# Dynamic Output Feedback Control for Nonlinear Networked Control Systems with Random Packet Dropout and Random Delay

Authors: Shuiqing Yu; Junmin Li; Yingde Tang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper investigates the dynamic output feedback control for nonlinear networked control systems with both random packet dropout and random delay. Random packet dropout and random delay are modeled as two independent random variables. An observer-based dynamic output feedback controller is designed based upon the Lyapunov theory. The quantitative relationship of the dropout rate, transition probability matrix, and nonlinear level is derived by solving a set of linear matrix inequalities. Finally, an example is presented to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Networked control systems (NCSs) are feedback control loops closed through a real time network. Compared with conventional point-to-point control systems, NCSs have many attractive advantages of low cost, simple installation, reduced wiring, easy maintenance, and so on. On the other hand, the introduction of network also presents some challenging problems such as quantization [1–4], packet dropout [5–8], and time delays [9–11], which bring difficulties for analysis and design of NCSs. Hence, NCSs have attracted considerable research interests in recent years [12–14].Generally, time delays and packet dropout are two essential issues that need careful consideration in NCSs design. They can be handled separately or simultaneously. Some results have been available. Markov chain and independent Bernoulli random process were commonly used to model the packet loss processes. The results applying the Bernoulli process assume that the packet losses are independent and identically distributed [5, 7], while those using the Markov chain assume that the packet losses are burst and occur according to a Markov chain [6, 8]. Based on a similar Bernoulli random delay model, the controller design problem for NCSs with both sensor-to-controller and controller-to-actuator delays has been studied in [9]. A delay-dependent approach for NCSs controller design has been provided in [10], where the feedback gain of a memoryless controller was determined by solving a set of linear matrix inequalities. In [1, 2, 11], the network-induced random delays were modeled as Markov chains such that the closed-loop system was a jump linear system. In [15], a nonlinear stochastic system model was proposed to describe the NCSs with both random packet dropout and network-induced time-varying delay. Based on this model, a sufficient condition was derived to the exponential stability of the nonlinear NCSs. However, [15] only studied the static output feedback controller and time-varying delay. In this paper, we extend the results in [15] to general random delay and dynamic output feedback controller.This paper proposes a new nonlinear networked control systems model with both random packet dropout and random delay. In this model, two random variables are included and they are independent of each other. The random delay is modeled as a Markov chain with partly known transition probability matrix, which covers the completely known and completely unknown transition probabilities as its two special cases. The random packet dropout is modeled as Bernoulli distributed white sequences. A sufficient condition is presented for the stabilization of the closed systems, and a dynamic output feedback controller is designed based upon the Lyapunov theory. An example is presented to illustrate the effectiveness of the proposed method.Notation.
Throughout the paper,Rn denotes the n-dimensional Euclidean space. The superscripts “T” and “–1” stand for the transpose and inverse of a matrix, respectively. diag{⋯} stands for a block-diagonal matrix. The symbol * denotes the symmetric part in a symmetric matrix. The notation P>0 means that P is real symmetric and positive definite matrix. In addition, E{x} and E{x∣y} denote expectation of x and x conditional on y, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.


## 2. Problem Statement and Preliminaries

The structure of the NCSs to be discussed in this paper is depicted in Figure1.Figure 1
Block diagram of the considered NCSs.The plant model is described by the following nonlinear discrete-time system:(1)x(k+1)=Ax(k)+Bu(k)+f(x(k))y(k)=Cx(k),
where x(k)∈Rn, u(k)∈Rm, and y(k)∈Rp are the state, control input, and measured output, respectively, and A, B, C are known real matrices with appropriate dimensions. The function f(x(k)) satisfies the following inequality:
(2)fT(x(k))f(x(k))≤β2xT(k)GTGx(k),
where β>0 is a scalar parameter and G is a constant matrix.Throughout the paper, the following assumptions are needed.Assumption 1.
The statex(k) is not measurable.Assumption 2.
For convenience of analysis, it is assumed that the network induced packet dropout and delay occur only in the sensor-to-controller side.In what follows, we model signal transmission delay and random packet dropout mathematically.


### 2.1. Signal Transmission Delay

Letdk represent the random delay from sensor to controller. Assume that dk take the positive integers between dm and dM. In real communication systems, current time delay is usually affected by previous time delays. It is reasonable to model the time delay dk as a Markov chain and its transition probability matrix is Π=[λrs]; that is, dk jump from mode r to s with probability λrs, which is defined by
(3)λrs=Pr(dk+1=s∣dk=r),
where λrs>0, ∑s=dmdMλrs=1, for all r, s∈Λ, where Λ={dm,dm+1,…,dM}. In reality, the transition probabilities of the Markov chain in this paper are usually partially available; namely, some elements in matrix Π are unknown. For instance, dk with four modes will have the transition probability matrix Π as
(4)Π=[λ11?λ13????λ24λ31??λ34??λ43λ44],
where “?” represents the unavailable elements. For notational clarity, for all r∈Λ, we denote Iκr={s:λrsisknown}, Iuκr={s:λrsisunknown}. Moreover, if Iκr≠∅ and have m elements, Iκr and Iuκr can be further described as Iκr={κ1r,…,κmr}, Iuκr={κ-1r,…,κ-M-mr}. Where κmr∈Z+ represents the mth known element with the index κmr in the rth row of matrix Π and κ-M-mr∈Z+ represents the (M-m)th unknown element with the index κ-M-mr in the rth row of matrix Π.


### 2.2. Random Packet Dropout

The Bernoulli distributed white sequenceαk is introduced to reflect the random packet dropout from the sensor to the controller, which takes value in {0,1} and satisfies
(5)Prob{αk=1}=E{αk}=α-,0≤α-≤1,Prob{αk=0}=1-E{αk}=1-α-,var{αk}=E{(αk-α-)2}=(1-α-)α-=α1.Assumption 3.
The random variablesαk and dk are independent of each other.
In this paper, the measurement with packet dropout and random delay is described by(6)y-(k)=(1-αk)y(k-dk)+αky-(k-1).Byαk=0, we mean that the delayed packet is correctly delivered; that is, y-(k)=y(k-dk). While by αk=1, we mean that the packet is lost; that is y-(k)=y-(k-1).Remark 4.
If the random delay is transformed into the time-varying delay, the model (6) is the same as the one used in [15]. If the transmission delay dk is not taken into account, then (6) reduces to the one used in [6, 7]. This, in turn, implies that the model (6) is more general than those in [6, 7, 15].
The output feedback controller to be designed is of the following form:(7)xc(k+1)=Axc(k)+Bu(k)+L(y-(k)-Cxc(k))u(k)=Kxc(k),
where the gain matrices L and K will be designed later.
Define that(8)e(k)=x(k)-xc(k).
Substituting (6), (7) into (8), we have
(9)e(k+1)=[α-Ly-(k-1)(A-LC)e(k)+LCx(k)-(1-α-)LCx(k-dk)-α-Ly-(k-1)]+(αk-α-)[LCx(k-dk)-Ly-(k-1)]+f(x(k)).
Setingξ(k)=[xT(k)eT(k)y-T(k-1)]T, we obtain the augmented NCSs with random packet dropout and random delay:
(10)ξ(k+1)=A-ξ(k)+B-ξ(k-dk)+C-f(x(k)),
where
(11)A-=A1+(αk-α-)A2,B-=[(1-α-)+(α--αk)]B1,A1=[A+BK-BK0LCA-LC-α-L00α-],A2=[00000-L00-I],B1=[000-LC00C00],C-=[II0].


## 2.1. Signal Transmission Delay

Letdk represent the random delay from sensor to controller. Assume that dk take the positive integers between dm and dM. In real communication systems, current time delay is usually affected by previous time delays. It is reasonable to model the time delay dk as a Markov chain and its transition probability matrix is Π=[λrs]; that is, dk jump from mode r to s with probability λrs, which is defined by
(3)λrs=Pr(dk+1=s∣dk=r),
where λrs>0, ∑s=dmdMλrs=1, for all r, s∈Λ, where Λ={dm,dm+1,…,dM}. In reality, the transition probabilities of the Markov chain in this paper are usually partially available; namely, some elements in matrix Π are unknown. For instance, dk with four modes will have the transition probability matrix Π as
(4)Π=[λ11?λ13????λ24λ31??λ34??λ43λ44],
where “?” represents the unavailable elements. For notational clarity, for all r∈Λ, we denote Iκr={s:λrsisknown}, Iuκr={s:λrsisunknown}. Moreover, if Iκr≠∅ and have m elements, Iκr and Iuκr can be further described as Iκr={κ1r,…,κmr}, Iuκr={κ-1r,…,κ-M-mr}. Where κmr∈Z+ represents the mth known element with the index κmr in the rth row of matrix Π and κ-M-mr∈Z+ represents the (M-m)th unknown element with the index κ-M-mr in the rth row of matrix Π.


## 2.2. Random Packet Dropout

The Bernoulli distributed white sequenceαk is introduced to reflect the random packet dropout from the sensor to the controller, which takes value in {0,1} and satisfies
(5)Prob{αk=1}=E{αk}=α-,0≤α-≤1,Prob{αk=0}=1-E{αk}=1-α-,var{αk}=E{(αk-α-)2}=(1-α-)α-=α1.Assumption 3.
The random variablesαk and dk are independent of each other.
In this paper, the measurement with packet dropout and random delay is described by(6)y-(k)=(1-αk)y(k-dk)+αky-(k-1).Byαk=0, we mean that the delayed packet is correctly delivered; that is, y-(k)=y(k-dk). While by αk=1, we mean that the packet is lost; that is y-(k)=y-(k-1).Remark 4.
If the random delay is transformed into the time-varying delay, the model (6) is the same as the one used in [15]. If the transmission delay dk is not taken into account, then (6) reduces to the one used in [6, 7]. This, in turn, implies that the model (6) is more general than those in [6, 7, 15].
The output feedback controller to be designed is of the following form:(7)xc(k+1)=Axc(k)+Bu(k)+L(y-(k)-Cxc(k))u(k)=Kxc(k),
where the gain matrices L and K will be designed later.
Define that(8)e(k)=x(k)-xc(k).
Substituting (6), (7) into (8), we have
(9)e(k+1)=[α-Ly-(k-1)(A-LC)e(k)+LCx(k)-(1-α-)LCx(k-dk)-α-Ly-(k-1)]+(αk-α-)[LCx(k-dk)-Ly-(k-1)]+f(x(k)).
Setingξ(k)=[xT(k)eT(k)y-T(k-1)]T, we obtain the augmented NCSs with random packet dropout and random delay:
(10)ξ(k+1)=A-ξ(k)+B-ξ(k-dk)+C-f(x(k)),
where
(11)A-=A1+(αk-α-)A2,B-=[(1-α-)+(α--αk)]B1,A1=[A+BK-BK0LCA-LC-α-L00α-],A2=[00000-L00-I],B1=[000-LC00C00],C-=[II0].


## 3. Main Results

In this section, we will present our main results. To this end, the following definition and lemmas will be introduced.Definition 5.
The closed-loop system (10) is said to be stochastically stable if for any initial condition x0∈Rn and d0∈Λ, the following inequality
(12)E{∑k=0∞∥x(k)∥2|x0,d0}<∞holds.Lemma 6 (see [16]).
For given scalarλi≥0 and matrix Pi≥0, i=1,2,…,N, one has
(13)∑i=1NλiPi≤∑i=1Nλi∑i=1NPi.Lemma 7 (see [17]).
For any positive definite matrixW∈Rn×n, two positive integers r and r0 satisfying r≥r0≥1, and vector function x(i)∈Rn, one has
(14)(∑i=r0rx(i))TW(∑i=r0rx(i))≤r~∑i=r0rxT(i)Wx(i),
where r~=r-r0+1.In the following theorem, we will derive a sufficient condition such that the closed-loop system (10) is stochastically stable.Theorem 8.
For the given scalars0≤α-≤1, β>0, and matrices K, L, the closed-loop system (10) is stochastically stable for each mode r∈Λ, if there exist matrices P1(r)>0, P2(r)>0, P3(r)>0, Z1>0, Z2>0, Q1>0, Q2>0, Q3>0 and scalar Γ>0 satisfying the following linear matrix inequality:
(15)[Φ11Φ12Φ13Φ14Z20Φ17*Φ22Φ23Φ2400Φ27**Φ33000Φ37***Φ44Φ45Φ46Φ47****Φ5500*****Φ660******Φ77]<0,
where
(16)Φ11=(A+BK)TP-1(s)(A+BK)-P1(r)+(LC)TP-2(s)(LC)-Z2+(dM-dm+1)Q1+Q2+Q3+(dM-dm)2(A+BK-I)TZ1(A+BK-I)+dm2(A+BK-I)TZ2(A+BK-I)+Γβ2GTG,Φ12=-(A+BK)TP-1(s)(BK)+(LC)TP-2(s)(A-LC)-dm2(A+BK-I)TZ2(BK)-(dM-dm)2(A+BK-I)TZ1(BK),Φ13=-α-(LC)TP-2(s)L,Φ14=-(1-α-)(LC)TP-2(s)(LC),Φ17=(A+BK)TP-1(s)+(LC)TP-2(s)+(dM-dm)2(A+BK-I)TZ1+dm2(A+BK-I)TZ2,Φ22=(BK)TP-1(s)(BK)+(A-LC)TP-2(s)(A-LC)+(dM-dm)2(BK)TZ1(BK)+dm2(BK)TZ2(BK)-P2(r),Φ23=-α-(A-LC)TP-2(s)L,Φ24=-(1-α-)(A-LC)TP-2(s)(LC),Φ27=-(BK)TP-1(s)+(A-LC)TP-2(s)-(dM-dm)2(BK)TZ1-dm2(BK)TZ2,Φ33=α-2LTP-2(s)L+α1LTP-2(s)L+α-2P-3(s)+α1P-3(s)-P3(r),Φ37=-α-LTP-2(s),Φ44=(1-α-)2(LC)TP-2(s)(LC)+α1(LC)TP-2(s)(LC)+(1-α-)2CTP-3(s)C+α1CTP-3(s)C-Q1-2Z1,Φ45=Z1,Φ46=Z1,Φ47=-(1-α-)(LC)TP-2(s),Φ55=-Q2-Z1-Z2,Φ66=-Q3-Z1,Φ77=P-1(s)+P-2(s)+(dM-dm)2Z1+dm2Z2-ΓI.Proof.
Choose the Lyapunov function candidate as(3)V(x(k),dk,αk)=∑i=15Vi(x(k),dk,αk),V1(x(k),dk,αk)=xT(k)P1(dk)x(k)+eT(k)P2(dk)e(k)+y-T(k-1)P3(dk)y-(k-1),V2(x(k),dk,αk)=∑i=k-dkk-1xT(i)Q1x(i)V3(x(k),dk,αk)=∑i=k-dmk-1xT(i)Q2x(i)+∑i=k-dMk-1xT(i)Q3x(i),V4(x(k),dk,αk)=∑j=-dM+1-dm∑i=k+jk-1xT(i)Q1x(i)V5(x(k),dk,αk)=∑j=-dM+1-dm∑i=k+j-1k-1(dM-dm)ηT(i)Z1η(i)+∑j=-dm-1∑i=k+jk-1dmηT(i)Z2η(i),
where η(k)=x(k+1)-x(k)=(A+BK-I)x(k)-BKe(k)+f(x(k)).
Then, fordk+1=s, dk=r, P-(s)=∑s=dmdMλrsP(s), we have
(18)E{ΔV1}=[(A+BK)x(k)-BKe(k)+f(x(k))]T×P-1(s)[(A+BK)x(k)-BKe(k)+f(x(k))]-xT(k)P1(r)x(k)+[α-Ly-(k-1)(A-LC)e(k)+LCx(k)-(1-α-)LCx(k-r)-α-Ly-(k-1)]T×P-2(s)[α-Ly-(k-1)(A-LC)e(k)+LCx(k)-(1-α-)LCx(k-r)-α-Ly-(k-1)]+2[α-Ly-(k-1)(A-LC)e(k)+LCx(k)-(1-α-)LCx(k-r)-α-Ly-(k-1)]T×P-2(s)f(x(k))+α1[LCx(k-r)-Ly-(k-1)]T×P-2(s)[LCx(k-r)-Ly-(k-1)]+fT(x(k))P-2(s)f(x(k))-eT(k)P2(r)e(k)-y-T(k-1)P3(r)y-(k-1)+[(1-α-)Cx(k-r)+α-y-(k-1)]T×P-3(s)[(1-α-)Cx(k-r)+α-y-(k-1)]+α1[y-(k-1)-Cx(k-r)]T×P-3(s)[y-(k-1)-Cx(k-r)],(19)E{ΔV2}=xT(k)Q1x(k)-xT(k-r)Q1x(k-r)+∑i=k+1-dk+1k-1xT(i)Q1x(i)-∑i=k+1-dkk-1xT(i)Q1x(i).
Note that(20)∑i=k+1-dk+1k-1xT(i)Q1x(i)=∑i=k+1-dmk-1xT(i)Q1x(i)+∑i=k+1-dk+1k-dmxT(i)Q1x(i)≤∑i=k+1-dkk-1xT(i)Q1x(i)+∑i=k+1-dMk-dmxT(i)Q1x(i).
Therefore, we have(21)E{ΔV2}≤xT(k)Q1x(k)-xT(k-r)Q1x(k-r)-∑i=k+1-dMk-dmxT(i)Q1x(i)(22)E{ΔV3}=xT(k)Q2x(k)-xT(k-dm)Q2x(k-dm)+xT(k)Q3x(k)-xT(k-dM)Q3x(k-dM)(23)E{ΔV4}=(dM-dm)xT(k)Q1x(k)-∑i=k+1-dMk-dmxT(i)Q1x(i)(24)E{ΔV5}=(dM-dm)2ηT(k)Z1η(k)-∑i=k-dMk-dm-1(dM-dm)ηT(i)Z1η(i)+dm2ηT(k)Z2η(k)-∑i=k-dmk-1dmηT(i)Z2η(i).
Using Lemma7, we have the following inequality
(25)-∑i=k-dmk-1dmηT(i)Z2η(i)-∑i=k-dMk-dm-1(dM-dm)ηT(i)Z1η(i)=-∑i=k-dmk-1dmηT(i)Z2η(i)-∑i=k-rk-dm-1(dM-dm)ηT(i)Z1η(i)-∑i=k-dMk-r-1(dM-dm)ηT(i)Z1η(i)≤-[x(k)-x(k-dm)]T×Z2[x(k)-x(k-dm)]-[x(k-dm)-x(k-r)]T×Z1[x(k-dm)-x(k-r)]-[x(k-r)-x(k-dM)]T×Z1[x(k-r)-x(k-dM)].
From (2), for any scalar Γ>0, we have
(26)Γ(β2xT(k)GTGx(k)-fT(x(k))f(x(k)))≥0.
Then, from (18), (21)–(26), we have
(27)E{ΔV(x(k),dk,αk)}≤ζTΦζ,
where
(28)ζ(k)=[ξT(k)xT(k-r)xT(k-dm)xT(k-dM)fT(x(k))]T(29)Φ=[Φ11Φ12Φ13Φ14Z20Φ17*Φ22Φ23Φ2400Φ27**Φ33000Φ37***Φ44Φ45Φ46Φ47****Φ5500*****Φ660******Φ77].
From (15), we conclude that Φ<0; thus we have
(30)E{ΔV(x(k),dk,αk)}≤ζTΦζ<0.
Following a similar line in the proof of Theorem  1 in [18], the stochastic stability of the closed-loop system (10) can be confirmed.
Now, we apply the result obtained in Theorem8 to solve the controller design problem.Theorem 9.
For the given scalars0≤α-≤1, β>0, εl>0, l=1,2,3,4,5, the closed-loop system (10) is stochastically stable for each mode r∈Λ, if there exist matrices X1(r)>0, X2(r)>0, X3(r)>0, Z1>0, Z2>0, Q1>0, Q2>0, Q3>0, K, L and scalar Γ>0, satisfying the following linear matrix inequality:(31)[Ω-0Ω1Ω2Ω3Ω-3Ω4Ω-4Ω5Ω-5Ω6Ω-6Ω7Ω-7*-2ε1I+ε12Z100000000000**-2ε2I+ε22Z20000000000***Ω8000000000****Ω-800000000*****Ω90000000******Ω-9000000*******Ω900000********Ω-90000*********Ω10000**********Ω-1000***********Ω100************Ω-10]<0,
where
(32)Ω-0=[Φ--11000Z200*-2ε4I+ε42X2(r)00000**-2ε5I+ε52X3(r)0000***-Q1-2Z1Φ45Φ460****Φ5500*****Φ660******-ΓI],Φ--11=-2ε3I+ε32X1(r)-Z2+(dM-dm+1)Q1+Q2+Q3+Γβ2GTG,P-1(r)=X(r),Θ1=[λrκ1r⋯λrκmr],Θ2=[υr⋯υr],Ω1=[(dM-dm)(A+BK-I)-(dM-dm)(BK)0000(dM-dm)I]T,Ω2=[dm(A+BK-I)-dm(BK)0000dmI]T,Ω3=Θ1[(A+BK)-BK0000I]T,Ω-3=Θ2[(A+BK)-BK0000I]T,Ω4=Θ1[LCA-LC-α-L-(1-α-)(LC)00I]T,Ω-4=Θ2[LCA-LC-α-L-(1-α-)(LC)00I]T,Ω5=Θ1[00α1L-α1(LC)000]T,Ω-5=Θ2[00α1L-α1(LC)000]T,Ω6=Θ1[00α-I(1-α-)C000]T,Ω-6=Θ2[00α-I(1-α-)C000]T,Ω7=Θ1[00α1I-α1C000]T,Ω-7=Θ2[00α1I-α1C000]T,Ω8=-diag{X1(κ1r)⋯X1(κmr)},Ω-8=-diag{X1(κ-1r)⋯X1(κ-M-mr)},Ω9=-diag{X2(κ1r)⋯X2(κmr)},Ω-9=-diag{X2(κ-1r)⋯X2(κ-M-mr)},Ω10=-diag{X3(κ1r)⋯X3(κmr)},Ω-10=-diag{X3(κ-1r)⋯X3(κ-M-mr)}.Proof.
Note that (15) is equivalent to
(33)[Φ-11000Z200*-P2(r)00000**-P3(r)0000***-Q1-2Z1Φ45Φ460****Φ5500*****Φ660******-ΓI]+[A+BK-BK0000I]TP-1(s)×[A+BK-BK0000I]+[LCA-LC-α-L-(1-α-)(LC)00I]TP-2(s)×[LCA-LC-α-L-(1-α-)(LC)00I]+[00α1L-α1(LC)000]TP-2(s)×[00α1L-α1(LC)000]+[00α-I(1-α-)C000]TP-3(s)×[00α-I(1-α-)C000]+[00α1I-α1C000]TP-3(s)×[00α1I-α1C000]+[dm(A+BK-I)-dm(BK)0000dmI]TZ2×[dm(A+BK-I)-dm(BK)0000dmI]+[(dM-dm)(A+BK-I)T-(dM-dm)(BK)T0000(dM-dm)I]Z1[(dM-dm)(A+BK-I)T-(dM-dm)(BK)T0000(dM-dm)I]T<0,
where Φ-11=-P1(r)-Z2+(dM-dm+1)Q1+Q2+Q3+Γβ2GTG.
Via Lemma6, we obtain
(34)∑s=1MλrsP(τs)=∑s∈IκrλrsP(τs)+∑s∈IuκrλrsP(τs)≤P(Iκr)+υrP(Iuκr),
where
(35)P(Iκr)=∑s∈IκrλrsP(τs),P(Iuκr)=∑s∈IuκrP(τs),υr=1-∑s∈Iκrλrs.
From (34) and Schur complement Lemma, (33) can be transformed into the following inequality:(36)[Ω0Ω1Ω2Ω3Ω-3Ω4Ω-4Ω5Ω-5Ω6Ω-6Ω7Ω-7*-Z1-100000000000**-Z2-10000000000***Ω8000000000****Ω-800000000*****Ω90000000******Ω-9000000*******Ω900000********Ω-90000*********Ω10000**********Ω-1000***********Ω100************Ω-10]<0,where
(37)Ω0=[Φ-11000Z200*-P2(r)00000**-P3(r)0000***-Q1-2Z1Φ45Φ460****Φ5500*****Φ660******-ΓI].
For scalarsεl, l=1,2,3,4,5, the following inequalities hold:
(38)(I-ε1Z1)TZ1-1(I-ε1Z1)≥0,(I-ε2Z2)TZ2-1(I-ε2Z2)≥0,(I-ε3P1-1(r))TP1(r)(I-ε3P1-1(r))≥0,(I-ε4P2-1(r))TP2(r)(I-ε4P2-1(r))≥0,(I-ε5P3-1(r))TP3(r)(I-ε5P3-1(r))≥0.
Then we have
(39)-Z1-1≤-2ε1I+ε12Z1,-Z2-1≤-2ε2I+ε22Z2,-P1(r)≤-2ε3I+ε32P1-1(r),-P2(r)≤-2ε4I+ε42P2-1(r),-P3(r)≤-2ε5I+ε52P3-1(r).
According to the inequalities (39), it is obtained that if (31) holds, then (36) holds.Remark 10.
In Theorem9, the inequality (31) is a linear matrix inequality, which can easily be solved by using the Matlab LMI Toolbox.Remark 11.
Compared with the results in [9, 10], partially unknown elements in the transition probability matrix do not require the complete knowledge or even the lower and upper bounds of the transition probabilities in the jump process, which means that it is more powerful and desirable.Remark 12.
The results in this paper derived based on the assumption that the network exists only in the sensor-to-controller side, but when the network exists both in the sensor-to-controller and controller-to-actuator sides, this needs further study.


## 4. Numerical Example

The system data of (1) is described as follows:
(40)A=[0.080.130.410.19],B=[10],C=[0.200.6-0.1].The random delay isdk∈{1,2,3}, and its transition probability matrix is given by
(41)Π1=[0.90.050.050.50.40.10.10.10.8],Π2=[0.9???0.4?0.10.10.8],Π3=[?????????].The maximum allowable nonlinear levelβ and the maximum packet dropout rate α- for three different transition probability matrices are given in Tables 1 and 2, respectively, in which “—” means that the results are not applicable to the corresponding cases. Thus, from Tables 1 and 2, it is easily seen that the more transition probability matrix we have, the larger maximum dropout rate α- for different β and the larger maximum allowable nonlinear level β for different α- can be obtained for ensuring stability.Table 1
The maximum dropout rateα- for different β and Π.
β
0.1
0.2
0.3
Π
1
0.9999
0.9865
0.9712
Π
2
0.8335
0.8309
0.8151
Π
3
0.2793
—
—Table 2
The maximum allowable nonlinear levelβ for different α- and Π.
α
-
0.15
0.35
0.55
Π
1
0.3671
0.3652
0.3640
Π
2
0.3302
0.3282
0.3258
Π
3
0.1608
—
—Give that(42)x(-2)=x(-1)=x(0)=[1-0.5]T,y-(-2)=y-(-1)=y-(0)=[00]T,α-=0.5,β=0.3,ε1=3.5,ε2=2.8,ε3=10,ε4=1.1,ε5=0.6,G=1.Assuming that the nonlinear function isf(x(k))=0.1sin(x(k)), by Theorem 9, we can obtain the controller gain and observer gain with Π2 as follows:
(43)K=[0.0017-0.0137],L=[0.00280.00200.01480.0361].Based on the designed controller and observer, the simulation results of the state responses are given in Figure2. It is obvious that the designed novel dynamic output feedback controller is feasible and ensures the stability of the closed-loop system despite the random packet dropout and random delay.Figure 2
State response of the closed-loop system.


## 5. Conclusion

In this paper, a novel dynamic output feedback controller is designed for nonlinear networked control systems with both random packet dropout and random delay. A sufficient condition is presented for the stabilization of the closed systems based upon the Lyapunov theory. The quantitative relationship of the maximum dropout rate, the transition probability matrix, and the maximum allowable nonlinear level is derived by solving a set of linear matrix inequalities. An example is presented to illustrate the effectiveness of the proposed method.



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*Source: 820280-2013-11-25.xml*