# H

Authors: Jufeng Wang; Chunfeng Liu; Kai Li
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper studies theH-infinity stochastic control problem for a class of networked control systems (NCSs) with time delays and packet dropouts. The state feedback closed-loop NCS is modeled as a Markovian jump linear system. Through using a Lyapunov function, a sufficient condition is obtained, under which the system is stochastically exponential stability with a desired H-infinity disturbance attenuation level. The designed H-infinity controller is obtained by solving a set of linear matrix inequalities with some inversion constraints. An numerical example is presented to demonstrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

In the past few years, the networked control systems (NCSs) whose control loops are connected via communication networks have received increasing attention due to their advantages, such as reduced cost, low weight, easier installation, and maintenance. Time delay and packet dropout are the two major causes of instability of system and deterioration of system performance. Therefore, the time delay and packet dropout problems have been investigated in the existing literature. In [1], time delays are time-varying in intervals. In [2, 3], the bounds were imposed on the maximum number of successive dropouts. In [4], the sufficient condition that establishes the quantitative relation between the packet-dropout rate and the stability of the NCS with a constant delay is obtained.Considering the disturbance attenuation problem, there has been much research effort onH∞ controller design. In [5–7], the controller dynamics is continuous, but in many NCSs, the system is controlled by a discrete-time controller with sample and hold devices. In [8–10], a discrete-time controller is designed; however, it should be pointed out that the packet dropout or the delay problem is studied separately.In [11, 12], H∞ control of a class of systems with random packet dropout is investigated. It is noticed that the plant is a discrete-time system and the delay is a multiple of the sampling time; therefore, the result of the papers cannot be applied to the NCSs when the plant is a continuous-time system and the delay is smaller than the sampling period. In [13], the plant studied is a continuous-time system; the delay takes values in a finite set at a fixed rate. In fact, the time delays and packet dropouts may be random and modeled as Markov chains in most cases. Unfortunately, they do not take into account the time delay and packet dropout with Markovian characterization in [13].In this paper, we investigate theH∞ stochastic control of a class of NCSs with time delays and packet dropouts. The random time delay and packet dropout are described by a Markov chain. By using a Lyapunov function, we obtain the system with exponential stability with a desired H∞ disturbance attenuation level. The designed H∞ stochastic controller is obtained by an iterative linear matrix inequality approach. To demonstrate the effectiveness of the method, an illustrative example is presented.


## 2. Model for Networked Control System

The structure of the NCS is shown in Figure1. Consider a continuous-time linear system described by
(1)x˙=Apx(t)+Bpu(t)+Epw(t),z(t)=Cx(t),
where x(t)∈Rn is the state, u(t)∈Rm is the plant input, w(t)∈Rq is the disturbance input, and z(t)∈Rl is the plant output. Ap, Bp, Ep, and C are constant matrices of appropriate dimensions.Figure 1
The structure of NCS.The following assumptions are made for the considered NCS throughout the paper [13].The controller is event-driven; both the sensor and the actuator are time-driven. The sampling period of the sensor isT. The actuator has a receiving buffer which contains the most recently updated packet from the controller. The actuator reads the buffer periodically at a smaller period than T, say T0=T/N for some integer N large enough. The sensor and the actuator are time synchronized. Upon reading a new value, the actuator with a zero-order-hold device will update the output value. The network-induced delay τ(k) satisfies 0≤τ(k)<T.Based on the above assumptions, the discrete-time state feedbackH∞ controller can be expressed as follows:
(2)u(k)=Kx^(k),
where
(3)x^(k)={x(k),ifx(k)issuccessfullytransmitted,x^(k-1),ifx(k)islostduringtransmission,
where x(k) is the value of x(t) at the sampling time kT.Consider(4)z(k)=Cx(k),
where z(k) is the value of the z(t) at the sampling time kT.During each sampling period, several different cases may arise, which leads to the following discrete-time switched system model [13]:
(5)x~(k+1)=Aσ(k)x~(k)+E~w(k),z(k)=C~x~(k),
where
(6)x~(k+1)=[x(k+1)x^(k)],E~=[E0],C~=[C0],(7)Aσ(k)=[A+A1σ(k)KA0σ(k)KI0],llforσ(k)=0,1,2…,N-1(8)Aσ(k)=[ABK0I],forσ(k)=N(9)A=exp⁡{ApT},A0σ(k)=∫T-σ(k)T0Texp⁡{Apτ}Bpdτ,B=∫0Texp⁡{Apτ}Bpdτ,A1σ(k)=∫0T-σ(k)T0exp⁡{Apτ}Bpdτ,E=∫0Texp⁡{Apτ}Epdτ.
The σ(k) is called a switching signal. Note that σ(k)=i, i=0,1,…,N-1, implies τ(k)=iT0, while σ(k)=N implies packet dropout.σ
(
k
) is modeled as Markov chain that takes values in {0,1,…,N-1,N}. The transition probability matrices of σ(k) are Π=[πij]. That means that σ(k) jump from mode i to mode j, from mode with probabilities πij:
(10)πij=Pr(σ(k+1)=j∣σ(k)=i),
where πij≥0 and ∑j=0Nπij=1.


## 3.H∞ Disturbance Attenuation Analysis

Definition 1.
System (5) is said to be stochastically and exponentially stable, if there exist constants C>0 and 0<λ<1, such that E(∥x~(k)∥2)≤CλkE(∥x~(0)∥2) for w(t)≡0.Definition 2.
System (5) is said to be stochastically and exponentially stable with an H∞ disturbance attenuation level γ, if system (5) is stochastically and exponentially stable and for the zero initial condition, ∑k=0∞E{zT(k)z(k)}≤γ2∑k=0∞E{wT(k)w(k)}.Lemma 3 (see [14]).
DefineVσ(k)(k)=x~T(k)Pσ(k)x~(k), where Pσ(k) is a positive definite matrix; then there exist constant scalars β1,β2>0 such that
(11)β1∥x~(k)∥2≤Vσ(k)(k)≤β2∥x~(k)∥2,σ(k)=0,1,2,…,N.Theorem 4.
For given positive scalarsπij(i,j=0,1,2,…,N), λ, and γ, if there exist matrices Pi>0, Qi>0, such that
(12)Ωi=[Ci11*Ci21Ci22]<0,i=0,1,…,N-1,(13)Ω¯=[C¯11*C¯21C¯22]<0,
where
(14)Ci11=[∑j=0NπijQj-λPi+CTC000-λQi000-γ2I],Ci21=[πi0(A+A1iK)πi0A0iKπi0Eπi1(A+A1iK)πi1A0iKπi1E⋮⋮⋮πiN(A+A1iK)πiNA0iKπiNE],Ci22=diag⁡{-P0-1-P1-1⋯-PN-1},C¯11=[-λPN+CTC0∑j=0NπNjPjE*∑j=0NπNjQj-λQN0**∑j=0NπNjETPjE-γ2I],C¯21=[πN0AπN0BK0πN1AπN1BK0⋮⋮⋮πNNAπNNBK0],C¯22=diag⁡{-P0-1-P1-1⋯-PN-1}
then system (5) is stochastically and exponentially stable with an H∞ disturbance attenuation level γ.Proof.
Let the Lyapunov function(15)Vi(k)=xT(k)Pix(k)+x^T(k-1)Qix^(k-1)=x~T(k)P~ix~(k)
correspond to the subsystem as follows:
(16)x~(k+1)=Aix~(k)+E~w(k),z(k)=C~x~(k),
where
(17)P~i=[Pi00Qi].
Whenσ(k)=i(i=0,1,2,…,N-1), we obtain
(18)E(Vσ(k+1)(k+1)-λVσ(k)(k)+zT(k)z(k)-γ2wT(k)w(k))=E(Vσ(k+1)(k+1)∣σ(k)=i)-λVi(k)+x~T(k)C~TC~x~(k)-γ2wT(k)w(k)=∑j=0Nπij(x~T(k)AiT+wT(k)E~T)P~j(Aix~(k)+E~w(k))-λx~T(k)P~ix~(k)+xT(k)C~TC~x(k)-γ2wT(k)w(k)=x~T(k)(∑j=0NπijAiTP~jAi)x~(k)+x~T(k)×(∑j=0NπijAiTP~jE~)w(k)+wT(k)(∑j=0NπijE~TP~jAi)x~(k)+wT(k)(∑j=0NπijE~TP~jE~)w(k)-λx~T(k)P~ix~(k)+xT(k)C~TC~x(k)-γ2wT(k)w(k)=[x~T(k)wT(k)]Θi[x~(k)w(k)],
where
(19)Θi=[∑j=0NπijAiTP~jAi-λP~i+C~TC~∑j=0NπijAiTP~jE~∑j=0NπijE~TP~jAi∑j=0NπijE~TP~jE~-γ2I].
From (6) and (7), it can be obtained that
(3)Θi=[Ai11Ai12Ai13*Ai22Ai23**Ai33].Ai11=∑j=0Nπij[(AT+KTA1iT)Pj(A+A1iK)+Qj]-λPi+CTC,Ai12=∑j=0Nπij(AT+KTA1iT)PjA0iK,Ai13=∑j=0Nπij(A+A1iK)TPjE,Ai22=∑j=0NπijKTA0iTPjA0iK-λQi,Ai23=∑j=0Nπij(A0iK)TPjE,Ai33=∑j=0NπijETPjE-γ2I.
Θ
i
<
0 can be rewritten as follows:
(21)Φi+[πi0(AT+KTA1iT)πi0KTA0iTπi0ET]×P0[πi0(A+A1iK)πi0A0iKπi0E]<0,
where
(22)Φi=[Bi11Bi12Bi13*Bi22Bi23**Bi33].Bi11=∑j=1Nπij(AT+KTA1iT)Pj(A+A1iK)+∑j=0NπijQj-λPi+CTC,Bi12=∑j=1Nπij(AT+KTA1iT)PjA0iK,Bi13=∑j=1Nπij(A+A1iK)TPjE,Bi22=∑j=1NπijKTA0iTPjA0iK-λQi,Bi23=∑j=1Nπij(A0iK)TPjE,Bi33=∑j=1NπijETPjE-γ2I.
From the Schur complement, we have that (21) is equivalent to
(23)Ψi=[Bi11Bi12Bi13πi0(AT+KTA10T)*Bi22Bi23πi0KTA0iT**Bi33πi0ET***-P0-1]<0.
Similarly, we can see thatΨi<0 is equivalent to
(24)Ωi=[Ci11*Ci21Ci22]<0.
It can be seen that if (12) holds, Θi<0 is true, which means
(25)E(Vσ(k+1)(k+1)-λVσ(k)(k)+zT(k)z(k)-γ2wT(k)w(k))<0.
Whenσ(k)=N,
(26)E(Vσ(k+1)(k+1)-λVσ(k)(k)+zT(k)z(k)-γ2wT(k)w(k))=[x~T(k)wT(k)]Θ¯[x~(k)w(k)],
where
(27)Θ¯=[∑j=0NπNjANTP~jAN-λP~N+C~TC~∑j=0NπNjANTP~jE~∑j=0NπNjE~TP~jAN∑j=0NπNjE~TP~jE~-γ2I].
From (6) and (8), it can be seen that
(28)Θ¯=[A¯11A¯12A¯13*A¯22A¯23**A¯33],A¯11=∑j=0NπNjATPjA-λPN+CTC,A¯12=∑j=0NπNjATPjBK,A¯13=∑j=0NπNjPjE,A¯22=∑j=0NπNjKTBTPjBK+∑j=0NπNjQj-λQN,A¯23=0,A¯33=∑j=0NπNjETPjE-γ2I.
Θ
¯
<
0 can be rewritten as follows:
(29)Φ¯+[πN0ATπN0KTBT0]P0[πN0AπN0BK0]<0,
where
(30)Φ¯=[B¯¯11B¯12B¯13*B¯22B¯23**B¯33],B¯11=∑j=1NπNjATPjA-λPN+CTC,B¯12=∑j=1NπNjATPjBK,B~13=∑j=0NπNjPjE,B¯22=∑j=1NπNjKTBTPjBK+∑j=0NπNjQj-λQN,B¯23=0,B¯33=∑j=0NπNjETPjE-γ2I.
From the Schur complement, we have that (29) is equivalent to
(31)Ψ¯=[B¯11B¯12B¯13πN0AT*B¯22B23πN0KTBT**B¯330***-P0-1]<0.
Similarly, it is easy to see thatΨ¯<0 is equivalent to
(32)Ω¯=[C¯11*C¯21C¯22]<0.
It can be seen that if (13) holds, Θ¯<0 is true, which means
(33)E(Vσ(k+1)(k+1)-λVN(k)+zT(k)z(k)-γ2wT(k)w(k))<0.
It follows from (25) and (33) that
(34)E(Vσ(k+1)(k+1)-λVσ(k)(k)+zT(k)z(k)-γ2wT(k)w(k))<0
which means
(35)E(Vσ(k+1)(k+1))<E(Vσ(k)(k))-E(zT(k)z(k))+γ2E(wT(k)w(k)),E(Vσ(∞)(k+1))<E(Vσ(0)(0))-∑k=0∞E(zT(k)z(k))+γ2∑k=0∞E(wT(k)w(k)),∑k=0∞E(zT(k)z(k))≤γ2∑k=0∞E(wT(k)w(k)).
Next, we prove the stochastically and exponentially stable system (5). The perturbation w(t) is assumed to be zero.
Whenσ(k)=i,(i=0,1,2,…,N-1), we obtain
(36)E(Vσ(k+1)(k+1)-λVσ(k)(k))=E(Vσ(k+1)(k+1)∣σ(k)=i)-λVi(k)=x~T(k)(∑j=0NπijAiTP~jAi-λP~i)x~.
From (21), (23), and (24), it can be seen that Θ<0 is equivalent to Ω<0.
Then, it can be seen from (19) that if (12) holds, we have
(37)∑j=1NπijAiTP~jAi-λP~i+C~TC~<0
and then
(38)∑j=1NπijAiTP~jAi-λP~i<0
which means
(39)E(Vσ(k+1)(k+1)-λVσ(k)(k))<0.
Whenσ(k)=N,
(40)E(Vσ(k+1)(k+1)-λVN(k))=x~T(k)(∑j=0NπNjANTP~jAN-λP~N)x~(k).
From (27), (29), (31), and (32), it can be seen that Θ¯<0 is equivalent to Ω¯<0.
Then, it can be seen from (27) that if (13) holds, we have
(41)∑j=0NπNjANTP~jAN-λP~N+C~TC~<0
and then
(42)∑j=0NπNjANTP~jAN-λP~N<0
which means
(43)E(Vσ(k+1)(k+1)-λVN(k))<0.
From (36) and (43), we have
(44)E(Vσ(k+1)(k+1))<E(λVσ(k)(k)),E(Vσ(k)(k))<λkE(Vσ(0)(0)).
From Lemma3, we get
(45)E(∥x~(k)∥2)≤CλkE(∥x~(0)∥2).
Then, the result is established.The conditions in Theorem4 are a set of LMIs with some inversion constraints. K can be solved by an iterative LMI approach which is called the cone complementarity linearization algorithm [15, 16].


## 4. Numerical Example

Consider the following system [13]. Suppose γ=0.91, λ=0.9760. The transition probability matrices of σ(k) are taken as follow:
(46)[0.10.800.10.20.700.10.40.500.10.60.300.1]
which means
(47)π01=0.1,π02=0.8,π03=0,π04=0.1,π11=0.3,π12=0.7,π13=0,π14=0,π21=0.3,π22=0.7,π23=0,π24=0,π31=0.6,π32=0.3,π33=0,π34=0.1.
Using Theorem 4 and the cone complementarity linearization algorithm, we obtain
(48)K=[-1.4252-5.7880].
Figure 2 is the possible realizations of the mode σ(k). Under this mode sequence, the corresponding state trajectories of the closed-loop system are shown in Figure 3. It is shown that the closed-loop system is stochastically and exponentially stable.Figure 2
Random modeσ(k).Figure 3
State trajectories of the NCS.


## 5. Conclusions

In this paper, by modeling the random delays and packet dropouts as a Markov chain, a new Markovian jump system model is presented to describe the networked control system with disturbance attenuation. The criteria for the system are stochastically and exponentially stable with anH∞ disturbance attenuation level which is derived by an iterative LMI approach.



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*Source: 749574-2014-07-01.xml*