# H

Authors: Jinna Li; Lifeng Wei; Chao Liu; Haibin Yu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Compensation scheme-basedH∞ control is investigated for networked control systems with packet disordering and packet loss. Since the existence of packet disordering and packet loss inevitably degrades the control performance of networked control systems, it is worth studying a control scheme to compensate for them, such that the control performance can be improved. Thus, a compensation control strategy is first proposed following this direction. Next, a mathematical model of networked control systems with Markovian property is constructed due to the signals executed by the plant subject to Markovian chain. Based on it, a sufficient condition for stochastic stability of networked control systems with uncertain parameters as well as compensation strategy is presented, and an adaptive controller is designed based on linear matrix inequality (LMI) technique. Finally, a numerical example and simulations are given to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Networked control systems (NCSs), in which nodes communicate over communication networks, have attracted lots of researchers’ attention [1–4]. Since networks-based control gives rise to many advantages including low cost, easy maintenance, and flexible system structure, the successful application of NCSs can be found in a wide range of areas such as industrial automation, intelligent transportation system, and smart grid. However, packet disordering and packet dropout inevitably exist in the transmission of signals. They are recognized to be two main causes for performance deterioration or even instability of NCSs, hence, considerable research has been done (see, e.g. [5–21] and the references therein).So far, the majority of NCSs research has focused on controller design to provide sufficient stability conditions for NCSs with packet loss. A lot of effort has also been taken for modeling NCSs in presence of packet losses as asynchronous dynamic systems or Markovian jumping systems [5, 6]. With further study on packet loss, some effective compensation strategies for packet loss that occurred during communication are proposed to improve control performance of NCSs. Predictive control is a typical method with wich the control prediction generator provides a set of future control predictions to enable the closed-loop system to achieve the desired control performance leading to removing the effects of data dropout [7–11]. Another typical compensation methodology for packet loss is observer-based state estimation [12]. In addition, [13] proposed a packet dropout-based compensation scheme, namely, the latest control signal is used for compensation if the ideal control input is missing. However, note that, in all of the aforementioned literature, packet disordering is not considered, but packet disordering and packet loss coexist in packets delivered network communication.Packet disordering means that a packet sent earlier may arrive at the destination node later or vice versa. Packet disordering of NCSs has drawn an increasing attention [10, 14–21]. In [14], packets that arrived late at control nodes were discarded, and stability and H∞ compensation control were investigated. However, the packet disordering is not described clearly. In [15, 16], the sampling instants of received signals were compared to describe packet disordering, and stability analysis and synthesis were studied. Some literature using the similar method can be found in the existing reported results (e.g., [17]). Recently, [10, 18] proposed an active compensation for packet disordering; that is, the latest control actions applied to the plant are available by comparing the time stamps of packets. The so called compensation method has been also presented in [19–21], where the latest signals are executed by the plant by defining an operator constructing a mapping between the newest signals and packet displacement values. However, note that a situation where no new control actions arrive at the actuator may occur due to packet disordering and packet loss during a sampling interval. In this case, it is critical how to control the plant. To the best of the authors’ knowledge, this problem has been not fully investigated to date, which motivates this work for proposing a new compensation scheme.The specific problem addressed in this paper is the compensation control when the newest signal is not available for NCSs due to the packet disordering and packet loss. The highlighted method is that control inputs are determined by defining some operators associated with packet displacements; that is, the latest control input is chosen when no new signals arrive at the actuator during the sampling interval((k-1)T,kT]; otherwise, the newest signal controls the plant. After that, a Markovian jumping model of NCSs is put forward. Stability analysis and the controller synthesis are thoroughly investigated and the adaptive controller design is obtained in terms of linear matrix inequalities (LMI). A numerical example is provided to demonstrate the effectiveness of the proposed approach.The rest of the paper is organized as follows. Section2 is concerned with problem statement. In this section, a compensation control scheme is proposed, and a model of networked control systems is constructed. Section 3 investigates the stability and H∞ control for NCSs with packet disordering and packet losses. The results of numerical simulation are presented in Section 4. Conclusions are stated in Section 5.NotationR
n denotes the n dimensional Euclidean space. P>0 means that matrix P is real symmetric and positive definite, and I is the identity matrix of appropriate dimensions. The subscript “T" denotes the matrix transpose. In symmetric block matrices, we use “*” to represent a term that is induced by symmetry, and diag(⋯) stands for a block-diagonal matrix. ∥x∥ stands for the standard l2 norm of vector x; that is, ∥x∥=(xTx)1/2.


## 2. Problem Statement

The following system is considered:(1)xk+1=A¯xk+B¯uk+D1wkzk=Cxk+D2wk,
where xk=x(kT)∈ℜn, uk=u(kT)∈ℜM, and zk=z(kT)∈ℜP are the state vector, control input vector, and controlled output vector, respectively. T is the sampling period. A¯=A+ΔA,  B¯=B+ΔB,  (ΔAΔB)=DF(k)(E1E2),  A,B,C,D,D1,D2,E1, and E2 are some constant matrices of appropriate dimensions, and FT(k)F(k)≤I.  wk=w(kT)∈ι2[0,∞] denotes the exogenous disturbance signal.The state feedback controller can be expressed as(2)uk=Kxk,
where K is some constant matrix of appropriate dimensions.Here, we assume that sensor and actuator are time-driven synchronously, the period is identical and equal toT, and the controller is event-driven. Since the states of systems and control signals are transmitted over the communication networks with limited bandwidth, the packet disordering and intermittent packet dropouts are inevitable in the communication channels. To describe the phenomenon of packet disordering and design compensation scheme, we define the displacement values of packets and some operators determining the control actions. The details are as follows.Without loss of generalization, we consider a sequence of packetsxk-h,xk-h+1,…,xk transmitted over the network from the sensor, where h is a given integer. The maximum delay bound is an alternative solution to h. For xk-h,xk-h+1,…,xk, it is well known that the corresponding expected arrival sequence numbers are 1,2,…,h+1. Then, the expected arrival sequence number of packet xk-i is h+1-i(i=0,1,…,h) is easily obtained. A receive_index l(l=1,2,…,h+1) is assigned to each nonduplicate packet as it arrives at the point of measurement, which we refer to as the destination (actuator) since the control is event-driven. To describe the newest signal executed by the plant, we assume that packets which have not appeared or lost during the sampling interval ((k-1)T,kT] arrive at the actuator in order after the kT time instant, and their receive_index values are 1 more than the real values. Moreover, if the sampled packets behind p (p is some positive integer) lost packets arrive at the plant before the kT time instant (including kT time instant), their receive index values are p more than the real values. Rk(i) and dk(h+1-i) denote the receive_index and displacement value of packet xk-i, respectively. For packet xk-i arriving at the actuator before the kT instant (including kT instant), if dk(h+1-i)≠0, then a “disordering event” has occurred in communication. Packet xk-i is late if dk(h+1-i)>0, early if dk(h+1-i)<0, and in order if dk(h+1-i)=0 (see [19–21]). To guarantee the newest signals being executed by the plant, the packets that arrive at the actuator late are discarded. Define the following operators:
(3)δ(dk(h+1-i))={1dk(h+1-i)≤00dk(h+1-i)>0,(4)θk(i)=∏j=0i-1(1-δ(dk(h+1-j)))δ(dk(h+1-i)),
where ∏j=0-1(1-δ(dk(h+1-j)))=1(i=0,1,…,h). The function of θk(i)(i=0,1,…,h) is to guarantee that the newest signal is executed if it has arrived at the actuator during the interval ((k-1)T,kT]. Moreover, note that it may happen not to receive new signal due to late coming packet or packet loss; thus we design the following controller:
(5)uk=∑i=0h‍θk(i)Kxk-i+θk(-1)uk-1,
where
(6)θk(-1)={1dk(h+1-i)>0,∀i(i=0,1,…,h)0otherwise.Remark 1.
As a matter of fact, the operatorsθk(i)(i=-1,0,…,h) are defined for the purpose of selecting control input. Note that θk(i)=1 or 0, and ∑i=-1h‍θk(i)=1. θk(i)=1 or θk(i)=0(i=0,1,…,h), which is determined in terms of the displacement values of packets. More detailed explanations can be found in the examples in Figures 1 and 2, Tables 1 and 2. Since we choose event-triggered controller, once xk-i arrives at the controller, control action Kxk-i is sent to the actuator. When the displacement values of all packets dk(h+1-i)>0(i=0,1,…,h); that is, there are no new signals arriving the actuator during sampling interval ((k-1)T,kT], θk(-1) is set to be 1 by (6) and other θk(i)=0(i=0,1,…,h) hold due to (4), which indicates the control input uk-1 to act on the plant (see Figure 1 and Table 1). Otherwise, θk(-1)=0 and there exists one θk(i)=1(i=0,1,…,h). In this context, the newest control signal Kxk-i is regarded as the control action uk. It is worth pointing out that the packet disordering, random time-varying transmission delay, and packet loss are taken into account in the suggested compensation control scheme (5), simultaneously. From this point of view, it is readily seen that the proposed control scheme (5) is quite general.Table 1
An example in Figure1.
x
k
-
2
x
k
-
1
x
k
Exception values
1
2
3
Receive_index
2
3
4
Displacement values
1
1
1
Parameters
θ
k
(
-
1
)
=
1
θ
k
(
0
)
=
0
θ
k
(
1
)
=
0
θ
k
(
2
)
=
0Table 2
An example in Figure2.
x
k
-
1
x
k
-
1
x
k
Exception values
1
2
3
Receive_index
1
3
4
Displacement values
0
1
1
Parameters
θ
k
(
-
1
)
=
0
θ
k
(
0
)
=
0
θ
k
(
1
)
=
0
θ
k
(
2
)
=
1Figure 1
Timing diagram of signals transmitting in the example with packet disordering.Figure 2
Timing diagram of signals transmitting in the example with packet loss.For further understanding compensation control scheme (5), we study two examples shown in Figures 1 and 2 which illustrate the arriving timing of signals transmitted. Expected arrival sequence numbers, assigned receive_index values, and displacement values for a sequence transmitted, corresponding to Figures 1 and 2, are given in Tables 1 and 2, respectively. In the two examples, we choose h=2, which means that a group of 3 packets is used as the studying object. From Table 1, we find that, packets xk-2, xk-1 and xk are displaced by one unit from their positions, then all of displacement values are equal to 1. In this context, the actuator does not receive new signal during the sampling interval ((k-1)T,kT], which can be seen in Figure 1. By (3), (4), and (6), we obtain θk(-1)=1 and θk(0)=θk(1)=θk(2)=0. And by (5), the control input uk-1 acts on the plant, which is entirely consistent with the proposed compensation control scheme that the latest control action is utilized to control the plant when no new signal is available during the sampling interval ((k-1)T,kT]. In Figure 2, note that packet xk-1 has lost. Similarly, by calculating, we obtain θk(2)=1 and θk(0)=θk(1)=θk(-1)=0. Then, Kxk-2 is used as the newest control input by virtue of (5). Obviously, this result accords with the actual situation shown in Figure 2.Remark 2.
Similar to [13], we execute the compensation control scheme. However, there are two distinct differences from [13]. The first one is that the stability analysis and compensation strategy are investigated in the presence of both packet loss and packet disordering simultaneously. In this sense, the theory method presented in this paper extends the results presented in [13], since packet disordering is not taken into account in [13]. The second difference is that an operator deciding how to choose control actions is clearly defined based on the displacement values of packets, while [13] control strategy is proposed in terms of delay information.Remark 3.
Compared with the existing studies on NCSs with packet disordering [10, 14–21], a key difference is that a compensation control scheme is proposed when no new signal arrives at the actuator.
The closed-loop system can be obtained by substituting (5) into (1):
(7)xk+1=A¯xk+B¯(∑i=0h‍θk(i)Kxk-i+θk(-1)uk-1)+D1wkzk=Cxk+D2wk.
LettingξkT=[xkTxk-1T⋯xk-hT] and ηkT=[ξkTuk-1T], (7) is expressed as
(8)ξk+1=A11,kξk+A12,kuk-1+D~1wkuk=A21,kξk+θk(-1)uk-1,
where
(9)A11,k=[A¯+M0M1⋯Mh-1MhI0⋯00⋮⋮⋱⋮⋮00⋯I0],A12,k=[B¯θk(-1)0⋮0],D~1=[D10⋮0],A21,k=[θk(0)Kθk(1)K⋯θk(h)K],Mi=B¯ik(i)K(i=0,1,…,h).
Further, we have
(10)ηk+1=A¯kηk+D¯1wkzk=C^ηk+D2wk,
where
(11)A¯k=[ΓRΛ(θk(0),…,θk(h))K¯θk(-1)I],R=D^F(k)E2θk(-1)+B^θk(-1),Γ=A^+B^Λ(θk(0),…,θk(h))K¯+D^F(k)(E^1+E2Λ(θk(0),…,θk(h))K¯),Λ(θk(0),…,θk(h))=[θk(0)I⋯θk(h)I],K¯=diag{K,K,…,K︸(h+1)},C^=[C0⋯0],A^=[A0⋯00I0⋯00⋮⋮⋱⋮⋮00⋯I0],B^=[B0⋮0],E^1=[E0⋯0],D^=[D0⋮0],D¯1=[D~10].
It is well known that the newest signals executed by plant may be subject to some probability distribution [6]; here we will assume that the newest signals transmitted over communication network are subject to Markovian chain. Define dk=[θk(-1),θk(0),…,θk(h)]T; since θk(i)=1 or 0(i=-1,0,…,h) and ∑i=-1h‍θk(i)=1, then, there are h+2 possible values for dk, obviously. Similar to our prior effort [21], for ease of notation, we define a vector-valued function f:dk→σ(k) to map the vector dk into a scalar number σ(k)∈ℑ={1,2,…,r}, where r=h+2. σ(k)=i also denotes the No. i(i∈ℑ) subsystem of NCS (10). Moreover, transition probability associated with the newest signals executed is defined as πij=Prob(σ(k+1)=j∣σ(k)=i), where σ(k)=i denotes dk=[0,…,1,…,0]T, namely, θk(i-2)=1. Obviously, ∑j∈ℑ‍πij=1. Thus, (10) is a Markovian jumping system.Remark 4.
In this paper, the stability analysis andH∞ controller design are investigated on the premise that transition probability matrix is fully known. Actually, the analysis and control methods for NCSs with uncertain transition probabilities have been developed, and we can refer to [22, 23].


## 3. Stability Analysis andH∞ Controller Design

In this section, we will present a sufficient condition forH∞ control and the design of the controller gains Ki(i=1,2,…,r) adapting to No. i switched subsystem.Lemma 5 (see [24]).
For any matricesW, M, N, F(k) with FT(k)F(k)<I, and any scalar ε>0, the following inequality holds:
(12)W+MF(k)N+NTFT(k)MT≤W+εMMT+ε-1NTN.Theorem 6.
For given scalarsh and γ>0, matrices Ki, if there exist matrices Pi>0,  Qi>0(i∈ℑ), such that
(13)[Υ1Υ2*Υ3]<0,
then the closed-loop system (10) is stochastic stable with an H∞ norm bound γ, where Υ1=∑j∈ℑ‍πijA¯kTWjA¯k+γ-1C^TC^-Wi, Υ2=∑j∈ℑ‍πijA¯kTWjD¯1+γ-1C^TD2,Υ3=∑j∈ℑ‍πijD¯1TWjD¯1+γ-1D2TD2-γI.Proof.
Without loss of generalization, we setσ(k) to be i. Choosing a Lyapunov-Krasovskii functional candidate which is given by
(14)Vk=ηkTWiηk,
where
(15)Wi=[Pi00Qi],
we can obtain
(16)EVk+1-Vk=∑j∈ℑ‍πij(A¯kηk+D¯1wk)TWj×(A¯kηk+D¯1wk)-ηkTWiηk.
Let ek=[ηkTwkT]T; we can obtain EVk+1-Vk=ekTΘiek, where
(17)Θi=[-Wi+∑j∈ℑπijA¯kTWjA¯k∑j∈ℑπijA¯kTWjD¯1*∑j∈ℑπijD¯1TWjD¯1].
And
(18)γ-1zkTzk-γwkTwk+EVk+1-Vk=ηkT(Θi+Λ)ηk,
where
(19)Λ=[γ-1C^TC^γ-1C^TD2*γ-1D2TD2-γI].
If (13) is satisfied, we have (Θi+Λ)<0. Further, we can obtain
(20)γ-1zkTyk-γwkTwk+EVk+1-Vk<0.
Then, γ-1zkTyk-γwkTwk<-(EVk+1-Vk). Summing up both sides of the above inequality from k=0 to k=n, using the zero initial condition, we have ∑k=0n‍∥zk∥2<γ2∑k=0n‍∥wk∥2-γEVn+1 for all n. Letting n→∞, we have
(21)∥zk∥22<γ2∥wk∥22.
Ifwk≡0 and (13) holds, it is clear that Γi=-Wi+∑j∈ℑ‍πijA¯kTWjA¯k<0 can be implied by (13), then
(22)EVk+1-Vk≤-βxkTxk,
where β=min{λmin(-Γi),i∈ℑ}. Summing up both sides of the above inequality from k=0 to k=ℏ and Letting ℏ→∞, we can see that for any ℏ>1(23)limℏ→∞EV(ℏ+1)-V(φ0,s0)≤-βlimℏ→∞∑k=0ℏ‍E(xkTxk)
or
(24)limℏ→∞∑k=0ℏ‍E(xkTxk)≤limℏ→∞1βV(φ0,s0)<∞,
where φ0 and s0 are the initial condition of the system. The stochastic stability is obtained. This completes the proof.For the purpose of controller design, we give Theorem7.Theorem 7.
For a given scalarh1, if there exist matrices Xi>0, Yi, and Fi(i∈ℑ), such that
(25)[-Si0Ξi1SiC^TH^iT*-γIΞi2D2T0**Ξi300***-γI0****-εI]<0,
then Ki(K¯i=FiXi-1,K¯i=diag(Ki,Ki,…,Ki)) are adaptive controller gains with an H∞ norm bound γ, where
(26)Ξi1=[ρi1NiTρi2NiT⋯ρirNiT],Ξi2=[ρi1D¯1Tρi2D¯1T⋯ρirD¯1T],Ψi=[Ψi,11Ψi,12⋯Ψi,1r*Ψi,22⋯Ψi,2r**⋱⋮***Ψi,rr],Ξi3=diag(-S1,-S2,…,-Sr)+Ψi,ρij=πij(j=1,2,…,r),Ni=[A^kXi+B^ΛiFiB^θk(-1)YiΛiFiθk(-1)Yi],H^i=[E^1Xi+E2ΛiFiE2θk(-1)Yi],Ψi,lj=ρil*ρijεD~D~T(l,j=1,2,…,r).Proof.
By Schur complement and Lemma5, (13) is equivalent to
(27)[-Wi0Λi1C^THiT*-γIΞi2D2T0**Λi300***-γI0****-εI]<0,
where
(28)Λi1=[ρi1A^kTρi2A^kT⋯ρirA^kT],Λi3=diag(-W1,-W2,…,-Wr)+Ψi,Hi=[E^1+E2ΛiK¯iE2θk(-1)].
Pre- and postmultiplying both sides of (27) with diag(Wi-1,I,…,I︸r+3) and its transpose, letting Xi=Pi-1, Yi=Qi-1, Si=Wi-1, K¯iXi=Fi(i=1,2,…r), (25) can be derived. Theorem 7 is completed.


## 4. Numerical Examples

In this section, we verify the effectiveness of the control strategy proposed for NCSs with packet disordering and packet loss. First, we show the control results for NCSs with packet disordering and packet loss under two cases. One is in the absence of structural uncertainty, and the other is with uncertain structure. One can clearly see that the method proposed has good robustness. Second, comparative studies are performed to demonstrate clearly the advantages enjoyed by the suggested compensation scheme described in this paper.Example 1.
Consider the following unstable system:(29)xk+1=([0.0885-0.0659-0.15380.2977]+ΔA)xk+([0.5234-0.0990]+ΔB)uk+[1.25440.5317]wkzk+1=[0.0690-0.3554]xk+0.3304wk,
where
(30)D=[0.38850.3112],E1=[0.3237-0.2128]E2=0.5243.
The transmission delays and displacement values of packets delivered over network are shown in Figure 3. It should be explained that delay determines the arrival orders of packets, based on which displacement values of packets are calculated. Clearly, the bound of transmission delay h=2 under assuming sampling period 0.15s. Based on the displacement values of packets, thus the jumping process taking values in a finite set ℑ={1,2,3,4}, standing for ℑ={[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]}, governs the switching among the different system modes, then r=4. [1,0,0,0]→1 means no new signal arrives at the actuator, u(k-1) acts on the plant; [0,1,0,0]→2 means the newest signal K2xk transmitted over network is executed by the plant at kT time instant, the rest may be deduced by analogy; [0,0,0,1]→4 means the newest signal K4xk-2 transmitted over network is executed by the plant at kT time instant.Transmission delay (a) and displacement values (b).
(a)
(b)Thus, the newest signals executed by the plant are subject to Markovian process, whose switched states are shown in Figure4 and transition probability matrix is given as follows:
(31)[0.20000.500000.300000.29410.7059000.416700.58330.25000.281300.4688].Figure 4
Markovian process of NCS with packet disordering and dropouts competition.


### 4.1. Verification of Compensation Scheme

First, we consider the systems without structural uncertainty in NCS. By Theorem7, we obtain the following adaptive controller gains
(32)K2=[-0.32710.5110]K3=1.0e-03*[0.1137-0.1898]K4=1.0e-03*[-0.04740.1103].
Second, if the uncertainty exists in the NCSs, by Theorem 7, we also obtain the corresponding adaptive controller gains
(33)K2=[-0.39030.5512]K3=[-0.20380.3679]K4=[-0.28690.3902].
We choose the uncertain parameter F(k)=sin(k). At the initial state value x0=[-1-3]T, the states and output response of the NCS, without uncertainty and with uncertainty under the network environment in the presence of packet disordering and packet loss competition, are shown in Figures 5 and 6, respectively. Compared with the result given in Figure 5, the NCS with uncertainty can be also stabilized quickly using the competition controller designed in this paper though there exist packet disordering and packet dropout in communication network. This makes it clear that the proposed control scheme has a good robustness.Figure 5
State and output response of the system.Figure 6
State and output response of the system.


### 4.2. Control Performance Comparison

It should be pointed out that the discrete-time system in this example can be inverted into a continuous-time system in [14, 20] if sampling period T=0.1s is given. If wk≠0, the H∞ norm bounds and corresponding controller gains are shown in Table 3 (“—” denotes that the conditions are infeasible). Obviously, a more optimal H∞ norm bound is obtained in this paper than those in [14, 20] since the competition scheme is performed when no signal is available by the plant due to packet disordering and packet loss. At the initial state value x0=[-1-3]T, the disturbance input w(t) is as follows:
(34)w(t)={sin(t)5s≤t<20s0otherwise,
the state and output response of the NCS in the presence of packet disordering and packet loss are shown in Figure 7.Table 3
Comparison of convergence time.
H
∞ norm  bound
Controller  gain
[14]
—
—
[20]
1.5
K
=
1.0
*
1
0
-
7 [0.1278-0.0714]
This paper
0.6977
K
2
=
[
-
0.4867
0.7869
] K3=[00] K4=[00]Figure 7
Curves of state and output response of the system subject to exogenous disturbance.


## 4.1. Verification of Compensation Scheme

First, we consider the systems without structural uncertainty in NCS. By Theorem7, we obtain the following adaptive controller gains
(32)K2=[-0.32710.5110]K3=1.0e-03*[0.1137-0.1898]K4=1.0e-03*[-0.04740.1103].
Second, if the uncertainty exists in the NCSs, by Theorem 7, we also obtain the corresponding adaptive controller gains
(33)K2=[-0.39030.5512]K3=[-0.20380.3679]K4=[-0.28690.3902].
We choose the uncertain parameter F(k)=sin(k). At the initial state value x0=[-1-3]T, the states and output response of the NCS, without uncertainty and with uncertainty under the network environment in the presence of packet disordering and packet loss competition, are shown in Figures 5 and 6, respectively. Compared with the result given in Figure 5, the NCS with uncertainty can be also stabilized quickly using the competition controller designed in this paper though there exist packet disordering and packet dropout in communication network. This makes it clear that the proposed control scheme has a good robustness.Figure 5
State and output response of the system.Figure 6
State and output response of the system.


## 4.2. Control Performance Comparison

It should be pointed out that the discrete-time system in this example can be inverted into a continuous-time system in [14, 20] if sampling period T=0.1s is given. If wk≠0, the H∞ norm bounds and corresponding controller gains are shown in Table 3 (“—” denotes that the conditions are infeasible). Obviously, a more optimal H∞ norm bound is obtained in this paper than those in [14, 20] since the competition scheme is performed when no signal is available by the plant due to packet disordering and packet loss. At the initial state value x0=[-1-3]T, the disturbance input w(t) is as follows:
(34)w(t)={sin(t)5s≤t<20s0otherwise,
the state and output response of the NCS in the presence of packet disordering and packet loss are shown in Figure 7.Table 3
Comparison of convergence time.
H
∞ norm  bound
Controller  gain
[14]
—
—
[20]
1.5
K
=
1.0
*
1
0
-
7 [0.1278-0.0714]
This paper
0.6977
K
2
=
[
-
0.4867
0.7869
] K3=[00] K4=[00]Figure 7
Curves of state and output response of the system subject to exogenous disturbance.


## 5. Conclusions

In this paper, we are concerned withH∞ control of NCSs with compensation scheme. The aim of devised control scheme is that the effect of packet disordering and packet loss on control performance is eliminated. The main idea is that we first describe the packet disordering and give the compensation scheme when there are no new signals executed by the plant during the sampling interval ((k-1)T,kT]. Second, a model of NCSs with Markovian jumping property is presented. Furthermore, the stochastic stability and controller design are discussed. Finally, a numerical example and simulations are given to illustrate the advantages and the effectiveness of the developed theory.



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*Source: 836314-2013-06-13.xml*