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Research ArticleDistributedπ»
βConsensus Control of Networked Control
Systems with Time-Triggered Protocol
Ronghua Xie, Weihua Fan, and Qingwei Chen
School of Automation, Nanjing University of Science and Technology, 200 Xiaolingwei Street, Nanjing 210094, China
The analysis and design of traditional networked control systems focused on single closed-loop scenario. This paper introduces adistributed control approach for the networked control systems (NCSs)withmultiple subsystems based on a time-triggered networkprotocol. Firstly, some basic ideas of the time-triggered protocol are introduced and a time schedule scheme is employed for theNCS. 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 toπ»
βconsensus control is present based on the
Lyapunov-Krasovskii function. Also, the controller design approach towards the given π»β
performance index is given by a conecomplement linearization and iterative algorithm. Finally, numerical examples are given to validate the approach.
1. Introduction
Networked control systems (NCSs) are a class of closed-loop control systems in which sensors, controllers, and actu-ators are connected over network (see [1]). In recent years,NCSs have received increasing attention due to the broadapplication in industrial areas. The induced network bringsabout many advantages, such as low installation and mainte-nance costs, high reliability, and increased system flexibility.But, simultaneously, network-induced imperfections, such astime delays, packet losses and disorder, time-varying packettransmission/sampling intervals, and competition ofmultiplenodes accessing network, will decrease the performance ofNCSs (see [2]). More seriously, some imperfections maycause instability. During the past decades, many researchershave studied the NCSs, and variousmethodologies have beenproposed on themodeling (see [3, 4]), scheduling (see [5, 6]),analysis, and control design (see [7β11]).
When the plant ismultiple-input-multiple-output (MIMO),the NCS is called MIMO NCSs. Because the nodes shouldcompete to access not only outside nodes, but also othernodes inside, the research of MIMO NCSs is a more chal-lenging job when compared with the so-called single-input-single-output (SISO) NCSs. Yan et al. [12] presented a con-tinuous time model of MIMO NCSs with distributed time
delays and uncertainties and gave delay-dependent stabilitycriteria in terms of linear matrix inequalities (LMIs). Xiaet al. [13] presented a discrete-time model of MIMO NCSswith multiple time-varying delays, and the design of outputfeedback controllers is proposed in terms of matrix inequal-ities, together with an iterative algorithm. Okajima et al.[14] proposed a design method for feedback-type dynamicquantization in a MIMO NCS, which is extended from SISONCSs. Guan et al. [15] studied the optimal tracking perform-ance for MIMO LTI discrete-time control systems with com-munication constraints in feedback path and how the band-width and AWGN of the communication channel affectedthe tracking capability. Jiang et al. [16] studied the optimaltracking performance of MIMO NCSs with AWGN channelbetween the controller and the plant and concluded thatthe optimal tracking performance was closely dependent onnonminimum phase zeros, unstable poles of the plant, andcharacteristics of the signals and channel. Cao et al. [17]presented delay dependant stability criteria for MIMO NCSswith nonlinear perturbation and delay, which gave much lessconservative maximum allowable delay bound. Li et al. [18]modeled theMIMONCSwithmultichannel packet disorder-ing, packet dropout, and bounded time-varying transmissiondelay, as a jump linear system subject to Markovian chains,and a real-time controller was proposed such that the cost
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 963282, 13 pageshttp://dx.doi.org/10.1155/2015/963282
Figure 1: Structure of the networked control systems.
Figure 2: The Baggage Handling System.
function value is lower than a specified upper bound. Du etal. [19]modeledMIMONCSs as unknown switched sequenceand proposed a sufficient condition to be asymptoticallystable in terms of a set of bilinear matrix inequalities. Inthe 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 controllernodes 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 forNCS toachieve finite gain L2 stability independent of constant timedelay, which consisted two parts. One was a local controllerdesigned without network; the other was a remote part tocompensate the network-induced delay to keep stability. But,actually, here the controller was divided into two separatedparts, which was not so-called distributed control. In thispaper, we consider a class of MIMO networked controlsystems with multiple subsystems, multiple sensor nodes,controller nodes, and actuator nodes, whose subsystemsexchange information through network. Figure 1 gives thecommon structure of the NCSs. It is clear that the NCS isa distributed system over the network channel. This kind ofsystem can be easily found, such as Baggage Handling Sys-tems (BHS, as in Figure 2), product line systems, and Multi-joint Robots. For these systems, all subsystems are essentially
required to be stable. Furthermore, there are usually somespecial requirements. Take BHS as an example; one BHS oftenhas a few branches, which contain dozens of motors. Whiletransplanting baggage, the velocity of motors in each branchshould be consistent. Otherwise, the baggage may collidebecause of different velocity. For more details, the consensuscontrol in NCSs means consensus not only in steady state,but also in transient response; that is, when the referencesignal changes or disturbance occurs, the output signal of thesubsystems is required to change simultaneously.
The consensus control of NCSs is an interesting prob-lem and is different from that in multivehicle cooperativecontrol, because the dynamic of each subsystem is differentfrom the others. Some effective conclusions in multivehiclecooperative 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 consensuscontrol of the NCSs with time-triggered protocol and dis-tributed control law. The contributions are as follows:
(i) Firstly, the time-triggered protocol is introduced andemployed to the NCSs; a scheduling scenario whichreduces the network-induced delay within each sub-system is introduced.
(ii) Secondly, a model for the NCSs with time-triggeredprotocol and short time-varying network-induceddelays is proposed, while the distributed controllerswhich use the feedback information from the subsys-tem neighbors are used.
(iii) Thirdly, the sufficient conditions for asymptotical sta-bility and π»
βconsensus control of the NCSs are
obtained by Lyapunov-Krasovskii function. The con-ditions guarantee all subsystems reach consensuswhile satisfying the desired π»
βperformance on the
fixed time-triggered protocol. Also, an iterative algo-rithm is given for distributed controller gain matrix.
The rest of this paper is organized as follows. Section 2 in-troduces the protocol of theNCSs, the feature of the network-induced delay, and the mathematical model. Section 3 dealswithπ»
βconsensus control problem for NCSs. Some numer-
ical examples are given in Section 4 to demonstrate the effec-tiveness of the proposed design technique. The conclusion isprovided in Section 5.
Mathematical Problems in Engineering 3
Reference message
Exclusive window
Exclusive window
Arbitrating window
Free window
Exclusive window
Reference message
Basic cycle time
Figure 3: Basic cycle of TTCAN.
2. Modeling of the NCSs
2.1. Protocol of theNetwork. Consider theNCSswithmultiplesubsystems in Figure 1. The sensor nodes and the controllernodes will access the network after sampling or calculation,respectively. As we know, their authorities to access networkdepend 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-triggeredprotocol or event-triggered protocol. Time-triggered proto-col allows the node to access network in certain time slot,such as Ether-CAT, FlexRay, and Time-Triggered CAN, whilethe nodes in event-triggered protocol access the networkwhenever they are ready for transmission, such as TCP/IP,CAN. Most communication networks adopt event-triggeredprotocol, because the protocol is efficient while nodes joinand quit frequently. But the situation is different in con-trol systems. Few nodes in control systems will join orexit frequently while working, unless the node is down orcrashed. Moreover, event-triggered protocol brings aboutmany uncertainties to the system, because of the randomaccess. In [21], it is concluded that, compared with event-trig-gered protocol, the time-triggered protocol brought aboutmore convenience to design and analysis. So we discuss theNCSs based on the time-triggered protocol. In order tomodelthe NCSs, we introduce some important features of time-triggered protocol.
Usually, a basic cycle exists in time-triggered protocolnetwork, which means that a basic period for all importantnodes has at least one chance to transfer data. For example,a basic cycle of TTCAN [22] is shown in Figure 3. A basiccycle begins with a reference message, which is sent by aspecial node and can be identified by all participants. A basiccycle usually consists of several time windows (or slots) ofdifferent length and offers the necessary time for the mes-sage to be transmitted. The exclusive window is a time slotfor periodic messages, while the arbitrating window is foraperiodic messages. Free window is reserved for furtherextensions. An exclusive window allows only one node tosend a frame. In the arbitrating windows, these nodes thatneed to send frames are allowed to compete for networkaccess as in event-triggered protocol. The end of an arbitrat-ing window is always predictable. Thus, the advantages ofevent-triggered communication can be combined with thoseof time-triggered communication.
Of course, the sequence of these windows in a basiccycle can be designed according to scheduling strategy. Forexample, the sequence can be designed as in Figure 4 toreduce network-induced delay within each subsystem whenit is used for NCS with four subsystems.
Ref. Sen. Sen.CTD#1#1
CTD#2#2
Sen. CTD#3#3
Sen. CTD#4#4 Arbitration
Figure 4: Basic cycle of NCS with four subsystems based onTTCAN.
Furthermore, the reference message also gives someimportant information, including a global time stamp, whereparticipants can achieve a synchronization accuracy of 1πs.Thatmeans the time jitter between all nodes can be negligible,unless the main time constant of the system is shorter thanmicroseconds.
2.2. Modeling of the NCSs. According to the facts inSection 2.1, we can give the following reasonable assump-tions.
Assumption 1. The sensor nodes and controller nodes are alltime triggered.
Remark 2. According to the time-triggered protocol, theintelligent nodes access the network in appointed time slots.So it is reasonable to set sensor nodes and controller nodesto be time triggered. And they should be idle at rest timeto reduce power consumption. The actuator nodes are eithertime triggered or event triggered, because the data packetsfrom controller nodes arrive at almost the same time in eachbasic cycle.
Remark 3. For time-triggered protocol, little conflictionsoccur during transmission. It is reasonable to assume littledata loss. So, we do not consider data loss in this paper.
By Assumption 1, the network-induced time delay ππ=
ππ,π π+ ππ,ππ
is constant, where ππ,π π
is time delay between thesensor node and controller node of the πth subsystem and π
π,ππ
is time delay between the controller node and actuator node.Also, we have π
π< π, where π is the basic cycle time.
Suppose that the plant of any subsystem is LTI and isdescribed as space state equation:
οΏ½οΏ½π(π‘) = π΄
πππ₯π(π‘) + π΅
πποΏ½οΏ½π(π‘) + π
π(π‘) ,
π¦π(π‘) = πΆ
πππ₯π(π‘) ,
(1)
where π₯π(π‘) β Rπ is the state vector of the πth plant, π =
1, . . . , π. οΏ½οΏ½π(π‘) β Rπ is the control input vector, π¦
π(π‘) β Rπ
is the output vector, ππ(π‘) β Rπ is the external disturbance,
and π΄ππβ RπΓπ, π΅
ππβ RπΓπ, and πΆ
ππβ RπΓπ are known real
constant matrices.
4 Mathematical Problems in Engineering
t
Neighbor sensor node 1
Neighbor sensor node Ni
...
t k+1t k
t k+π1,sc
t k+πi,sc
t k+πi,c
t k+πi
Actuator node iController node i
Sensor node i
Figure 5: Data flow of πth subsystem.
We use π₯πto denote the sampled data in the receiver of the
πth controller node; π’πdenotes the control variable calculated
by the πth controller node.By (1), the NCS can be described in discrete time as
ππ΄πππ ππ π΅ππ, and π΅
π2=
β«
π
ππ
ππ΄πππ ππ π΅ππ. π΄π, π΅ππ, π = 1, . . . , π, π = 1, 2, are known
matrices for ππis constant.
Since NCSs are usually large scale, it is not advisableto employ centralized control. In this paper, we employ adistributed control law as (3) for the NCS:
π’π= πΎπβ
πβππ
πππ(π₯πβ π₯π) , (3)
where πΎπβ RπΓπ is gain of πth controller node. The control
law (3) means each subsystem controller uses both its ownfeedback and also datum from its neighbors.
The data flow of the πth subsystem is shown in Figure 5;π‘π+ ππ,πis the moment the controller node calculates control
are the πth and (π + 1)th sampling time, π‘π+1β
π‘π= π. Suppose the cycle time in Figure 4 is used in the
NCS; when the πth subsystem calculates the control variable,(π+1)th, . . . , πth subsystems have not sent their data packets.That means the neighbor of the πth subsystem isπ
π= {Vπ, Vπβ
πΈ, π < π}.Using (2) and (4), the closed-loop system can be de-
Hence, (7) is stable if the following matrix inequality holds:
Ξ¦ + Ξ + [
ππ
1
ππ
1
]πβ1
[π1π2] < 0. (18)
By Schur complements, inequality (18) is equivalent to
[
[
[
[
π11+ππ
1+π1π12βππ
1+π2ππ
1
β π22βππ
2βπ2ππ
2
β β βπ
]
]
]
]
< 0. (19)
That is equivalent to
[
[
[
[
[
(π΄ + π΅2πΎπΏ)
π
π (π΄ + π΅2πΎπΏ) β π + π +π
π
1+π1(π΄ + π΅
2πΎπΏ)
π
π (π΅1πΎπΏ) βπ
π
1+π2ππ
1
β (π΅1πΎπΏ)
π
π (π΅1πΎπΏ) β π βπ
π
2βπ2ππ
2
β β βπ
]
]
]
]
]
+
[
[
[
[
[
(π΄ + π΅2πΎπΏ β πΌ)
π
(π΅1πΎπΏ)
π
0
]
]
]
]
]
π [(π΄ + π΅2πΎπΏ β πΌ) (π΅
1πΎπΏ) 0] < 0.
(20)
By Schur complements, we have
[
[
[
[
[
[
[
[
[
[
[
[
[
βπ + π +ππ
1+π1
βππ
1+π2
ππ
1(π΄ + π΅
2πΎπΏ β πΌ)
π
β βπ βππ
2βπ2ππ
2(π΅1πΎπΏ)
π
β β βπ 0
β β β βπβ1
]
]
]
]
]
]
]
]
]
]
]
]
]
+
[
[
[
[
[
[
[
[
(π΄ + π΅2πΎπΏ)
π
(π΅1πΎπΏ)
π
0
0
]
]
]
]
]
]
]
]
π [(π΄ + π΅2πΎπΏ) (π΅
1πΎπΏ) 0 0] < 0.
(21)
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 π»
βdisturbance attenuation
index πΎ, if there exist symmetric positive definite matrices π,π,and π and matricesπ
1,π2, and πΎ, such that
[
[
[
[
[
[
[
[
[
[
[
[
[
[
βπ + π +ππ
1+π1+ π»π
π» βππ
1+π2
0 ππ
1(π΄ + π΅
2πΎπΏ β πΌ)
π
(π΄ + π΅2πΎπΏ)
π
β βπ βππ
2βπ2
0 ππ
2(π΅1πΎπΏ)
π
(π΅1πΎπΏ)
π
β β βπΎ2
πΌ 0 πΌ πΌ
β β β βπ 0 0
β β β β βπβ1
0
β β β β β βπβ1
]
]
]
]
]
]
]
]
]
]
]
]
]
]
< 0. (22)
And if the matrix inequality is feasible, the feedback matrix ofthe consensus protocol is πΎ.
Proof. Let Lyapunov-Krasovskii function as (12). Using (7)and Lemmas 4 and 5, we have
Mathematical Problems in Engineering 7
Ξπ (π) β€ [ππ
(π) ππ
(π β 1) ππ
(π)] Ξ¦[
[
[
π (π)
π (π β 1)
π (π)
]
]
]
,
Ξ¦ =
[
[
[
[
π11+ππ
1+π1+ππ
1πβ1
π1π12βππ
1+π2+ππ
2πβ1
π1π13
β π22βππ
2βπ2+ππ
2πβ1
π2π23
β β π + π
]
]
]
]
,
π13= (π΄ + π΅
2πΎπΏ)
π
π + (π΄ + π΅2πΎπΏ β πΌ)
π
π,
π23= (π΅1πΎπΏ)
π
(π + π) .
(23)
Firstly, from condition (22), we have (11), so the NCS (7) isasymptotically stable. Then, we have
limπββ
(π₯π(π) β π₯
π(π)) = 0. (24)
Then, we findπ»β
performance index.For any π > 0, consider the following cost function:
π½ =
β
β
π=0
[π§π
(π) π§ (π) β πΎ2
ππ
(π) π (π)] . (25)
By the zero initial condition (π(0) = 0), we have
π½ =
β
β
π=0
[π§π
(π) π§ (π) β πΎ2
ππ
(π) π (π) + Ξπ (π)]
β π (β) + π (0)
β€
β
β
π=0
[π§π
(π) π§ (π) β πΎ2
ππ
(π) π (π) + Ξπ (π)] ,
(26)
for
π§π
(π) π§ (π) β πΎ2
ππ
(π) π (π) + Ξπ (π)
= [ππ
(π) ππ
(π β 1) ππ
(π)]Ξ[
[
[
π (π)
π (π β 1)
π (π)
]
]
]
,
(27)
where
Ξ =
[
[
[
[
π11+ππ
1+π1+ππ
1πβ1
π1+ π»π
π» π12βππ
1+π2+ππ
2πβ1
π1
π13
β π22βππ
2βπ2+ππ
2πβ1
π2
π23
β β π + π β πΎ2
πΌ
]
]
]
]
. (28)
According to Schur complements, condition (22) is equiva-lent to Ξ < 0. That is,
β
β
π=0
π§π
(π) π§ (π) < πΎ2
β
β
π=0
ππ
(π) π (π) , (29)
so the robust π»β
consensus control is achieved. This com-pletes the proof.
Note that condition (22) is nonconvex as it contains π,πβ1, π, and πβ1. 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 π»
βdisturbance attenuation
index πΎ, if there exist symmetric positive definite matrices π,π,π, π, and π and matricesπ
1,π2, and πΎ, such that
minπ,π,π1,π2 ,πΎ
[tr (ππ + ππ)]
π .π‘.
[
[
[
[
[
[
[
[
[
[
[
[
[
βπ + π +ππ
1+π1+ π»π
π» βππ
1+π2
0 ππ
1(π΄ + π΅
2πΎπΏ β πΌ)
π
(π΄ + π΅2πΎπΏ)
π
β βπ βππ
2βπ2
0 ππ
2(π΅1πΎπΏ)
π
(π΅1πΎπΏ)
π
β β βπΎ2
πΌ 0 πΌ πΌ
β β β βπ 0 0
β β β β βπ 0
β β β β β βπ
]
]
]
]
]
]
]
]
]
]
]
]
]
< 0,
(30)
8 Mathematical Problems in Engineering
[
π πΌ
πΌ π
] β₯ 0, (31)
[
π πΌ
πΌ π
] β₯ 0. (32)
If the matrix inequality is feasible, then the feedback matrix ofthe consensus protocol is πΎ.
The proof is omitted.In order to design distributed control law (3), we present
an iterative algorithm as follows.
Algorithm 8. (1) For (30)β(32), find a feasible solution:π0,π0,π0,π01,π02, πΎ0, π0, and π0, and let β = 0.
(2) Set πβ+1 = πβ, πβ+1 = πβ, πβ+1 = πβ, and πβ+1 = πβ,and solve the following optimal problem:
min [tr (πβπ + πβπ + πβπ + πβπ)]
subject to (30) , (31) , (32) .
(33)
(3) If a stopping criterion given in advance is satisfied, theiteration ends.
Otherwise, go to Step (2).
Remark 9. A simple stopping criterion is that πβ, πβ providea feasible solution to inequality (30). And using LMI toolbox,it is easily confirmed.
4. Numerical Examples
Example 1. We use the DCmotors of [25] as the plants of theNCS in Figure 1, where the transfer function of the DCmotoris
πΊ (π ) =
2029.826
(π + 26.29) (π + 2.296)
. (34)
The DC motor model is rewritten in state space as
οΏ½οΏ½ (π‘) = [
0 1
β60.3756 β28.586
] π₯ (π‘)
+ [
0
2029.826
] π’ (π‘) ,
π¦ (π‘) = [1 0] π₯ (π‘) .
(35)
And, in this example, the basic cycle time is chosen as π =0.005 s, there are three DC motors connected by the Time-Triggered CAN, and the total network-induced delay of eachclosed loop is π
1= 0.001 s, π
2= 0.002 s, and π
3= 0.003 s.
Then, we have the parameters in (7) as follows:
π΄ =
[
[
[
[
[
[
[
[
[
[
[
[
0.9993 0.004658
β0.2812 0.8861
0.9993 0.004658
β0.2812 0.8861
0.9993 0.004658
β0.2812 0.8861
]
]
]
]
]
]
]
]
]
]
]
]
,
π΅1=
[
[
[
[
[
[
[
[
[
[
[
[
0.001
2.001
0.0040
3.946
0.0089
5.836
]
]
]
]
]
]
]
]
]
]
]
]
,
π΅2=
[
[
[
[
[
[
[
[
[
[
[
[
0.0231
7.454
0.202
5.509
0.0153
3.6190
]
]
]
]
]
]
]
]
]
]
]
]
,
Mathematical Problems in Engineering 9
πΏ =
[
[
[
[
[
[
[
[
[
[
[
[
1
1
β1 1
β1 1
β1 1
β1 1
]
]
]
]
]
]
]
]
]
]
]
]
.
(36)
Here, we just consider the stability of the whole NCS;then using the matrix inequality in Lemma 5 together withAlgorithm 8, the following feasible solution can be obtainedby MATLAB LMI toolbox:π
Using above πΎ, we have the state response curves shown inFigure 6(a), where the initial state π₯π
π(0) = [β1 1], π₯π
π(βπ‘) =
[0 0], π = 1, 2, 3. It is clear that the system is asymptoticallystable.
When we set πΏ = [
[
1
1
β1 1
β1 1
β1 0 1
β1 0 1
]
]
, the control gain matrix
is πΎ = [ 0.0228 β0.0340 0.0107 β0.01500.0105 β0.0144
]. The resultingstate response curve is shown in Figure 6(b), with the sameinitial condition as in Figure 6(a).
For we are concerned with the stability only, the con-trolled output π§(π‘) cannot converge to zero. That meansconsensus is not achieved with above control gains, as shownin Figure 7, where the input signal is π(π‘) = [ 0
1], π‘ > 0.
Example 2. Each subsystem of the NCS in Figure 1 isdescribed as follows (see [1]):
οΏ½οΏ½ (π‘) = [
0 1
0 β0.1
] π₯ (π‘) + [
0
0.1
] οΏ½οΏ½ (π‘) ,
π¦ (π‘) = [1 0] π₯ (π‘) .
(38)
The basic cycle time is chosen as π = 1 s; there arethree subsystems in the NCS. The network protocol is Time-Triggered CAN. And the total network-induced delay of eachsubsystem is π
1= 0.1 s, π
2= 0.2 s, and π
3= 0.3 s.
We have the parameters in (7) as follows:
π΄ =
[
[
[
[
[
[
[
[
[
[
[
[
1 0.9516
0 0.9048
1 0.9516
0 0.9048
1 0.9516
0 0.9048
]
]
]
]
]
]
]
]
]
]
]
]
,
10 Mathematical Problems in Engineering
4010 15 20 25 30 35 500 455t (s)
β1
β0.5
0
0.5
1
1.5
2
x6
x5
x4
x3
x2
x1
(a)
4010 15 20 25 30 35 500 455t (s)
β1
β0.5
0
0.5
1
1.5
x6
x5
x4
x3
x2
x1
(b)
Figure 6: The state response curve.
π΅1=
[
[
[
[
[
[
[
[
[
0.0005
0.0100
0.0020
0.0198
0.0045
0.0296
]
]
]
]
]
]
]
]
]
,
π΅2=
[
[
[
[
[
[
[
[
[
0.0479
0.0852
0.0464
0.0754
0.0439
0.0656
]
]
]
]
]
]
]
]
]
,
πΏ = 0.5 β
[
[
[
[
[
[
[
[
[
[
[
[
1
1
β1 1
β1 1
β1 1
β1 1
]
]
]
]
]
]
]
]
]
]
]
]
.
(39)
Set πΎ = 0.8; then using Corollary 7 and solving LMIs (30),(31), and (32) with MATLAB YALMIP tools box, it is foundthat
Using above πΎ, we have the state response curves shown inFigures 8 and 9.
In Figure 8, the disturbance signal π(π‘) = 0, and theinitial state π₯π
π(0) = [β1 1], π₯π
π(βπ‘) = [0 0], π = 1, 2, 3.
It is clear that the system is asymptotically stable. And,
in Figure 9, π(π‘) = {[1 1]
π
, π‘ β [10, 20]; 0, others};we can see the state will converge to a certain value; π»
β
performance index is satisfied. In Figure 10, the controlledoutputs converge to zero with π(π‘) = {[1 1]
π
, π‘ β
[10, 20]; 0, others}. When π(π‘) = [ 01], π‘ > 0, the controlled
12 Mathematical Problems in Engineering
4510 15 20 25 30 35 40 500 5t (s)
β9
β8
β7
β6
β5
β4
β3
β2
β1
0
1
z4
z3z2
z1
Figure 7: The consensus of Example 1.
4010 15 20 25 30 35 500 455t (s)
β1
β0.5
0
0.5
1
1.5
2
2.5
x6
x5
x4
x3
x2
x1
Figure 8: The state response curve without input signal.
output is shown in Figure 11, which means the consensus isachieved.
5. Conclusions
In this paper, a class of networked control systems withmultiple subsystems is studied, including the modeling andstability analysis for the massive networked control systemswith multiple subsystems and distributed control law. Firstly,the time-triggered protocol is introduced to the system,and the executive windows are scheduled to each intelligentnode, which need to access network for real-time control.Secondly, while considering the delay, the model of the NCS
4010 15 20 25 30 35 500 455t (s)
β2
β1
0
1
2
3
4
5
x6
x5
x4
x3
x2
x1
Figure 9: The state response curve with input signal.
4010 15 20 25 30 35 500 455t (s)
β3
β2
β1
0
1
2
3
z4
z3z2
z1
Figure 10: The consensus of Example 2.
with distributed controller is presented.Then,π»βconsensus
control problem is studied. With the Lyapunov-Krasovskiifunctional method, the consensus is analyzed, and the suf-ficient condition with matrix inequalities is given. Finally,simulations are given to validate the proposed approach. Ourfurther work will focus on such problems as tracking controlproblems and synchronous coordinative problems.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Mathematical Problems in Engineering 13
4010 15 20 25 30 35 500 455t (s)
β2
β1.5
β1
β0.5
0
0.5
1
z4
z3z2
z1
Figure 11: The consensus of Example 2 when input is not zeros.
Acknowledgments
This work was supported by NSFC (Grant no. 61074023), theJiangsu Postdoctoral Fund, and the Jiangsu Overseas (Grantno. 1001003B) Research & Training Program for UniversityProminent Young & Middle-Aged Teachers and Presidents.
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