many cases [9–14]. So it is very important to discuss the problem of impulsive control. For the stability of the impulsive control system, nonlinear impulsive control was put forward and the concept of asymptotic stability condition was provided in [11]. Asymptotic stability condition for a class of uncertain impulsive system was established through the comparison theorem in [12]. Switch control method was used to research the stability of singular impulsive system, robust stabilization, and H∞ control problem in [13]. Linear approximation and the LMI method were used, respectively, to study the problem of system stability and the sufficient conditions for asymptotic stability in [14].In network impulsive control system packet dropouts and time-delay exist which will influence the stability of singular system. It is necessary to analyze stability condition and the method of controller design. That is the problem focussed in this study. According to the Lyapunov function theory and comparison theorem, the sufficient conditions for the global exponential stability of the system is obtained. The detailed design process of impulsive controller is given in the paper. System will be stable in accordance with the decay rate to achieve exponential stability. A numerical example is provided to illustrate the correctness of theoretical and the effectiveness of design method.


## 2. The Mathematic Model of CNSINCS

The mathematic model of CNSINCS can be described as(1)Ex˙i(t)=Axi(t)+f(xi(t))+∑j=1N‍GijΓxj(t-τ(t)),Ex˙i(t)=Axi(t)+fi=1,…,N,t∈(tk-1,tk]xi(tk+)=cixi(tk)+u¯i(tk),t=tk+,k=1,2,…xi(t)=φi(t),t∈[-τ¯,0],
where xi(t)∈Rn is the state vector of the ith node. A is a constant matrix of n×n. ci is known scalar. E∈Rn×n is a singular constant matrix, and 0<rankE=r≤n, without loss of generality; we hypotheses E=[Ir000]. f(·) is a nonlinear function. Γ is the internal coupling matrix. G=Gij∈Rn×n is the coupling matrix of the whole network structure and weights. τ(t) is network transmission delay and is assumed to satisfy 0≤τ(t)≤τ¯.In the process of data transmitting, the buffer’s model can be described as:(2)u¯i(tk)={ui(tk),k=1,2,…,iftransmittedsuccessfully,u¯i(tk-1),k=1,2,…,otherwise,x¯i(tk)={xi(tk),k=1,2,…,iftransmittedsuccessfully,x¯i(tk-1),k=1,2,…,otherwise.The impulsive controller can be designed as(3)u¯i(tk)=Kix¯i(tk),
where u¯i(tk)∈Rm. Substituting (2) and (3) into (1), the closed-loop nonlinear singular impulsive networked system model is obtained as follows:
(4)Ex˙i(t)=Axi(t)+f(xi(t))+∑j=1N‍GijΓxj(t-τ(t)),Ex˙i(t)=Axi(t)+fi=1,…,N,t∈(tk-1,tk],xi(tk+)=cixi(tk)+(1-σi(tk))Kixi(tk)+σi(tk)Kix¯i(tk-1),t=tk+,k=1,2,…,xi(t)=φi(t),t∈[-τ¯,0],
where σi(tk)=1 denotes that there are data dropouts and, σi(tk)=0, there are no packet dropouts.Lemma 1.
LetP∈Rn×n be a symmetric positive definite matrix and P=QTQ. For any x,y∈Rn×n and A∈Rn×n, then (1)
xT(ATPE+ETPTA)x≤2λmax(A)xTETPEx,(2)
xT(ATP+PA)x≤2μ(QAQ-1)xTPx,(3)
|xTPy|≤xTPxyTPy.Lemma 2.
According to the definition of Kronecker product, for a given matrixA,B, and scalar α, the following equality can be achieved: (1)
(αA)⊗B=A⊗(αB),(2)
(A+B)⊗C=(A⊗C)+(B⊗C),(3)
(A⊗B)⊗(C⊗D)=(AC)⊗(BD).IfX(t)=(x1T(t),…,xNT(t))T, C=diag{c1,…,cN}, and F(X(t))=(fT(x1T(t),…,fT(xNT(t))T, according to Lemma 1, the complex nonlinear singular system can be expressed as
(5)(IN⊗E)X˙(t)=(IN⊗A)X(t)+F(X(t))(IN⊗E)X˙(t)=+(G⊗Γ)X(t-τ(t)),t∈(tk-1,tk],X(tk+)=CX(tk)+(1-σ(tk))KX(tk)X(tk+)=+σ(tk)X¯(tk-1),t=tk+.
Suppose
(6)E¯=[E000],I¯N=[IN000],Z(tk)=[X(tk)X(tk-1)],A¯=[A000].
Then (5) is equivalent to the following system:
(7)(I¯N⊗E¯)z˙(t)=(I¯N⊗A¯)z(t)+[F(x(t))0](I¯N⊗E¯)z˙(t)=+[G⊗Γ000]z(t-τ(t)),(I¯N⊗E¯)z(tk+)=C(I¯N⊗E¯)z(tk)+[(1-σ(tk))Kσ(tk)K00](I¯N⊗E¯)z(tk).Lemma 3 (see [15]).
IfL(t,z(t)) and Uk(z(t)) satisfy the Lipchitz condition, there exists a uniqueness of solution to nonlinear singular impulsive differential equation which is written as
(8)z˙(t)=L(x,t),t∈(tk-1,tk],Δz(t)=Uk(z(t)),t=tk+,k=1,2,…,
where z(t)∈Rn,L:R+×Rn→Rn,Uk:Rn→Rn.


## 3. The Design of CNSINCS

For the nonlinear singular networked impulsive control system (4), we have the following theorem.Theorem 4.
If there exist0<ρ=supk∈N{tk-tk-1}<∞ and a nonsingular matrix K∈Rn×n, such that
(9)2lnβρ+(2λmax(A¯)+2Lλmax((I¯N⊗E¯)TP)λmin((I¯N⊗E¯)TP)+1)+λmax2(G⊗Γ)<1(10)0<β<1,
where
(11)β=[λmax2(C+[(1-σ(tk))Kσ(tk)K00])]1/2
then the nonlinear singular networked impulsive control system (4) is asymptotically stable:
(12)∥x(t)∥≤1βλmax(P)λmin(P)sup-τ≤θ¯≤0{∥ϕ(θ)∥}e-(λ/2)t,
where λ is the positive solution of λ+p+qeλτ¯=0, and q>0,p+q<0:
(13)p=2lnβρ+(2λmax(A¯)+2Lλmax((I¯N⊗E¯)TP)λmin((I¯N⊗E¯)TP)+1)+λmax2(G⊗Γ).Proof.
From [16], we know that λ+p+qeλτ¯=0 must have a solution. Set V(t)=zT(t)(I¯N⊗E¯)TP(I¯N⊗E¯)z(t), where (I¯N⊗E¯)TP(I¯N⊗E¯)≥0.
Whent∈(tk-1,tk], the derivative of V(t) along the trajectories of the CNSINCS (4) is
(14)V˙(t)=((I¯N⊗E¯)z˙(t))TP(I¯N⊗E¯)z(t)+((I¯N⊗E¯)z(t))TPT(I¯N⊗E¯)z˙(t)=((I¯N⊗A¯)z(t)+[F(x(t))0]+[G⊗Γ000]×z(t-τ(t))e899(I¯N⊗A¯)z(t)+[F(x(t))0]+[G⊗Γ000])TP(I¯N⊗E¯)z(t)+((I¯N⊗E¯)z(t))TP(I¯N⊗A¯)z(t)[F(x(t))0]+[G⊗Γ000]z(t-τ(t))=zT(t)(A¯TP(I¯N⊗E¯)+(I¯N⊗E¯)TPTA¯)×z(t)+FT(x(t))P(I¯N⊗E¯)z(t)+zT(t)(I¯N⊗E¯)TPF(x(t))+zT(t-τ(t))GTPEz(t)+zT(t)(I¯N⊗E¯)TPTGz(t-τ(t)).
According to Lemma 3 we have
(15)V˙(t)≤2λmax(A¯)zT(t)(I¯N⊗E¯)TP(I¯N⊗E¯)z(t)+2Lλ(I¯N⊗E¯)TP∥z(t)∥+2zT(t)P(I¯N⊗E¯)z(t)×zT(t-τ(t))(G⊗Γ)T(I¯N⊗E¯)TP(G⊗Γ)z(t-τ(t))≤λmax2(G⊗Γ)zT(t-τ(t))(I¯N⊗E¯)TP(I¯N⊗E¯)z(t-τ(t))+2Lλmax((I¯N⊗E¯)TP)λmin((I¯N⊗E¯)TP)zT(t)(I¯N⊗E¯)TP(I¯N⊗E¯)z(t)+2λmax(A¯)zT(t)(I¯N⊗E¯)TP(I¯N⊗E¯)z(t)+zT(t)(I¯N⊗E¯)TP(I¯N⊗E¯)z(t).
On the other hand, whent=tk+,
(16)V˙(tk+)=zT(tk+)(I¯N⊗E¯)TP(I¯N⊗E¯)z(tk+)=zT(tk+)(I¯N⊗E¯)T(I¯N⊗E¯)T×P(I¯N⊗E¯)(I¯N⊗E¯)z(tk+)=[(I¯N⊗E¯)zT(tk+)]T(I¯N⊗E¯)TP(I¯N⊗E¯)×[(I¯N⊗E¯)z(tk+)]=([(1-σ(tk))Kσ(tk)K00]C(I¯N⊗E¯)z(tk)+[(1-σ(tk))Kσ(tk)K00](I¯N⊗E¯)z(tk))T,(I¯N⊗E¯)TP(I¯N⊗E¯)×(C+[(1-σ(tk))Kσ(tk)K00])(I¯N⊗E¯)z(tk)=zT(tk)(I¯N⊗E¯)T×(C+[(1-σ(tk))Kσ(tk)K00])T×(I¯N⊗E¯)T,P(I¯N⊗E¯)z(tk)≤λmax2(C+[(1-σ(tk))Kσ(tk)K00])×zT(tk)(I¯N⊗E¯)T(I¯N⊗E¯)TP,(I¯N⊗E¯)(I¯N⊗E¯)z(tk)=λmax2(C+[(1-σ(tk))Kσ(tk)K00])×zT(tk)(I¯N⊗E¯)TP(I¯N⊗E¯)z(tk)=β2V(tk),
where β2=λmax2(C+[(1-σ(tk))Kσ(tk)K00]), supposing that ɛ>0 is random constant, a comparison system can be established as follows:
(17)v˙(t)=(2λmax(A¯)+2Lλmax((I¯N⊗E¯)TP)λmin((I¯N⊗E¯)TP)+1)v(t)+λmax2(G⊗Γ)v(t-τ(t))+ɛt≠tk,v(tk+)=βv(tk),t=tk,v(θ)=λmax(P)∥ϕ(θ)∥2,-τ¯≤θ≤0.
It is clear that V(t)≤v(t) when -τ¯≤θ≤0, according to [16], and we have V(t)≤v(t) when t≥0; the trivial solution of the comparison system is
(18)v(t)=W(t,0)v(0)+∫0tW(t,s)λmax2(G⊗Γ)v(s-τ(s)+ɛ)ds,t≥0,
where W(t,s) is Cauchy matrix which satisfies
(19)W(t,s)=β2η(t,s)×e{(2λmax(A¯)+2L(λmax((I¯N⊗E¯)TP)/λmin((I¯N⊗E¯)TP))+1)(t-s)}≤β2(t-s/ρ)-1e(p-(2lnβ/ρ))(t-s)≤β-2ep(t-s)
in which η(t,s) is the number of control impulses in the interval (s,t], 0<ρ=supk∈N{tk-tk-1}<∞, for t≥0; we have
(20)v(t)≤β-2λmax((I¯N⊗E¯)TP(I¯N⊗E¯))∥φ(0)∥2+∫0tβ-2ep(t-s)λmax2(G⊗Γ)v(s-τ(s)+ɛ)ds≤γept+ep(t-s)(qv(s-τ(s)+ɛ)ds,
where
(21)γ=β-2λmax((I¯N⊗E¯)TP(I¯N⊗E¯))sup-τ¯≤s≤0∥φ(s)∥2=β-2λmax(P)sup-τ¯≤s≤0∥φ(s)∥2.
In the following, we will prove that the following inequality holds:(22)v(t)≤γe-λt-ɛβ2p,t≥0
Since ɛ>0,p<0, so ɛ/β2p<0. If there exists t*>0 which satisfies
(23)v(t*)≥γe-λt*-ɛβ2p,(24)v(t)>γe-λt-ɛβ2p,t<t*.
From (16) and (24) we have
(25)v(t*)≤γe-λt*+∫0t*ep(t*-s)[qv(s-τ(s))+ɛ]ds<ept*{γ-ɛβ2(p+q)+∫0t*e-ps[γqe-λ(s-τ(s))-ɛqβ2(p+q)+ɛβ2]ds}<ept*{γ-ɛβ2(p+q)+γqeλτ×∫0t*e-(p+λ)sds+ɛpβ2(p+q)∫0t*e-psds}<ept*{γ-ɛβ2(p+q)+γ[e-(p+λ)t*-1]-ɛβ2(p+q)(e-pt*-1)}=γe-λt*-ɛβ2(p+q)
which contradicts with (23), and consequently (22) holds. Let ɛ→0; then
(26)V(t)≤v(t)≤γe-λt,t≥0.
Moreover
(27)V(t)≥λmin(P)∥x(t)∥2,t≥0.
Combining the inequality (26) and (27),
(28)∥x(t)∥≤(1β)λmax(P)λmin(P)sup-τ≤θ¯≤0{∥ϕ(θ)∥}e-(λ/2)t
which implies conclusion (22) and this completes the proof.Remark 5.
For the caseβ≥1, we can replace the condition 0<ρ=supk∈N{tk-tk-1}<∞ with 0<ς=infk∈N{tk-tk-1}<∞; then the conclusion of Theorem 4 still holds except that now inequality (9) becomes
(29)2lnβς+(2λmax(A¯)+2Lλmax((I¯N⊗E¯)TP)λmin((I¯N⊗E¯)TP)+1)+λmax2(G⊗Γ)<1.
For β≥1, we have
(30)∥xi(t)-xj(t)∥≤1βsup-τ¯≤t≤0{∥ϕ(t)∥}e-(λ/2)t
in which λ=maxk∈N{lnβ/(tk-tk-1)}.The proof of the above conclusion remains largely the same as Theorem4, so we omitted it to avoid repetition.


## 4. Design Procedure of Impulsive Control for Complex Network

According to Theorem4, the design process of impulsive control is given as follows.(1)
Calculate the parametersL,m.(2)
Choose a matrixP which satisfies (I¯N⊗E¯)TP(I¯N⊗E¯)≥0.(3)
For a given parameterλ0, we can determine the control sequence {tk}, t∈N as follows. If 0<β<1, let Θ:=(2λmax(A¯)+2L(λmax((I¯N⊗E¯)TP)/λmin((I¯N⊗E¯)TP))+1)+λmax2(G⊗Γ); then the upper bounds of time can be taken as 0<ρ=supk∈N{tk-tk-1}=-(lnβ)/Θ; if β≥1, let Θ:=(2λmax(A¯)+2L(λmax((I¯N⊗E¯)TP)/λmin((I¯N⊗E¯)TP))+1)+λmax2(G⊗Γ); then the lower bounds of control intervals is 0<ς=infk∈N{tk-tk-1}=(lnβ)/Θ.


## 5. Numerical Simulation

In this section, a numerical example is presented to illustrate the effectiveness of derived results.Example 6.
Consider the following complicated nonlinear singular system:(31)Ex˙i(t)=Axi(t)+f(xi(t))+∑j=1N‍GijΓxj(t-τ(t)),Ex˙i(t)=Axi(t)+fi=1,…,N,t∈(tk-1,tk],xi(tk+)=cixi(tk)+(1-σi(tk))Kixi(tk)+σi(tk)Kix¯i(tk-1),t=tk+,k=1,2,…,xi(t)=φi(t),t∈[-τ¯,0].
The parameters are given as follows:
(32)E=[1000010000100000],c1=c2=1.2,A=[-10100083-10000-8300021],N=2,f(t,xi(t))=(0(-xi1(t)xi3(t))T(xi1(t)xi2(t))T2xi3(t))T.
For simplicity, consider the system with 2 nodes. Assume that the external coupling matrix is G=[-733-4] and the internal coupling matrix is
(33)Γ=[0.1-0.2-0.1000.10.20-0.200.100.10.20.1-0.1].
Supposing that τ(t)=0.02sint. According to Lemma 1, we can choose P=I8×8, T=tk+1-tk=0.005. The region of parameters of chaotic system is L=80, and the gain of impulsive controller is -1.8653≤K≤-0.5347. The state trajectory diagram of system is depicted in Figure 1. For the case of packet dropouts probability is Pr(σ(k)=0∣0.8) and initial condition is x1(t)=[32-12]T, x2(t)=[65-48]T, and  t∈(-τ,0).Figure 1
The state response of CNSINCS via impulsive control (color online).Figure1 shows that the asymptotic stability of the closed-loop uncertain system can be guaranteed using the networked impulsive controller designed in this paper.


## 6. Conclusion

In this paper, the global exponential stability CNSINCS via impulsive control is investigated. According to the Lyapunov stability theory, the mathematic model of CNSINCS is established. A general model of network consisting of time-delay and packet dropouts has been formulated and the globally exponential stable sufficient conditions have been established. Impulsive controller, which may ensure the system achieves exponential stability with a given decay rate is designed. Therefore our control scheme is efficient and practical in dealing with problems of data transmission with time-delay and packet dropouts. As an application, a numerical simulation is given to demonstrate the usefulness and practicability of proposed theoretical results.



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