# DistributedH∞ Consensus Control of Networked Control Systems with Time-Triggered Protocol

Authors: Ronghua Xie; Weihua Fan; Qingwei Chen
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The analysis and design of traditional networked control systems focused on single closed-loop scenario. This paper introduces a distributed control approach for the networked control systems (NCSs) with multiple subsystems based on a time-triggered network protocol. Firstly, some basic ideas of the time-triggered protocol are introduced and a time schedule scheme is employed for the NCS. Then, a novel model is proposed to the NCS regarding the network-induced delay. The resulting closed-loop system is time-delay linear system considering a distributed control law. A sufficient condition toH∞ consensus control is present based on the Lyapunov-Krasovskii function. Also, the controller design approach towards the given H∞ performance index is given by a cone complement linearization and iterative algorithm. Finally, numerical examples are given to validate the approach.


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## Body

## 1. Introduction

Networked control systems (NCSs) are a class of closed-loop control systems in which sensors, controllers, and actuators are connected over network (see [1]). In recent years, NCSs have received increasing attention due to the broad application in industrial areas. The induced network brings about many advantages, such as low installation and maintenance costs, high reliability, and increased system flexibility. But, simultaneously, network-induced imperfections, such as time delays, packet losses and disorder, time-varying packet transmission/sampling intervals, and competition of multiple nodes accessing network, will decrease the performance of NCSs (see [2]). More seriously, some imperfections may cause instability. During the past decades, many researchers have studied the NCSs, and various methodologies have been proposed on the modeling (see [3, 4]), scheduling (see [5, 6]), analysis, and control design (see [7–11]).When the plant is multiple-input-multiple-output (MIMO), the NCS is called MIMO NCSs. Because the nodes should compete to access not only outside nodes, but also other nodes inside, the research of MIMO NCSs is a more challenging job when compared with the so-called single-input-single-output (SISO) NCSs. Yan et al. [12] presented a continuous time model of MIMO NCSs with distributed time delays and uncertainties and gave delay-dependent stability criteria in terms of linear matrix inequalities (LMIs). Xia et al. [13] presented a discrete-time model of MIMO NCSs with multiple time-varying delays, and the design of output feedback controllers is proposed in terms of matrix inequalities, together with an iterative algorithm. Okajima et al. [14] proposed a design method for feedback-type dynamic quantization in a MIMO NCS, which is extended from SISO NCSs. Guan et al. [15] studied the optimal tracking performance for MIMO LTI discrete-time control systems with communication constraints in feedback path and how the bandwidth and AWGN of the communication channel affected the tracking capability. Jiang et al. [16] studied the optimal tracking performance of MIMO NCSs with AWGN channel between the controller and the plant and concluded that the optimal tracking performance was closely dependent on nonminimum phase zeros, unstable poles of the plant, and characteristics of the signals and channel. Cao et al. [17] presented delay dependant stability criteria for MIMO NCSs with nonlinear perturbation and delay, which gave much less conservative maximum allowable delay bound. Li et al. [18] modeled the MIMO NCS with multichannel packet disordering, packet dropout, and bounded time-varying transmission delay, as a jump linear system subject to Markovian chains, and a real-time controller was proposed such that the cost function value is lower than a specified upper bound. Du et al. [19] modeled MIMO NCSs as unknown switched sequence and proposed a sufficient condition to be asymptotically stable in terms of a set of bilinear matrix inequalities. In the above reference, only one controller node is in the NCS, which is not a good choice for many real applications.Actually, the distributed control using multiple controller nodes is a more interesting topic on the MIMO NCSs. But, until now, there are few articles published. Hirche et al. [20] introduced a novel distributed controller approach for NCS to achieve finite gain L2 stability independent of constant time delay, which consisted two parts. One was a local controller designed without network; the other was a remote part to compensate the network-induced delay to keep stability. But, actually, here the controller was divided into two separated parts, which was not so-called distributed control. In this paper, we consider a class of MIMO networked control systems with multiple subsystems, multiple sensor nodes, controller nodes, and actuator nodes, whose subsystems exchange information through network. Figure 1 gives the common structure of the NCSs. It is clear that the NCS is a distributed system over the network channel. This kind of system can be easily found, such as Baggage Handling Systems (BHS, as in Figure 2), product line systems, and Multijoint Robots. For these systems, all subsystems are essentially required to be stable. Furthermore, there are usually some special requirements. Take BHS as an example; one BHS often has a few branches, which contain dozens of motors. While transplanting baggage, the velocity of motors in each branch should be consistent. Otherwise, the baggage may collide because of different velocity. For more details, the consensus control in NCSs means consensus not only in steady state, but also in transient response; that is, when the reference signal changes or disturbance occurs, the output signal of the subsystems is required to change simultaneously.Figure 1
Structure of the networked control systems.Figure 2
The Baggage Handling System.The consensus control of NCSs is an interesting problem and is different from that in multivehicle cooperative control, because the dynamic of each subsystem is different from the others. Some effective conclusions in multivehicle cooperative control cannot be used directly in this situation. Conditions to consensus control need to be investigated.In this paper, we focus on the modeling and consensus control of the NCSs with time-triggered protocol and distributed control law. The contributions are as follows:(i)
Firstly, the time-triggered protocol is introduced and employed to the NCSs; a scheduling scenario which reduces the network-induced delay within each subsystem is introduced.(ii)
Secondly, a model for the NCSs with time-triggered protocol and short time-varying network-induced delays is proposed, while the distributed controllers which use the feedback information from the subsystem neighbors are used.(iii)
Thirdly, the sufficient conditions for asymptotical stability andH∞ consensus control of the NCSs are obtained by Lyapunov-Krasovskii function. The conditions guarantee all subsystems reach consensus while satisfying the desired H∞ performance on the fixed time-triggered protocol. Also, an iterative algorithm is given for distributed controller gain matrix.The rest of this paper is organized as follows. Section2 introduces the protocol of the NCSs, the feature of the network-induced delay, and the mathematical model. Section 3 deals with H∞ consensus control problem for NCSs. Some numerical examples are given in Section 4 to demonstrate the effectiveness of the proposed design technique. The conclusion is provided in Section 5.


## 2. Modeling of the NCSs

### 2.1. Protocol of the Network

Consider the NCSs with multiple subsystems in Figure1. The sensor nodes and the controller nodes will access the network after sampling or calculation, respectively. As we know, their authorities to access network depend on the protocol. In addition, the features of network-induced delay and data loss also depend on the protocol.Network protocols can be classified into time-triggered protocol or event-triggered protocol. Time-triggered protocol allows the node to access network in certain time slot, such as Ether-CAT, FlexRay, and Time-Triggered CAN, while the nodes in event-triggered protocol access the network whenever they are ready for transmission, such as TCP/IP, CAN. Most communication networks adopt event-triggered protocol, because the protocol is efficient while nodes join and quit frequently. But the situation is different in control systems. Few nodes in control systems will join or exit frequently while working, unless the node is down or crashed. Moreover, event-triggered protocol brings about many uncertainties to the system, because of the random access. In [21], it is concluded that, compared with event-triggered protocol, the time-triggered protocol brought about more convenience to design and analysis. So we discuss the NCSs based on the time-triggered protocol. In order to model the NCSs, we introduce some important features of time-triggered protocol.Usually, a basic cycle exists in time-triggered protocol network, which means that a basic period for all important nodes has at least one chance to transfer data. For example, a basic cycle of TTCAN [22] is shown in Figure 3. A basic cycle begins with a reference message, which is sent by a special node and can be identified by all participants. A basic cycle usually consists of several time windows (or slots) of different length and offers the necessary time for the message to be transmitted. The exclusive window is a time slot for periodic messages, while the arbitrating window is for aperiodic messages. Free window is reserved for further extensions. An exclusive window allows only one node to send a frame. In the arbitrating windows, these nodes that need to send frames are allowed to compete for network access as in event-triggered protocol. The end of an arbitrating window is always predictable. Thus, the advantages of event-triggered communication can be combined with those of time-triggered communication.Figure 3
Basic cycle of TTCAN.Of course, the sequence of these windows in a basic cycle can be designed according to scheduling strategy. For example, the sequence can be designed as in Figure4 to reduce network-induced delay within each subsystem when it is used for NCS with four subsystems.Figure 4
Basic cycle of NCS with four subsystems based on TTCAN.Furthermore, the reference message also gives some important information, including a global time stamp, where participants can achieve a synchronization accuracy of 1μs. That means the time jitter between all nodes can be negligible, unless the main time constant of the system is shorter than microseconds.


### 2.2. Modeling of the NCSs

According to the facts in Section2.1, we can give the following reasonable assumptions.Assumption 1.
The sensor nodes and controller nodes are all time triggered.Remark 2.
According to the time-triggered protocol, the intelligent nodes access the network in appointed time slots. So it is reasonable to set sensor nodes and controller nodes to be time triggered. And they should be idle at rest time to reduce power consumption. The actuator nodes are either time triggered or event triggered, because the data packets from controller nodes arrive at almost the same time in each basic cycle.Remark 3.
For time-triggered protocol, little conflictions occur during transmission. It is reasonable to assume little data loss. So, we do not consider data loss in this paper.By Assumption1, the network-induced time delay τi=τi,sc+τi,ca is constant, where τi,sc is time delay between the sensor node and controller node of the ith subsystem and τi,ca is time delay between the controller node and actuator node. Also, we have τi<T, where T is the basic cycle time.Suppose that the plant of any subsystem is LTI and is described as space state equation:(1)x˙it=Apixit+Bpiu^it+ωit,yit=Cpixit,where xi(t)∈Rn is the state vector of the ith plant, i=1,…,N. u^i(t)∈Rm is the control input vector, yi(t)∈Rr is the output vector, ωi(t)∈Rq is the external disturbance, and Api∈Rn×n, Bpi∈Rn×m, and Cpi∈Rr×n are known real constant matrices.We usex^i to denote the sampled data in the receiver of the ith controller node; ui denotes the control variable calculated by the ith controller node.By (1), the NCS can be described in discrete time as(2)xitk+1=Aixitk+Bi1u^itk+Bi2u^itk+τi+ωitk,yitk=Cixitk,where Ai=eApiT, Bi1=∫0τieApisdsBpi, and Bi2=∫τiTeApisdsBpi. Ai, Bij, i=1,…,N, j=1,2, are known matrices for τi is constant.Since NCSs are usually large scale, it is not advisable to employ centralized control. In this paper, we employ a distributed control law as (3) for the NCS:(3)ui=Ki∑j∈Niaijx^i-x^j,where Ki∈Rm×n is gain of ith controller node. The control law (3) means each subsystem controller uses both its own feedback and also datum from its neighbors.The data flow of theith subsystem is shown in Figure 5; tk+τi,c is the moment the controller node calculates control variable. So the plant input is(4)
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,where tk,tk+1 are the kth and (k+1)th sampling time, tk+1-tk=T. Suppose the cycle time in Figure 4 is used in the NCS; when the ith subsystem calculates the control variable, (k+1)th,…,Nth subsystems have not sent their data packets. That means the neighbor of the ith subsystem is Ni=vl,vj∈E,j<i.Figure 5
Data flow ofith subsystem.Using (2) and (4), the closed-loop system can be described as(5)xitk+1=Ai+Bi2Ki∑j∈Niaijxitk+Bi1Ki∑j∈Niaijxitk-1-Bi2Ki∑j∈Niaijxjtk-Bi1Ki∑j∈Niaijxjtk-1+ωik.Considering that the consensus control is also an objective, we define a novel controlled output as follows:(6)zitk=xi+1tk-x1tk,i=2,…,N.Letξ(k)=x1T(tk),…,xNT(tk)T∈RNn, z(k)=z1T(tk),…,zNT(tk)T; we have the following model by (5):(7)ξk+1=A-+B-2K-Lξk+B-1K-Lξk-1+ωk,zk=Hξk,where A-=diag(A1,…,AN), B-i=diag(B1i,…,BNi), i=1,2, H=-II0⋮⋱-I0I, K-=diag(K1,…,KN), and L=lij is the Laplacian matrix, lij=-aij,j≠i;∑j∈Niaij,j=i.From the above, the objective is to design distributed controller (3) such that(1)
the control law asymptotically solves the consensus problem; that is, the states of subsystems satisfy(8)limk→∞⁡xik-xjk=0;(2)
the closed-loop system satisfies the following dissipation inequality:(9)zk22ωk22=∑k=0∞zTkzk∑k=0∞ωTkωk<γ2,ωk∈L20,∞.


## 2.1. Protocol of the Network

Consider the NCSs with multiple subsystems in Figure1. The sensor nodes and the controller nodes will access the network after sampling or calculation, respectively. As we know, their authorities to access network depend on the protocol. In addition, the features of network-induced delay and data loss also depend on the protocol.Network protocols can be classified into time-triggered protocol or event-triggered protocol. Time-triggered protocol allows the node to access network in certain time slot, such as Ether-CAT, FlexRay, and Time-Triggered CAN, while the nodes in event-triggered protocol access the network whenever they are ready for transmission, such as TCP/IP, CAN. Most communication networks adopt event-triggered protocol, because the protocol is efficient while nodes join and quit frequently. But the situation is different in control systems. Few nodes in control systems will join or exit frequently while working, unless the node is down or crashed. Moreover, event-triggered protocol brings about many uncertainties to the system, because of the random access. In [21], it is concluded that, compared with event-triggered protocol, the time-triggered protocol brought about more convenience to design and analysis. So we discuss the NCSs based on the time-triggered protocol. In order to model the NCSs, we introduce some important features of time-triggered protocol.Usually, a basic cycle exists in time-triggered protocol network, which means that a basic period for all important nodes has at least one chance to transfer data. For example, a basic cycle of TTCAN [22] is shown in Figure 3. A basic cycle begins with a reference message, which is sent by a special node and can be identified by all participants. A basic cycle usually consists of several time windows (or slots) of different length and offers the necessary time for the message to be transmitted. The exclusive window is a time slot for periodic messages, while the arbitrating window is for aperiodic messages. Free window is reserved for further extensions. An exclusive window allows only one node to send a frame. In the arbitrating windows, these nodes that need to send frames are allowed to compete for network access as in event-triggered protocol. The end of an arbitrating window is always predictable. Thus, the advantages of event-triggered communication can be combined with those of time-triggered communication.Figure 3
Basic cycle of TTCAN.Of course, the sequence of these windows in a basic cycle can be designed according to scheduling strategy. For example, the sequence can be designed as in Figure4 to reduce network-induced delay within each subsystem when it is used for NCS with four subsystems.Figure 4
Basic cycle of NCS with four subsystems based on TTCAN.Furthermore, the reference message also gives some important information, including a global time stamp, where participants can achieve a synchronization accuracy of 1μs. That means the time jitter between all nodes can be negligible, unless the main time constant of the system is shorter than microseconds.


## 2.2. Modeling of the NCSs

According to the facts in Section2.1, we can give the following reasonable assumptions.Assumption 1.
The sensor nodes and controller nodes are all time triggered.Remark 2.
According to the time-triggered protocol, the intelligent nodes access the network in appointed time slots. So it is reasonable to set sensor nodes and controller nodes to be time triggered. And they should be idle at rest time to reduce power consumption. The actuator nodes are either time triggered or event triggered, because the data packets from controller nodes arrive at almost the same time in each basic cycle.Remark 3.
For time-triggered protocol, little conflictions occur during transmission. It is reasonable to assume little data loss. So, we do not consider data loss in this paper.By Assumption1, the network-induced time delay τi=τi,sc+τi,ca is constant, where τi,sc is time delay between the sensor node and controller node of the ith subsystem and τi,ca is time delay between the controller node and actuator node. Also, we have τi<T, where T is the basic cycle time.Suppose that the plant of any subsystem is LTI and is described as space state equation:(1)x˙it=Apixit+Bpiu^it+ωit,yit=Cpixit,where xi(t)∈Rn is the state vector of the ith plant, i=1,…,N. u^i(t)∈Rm is the control input vector, yi(t)∈Rr is the output vector, ωi(t)∈Rq is the external disturbance, and Api∈Rn×n, Bpi∈Rn×m, and Cpi∈Rr×n are known real constant matrices.We usex^i to denote the sampled data in the receiver of the ith controller node; ui denotes the control variable calculated by the ith controller node.By (1), the NCS can be described in discrete time as(2)xitk+1=Aixitk+Bi1u^itk+Bi2u^itk+τi+ωitk,yitk=Cixitk,where Ai=eApiT, Bi1=∫0τieApisdsBpi, and Bi2=∫τiTeApisdsBpi. Ai, Bij, i=1,…,N, j=1,2, are known matrices for τi is constant.Since NCSs are usually large scale, it is not advisable to employ centralized control. In this paper, we employ a distributed control law as (3) for the NCS:(3)ui=Ki∑j∈Niaijx^i-x^j,where Ki∈Rm×n is gain of ith controller node. The control law (3) means each subsystem controller uses both its own feedback and also datum from its neighbors.The data flow of theith subsystem is shown in Figure 5; tk+τi,c is the moment the controller node calculates control variable. So the plant input is(4)
u
^
i
t
=
K
i
∑
j
∈
N
i
a
i
j
x
i
t
k
-
x
j
t
k
,
t
∈
t
k
+
τ
i
,
t
k
+
1
u
^
i
t
k
,
t
∈
t
k
,
t
k
+
τ
i
,where tk,tk+1 are the kth and (k+1)th sampling time, tk+1-tk=T. Suppose the cycle time in Figure 4 is used in the NCS; when the ith subsystem calculates the control variable, (k+1)th,…,Nth subsystems have not sent their data packets. That means the neighbor of the ith subsystem is Ni=vl,vj∈E,j<i.Figure 5
Data flow ofith subsystem.Using (2) and (4), the closed-loop system can be described as(5)xitk+1=Ai+Bi2Ki∑j∈Niaijxitk+Bi1Ki∑j∈Niaijxitk-1-Bi2Ki∑j∈Niaijxjtk-Bi1Ki∑j∈Niaijxjtk-1+ωik.Considering that the consensus control is also an objective, we define a novel controlled output as follows:(6)zitk=xi+1tk-x1tk,i=2,…,N.Letξ(k)=x1T(tk),…,xNT(tk)T∈RNn, z(k)=z1T(tk),…,zNT(tk)T; we have the following model by (5):(7)ξk+1=A-+B-2K-Lξk+B-1K-Lξk-1+ωk,zk=Hξk,where A-=diag(A1,…,AN), B-i=diag(B1i,…,BNi), i=1,2, H=-II0⋮⋱-I0I, K-=diag(K1,…,KN), and L=lij is the Laplacian matrix, lij=-aij,j≠i;∑j∈Niaij,j=i.From the above, the objective is to design distributed controller (3) such that(1)
the control law asymptotically solves the consensus problem; that is, the states of subsystems satisfy(8)limk→∞⁡xik-xjk=0;(2)
the closed-loop system satisfies the following dissipation inequality:(9)zk22ωk22=∑k=0∞zTkzk∑k=0∞ωTkωk<γ2,ωk∈L20,∞.


## 3. Main Results

Firstly, some useful lemmas are given.Lemma 4 (see [23]).
Assumex(k)∈Rn; then, for any matrices X>0, M1, M2∈Rn×n and a scalar function h∶=h(k)≥0, the following inequality holds:(10)-∑i=k-hk-1ηTiXηi≤ζTkΛ+hM1TM2TX-1M1M2ζk,where Λ=M1T+M1-M1T+M2∗-M2T-M2, ζ(k)=x(k)x(k-h), and η(i)=x(i+1)-x(i).Using Lemma4, we have the following lemma.Lemma 5.
Assumeω(k)=0, if there exist symmetric positive definite matrices P, Q, and Z and matrices M1, M2, and K- such that(11)-P+Q+M1T+M1-M1T+M2M1TA-+B-2K-L-ITA-+B-2K-LT∗-Q-M2T-M2M2TB-1K-LTB-1K-LT∗∗-Z00∗∗∗-Z-10∗∗∗∗-P-1<0.
The NCS (7) is asymptotically stable.Proof.
For system (7), define y(k)=ξ(k)-ξ(k-1). Choose Lyapunov-Krasovskii function as follows:(12)Vk=ξTkPξk+ξTk-1Qξk-1+yTkZyk.Then, the difference of V(k) is(13)ΔVk=Vk+1-Vk=ξTk+1Pξk+1+ξTkQξk+yTk+1Zyk+1-ξTkPξk-ξTk-1Qξk-1-yTkZyk=ξTkA-+B-2K-LTPA-+B-2K-Lξk+2ξTkA-+B-2K-LTPB-1K-Lξk-1+ξTk-1B-1K-LTPB-1K-Lξk-1+ξTkQξk+yTk+1Zyk+1-ξTkPξk-ξTk-1Qξk-1-yTkZyk.Aimed at the term yT(k+1)Zy(k+1), using (7),(14)yTk+1Zyk+1=ξk+1-ξkTZξk+1-ξk=ξTkA-+B-2K-L-ITZA-+B-2K-L-Iξk+2ξTkA-+B-2K-L-ITZB-1K-Lξk-1+ξTk-1B-1K-LTZB-1K-Lξk-1.By Lemma 4, the inequality holds:(15)-yTkZyk≤ξTkξTk-1Λ+M1TM1TZ-1M1M2ξkξk-1,where Λ=M1T+M1-M1T+M2∗-M2T-M2. Then, we have(16)ΔVk≤ξTkA-+B-2K-LTPA-+B-2K-L+A-+B-2K-L-ITZA-+B-2K-L-I-P+Qξk+2ξTkA-+B-2K-LTPB-1K-L+A-+B-2K-L-ITZB-1K-Lξk-1+ξTk-1B-1K-LTPB-1K-L+B-1K-LTZB-1K-L-Qξk-1+ξTkξTk-1Λ+M1TM1TZ-1M1M2ξkξk-1=ξTkξTk-1Φ+Λ+M1TM1TZ-1M1M2ξkξk-1,where Φ=ϕ11ϕ12ϕ12Tϕ22, and(17)ϕ11=A-+B-2K-LTPA-+B-2K-L+A-+B-2K-L-ITZA-+B-2K-L-I-P+Q,ϕ12=A-+B-2K-LTPB-1K-L+A-+B-2K-L-ITZB-1K-L,ϕ22=B-1K-LTPB-1K-L+B-1K-LTZB-1K-L-Q.Hence, (7) is stable if the following matrix inequality holds:(18)Φ+Λ+M1TM1TZ-1M1M2<0.By Schur complements, inequality (18) is equivalent to(19)ϕ11+M1T+M1ϕ12-M1T+M2M1T∗ϕ22-M2T-M2M2T∗∗-Z<0.That is equivalent to(20)A-+B-2K-LTPA-+B-2K-L-P+Q+M1T+M1A-+B-2K-LTPB-1K-L-M1T+M2M1T∗B-1K-LTPB-1K-L-Q-M2T-M2M2T∗∗-Z+A-+B-2K-L-ITB-1K-LT0ZA-+B-2K-L-IB-1K-L0<0.By Schur complements, we have(21)-P+Q+M1T+M1-M1T+M2M1TA-+B-2K-L-IT∗-Q-M2T-M2M2TB-1K-LT∗∗-Z0∗∗∗-Z-1+A-+B-2K-LTB-1K-LT00PA-+B-2K-LB-1K-L00<0.Then, we have inequality (11). This completes the proof.Theorem 6.
With the distributed control law (3), the NCS (7) achieves consensus with a given H∞ disturbance attenuation index γ, if there exist symmetric positive definite matrices P, Q, and Z and matrices M1, M2, and K-, such that(22)-P+Q+M1T+M1+HTH-M1T+M20M1TA-+B-2K-L-ITA-+B-2K-LT∗-Q-M2T-M20M2TB-1K-LTB-1K-LT∗∗-γ2I0II∗∗∗-Z00∗∗∗∗-Z-10∗∗∗∗∗-P-1<0.And if the matrix inequality is feasible, the feedback matrix of the consensus protocol is K-.Proof.
Let Lyapunov-Krasovskii function as (12). Using (7) and Lemmas 4 and 5, we have(23)ΔVk≤ξTkξTk-1ωTkΦ^ξkξk-1ωk,Φ^=ϕ11+M1T+M1+M1TZ-1M1ϕ12-M1T+M2+M2TZ-1M1ϕ13∗ϕ22-M2T-M2+M2TZ-1M2ϕ23∗∗P+Z,ϕ13=A-+B-2K-LTP+A-+B-2K-L-ITZ,ϕ23=B-1K-LTP+Z.Firstly, from condition (22), we have (11), so the NCS (7) is asymptotically stable. Then, we have(24)limk→∞⁡xik-xjk=0.Then, we find H∞ performance index.
For anyk>0, consider the following cost function:(25)J=∑k=0∞zTkzk-γ2ωTkωk.By the zero initial condition (V(0)=0), we have(26)J=∑k=0∞zTkzk-γ2ωTkωk+ΔVk-V∞+V0≤∑k=0∞zTkzk-γ2ωTkωk+ΔVk,for (27)zTkzk-γ2ωTkωk+ΔVk=ξTkξTk-1ωTkΘξkξk-1ωk,where(28)Θ=ϕ11+M1T+M1+M1TZ-1M1+HTHϕ12-M1T+M2+M2TZ-1M1ϕ13∗ϕ22-M2T-M2+M2TZ-1M2ϕ23∗∗P+Z-γ2I.According to Schur complements, condition (22) is equivalent to Θ<0. That is,(29)∑k=0∞zTkzk<γ2∑k=0∞ωTkωk,so the robust H∞ consensus control is achieved. This completes the proof.Note that condition (22) is nonconvex as it contains P, P-1, Z, and Z-1. Using cone complement linearization [24], we have the following corollary.Corollary 7.
With the distributed control law (3), the NCS (7) achieves consensus with a given H∞ disturbance attenuation index γ, if there exist symmetric positive definite matrices P, Q, Z, S, and T and matrices M1, M2, and K-, such that(30)min⁡P,Q,M1,M2,K-tr⁡ZS+PTs.t.-P+Q+M1T+M1+HTH-M1T+M20M1TA-+B-2K-L-ITA-+B-2K-LT∗-Q-M2T-M20M2TB-1K-LTB-1K-LT∗∗-γ2I0II∗∗∗-Z00∗∗∗∗-S0∗∗∗∗∗-T<0,(31)ZIIS≥0,(32)PIIT≥0.If the matrix inequality is feasible, then the feedback matrix of the consensus protocol is K-.The proof is omitted.In order to design distributed control law (3), we present an iterative algorithm as follows.Algorithm 8.
(1) For (30)–(32), find a feasible solution: P0, Q0, Z0, M10, M20, K-0, S0, and T0, and let h=0.
(2) SetPh+1=Ph, Zh+1=Zh, Sh+1=Sh, and Th+1=Th, and solve the following optimal problem:(33)min⁡tr⁡⁡ZhS+ShZ+PhT+ThPsubject  to30,31,32.
(3) If a stopping criterion given in advance is satisfied, the iteration ends.
Otherwise, go to Step (2).Remark 9.
A simple stopping criterion is thatPh, Zh provide a feasible solution to inequality (30). And using LMI toolbox, it is easily confirmed.


## 4. Numerical Examples

Example 1.
We use the DC motors of [25] as the plants of the NCS in Figure 1, where the transfer function of the DC motor is(34)Gs=2029.826s+26.29s+2.296.The DC motor model is rewritten in state space as (35)x˙t=01-60.3756-28.586xt+02029.826ut,yt=10xt.And, in this example, the basic cycle time is chosen as T=0.005 s, there are three DC motors connected by the Time-Triggered CAN, and the total network-induced delay of each closed loop is τ1=0.001 s, τ2=0.002 s, and τ3=0.003 s.
Then, we have the parameters in (7) as follows:(36)A-=0.99930.004658-0.28120.88610.99930.004658-0.28120.88610.99930.004658-0.28120.8861,B-1=0.0012.0010.00403.9460.00895.836,B-2=0.02317.4540.2025.5090.01533.6190,L=11-11-11-11-11.Here, we just consider the stability of the whole NCS; then using the matrix inequality in Lemma 5 together with Algorithm 8, the following feasible solution can be obtained by MATLAB LMI toolbox:(37)P=0.73310.0011-0.00220.0111-0.0001-0.00010.00110.87640.0037-0.01400.00010.0002-0.00220.00370.72520.0253-0.00170.00640.0111-0.01400.02530.81170.0024-0.0097-0.00010.0001-0.00170.00240.72390.0301-0.00010.00020.0064-0.00970.03010.7989,Q=0.45660.0114-0.01210.01420.0001-0.00070.01140.59850.0164-0.0211-0.00010.0010-0.01210.01640.44840.0223-0.01140.01340.0142-0.02110.02230.57720.0150-0.01870.0001-0.0001-0.01140.01500.45220.0184-0.00070.00100.0134-0.01870.01840.5796,Z=1.01640.00030.0002-0.00020.00000.00000.00030.9573-0.00030.0156-0.00000.00030.0002-0.00031.0174-0.00110.0000-0.0000-0.00020.0156-0.00110.9825-0.00010.00530.0000-0.00000.0000-0.00011.0178-0.00160.00000.0003-0.00000.0053-0.00160.9887,M1=-0.9176-0.0010-0.00350.00490.0000-0.0001-0.0001-0.86790.0047-0.01910.0000-0.00010.0036-0.0050-0.91930.0017-0.00430.0058-0.0051-0.00560.0019-0.89030.0057-0.0120-0.00000.00000.0054-0.0072-0.92050.00400.0001-0.0004-0.00720.00520.0029-0.8969,M2=0.8879-0.00050.0045-0.00600.00000.0001-0.00110.8353-0.00610.0200-0.00000.0001-0.00370.00500.8897-0.00320.0055-0.00710.00530.0047-0.00330.8570-0.00720.01340.00000.0000-0.00570.00750.8899-0.0041-0.00010.00030.0078-0.0060-0.00310.8618,K-=0.0229-0.03410.0117-0.01640.0108-0.0146.Using above K¯, we have the state response curves shown in Figure 6(a), where the initial state xiT(0)=-11, xiT(-t)=00, i=1,2,3. It is clear that the system is asymptotically stable.
When we setL=11-11-11-101-101, the control gain matrix is K-=0.0228-0.03400.0107-0.01500.0105-0.0144. The resulting state response curve is shown in Figure 6(b), with the same initial condition as in Figure 6(a).
For we are concerned with the stability only, the controlled outputz(t) cannot converge to zero. That means consensus is not achieved with above control gains, as shown in Figure 7, where the input signal is r(t)=01, t>0.Figure 6
The state response curve.
(a)
(b)Figure 7
The consensus of Example1.Example 2.
Each subsystem of the NCS in Figure1 is described as follows (see [1]):(38)x˙t=010-0.1xt+00.1u^t,yt=10xt.
The basic cycle time is chosen asT=1 s; there are three subsystems in the NCS. The network protocol is Time-Triggered CAN. And the total network-induced delay of each subsystem is τ1=0.1 s, τ2=0.2 s, and τ3=0.3 s.
We have the parameters in (7) as follows:(39)A-=10.951600.904810.951600.904810.951600.9048,B-1=0.00050.01000.00200.01980.00450.0296,B-2=0.04790.08520.04640.07540.04390.0656,L=0.5∗11-11-11-11-11.Set γ=0.8; then using Corollary 7 and solving LMIs (30), (31), and (32) with MATLAB YALMIP tools box, it is found that(40)P=27.752960.17071.01878.0159-1.34270.351260.1707151.66293.26519.8879-0.2402-2.06601.01873.26515.329111.7535-0.09310.00568.01599.887911.753540.4633-0.5335-2.2785-1.3427-0.2402-0.0931-0.53352.36332.46110.3512-2.06600.0056-2.27852.461111.1664,Q=5.546013.38150.52421.8687-0.01570.061313.381534.20540.24861.1222-0.2377-0.89330.52420.24861.52054.8289-0.0560-0.22461.86871.12224.828916.9049-0.5001-2.0982-0.0157-0.2377-0.0560-0.50010.46031.46830.0613-0.8933-0.2246-2.09821.46836.3043,Z=0.18210.09550.14540.09700.14460.09890.09550.10050.09660.10040.09760.10030.14540.09660.20060.09670.12350.09910.09700.10040.09670.10040.09830.10030.14460.09760.12350.09830.19970.09890.09890.10030.09910.10030.09890.1002,T=4.0623-0.93251.6962-0.80171.6925-0.6702-0.932519.6255-0.6937-13.4188-0.64073.61911.6962-0.69374.3974-0.93711.3467-0.6085-0.8017-13.4188-0.937138.6762-0.6207-18.40341.6925-0.64071.3467-0.62074.4193-0.6187-0.67023.6191-0.6085-18.4034-0.618787.4037,S=0.00100.0229-0.0001-0.0089-0.0002-0.01460.02290.9120-0.0065-0.8100-0.0014-0.1140-0.0001-0.00650.00040.0162-0.0000-0.0099-0.0089-0.81000.01621.3595-0.0021-0.5577-0.0002-0.0014-0.0000-0.00210.00020.0037-0.0146-0.1140-0.0099-0.55770.00370.7012×105,M1=-0.1821-0.0955-0.1454-0.0970-0.1446-0.0989-0.0955-0.1005-0.0966-0.1004-0.0976-0.1003-0.1454-0.0966-0.2006-0.0967-0.1235-0.0991-0.0970-0.1004-0.0967-0.1004-0.0983-0.1003-0.1446-0.0976-0.1235-0.0983-0.1997-0.0989-0.0989-0.1003-0.0991-0.1003-0.0989-0.1002,M2=0.18210.09550.14540.09700.14460.09890.09550.10050.09660.10040.09760.10030.14540.09660.20060.09670.12350.09910.09700.10040.09670.10040.09830.10030.14460.09760.12350.09830.19970.09890.09890.10030.09910.10030.09890.1002,K-=-4.2494-10.9620-1.98-6.7336-1.328-5.7161.Using above K¯, we have the state response curves shown in Figures 8 and 9.
In Figure8, the disturbance signal ω(t)=0, and the initial state xiT(0)=-11, xiT(-t)=00, i=1,2,3. It is clear that the system is asymptotically stable. And, in Figure 9, ω(t)=11T,t∈10,20;0,others; we can see the state will converge to a certain value; H∞ performance index is satisfied. In Figure 10, the controlled outputs converge to zero with ω(t)=11T,t∈10,20;0,others. When ω(t)=01, t>0, the controlled output is shown in Figure 11, which means the consensus is achieved.Figure 8
The state response curve without input signal.Figure 9
The state response curve with input signal.Figure 10
The consensus of Example2.Figure 11
The consensus of Example2 when input is not zeros.


## 5. Conclusions

In this paper, a class of networked control systems with multiple subsystems is studied, including the modeling and stability analysis for the massive networked control systems with multiple subsystems and distributed control law. Firstly, the time-triggered protocol is introduced to the system, and the executive windows are scheduled to each intelligent node, which need to access network for real-time control. Secondly, while considering the delay, the model of the NCS with distributed controller is presented. Then,H∞ consensus control problem is studied. With the Lyapunov-Krasovskii functional method, the consensus is analyzed, and the sufficient condition with matrix inequalities is given. Finally, simulations are given to validate the proposed approach. Our further work will focus on such problems as tracking control problems and synchronous coordinative problems.



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*Source: 963282-2015-11-10.xml*