# Control Method for Sampled-Data Systems with Multiple Channels Based on Deadband-Triggered Scheme

Authors: Ying-Ying Liu; Yun-kai Chu
Journal: Security and Communication Networks
 (2018)

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## Abstract

A new deadband-triggered scheme is proposed to investigate the control problems for sampled-data systems with multiple transmitting channels. Sampled-data systems simultaneously contain continuous-time and discrete-time signals, which make the systems hybrid. In the sampled-data systems with multiple channels, the every state signals are transmitting at different channels. The deadband communication constraint is adopted to reduce the usage of communication resources. When the difference between the previous value and the most present value is lager than a given threshold of deadband, then the node of channels transmits the most present value. Furthermore, by use of Lyapunov functional method and input delay approach, the new stability analysis and stabilization conditions for the sampled-data with multiple channels on the basis of deadband-triggered scheme are proposed. Numerical simulations and experiments show the validity and usefulness of the derived conditions. The proposed deadband-triggered scheme is beneficial to further reduce the load of the communication data.


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## Body

## 1. Introduction

With the rapid development of digital technology, sensor technology, and the huge performance of these technologies, sampled-data systems has become one of the most widely used systems including industrial processes [1–3]. In this class of system, the object is running in a continuous domain, and the output of the object is a discrete signal measured by a sensor. This makes the controller only rely on discrete sampling signals to monitor objects. So the sampled-data system becomes a hybrid system containing continuous and discrete signals [3–6].Compared to a generally pure continuous or pure discrete system, the sampled-data system can achieve a more essential description of most of the existing industrial systems or devices. And because of its large use of digital and sensor technologies, this kind of system is more automated and more complex. A typical sampled-data system is networked control system [7–9]. Because of the wide application of the digital hardware technologies, the sampled-data control method has been more important than other control approaches. Recently, some researchers have investigated the sampled-data control [10–12]. However, all the above literature considered that the signals of sampled-data systems are transmitted by one channel. In the actual system, sensor and actuator are sometimes distributed in different areas, the signals require communication through multiple channels, each channel will lead to uncertainties is not the same, and this assumption is not suitable for single channel mode. So it is necessary to study the control problem of sampled-data systems with multiple channels. At present, [13] studied the control problems of the networked control systems based on the channels sharing method. In fact, in many sampled-data systems, the bandwidth of data transmitting channel is limited. Therefore, to avoid the unnecessary waste of communication resources and achieve better performance of sampled-data systems, it is necessary to consider a potential improvement of the traditional sampled-data control methods, namely, the data communication triggered scheme. So far, many results have been obtained about event-triggered control [14–21]. Inspired by these results, this paper gives a deadband-triggered scheme, which controls the account of the data transmitting, according to the difference of the present value and the previous value at the deadband-triggered processor. However, the control problems of the sampled-data systems with multiple channel deadband-triggered communication scheme have not been studied in full detail. This motivates the present study.In response to the above discussion, in this paper, we aim to investigate the stabilization problem for sampled-data systems with multiple communication channels based on deadband-triggered scheme. The organization of this paper is as follows. Section2 presents the problem of the sampled-data systems under the consideration of deadband-triggered scheme and multiple communication channels. Section 3 presents the stability analysis condition of the sampled-data systems. And, a design method of stabilization for the sampled-data systems is proposed. In Section 4, an example is presented to illustrate the effectiveness of the proposed method. Finally, Section 5 gives some concluding remarks.Notation. Given a matrix E, ET, and E-1 denote its transpose and inverse when it exists, respectively. The notation P>0 means that P is real symmetric and positive definite. In denotes the n×n identity matrix. In symmetric block matrices, we use an asterisk ∗ to represent a term that is induced by symmetry. Rn and Rm×n stand for the set of n dimension vectors, m×n matrices, respectively. The notations throughout this paper are standard. diag{···} denotes a block-diagonal matrix.


## 2. Problem Formulation

### 2.1. Sampled-Data System with Time-Varying Sampling Intervals

Consider sampled-data system in the form(1)x.t=Axt+But,where x(t)∈Rn is the system state, u(t)∈Rp is the control input, and A, B are some constant matrices of appropriate dimensions.The sequence of sampling instantstkk=0∞ with tk+1>tk is increasing and divergent with t0=0. Without loss generality, it is assumed that t0=0 and the first sampling happens at time t0. The initial state is given as x(t0)=x0. The sampling interval Tk=tk+1-tk is time-varying and its lower bound and upper bound are known:(2)0<Tm≤Tk≤TM,where Tm and TM denote the lower bound of sampling interval and the upper bound of sampling interval, respectively.We assume that the state of (1) is measurable at tk, where the output signal of the controller u′(tk) is determined from the sampled data xi′(tk) and a constant feedback gain K  (K∈Rm×n).


### 2.2. Multiple Communication Channels

In the sampled-data system, it transmits the useful information through multiple channels. As shown in Figure1, the closed-loop system is composed of a plant, some sensors, some deadband-triggered processors, a controller, and an actuator. The state signals of the plant are sampled at the sensor sides and then released to the communication channels. xi(tk)(i=1…n) are the state signals of the plant sampled by the sensors. At the same time, it is assumed that the sensor and sampler are clock-driven, while the controller and actuator are event-driven. As shown in Figure 1, unlike the widely researched state-feedback control system with single channel environment, the signals from sensors are transmitted to the controller via n different communication channels. The spatially distributed sensors can be synchronized by using some existing approaches such as time synchronization based on the linear consensus algorithm or slow-flooding [19].Figure 1
Structure of the sampled-data control systems with multiple communication channels.In the sampled-data systems, if the sampling interval is too small, the number of data packets will increase quickly, which can lead to an overloaded communication bandwidth. However, fewer sampled signal are must be transmitted for maintaining the running of the controlled systems. As an alternative, this investigation explores a deadband-triggered scheme as a solution to reduce data transmission in the system.


### 2.3. Deadband-Triggered Communication Scheme

In the sampled-data system (see Figure1), a deadband-triggered processor (DTP) is set at the every sensor nodes. For the ith channel, the DTP of the sensor to controller is denoted as Deadbandisc  (i=1,2…n). The node does not transfer a new message if the node signal is within the deadband. A number of sampled-data packets will be discarded under the deadband-triggered mechanism, which will make the channel loads reduced greatly. Thus, a good quality of the channel can be possibly ensured.In Figure1, at the sampling instant tk(k=0,1,…n), both the state and the control input signals are processed by deadbands before they are sent to the controller and actuator, respectively. Each state signal xi(tk)(i=1,2,⋯,n) is sampled by sensor i. Whether or not the sampled signal xi(tk) should be transmitted is determined by the ith DTP. The mechanism of Deadbandisc is precisely described as follows: (3)xi′tk+1=xitk+1xi′tk-xitk+1>δ1ixitk+1xi′tkxi′tk-xitk+1≤δ1ixitk+1where we denote the input signal of deadband-triggered processor Deadbandisc as xi(tk) and its output signal as xi′(tk). The deadband node i in Deadbandisc compares the previous value xi′(tk) to the most recent value xi(tk+1). The data packet is sent over the channels only if the difference value changes more than a given threshold, where δ1ixi(tk+1) is the threshold of DTP Deadbandisc. δ1i is the weighting of threshold.Remark 1.
If the absolute value of the difference betweenxi(tk+1) and xi′(tk) is within the threshold of deadband then the signal data is not sent.
DefineΔ1i(tk+1)∈[-δ1i(tk+1),δ1i(tk+1)], then (3) can be included in following equality:(4)xi′tk+1=xitk+1+Δ1itk+1xitk+1Remark 2.
Because of the introduction of the deadband-triggered scheme, some uncertain will be brought to the sampled-data system. We furthermore use (4) to describe deadband mechanism (3) in the model of systems (1). We can clearly see that there exists xi′(tk+1)=xi(tk+1) when Δ1i(tk+1)=0 in (4), and there exists xi′(tk+1)=xi′(tk) when Δ1i(tk+1)=0 in (4).For the sake of simplicity, in the following, we useδ1i to denote δ1i(tk+1). Δ1(tk+1)=diag{Δ11(tk+1),Δ12(tk+1),⋯,Δ1n(tk+1)}. Thus, for the deadband-triggered processor 1, its input signal and output signal show the following:(5)x′tk+1=I+Δ1tk+1xtk+1At the same time, the input signal and the output signal of deadband-triggered processor 2 are u′ and u, respectively. The sampling instants are denoted as tk, k=0,1,⋯,∞. In the sampled-data system (1), the digital controller is used, and it satisfies the following:(6)u′tk=Kx′tk,k=0,1,2…K is the state-feedback gain to be determined later. Following, the relation between the input signal u′ and the output signal u in deadband-triggered processor 2 can be described as(7)ujtk+1=u′jtk+1ujtk-u′jtk+1>δ2ju′jtk+1ujtkujtk-u′jtk+1≤δ2ju′jtk+1where j=1,2,⋯,m, δ2juj(sk+1) is the threshold of event-triggered generator 2. δ2j is the weighting of threshold.Similar to (3), (4), then (7) can be included in(8)ujtk+1=uj′tk+1+Δ2jtk+1uj′tk+1Define Δ2(tk+1)=diag{Δ21(tk+1),Δ22(tk+1),⋯,Δ2m(tk+1)}, Δ2j(tk+1)∈[-δ2j,δ2j]. Combine (4) and (8); the control input signal can be written as(9)ut=I+Δ2tkKI+Δ1tkxtktk≤t<tk+1,where I denotes the identity matrix of appropriate dimensions. Δ1i(tk)≤δ1i and Δ2j(tk)≤δ2j. For the sake of simplicity, it is assumed that δ1i=δ1 and δ2j=δ2. Then, it is not difficult to get Δ1i(tk)∈-δ1,δ1 and Δ2j(tk)∈-δ2,δ2. In the following, for the sake of simplicity, we denote Δ1(tk), Δ2(tk) as Δ1, Δ2, respectively.For the sampled-data system, the digital control law may be represented as follows by using input delay approach [12]:(10)ut=I+Δ2KI+Δ1xt-t-tktk≤t<tk+1,Denote d(t)=t-tk; the piecewise-constant control law (10) can be represented as a continuous-time controller with a time-varying piecewise continuous delay:(11)ut=I+Δ2KI+Δ1xt-dt,tk≤t<tk+1where d(t)=t-tk is piecewise linear with the derivative d˙(t)=1 for t≠tk. It is assumed that d(t) is bounded that satisfies Tm<d(t)<TM.Substituting the controller (11) into the sampled-data systems (1), we can obtain the closed-loop sampled-data system on the basis of deadband-triggered scheme:(12)x˙t=Axt+BK+ΔKxt-dttk≤t<tk+1 where Δ(K)=Δ2K+KΔ1+Δ2KΔ1.Our objective is to find a state-feedback controllerK, which stabilizes system (1).


## 2.1. Sampled-Data System with Time-Varying Sampling Intervals

Consider sampled-data system in the form(1)x.t=Axt+But,where x(t)∈Rn is the system state, u(t)∈Rp is the control input, and A, B are some constant matrices of appropriate dimensions.The sequence of sampling instantstkk=0∞ with tk+1>tk is increasing and divergent with t0=0. Without loss generality, it is assumed that t0=0 and the first sampling happens at time t0. The initial state is given as x(t0)=x0. The sampling interval Tk=tk+1-tk is time-varying and its lower bound and upper bound are known:(2)0<Tm≤Tk≤TM,where Tm and TM denote the lower bound of sampling interval and the upper bound of sampling interval, respectively.We assume that the state of (1) is measurable at tk, where the output signal of the controller u′(tk) is determined from the sampled data xi′(tk) and a constant feedback gain K  (K∈Rm×n).


## 2.2. Multiple Communication Channels

In the sampled-data system, it transmits the useful information through multiple channels. As shown in Figure1, the closed-loop system is composed of a plant, some sensors, some deadband-triggered processors, a controller, and an actuator. The state signals of the plant are sampled at the sensor sides and then released to the communication channels. xi(tk)(i=1…n) are the state signals of the plant sampled by the sensors. At the same time, it is assumed that the sensor and sampler are clock-driven, while the controller and actuator are event-driven. As shown in Figure 1, unlike the widely researched state-feedback control system with single channel environment, the signals from sensors are transmitted to the controller via n different communication channels. The spatially distributed sensors can be synchronized by using some existing approaches such as time synchronization based on the linear consensus algorithm or slow-flooding [19].Figure 1
Structure of the sampled-data control systems with multiple communication channels.In the sampled-data systems, if the sampling interval is too small, the number of data packets will increase quickly, which can lead to an overloaded communication bandwidth. However, fewer sampled signal are must be transmitted for maintaining the running of the controlled systems. As an alternative, this investigation explores a deadband-triggered scheme as a solution to reduce data transmission in the system.


## 2.3. Deadband-Triggered Communication Scheme

In the sampled-data system (see Figure1), a deadband-triggered processor (DTP) is set at the every sensor nodes. For the ith channel, the DTP of the sensor to controller is denoted as Deadbandisc  (i=1,2…n). The node does not transfer a new message if the node signal is within the deadband. A number of sampled-data packets will be discarded under the deadband-triggered mechanism, which will make the channel loads reduced greatly. Thus, a good quality of the channel can be possibly ensured.In Figure1, at the sampling instant tk(k=0,1,…n), both the state and the control input signals are processed by deadbands before they are sent to the controller and actuator, respectively. Each state signal xi(tk)(i=1,2,⋯,n) is sampled by sensor i. Whether or not the sampled signal xi(tk) should be transmitted is determined by the ith DTP. The mechanism of Deadbandisc is precisely described as follows: (3)xi′tk+1=xitk+1xi′tk-xitk+1>δ1ixitk+1xi′tkxi′tk-xitk+1≤δ1ixitk+1where we denote the input signal of deadband-triggered processor Deadbandisc as xi(tk) and its output signal as xi′(tk). The deadband node i in Deadbandisc compares the previous value xi′(tk) to the most recent value xi(tk+1). The data packet is sent over the channels only if the difference value changes more than a given threshold, where δ1ixi(tk+1) is the threshold of DTP Deadbandisc. δ1i is the weighting of threshold.Remark 1.
If the absolute value of the difference betweenxi(tk+1) and xi′(tk) is within the threshold of deadband then the signal data is not sent.
DefineΔ1i(tk+1)∈[-δ1i(tk+1),δ1i(tk+1)], then (3) can be included in following equality:(4)xi′tk+1=xitk+1+Δ1itk+1xitk+1Remark 2.
Because of the introduction of the deadband-triggered scheme, some uncertain will be brought to the sampled-data system. We furthermore use (4) to describe deadband mechanism (3) in the model of systems (1). We can clearly see that there exists xi′(tk+1)=xi(tk+1) when Δ1i(tk+1)=0 in (4), and there exists xi′(tk+1)=xi′(tk) when Δ1i(tk+1)=0 in (4).For the sake of simplicity, in the following, we useδ1i to denote δ1i(tk+1). Δ1(tk+1)=diag{Δ11(tk+1),Δ12(tk+1),⋯,Δ1n(tk+1)}. Thus, for the deadband-triggered processor 1, its input signal and output signal show the following:(5)x′tk+1=I+Δ1tk+1xtk+1At the same time, the input signal and the output signal of deadband-triggered processor 2 are u′ and u, respectively. The sampling instants are denoted as tk, k=0,1,⋯,∞. In the sampled-data system (1), the digital controller is used, and it satisfies the following:(6)u′tk=Kx′tk,k=0,1,2…K is the state-feedback gain to be determined later. Following, the relation between the input signal u′ and the output signal u in deadband-triggered processor 2 can be described as(7)ujtk+1=u′jtk+1ujtk-u′jtk+1>δ2ju′jtk+1ujtkujtk-u′jtk+1≤δ2ju′jtk+1where j=1,2,⋯,m, δ2juj(sk+1) is the threshold of event-triggered generator 2. δ2j is the weighting of threshold.Similar to (3), (4), then (7) can be included in(8)ujtk+1=uj′tk+1+Δ2jtk+1uj′tk+1Define Δ2(tk+1)=diag{Δ21(tk+1),Δ22(tk+1),⋯,Δ2m(tk+1)}, Δ2j(tk+1)∈[-δ2j,δ2j]. Combine (4) and (8); the control input signal can be written as(9)ut=I+Δ2tkKI+Δ1tkxtktk≤t<tk+1,where I denotes the identity matrix of appropriate dimensions. Δ1i(tk)≤δ1i and Δ2j(tk)≤δ2j. For the sake of simplicity, it is assumed that δ1i=δ1 and δ2j=δ2. Then, it is not difficult to get Δ1i(tk)∈-δ1,δ1 and Δ2j(tk)∈-δ2,δ2. In the following, for the sake of simplicity, we denote Δ1(tk), Δ2(tk) as Δ1, Δ2, respectively.For the sampled-data system, the digital control law may be represented as follows by using input delay approach [12]:(10)ut=I+Δ2KI+Δ1xt-t-tktk≤t<tk+1,Denote d(t)=t-tk; the piecewise-constant control law (10) can be represented as a continuous-time controller with a time-varying piecewise continuous delay:(11)ut=I+Δ2KI+Δ1xt-dt,tk≤t<tk+1where d(t)=t-tk is piecewise linear with the derivative d˙(t)=1 for t≠tk. It is assumed that d(t) is bounded that satisfies Tm<d(t)<TM.Substituting the controller (11) into the sampled-data systems (1), we can obtain the closed-loop sampled-data system on the basis of deadband-triggered scheme:(12)x˙t=Axt+BK+ΔKxt-dttk≤t<tk+1 where Δ(K)=Δ2K+KΔ1+Δ2KΔ1.Our objective is to find a state-feedback controllerK, which stabilizes system (1).


## 3. Main Results

### 3.1. Stability Analysis for the Sampled-Data Control Systems with Multiple Deadband Channels

Theorem 3.
For given scalarsTm, TM, δ1>0, δ2>0, 0≤α<1, and a matrix K, the closed-loop sampled-data system (12) is stable if there exist matrices P=PT>0, Qm=QmT>0  (m=1,2,3), Zj=ZjT>0  (j=1,2,3), and Ni, Ti, Mi, Ei, Li  (i=1-5) such that(13)Γ=Γ1Γ2BTZ1BTZ2BTZ3∗Γ3000∗∗-1TM-TmZ100∗∗∗-1TMZ20∗∗∗∗-1TMZ3<0,where(14)Γ1=Γ11Γ12Γ13Γ14Γ15∗Γ22Γ23Γ24Γ25∗∗Γ33Γ34Γ35∗∗∗Γ44Γ45∗∗∗∗Γ55,Γ2=N1T1M1E1L1N2T2M2E2L2N3T3M3E3L3N4T4M4E4L4N5T5M5E5L5,Γ3=diag-1αTMZ1-11-αTMZ1-1TM-TmZ1+Z2-1TMZ2-1TMZ3,Γ11=PA+ATP+∑i=13Qi+N1+N1T+L1+L1T,Γ12=PBK+ΔK+N2T-T1+M1-E1+L2T,Γ13=E1+N3T+L3T,Γ14=-M1+N4T+L4T,Γ15=T1-N1+N5T+L5T,Γ22=M2+M2T-T2-T2T-E2-E2T,Γ23=E2+M3T-T3T-E3T,Γ24=-M2+M4T-T4T-E4T,Γ25=T2+N2,Γ33=-Q1+E3+E3T,Γ34=-M3+E4T-L3,Γ35=T3-N3+E5T,Γ44=-Q2-M4-M4T,Γ45=T4-N4-M5T-L5T,Γ55=-1-αQ3+T5-N5+T5T-N5T,B=ABK+ΔK000,Proof.
Consider the following Lyapunov-Krasovskii functional(15)Vt=V1t+V2t+V3t+V4t+V5t+V6t+V7t,where(16)V1t=xTtPxt,V2t=∫t-TmtxsTQ1xsds,V3t=∫t-TMtxsTQ2xsds,V4t=∫t-αdttxsTQ3xsds,V5t=∫-TM0∫t+βtx˙TsZ1x˙sdsdβ,V6t=∫-TM-Tm∫t+βtx˙TsZ2x˙sdsdβ,V7t=∫-TM0∫t+βtx˙TsZ3x˙sdsdβ,P=PT>0, Qm=QmT>0  m=1,2,3, and Zj=ZjT>0 (j=1, 2, 3).
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T
d
s
-
∫
t
-
d
t
t
-
T
m
H
4
Z
2
-
1
H
4
T
d
s
-
∫
t
-
T
M
t
H
5
Z
3
-
1
H
5
T
d
s
,where(18)H1=ζTtN+x˙TsZ1,H2=ζTtT+x˙TsZ1,H3=ζTtM+x˙TsZ1+Z2,H4=ζTtE+x˙TsZ2,H5=ζTtL+x˙TsZ3,ζt=xTtxTt-dtxTt-TmxTt-TMxTt-αdtT,N=N1TN2TN3TN4TN5TT,T=T1TT2TT3TT4TT5TT,M=M1TM2TM3TM4TM5TT,E=E1TE2TE3TE4TE5TT,L=L1TL2TL3TL4TL5TT.By the Schur complements, combine (13) to obtain V˙(t)<0 for all tk<t<tk+1.
( 2 ) It is noting that d(tk)=tk-sk, ∀k∈N, d(tk-)=tk-sk-1, ∀k∈N.
The value ofx before and after tk points remains unchanged (since x(t) is continuous). Then, we have Vi(tk-)=Vi(tk)  (i=1,2,3,5,6,7) in Lyapunov-Krasovskii functional (15). Moreover, for V4(t), there exists V4(tk-)≥V4(tk). Thus, V(tk-)≥V(tk) for k=0,1,2,3,⋯.
Fort∈tk,tk+1, we have(19)Vt-Vtk≤0. Since limk→∞tk=∞, we have ⋃k=0∞tktk+1=t0,∞. It follows that(20)Vt-Vt0≤0. It shows that V(t) decreases. This completes the proof.Remark 4.
Theorem3 presents a stability analysis condition of sampled-data systems under deadband-triggered communication scheme, which describes the relations of the deadband parameters and sampling intervals. The given condition is in terms of Linear Matrix Inequality (LMI) when the controller gain K is given.


### 3.2. Stabilization for the Sampled-Data Control Systems with Multiple Deadband Channels

This section is devoted to solve the problem of stabilization controller design for sampled-data systems (12). The following theorem presents the conditions for existence of the desired controller.Theorem 5.
Consider the sampled-data system in Figure1. For given scalars Tm, TM, δ1>0, δ2>0, 0≤α<1, and 0≤εi<1(i=1-3) the closed-loop system (12) is stable if there exist matrices X=XT>0, Q~i=Q~iT>0(i=1,2,3), Z~j=Z~jT>0(j=1,2,3), N~l, T~l, M~l, E~l, L~l(l=1-5), and Y of appropriate dimensions, such that(21)ΞB~Y~∗-1γ1I0∗∗-1γ2I<0,where(22)Ξ=Ξ1Ξ2ALALAL∗Ξ3000∗∗1σ¯Z~1∗00∗∗∗1σZ~2∗0∗∗∗∗1σZ~3∗,Ξ1=Ξ11Ξ12Ξ13Ξ14Ξ15∗Ξ22Ξ23Ξ24Ξ25∗∗Ξ33Ξ34Ξ35∗∗∗Ξ44Ξ45∗∗∗∗Ξ55,Ξ2=N~1T~1M~1E~1N~2T~2M~2E~2N~3T~3M~3E~3N~4T~4M~4E~4N~5T~5M~5E~5,Ξ11=AX+XAT+∑i=13Q~i+N~1+N~1T+L~1+L~1T,Ξ12=BY+N~2T-T~1+M~1-E~1+L~2T,Ξ13=E~1+N~3T+L~3T,Ξ14=-M~1+N~4T+L~4T,Ξ15=T~1-N~1+N~5T+N~5T,Ξ22=M~2+M~2T-T~2-T~2T-E~2-E~2T,Ξ23=E~2+M~3T-T~3T-E~3T,Ξ24=-M~2-L~2+M~4T-T~4T-E~4T,Ξ25=T~2-N~2Ξ33=-Q~1+E~3+E~3T,Ξ34=-M~3+E~4T-L~3,Ξ35=T~3-N~3+E~5T,,Ξ44=-Q~2-M~4-M~4T,Ξ45=T~4-N~4-M~5T,Ξ55=-1-αQ3+T~5-N~5+T~5T-N~5T,AL=AXTBY000T,Ξ3=diag-1αTMZ~1-11-αTMZ~1-1TM-TmZ~1+Z~2-1TM-TmZ~2-1TMZ~3,B~=BT000000000BTBTBTT,Y~=0Y00000000000T.γ2=ε1+ε2+ε3δ12,γ1=1ε1δ22+1ε2δ12+1ε3δ22,Z~1∗=Z~1-2X,Z~2∗=Z~2-2X,Z~3∗=Z~3-2X.In this case, the state-feedback gain is given by(23)K=YX-1.Proof.
Define(24)WB=BTP000000000BTZ1BTZ2BTZ3TWK=0K000000000000,Γ~=Γ1~Γ2B~TZ1B~TZ2B~TZ3∗Γ3000∗∗-1TM-TmZ100∗∗∗-1TMZ20∗∗∗∗-1TMZ3,Γ1~=Γ11Γ~12Γ13Γ14Γ15∗Γ22Γ23Γ24Γ25∗∗Γ33Γ34Γ35∗∗∗Γ44Γ45∗∗∗∗Γ55,Γ~12=PBK,B~=ABK000,From (13), we have(25)Γ=Γ~+WKTΔfWB+WBTΔfWK+ΔgWKTWB+WBTWKΔg+ΔgWKTΔfWB+WBTΔfWKΔg≤Γ~+ε1WKTWK+1ε1Δf2WBTWB+ε2WKTWK+1ε2Δg2WBTWB+ε3Δg2WKTWK+1ε3Δf2WBTWB≤Γ~+ε1+ε2+ε3δg2WKTWK+1ε1δf2+1ε2δg2+1ε3δf2WBTWB,(26)Γ~GBTGKT∗-1γ1I0∗∗-1εbI<0.It can be shown that if (13) holds, there exists Γ<0, that is, V˙(t)<0 for all sk+τk<t<sk+1+τk+1.
DefineJ=diag{J1,J2,J3}, J1=diag{P-1,P-1,P-1,P-1,P-1,P-1,P-1,P-1,P-1,P-1,}, J2=diag{Z1-1,Z2-1,Z3-1}, and J3=diag{I,I}.
Pre- and postmultiplying both sides of (26) by J and denoting X=P-1, N~l=XNlX, T~l=XTlX, M~l=XMlX, E~l=XElX(l=1-5), Q~i=XQiX  (i=1,2,3), Z~j=XZjX  (j=1,2,3), and Y=KX. Then, we have(27)Ξ~B~Y~∗-1γ1I0∗∗-1γ2I<0,where(28)Ξ~=Ξ1Ξ2ALALAL∗Ξ3000∗∗-1σ¯Z1-100∗∗∗-1σZ2-10∗∗∗∗-1σZ3-1,
The conditions in above inequality (27) are not LMI because of the nonlinear terms Zj-1. It is noted that Zj-1=XZ~j-1X  (j=1,2,3). In order to solve this nonconvex problem, the inequalities in following are needed.(29)Z~m-XZ~m-1Z~m-X≥0m=1,2,3,it is equivalent to (30)-XZ~j-1X≤Z~j-2Xj=1,2,3.Then, we can obtain (21). This completes the proof.Remark 6.
By using inequalities (30), the nonlinear terms in (26) are replaced with Z~j-2X(j=1,2,3), which makes the controller design conditions for the sampled-data systems (12) to be presented in LMI form. From Theorem 5, we can easily obtain the stabilization controller design conditions for the sampled-data systems (12).


## 3.1. Stability Analysis for the Sampled-Data Control Systems with Multiple Deadband Channels

Theorem 3.
For given scalarsTm, TM, δ1>0, δ2>0, 0≤α<1, and a matrix K, the closed-loop sampled-data system (12) is stable if there exist matrices P=PT>0, Qm=QmT>0  (m=1,2,3), Zj=ZjT>0  (j=1,2,3), and Ni, Ti, Mi, Ei, Li  (i=1-5) such that(13)Γ=Γ1Γ2BTZ1BTZ2BTZ3∗Γ3000∗∗-1TM-TmZ100∗∗∗-1TMZ20∗∗∗∗-1TMZ3<0,where(14)Γ1=Γ11Γ12Γ13Γ14Γ15∗Γ22Γ23Γ24Γ25∗∗Γ33Γ34Γ35∗∗∗Γ44Γ45∗∗∗∗Γ55,Γ2=N1T1M1E1L1N2T2M2E2L2N3T3M3E3L3N4T4M4E4L4N5T5M5E5L5,Γ3=diag-1αTMZ1-11-αTMZ1-1TM-TmZ1+Z2-1TMZ2-1TMZ3,Γ11=PA+ATP+∑i=13Qi+N1+N1T+L1+L1T,Γ12=PBK+ΔK+N2T-T1+M1-E1+L2T,Γ13=E1+N3T+L3T,Γ14=-M1+N4T+L4T,Γ15=T1-N1+N5T+L5T,Γ22=M2+M2T-T2-T2T-E2-E2T,Γ23=E2+M3T-T3T-E3T,Γ24=-M2+M4T-T4T-E4T,Γ25=T2+N2,Γ33=-Q1+E3+E3T,Γ34=-M3+E4T-L3,Γ35=T3-N3+E5T,Γ44=-Q2-M4-M4T,Γ45=T4-N4-M5T-L5T,Γ55=-1-αQ3+T5-N5+T5T-N5T,B=ABK+ΔK000,Proof.
Consider the following Lyapunov-Krasovskii functional(15)Vt=V1t+V2t+V3t+V4t+V5t+V6t+V7t,where(16)V1t=xTtPxt,V2t=∫t-TmtxsTQ1xsds,V3t=∫t-TMtxsTQ2xsds,V4t=∫t-αdttxsTQ3xsds,V5t=∫-TM0∫t+βtx˙TsZ1x˙sdsdβ,V6t=∫-TM-Tm∫t+βtx˙TsZ2x˙sdsdβ,V7t=∫-TM0∫t+βtx˙TsZ3x˙sdsdβ,P=PT>0, Qm=QmT>0  m=1,2,3, and Zj=ZjT>0 (j=1, 2, 3).
( 1 ) For tk<t<tk+1, calculating the derivative of V(t) with respect to t along the solutions of the system (12) and using Leibniz-Newton formula, it yields that(17)
V
˙
t
=
2
x
T
t
P
x
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t
+
∑
i
=
1
3
x
T
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Q
i
x
t
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-
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m
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1
x
t
-
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m
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T
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x
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1
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-
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+
x
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t
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1
+
h
12
Z
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+
T
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3
x
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t
-
∫
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1
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x
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-
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≤
2
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+
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+
Δ
K
x
t
-
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t
+
∑
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=
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x
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+
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+
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∫
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-
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3
x
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+
2
ζ
T
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x
t
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x
t
-
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-
∫
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+
2
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-
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t
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+
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ζ
T
t
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x
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t
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t
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m
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+
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T
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x
t
-
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t
-
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-
∫
t
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x
˙
s
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s
≤
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T
t
Γ
1
+
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T
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1
-
1
N
T
+
1
-
α
T
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T
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1
-
1
T
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+
h
12
M
Z
1
+
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2
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+
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12
E
Z
2
-
1
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+
T
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3
-
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L
T
+
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T
T
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Z
1
+
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M
-
T
m
Z
2
+
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M
Z
3
B
ζ
t
-
∫
t
-
α
d
t
t
H
1
Z
1
-
1
H
1
T
d
s
-
∫
t
-
d
t
t
-
α
d
t
H
2
Z
1
-
1
H
2
T
d
s
-
∫
t
-
T
M
t
-
d
t
H
3
Z
1
+
Z
2
-
1
H
3
T
d
s
-
∫
t
-
d
t
t
-
T
m
H
4
Z
2
-
1
H
4
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d
s
-
∫
t
-
T
M
t
H
5
Z
3
-
1
H
5
T
d
s
,where(18)H1=ζTtN+x˙TsZ1,H2=ζTtT+x˙TsZ1,H3=ζTtM+x˙TsZ1+Z2,H4=ζTtE+x˙TsZ2,H5=ζTtL+x˙TsZ3,ζt=xTtxTt-dtxTt-TmxTt-TMxTt-αdtT,N=N1TN2TN3TN4TN5TT,T=T1TT2TT3TT4TT5TT,M=M1TM2TM3TM4TM5TT,E=E1TE2TE3TE4TE5TT,L=L1TL2TL3TL4TL5TT.By the Schur complements, combine (13) to obtain V˙(t)<0 for all tk<t<tk+1.
( 2 ) It is noting that d(tk)=tk-sk, ∀k∈N, d(tk-)=tk-sk-1, ∀k∈N.
The value ofx before and after tk points remains unchanged (since x(t) is continuous). Then, we have Vi(tk-)=Vi(tk)  (i=1,2,3,5,6,7) in Lyapunov-Krasovskii functional (15). Moreover, for V4(t), there exists V4(tk-)≥V4(tk). Thus, V(tk-)≥V(tk) for k=0,1,2,3,⋯.
Fort∈tk,tk+1, we have(19)Vt-Vtk≤0. Since limk→∞tk=∞, we have ⋃k=0∞tktk+1=t0,∞. It follows that(20)Vt-Vt0≤0. It shows that V(t) decreases. This completes the proof.Remark 4.
Theorem3 presents a stability analysis condition of sampled-data systems under deadband-triggered communication scheme, which describes the relations of the deadband parameters and sampling intervals. The given condition is in terms of Linear Matrix Inequality (LMI) when the controller gain K is given.


## 3.2. Stabilization for the Sampled-Data Control Systems with Multiple Deadband Channels

This section is devoted to solve the problem of stabilization controller design for sampled-data systems (12). The following theorem presents the conditions for existence of the desired controller.Theorem 5.
Consider the sampled-data system in Figure1. For given scalars Tm, TM, δ1>0, δ2>0, 0≤α<1, and 0≤εi<1(i=1-3) the closed-loop system (12) is stable if there exist matrices X=XT>0, Q~i=Q~iT>0(i=1,2,3), Z~j=Z~jT>0(j=1,2,3), N~l, T~l, M~l, E~l, L~l(l=1-5), and Y of appropriate dimensions, such that(21)ΞB~Y~∗-1γ1I0∗∗-1γ2I<0,where(22)Ξ=Ξ1Ξ2ALALAL∗Ξ3000∗∗1σ¯Z~1∗00∗∗∗1σZ~2∗0∗∗∗∗1σZ~3∗,Ξ1=Ξ11Ξ12Ξ13Ξ14Ξ15∗Ξ22Ξ23Ξ24Ξ25∗∗Ξ33Ξ34Ξ35∗∗∗Ξ44Ξ45∗∗∗∗Ξ55,Ξ2=N~1T~1M~1E~1N~2T~2M~2E~2N~3T~3M~3E~3N~4T~4M~4E~4N~5T~5M~5E~5,Ξ11=AX+XAT+∑i=13Q~i+N~1+N~1T+L~1+L~1T,Ξ12=BY+N~2T-T~1+M~1-E~1+L~2T,Ξ13=E~1+N~3T+L~3T,Ξ14=-M~1+N~4T+L~4T,Ξ15=T~1-N~1+N~5T+N~5T,Ξ22=M~2+M~2T-T~2-T~2T-E~2-E~2T,Ξ23=E~2+M~3T-T~3T-E~3T,Ξ24=-M~2-L~2+M~4T-T~4T-E~4T,Ξ25=T~2-N~2Ξ33=-Q~1+E~3+E~3T,Ξ34=-M~3+E~4T-L~3,Ξ35=T~3-N~3+E~5T,,Ξ44=-Q~2-M~4-M~4T,Ξ45=T~4-N~4-M~5T,Ξ55=-1-αQ3+T~5-N~5+T~5T-N~5T,AL=AXTBY000T,Ξ3=diag-1αTMZ~1-11-αTMZ~1-1TM-TmZ~1+Z~2-1TM-TmZ~2-1TMZ~3,B~=BT000000000BTBTBTT,Y~=0Y00000000000T.γ2=ε1+ε2+ε3δ12,γ1=1ε1δ22+1ε2δ12+1ε3δ22,Z~1∗=Z~1-2X,Z~2∗=Z~2-2X,Z~3∗=Z~3-2X.In this case, the state-feedback gain is given by(23)K=YX-1.Proof.
Define(24)WB=BTP000000000BTZ1BTZ2BTZ3TWK=0K000000000000,Γ~=Γ1~Γ2B~TZ1B~TZ2B~TZ3∗Γ3000∗∗-1TM-TmZ100∗∗∗-1TMZ20∗∗∗∗-1TMZ3,Γ1~=Γ11Γ~12Γ13Γ14Γ15∗Γ22Γ23Γ24Γ25∗∗Γ33Γ34Γ35∗∗∗Γ44Γ45∗∗∗∗Γ55,Γ~12=PBK,B~=ABK000,From (13), we have(25)Γ=Γ~+WKTΔfWB+WBTΔfWK+ΔgWKTWB+WBTWKΔg+ΔgWKTΔfWB+WBTΔfWKΔg≤Γ~+ε1WKTWK+1ε1Δf2WBTWB+ε2WKTWK+1ε2Δg2WBTWB+ε3Δg2WKTWK+1ε3Δf2WBTWB≤Γ~+ε1+ε2+ε3δg2WKTWK+1ε1δf2+1ε2δg2+1ε3δf2WBTWB,(26)Γ~GBTGKT∗-1γ1I0∗∗-1εbI<0.It can be shown that if (13) holds, there exists Γ<0, that is, V˙(t)<0 for all sk+τk<t<sk+1+τk+1.
DefineJ=diag{J1,J2,J3}, J1=diag{P-1,P-1,P-1,P-1,P-1,P-1,P-1,P-1,P-1,P-1,}, J2=diag{Z1-1,Z2-1,Z3-1}, and J3=diag{I,I}.
Pre- and postmultiplying both sides of (26) by J and denoting X=P-1, N~l=XNlX, T~l=XTlX, M~l=XMlX, E~l=XElX(l=1-5), Q~i=XQiX  (i=1,2,3), Z~j=XZjX  (j=1,2,3), and Y=KX. Then, we have(27)Ξ~B~Y~∗-1γ1I0∗∗-1γ2I<0,where(28)Ξ~=Ξ1Ξ2ALALAL∗Ξ3000∗∗-1σ¯Z1-100∗∗∗-1σZ2-10∗∗∗∗-1σZ3-1,
The conditions in above inequality (27) are not LMI because of the nonlinear terms Zj-1. It is noted that Zj-1=XZ~j-1X  (j=1,2,3). In order to solve this nonconvex problem, the inequalities in following are needed.(29)Z~m-XZ~m-1Z~m-X≥0m=1,2,3,it is equivalent to (30)-XZ~j-1X≤Z~j-2Xj=1,2,3.Then, we can obtain (21). This completes the proof.Remark 6.
By using inequalities (30), the nonlinear terms in (26) are replaced with Z~j-2X(j=1,2,3), which makes the controller design conditions for the sampled-data systems (12) to be presented in LMI form. From Theorem 5, we can easily obtain the stabilization controller design conditions for the sampled-data systems (12).


## 4. Numerical Examples

Example 1.
Consider the following system [22]:(31)x˙t=-2-0.1-0.10.01xt+0.050.02utChoosing Tm=0.10s, TM=0.20s, δ1=0.08, and δ2=0.05. By using Theorem 5, solving the LMI problem in (21), one can obtain the following solution: (32)X=10.4216-0.0232-0.023210.8572,Y=11.5574-8.9500Hence, the state-feedback controller gain K=1.1072-0.8220. Figure 2 shows the state response and control input under deadband-triggered scheme, where the initial state of the NCS is x0=-0.10.1T. It can been seen that the closed-loop NCS is asymptotically stable with above obtained control gain. It shows the effectiveness of stabilization controller design method proposed in this paper.
Figure3 shows the transmission of the state signal in two channels δ1=0.08. In channel 1, for the output signals from deadband processor deadband1sc, the previous value is x1′(tk) and the most recent value is x1′(tk+1). In channel 2, for the output signals from deadband processor deadband2sc, the previous value is x2′(tk) and the most recent value is x2′(tk+1). In Figure 3, curve output shows the difference between most recent value and the previous value of the output signals from deadband processor deadband1sc and deadband2sc. When the curve output is not zero, that is, xi′(tk+1)≠xi′(tk)(i=1,2), it shows that the state signals are transmitting in the channel 1 and in the channel 2. From Figure 3, one can see that the quantity of data transmission can be reduced and be effectively controlled by adopting the deadband-triggered scheme. Figure 4 shows the transmission of control input at deadband processor deadband1ca for δ2=0.05. It can be seen that the quantity of data transmission can be reduced largely to avoid the heavy of the channels. These have shown that the proposed control approach based on deadband-triggered scheme can guarantee the system stable and effectively reduce the data transmission in the channels.Figure 2
State response and control input of the sampled-data system under deadband-triggered scheme.Figure 3
Transmission of state signals under deadband processor 1 forδ1=0.08,δ2=0.05.Figure 4
Transmission of control input signals under deadband processor 2 forδ1=0.08,δ2=0.05.


## 5. Conclusions

This paper investigates the control problem for a sampled-data system with multiple channels, whereby the transmitting of each signal detected by the sensor is depended on the deadband-triggered scheme. By using Lyapunov functional approach and Leibniz-Newton formula, the stability analysis and stabilization methods for the sampled-data system with multiple channels based on deadband-triggered scheme are proposed. Moreover, the proposed stability analysis and stabilization controller design conditions can be presented in terms of linear matrix inequalities (LMIs). A numerical example is used to demonstrate the effectiveness of the proposed methods. From the numerical results, it can be seen that the given method in this paper can guarantee the closed sampled-data system stable, and the proposed deadband-triggered scheme can significantly reduce the number of data transmitting in the channels.



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*Source: 8165383-2018-06-24.xml*