# H

Authors: Xiaobo Zhang; Qingwei Chen
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is concerned with the problem ofH∞ control for multirate networked control systems with deadband scheduling policy and transmission delay. The transmission deadbands are set at the sensor nodes and the controller nodes to reduce the network traffic, and then the systems are modeled as discrete-time switch system with uncertain parameters. Using the Lyapunov theory, a robust H∞ controller is designed to maintain the asymptotic stability of the closed-loop systems. Finally, a simulation example is presented to show the advantage of the deadband scheduling policy and the efficiency of the proposed theoretical results.


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## Body

## 1. Introduction

Networked control systems (NCSs) are the complex systems in which the controlled plants, sensors, controllers, and actuators are distributed in different places and connected through a communication network [1]. Compared with the traditional control systems, NCSs have many advantages such as the flexibility of structure, high reliability, and simple installation. As the rapid development of information technology, NCSs have been applied to various applications including mobile robots, unmanned aerial vehicles, and remote surgeries [2–5]. However, the data transmission through a network brings many challenges, such as time delay, packet dropout, and data disorder, which will cause the system performance degradation and even, make the systems unstable. In recent years, the stability analysis and controller design of NCSs have been researched by many scholars [6–10].When the bandwidth of NCSs is constrained, the packet collision becomes more severe. It is difficult to guarantee the stabilization and acceptable performance of systems only by means of controller design [11, 12]. In this case, the network scheduling becomes particularly important. The network nodes’ unnecessary data traffic could be reduced by introducing appropriate network scheduling strategy; thus, the network-induced delay and packet dropouts can be reduced remarkably [13–17]. In [13], the problem of integrated design of controller and communication sequences is addressed for a class of multiple inputs multiple outputs NCSs based on switch system theory, but not taking the network-induced delay into account, and the results are extended to NCSs with transmission delay in [14, 15]. In [16], a coordinate design scheme of the try-once-discard (TOD) dynamic scheduling strategy and robust  H∞  controller for a class of networked control systems with communication constraints and random time-delay is proposed. In [17], the wireless NCSs with communication delay and packet dropouts, which are based on the deadband scheduling policy, are modeled as continuous time-delay systems and an H∞ controller is designed. Meanwhile, the event-triggering NCSs attract many researchers’ attention [18–20]. Compared with the widely used time-triggering sampling scheme, the event-triggering sampling scheme, whose main property is that the signal is sampled only when some functions of the system state or output measurement exceeds a threshold, can also save network bandwidth for the NCSs.It is worth pointing that all of the aforementioned results are under the assumption that NCSs work in a single rate mode (either the actuators are event driven or the updating period of actuators is the same as the sampling period). The multirate NCSs, in which the updating rate of actuators is several times the sampling rate, have many advantages: (1) compared with the NCSs with event-driven actuators, the actuator updating moments are fixed in the multirate NCSs; thus, the analysis and synthesis of the multirate NCSs become more easier; (2) compared with the NCSs in which the actuator updating rate is the same as the sampling rate, the control input can be implemented to the plant more sooner in the multirate NCSs; therefore, better system performance can be ensured [21–23]. In [21], the multirate NCSs are models as the discrete switch systems, and an H∞ controller is designed by using the average dwell time method. The model-predictive control problem for multirate NCSs is proposed in [22]. To the best of the authors’ knowledge, H∞ control problem of multirate NCSs with deadband scheduling has not been fully investigated to date, which motivates the present study.In this paper, we focus on the problem ofH∞ control for multirate NCSs with deadband scheduling and transmission delay. By analyzing the influences of transmission deadband, a discrete-time switch system model with uncertain parameters is proposed to describe the multirate NCSs. The design method of a robust H∞ controller that stabilizes this class of networked control systems is proposed. A simulation example is presented to show the advantage of the deadband scheduling policy and the efficiency of the proposed method.This paper is organized as follows. In Section2, the problem is stated and some useful definitions and lemmas are given, and then the main results of this paper are given in Section 3. Section 4 provides a simulation example to illustrate the effectiveness of our results. Finally, Section 5 gives some concluding remarks.Notation. ℝn denotes the n-dimensional Euclidean space, and I is identity matrix. AT  stands for the transpose of the corresponding matrix  A. The notation A≥0(A>0) means that the matrix A is a positive semidefinite (positive definite) matrix. For an arbitrary matrix Y and two symmetric matrices  X  and  Z, [XY*Z]  denotes a symmetric matrix, where * denotes a block matrix entry implied by symmetry.


## 2. Problem Formulation and Preliminaries

Consider NCSs whose setup is depicted in Figure1. The controlled plant, sensor, and controller are connected with a communication network. The transmission deadbands are set at the sensor node and the controller node.Figure 1
Illustration of NCSs over communication network.The plant is described by(1)x˙(t)=Apx(t)+Bpu(t)+Bpww(t)y(t)=Cpx(t),
where x(t)∈ℝn  is the state, u(t)∈ℝm is the control input, y(t)∈ℝl is the output, and w(t)∈ℝp is the exogenous input. Ap,Bp,Bpw, and Cp are known real constant matrices with appropriate dimensions.The sensors are periodically sampled with the sampling intervalTs. The sensor-to-controller transmission delay is described as  τsc, and the controller-to-actuator transmission delay is described as τca. We make the following assumptions about NCSs.Assumption 1.
τ
=
τ
sc
+
τ
ca
,
0
<
τ
<
T
s.Assumption 2.
The controller is event driven; the sensors and actuators are time driven.Assumption 3.
The updating period of actuators isTa, Ts=NTa, N is positive integer, and N>1.First, the discrete-time model of multirate NCSs without transmission deadband is proposed.The controller outputu(kT) arrives at the actuator at the instants kT+τk; if  τk∈((n-1)Ta,nTa],(n∈{1,2,…,N}), then the actuator will update at the instances kTs+nTa. Therefore, in the sample interval [kTs,(k+1)Ts], system (1) is described as
(2)x˙(t)=Apx(t)+BpKx((k-1)Ts)+Bpww(t),t∈[kTs,kTs+nTa),x˙(t)=Apx(t)+BpKx(kTs)+Bpww(t),t∈[kTs+nTa,(k+1)Ts),(3)y(t)=Cpx(t).
Let A=eApTs, system (1) with sampling period Ts is discretized to
(4)x(k+1)=Ax(k)+∫0nTaeAp(Ts-s)Bpds·u(k-1)+∫nTaTseAp(Ts-s)Bpds·u(k)+Bωω(k)=Ax(k)+∫0nTaeAp(NTa-s)Bpds·u(k-1)+∫nTaNTaeAp(NTa-s)Bpds·u(k)+Bωω(k).
Let
(5)B1n={∑α=nN-1∫αTa(α+1)TaeAp(Ts-s)Bpds,n<N0n=N,B2n=∑α=0n-1∫αTa(α+1)TaeAp(Ts-s)Bpds.
Then
(6)x(k+1)=Ax(k)+B1nu(k)+B2nu(k-1)+Bωω(k).
Next, we introduce the deadband scheduling policy. The transmission deadbands located at the sensor node and controller node are based on such a deadband scheduling policy: if the difference between the present signal and the last transmitted signal exceeds the threshold of deadband, the present signal will be transmitted, otherwise it will not be transmitted for the purpose of reducing the network traffic and saving the network bandwidth.Letx(k)=[x1(k),x2(k),…,xn(k)]T, xi(k) is the component of x(k) and sampled separately by each senor at the sampling instant. If xi(k) enters into the deadband, according to the proposed scheduling policy, xi(k) will not be transmitted and the controller will hold the previous transmitted one, which is equivalent to transmitting the previous transmitted one again. Therefore, the deadband 1 is described as
(7)x-i(k)=fi(xi(k))={xi(k)|x-i(k-1)-xi(k)|>δ1i|xi(k)|x-i(k-1)|x-i(k-1)-xi(k)|≤δ1i|xi(k)|,
where δ1i is the deadband threshold coefficient, δ1i∈(0,1).According to (7),
(8)xi(k)-δ1i|xi(k)|≤x-i(k)≤xi(k)+δ1i|xi(k)|.
Define Δ-1=diag{Δ-11,Δ-12,…,Δ-1n},Δ-1j∈[-δ1j,δ1j],(j=1,2,…,n) and δ1=diag{δ11,δ12,…,δ1n}; therefore, one can obtain
(9)x-(k)=(I+Δ-1)x(k).Similarly, deadband 2 is described as(10)u-i(k)=gi(ui(k))={ui(k)|u-i(k-1)-ui(k)|>δ2i|ui(k)|u-i(k-1)|u-i(k-1)-ui(k)|≤δ2i|ui(k)|,
where δ2i is the deadband threshold coefficient, δ2i∈(0,1).DefineΔ-2=diag{Δ-21,Δ-22,…,Δ-2m},Δ-2j∈[-δ2j,δ2j],(j=1,2,…,m) and δ2=diag{δ21,δ22,…,δ2m}; therefore, one can obtain
(11)u-(k)=(I+Δ-2)u(k).Combining (6), (9), and (11), the discrete-time model of multirate NCSs is described as
(12)x(k+1)=Ax(k)+B1n(I+Δ1)K(I+Δ2)x(k)+B2n(I+Δ3)K(I+Δ4)x(k-1)+Bωω(k)y(k)=Cx(k).The following definition and lemmas will be essential for the proofs in the next section.Definition 1.
System (12)  with a given matrix K is said to be asymptotically stable with an H∞ norm bound γ if the following conditions hold:(1)
system (12) is asymptotically stable with ω(k)=0;(2)
under the zero initial condition, the controlled outputy(k) satisfies ∥y(k)∥2<γ∥ω(k)∥2, where w(t)∈L2[0,∞).Lemma 2 (see [24] Schur complement).
For matricesA,Q=QT,P=PT>0, the matrix inequality ATPA+Q<0 holds, if and only if the following matrix inequalities
(13)[QATA-P-1]<0or[-P-1AATQ]<0
hold.Lemma 3 (see [25]).
For some given matricesW, M, and N of appropriate dimensions and with W being symmetric, then
(14)W+NTFT(t)MT+MF(t)N<0.
holds, where F(t) satisfying FT(t)F(t)≤I, if and only if there exists a scalar ε>0 such that
(15)W+εMMT+ε-1NTN<0.


## 3. Main Results

In this section, we will find a state feedback control matrixK such that the system is asymptotically stable with an H∞ norm bound γ.Theorem 4.
For given state feedback control matrixK, system (12) is asymptotically stable with an H∞ norm bound γ, if there exist symmetric positive definite matrices P and Q, such that the following condition holds:
(16)[Q-P00CT(A+B1n(I+Δ1)K(I+Δ2))T*-Q00(B2n(I+Δ3)K(I+Δ4))T**-γ2I0Bω***-I*****-P-1]<0.Proof.
Consider the following Lyapunov candidate functional:(17)V(k)=xT(k)Px(k)+xT(k-1)Qx(k-1).
First, we proof that system (12) is asymptotically stable with ω(k)=0 under (16).
Let(18)A-=A+B1n(I+Δ1)K(I+Δ2),B-=B2n(I+Δ3)K(I+Δ4),
along the trajectory of the system (12), we have
(19)ΔV(k)=xT(k+1)Px(k+1)+xT(k)Qx(k)-xT(k)Px(k)-xT(k-1)Qx(k-1)=[xT(k+1)xT(k)][Q-P+A-TPA-*B-TPA-B-PB--Q]×[xT(k+1)xT(k)]T,
using Lemma 2 (Schur complement), we have ΔV(k)<0 if and only if
(20)[Q-P0A-T*-QB-T**-P-1]<0.
By Lemma2 (Schur complement), we know that (16) implies (20); Therefore, system (12) is asymptotically stable with ω(k)=0 being proved.
Now, we will consider theH∞ performance of system (12).
Whenω(k)≠0, define
(21)J=∥y(k)∥22-γ2∥ω(k)∥22=∑k=0∞[yT(k)y(k)-γ2ωT(k)ω(k)+ΔV(k)]-V(∞)+V(0);
under zero condition and V(∞)≥0, we have
(22)J≤∑k=0∞[yT(k)y(k)-γ2ωT(k)ω(k)+ΔV(k)].
Letη(k)=[xT(k)xT(k-1)ωT(k)]T, we obtain
(23)J≤∑k=0∞ηT(k)Hη(k),
where
(24)H=[Q-P+A-TPA-+CTC**B-TPA-B-PB--Q*BωTPA-BωTPB-BωTPBω-γ2].
By Lemma2 (Schur complement), (16) is equivalent to H<0, which implies that J<0; thus, we have ∥y(k)∥2<γ∥ω(k)∥2. This completes the proof.Next, the design method of the state feedback controller is proposed.Theorem 5.
For given diagonal matricesδ1, δ2 and a scalar  λ, if there exist positive definite matrices Q-, X, and W, scalars εi>0(i=1,2,…,6) and a matrix Y, such that the following linear matrix inequalities
(25)[-WYT*-λI]<0,(26)W<2X-I,(27)[Γ11Γ12*Γ22]<0,
hold, then the system is asymptotically stable with an H∞ norm bound γ; the desired controllers are given as follows:
(28)K=YX-1,
where(29)
Γ
11
=
[
Q
-
-
X
0
0
(
C
X
)
T
(
A
X
+
B
1
n
Y
)
T
*
-
Q
-
0
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(
B
2
n
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)
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*
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γ
2
I
0
B
ω
T
*
*
*
-
I
0
*
*
*
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-
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]
,
Γ
22
=
diag
{
-
ε
1
,
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ε
1
,
-
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2
,
-
λ
ε
2
,
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3
,
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λ
ε
3
,
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ε
4
,
-
ε
4
,
-
ε
5
,
-
λ
ε
5
,
-
ε
6
,
-
λ
ε
6
}
,
Γ
12
=
[
ε
1
L
1
δ
1
L
2
Y
T
ε
2
L
1
λ
L
2
X
δ
2
ε
3
L
1
δ
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λ
L
2
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δ
2
ε
4
L
3
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1
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ε
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6
L
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λ
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δ
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,
L
1
=
[
0
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L
4
=
[
0
I
0
0
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]
T
.Proof.
Rewrite (16) as
(30)F=F1+F2<0,
where
(31)F1=[Q-P00CT(A+B1nK)T*-Q00(B2nK)T**-γ2I0Bω***-I*****-P-1],F2=L1Δ1KL2T+(L1Δ1KL2T)T+L1KΔ2L2T+(L1KΔ2L2T)T+L1Δ1KΔ2L2T+(L1Δ1KΔ2L2T)T+L3Δ3KL4T+(L3Δ3KL4T)T+L3KΔ4L4T+(L3KΔ4L4T)T+L3Δ3KΔ4L4T+(L3Δ3KΔ4L4T)T,L1,L2,L3, and L4 are defined above in Theorem 5.
By Lemma3, F<0 is equivalent to
(32)F-=F1+F-2<0,
where
(33)F-2=ε1L1Δ1Δ1TL1T+ε1-1L2KTKL2T+ε2L1L1T+ε2-1L2Δ2TKTKΔ2L2T+ε3L1Δ1Δ1TL1T+ε3-1L2Δ2TKTKΔ2L2T+ε4L3Δ3Δ3TL3T+ε4-1L4KTKL4T+ε5L3L3T+ε5-1L4Δ4TKTKΔ4L4T+ε6L3Δ3Δ3TL3T+ε6-1L4Δ4TKTKΔ4L4T.
By the definitions of Δ1,Δ2,Δ3, and Δ4, we have
(34)Δ1Δ1T=Δ1TΔ1≤δ1Tδ1,Δ3Δ3T=Δ3TΔ3≤δ1Tδ1,Δ2Δ2T=Δ2TΔ2≤δ2Tδ2,Δ4Δ4T=Δ4TΔ4≤δ1Tδ1,
using Lemma 2 (Schur complement), (25) is equivalent to YTY≤λW. Define K=YX-1, (26) implies that
(35)KTK≤λI.
Combining (33), (34), and (35), we have
(36)F-2≤F~2=ε1L1δ1δ1TL1T+ε1-1L2KTKL2T+ε2L1L1T+λε2-1L2δ2Tδ2L2T+ε3L1δ1δ1TL1T+λε3-1L2δ2Tδ2L2T+ε4L3δ3δ3TL3T+ε4-1L4KTKL4T+ε5L3L3T+λε5-1L4δ4Tδ4L4T+ε6L3δ3δ3TL3T+λε6-1L4δ4Tδ4L4T.
Therefore,
(37)F~=F-1+F~2<0
Guarantees (16).
By Lemma2 (Schur complement), (37) is equivalent to
(38)F^=[F1F^12*F^22]<0,
where (39)F^12=[ε1L1δ1L2KTε2L1λL2δ2ε3L1δ1λL2δ2ε4L3δ1L4KTε5L3λL4δ2ε6L3δ1λL4δ2],F^22=diag{-ε1,-ε1,-ε2,-λε2,-ε3,-λε3,-ε4,-ε4,-ε5,-λε5,-ε6,-λε6},
LetX=P-1, we pre- and postmultiply both sides of (38) with diag{X,X,I,I,…,I︷15} we have (27). This completes the proof.


## 4. Simulations

In this section, we will give an example to show the usefulness of the derived results and the effectiveness of the proposed methods.The controlled plant parameters are given as follows:(40)Ap=[-30.50.6-2],Bp=[10.50.31],Bpw=[0.1000.1],Cp=[1001].Given the sampling periodTs=50ms, and the actuator updating period Ta=Ts/3, following (6), we have
(41)A=[0.8610.02210.02660.9052],Bw=[0.00460.00010.00010.0048],B11=[0.03180.01610.01000.0324],B21=[0.01480.00770.00500.0155],B12=[0.01630.00820.00500.0164],B22=[0.03030.01560.01000.0315],B13=[0000],B23=[0.04660.02380.01500.0479].The deadband threshold coefficients of deadbands 1 and 2 are given asδ1=diag{0.3,0.3}, δ2=diag{0.2,0.2}, and the disturbance attenuation γ=0.3. According to Theorem 5, using the LMI toolbox of MATLAB, the controller parameters can be calculated as follows:
(42)K=[-2.726-0.8501-0.8501-3.897].The exogenous input is chosen asw(t)=5sin(t), the states responses of the closed-loop multirate NCSs are shown in Figure 2, which confirms that the closed-loop multirate NCSs are robustly asymptotically stable under the exogenous input.Figure 2
State trajectories of the closed-loop system.The packet dropout rates of the S-to-C channel and the C-to-A channel are shown in Figure3. The packet dropout mentioned here is the data packets which are lost “actively” when the multirate NCSs enter into the transmission deadband; thus, Figure 3 confirms that the proposed deadband scheduling policy could reduce the network traffic remarkably.Figure 3
Packet dropout rate curve of the closed-loop system.


## 5. Conclusions

In this paper,H∞ control problem for multirate networked control systems with deadband scheduling policy and transmission delay is considered. The transmission deadbands are set at the sensor node and controller node to reduce the network traffic. The systems are modeled as the discrete switch system with uncertain parameters and the design method of H∞ controller is established by using the Lyapunov theorem. Finally, the simulation example is presented to show the advantage of the deadband scheduling policy and the efficiency of the proposed theoretical results.



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*Source: 715484-2013-05-15.xml*