Research ArticleJoint Design of Control and Power Efficiency inWireless Networked Control System

Yan Wang and Zhicheng Ji

Department of IoT Engineering Jiangnan University Wuxi 214122 China

Correspondence should be addressed to Yan Wang violetwang0929gmailcom

Received 29 April 2014 Accepted 10 July 2014 Published 6 August 2014

Academic Editor Xi-Ming Sun

Copyright copy 2014 Y Wang and Z Ji This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a joint design method for wireless networked control system (WNCS) to balance both the demands of networkservice and the control performance Since the problems of power consumption communication reliability and system stabilityexist simultaneously and interdependently in WNCS most of the achieved results in the wireless network and wired networkedcontrol system cannot be used directly To coordinate the three problems sampling period is found to be the linking bridge Anadaptive sampling power efficiency algorithm is proposed to manage the power consumption such that it can meet the demands ofnetwork life span The sampling period is designed to update periodically on the constraints of network schedulability and systemstability The convergence of the power efficiency algorithm is further proved The sampling period is no longer a fixed valuehowever thus increasing the difficulty in modeling and controlling such a complicated time-varying system remains In this worka switched control system scheme is applied tomodel such aWNCS and the effect of network-induced delay is considered Switchedfeedback controllers are introduced to stabilize the WNCS and some considerations on stability condition and the bounds of theupdate circle for renewing sampling period are discussed A numerical example shows the effectiveness of the proposed method

1 Introduction

Wireless networked control systems are composed of dis-tributed fields and plant devices (sensors actuators andcontrollers) interconnected via a wireless network [1] Thesensors controllers and actuators exchange informationwithone another through the wireless network Sensors collect thestatus or outputs of plants at every sampling instant packetthe data with time-stamp and then send the informationto each corresponding controller via the wireless networkControllers then compute the control variables as soonas they receive the newest data from their plants Afterthat the control variables are sent to their correspondingactuators Plants will not update their status until they receivethe newest command from controllers Wireless capabilitiesclearly provide opportunities to be more inventive in systemorganization [2] The use of wireless network offers severaladvantages compared with a conventional wired networkedcontrol system in terms of cost maintenance scalability and

implementation flexibility However wireless nodes also haveobviousweak points such as reliability and availability partic-ularly the power consumption Power efficiency technologiesare important research areas in wireless network applications

Most studies onWNCS analyze the effects of the wirelessmedium on overall closed-loop control systems in [3ndash11]The first critical analysis on the use of wireless control isperformed by Kumar in [3] Kumar explores the impactof different protocol layers from routing to physical oncontrol performance In [4] Liu and Goldsmith analyzedthe performance of a control system in terms of variationin data rate error correcting codes and different maximumbounds in the number of retransmissions They also studiedthe impact of IEEE 80211b medium access control In [5]Colandairaj et al discuss the impact of data flow on thestability of a closed-loop wireless network control systembased on IEEE 80211 Sampling rate adaption is proposedas a codesign solution to enhance control and wirelessnetwork performance Different from our work in this paper

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 257079 10 pageshttpdxdoiorg1011552014257079

2 Mathematical Problems in Engineering

the purpose of adapting the sampling rate is to optimizebandwidth utilization not to save power A robust anddynamic cross-layer communication architecture for wirelessnetworked control system is presented in [6] by Israr et alThe protocol stack for WNCS comprises five layers Eachlayer contributes to the overall goal of reliable power-efficientcommunication However the control performance is nottaken into account in their work

Anumber of studies related to power efficiency inwirelesssensor networks (WSNs) and wireless networks have alsobeen conducted in [12ndash18] Current power efficiency researchalways falls into two categories one is reducing the powerconsumption of each single node in the network and theother is balancing the power consumption of all the nodes innetwork In [12] Colandairaj et al present a dynamic powercontrol strategy to minimize the communication powerconsumption of nodes by varying the transmission rate Aprotocol intending to balance power consumption from theremaining battery power of the node-based routing policy isproposed by Liang and Yang in [13] The nodes with greaterremaining power are allocated with more communicationtasks In [15] Kim et al propose a lifetime-based routingstrategy in which the survival time is estimated according tothe residual power and current ratio of power consumptionThe path with the longest node survival time is selected fordata transmission

Recently limited studies in [19ndash21] are conducted oneffective power saving strategies that specifically target atWNCS Fischione et al [20] propose a trade-off betweenwireless output power related to reliability and power con-sumption where a physical characteristic model revealedquantitative relations with communication outage probabil-ity They also focus on the lower layer optimal protocoldesign by considering the application layer requirementsLino [21] discusses the optimal sleepmode control of wirelessnetwork nodes and proposes a trade-off method betweencontrol performance and power consumption An optimalcontrol strategy is applied to optimize the control period In[22] event-predictive control for power saving of wirelessnetworked control system is discussed The key idea is tosave power by maximizing the control interval with con-strains of appropriate control performance The proposedcontrol method is rather complicated and requires onlineoptimizationmixed integer programming which reduces thepracticability Thus a simpler trade-off method for WNCS isrequired

Power consumption communication reliability and sys-tem stability exist simultaneously and react with one anotherin wireless networked control systems Supposed that thethree factors are interdependent most results achieved inwireless network power management and wired networkedcontrol systems cannot be directly applied to WNCS Thusthe motivation of this paper is to find a bridge which canlink the three factors and make a balance among thesefactors through the bridge parameter such that the overallsatisfactory performance can be achieved Fortunately thesampling period of sensor node is found to be the bridgeparameter From this point a joint designmethod of adaptivesampling power efficiency algorithm and coordinated control

method are discussed in this paper An updating rule ofsampling period is presented to satisfy the demands of wire-less life span under constrains of network schedulability andcontrol system stability Convergence of the power efficiencyalgorithm is further proved Subsequently the control systemis a varying-period system since the sampling periods ofsensors are time-varying It is then modeled as a class ofswitched control system with two types of behavior in eachupdate periodThe switched control law is applied to stabilizethe control system and stability conditions are discussedAlso the choosing rule of update period is given

The remaining sections are organized as follows Section 2is the problem formulation Section 3 presents the adaptivepower efficiency algorithm Section 4 discusses the coor-dinated wireless networked control system modeling anddesign method Numerical simulation is given in Section 5Section 6 is the conclusion of this paper

2 Problem Formulation

21 Description of WNCSs Consider the wireless networkedcontrol systems shown in Figure 1 There are three kindsof node in the system Power consumption varies for thedifferent kinds of node

Besides some necessary assumptions are made in thispaper as the follows

Assumption 1 The power of sensor and actuator nodes issupplied by battery while the power of controller is suppliedby base station

Assumption 2 The sensor and the actuator are clock drivenwhile the controller is event driven The sampling data ispacked in one packet for transmission with time stamp

Assumption 3 There exists transmission delay in the controlloop and it is assumed to be less than one sampling period

22 Analysis of Power Consumption inWNCSs Sensor poweris consumed by three processes data sampling sampledata reading by the ADC and data transfer The powerconsumption of the controller node is also consumed by threeprocesses receiving data calculating control variables andsending data packet The power of the actuator is consumedby two processes receiving data and DA conversion Thepower consumption of different tasks is shown in Table 1 (seein [6])

From the table we get the following two conclusions

(1) when the sensor transfers the same amount of data asthe actuator receives it will consume 25 times morepower than the actuator will consume

(2) data transfer consumes over 90 of the total sensorpower consumption

Given that the power required by control nodes can besupplied by the base station in most situations the powerrequired by the sensor nodes and actuators can be providedby batteries Thus sensors utilize the maximum amount

Mathematical Problems in Engineering 3

Actuator 1

Remote controller 1

Remote controller n

Actuator n

Plant 1

Plant n

Sensor 1

QoS data

Sensor node 1

Sensor node n

Adaptivesamplingalgorithm

modul

Adaptivesamplingalgorithm

modulQoS data

Sensor n

u1(j)

x1(j)

xn(j)

un(j) xn(j)

x1(j)

u1(j)

un(j)

T1(j + 1)

Tn(j + 1)

Figure 1 Structure of wireless networked control systems

Table 1 Power consumption of different tasks

Task Energy consumption (nAh)Receive data 8Transfer data 20Read data 0011Sample data 108

of power consumption in WNCS Managing the powerconsumption of sensors is the key to prolonging the survivaltime of the wireless network A direct and effective methodis to reduce as much of the transmission consumption aspossible by properly adjusting the amount of sample dataThis principle is the basis of our power control algorithm

In wireless networks the average power consumption ofsending a packet can be described as [3]

119864 = 119887 times packet size + 119888 (1)

where 119887 is the coefficient of power consumption and 119888 is thefixed power consumption of the node sending a data packetAccording to Assumptions 1 and 2 a sensor node sends apacket to the corresponding controller at every samplingtime Given the lack of packet retransmission the survivaltime of the sensor can be described as

119871 =

sumsum

119894=1

(119905 (119894) minus 119905 (119894 minus 1)) (2)

where 119871 is the survival time of the sensor node sum is themaximum number of packets sent by the sensor given aninitial power sum = lfloor119864init119864rfloor lfloor119909rfloor is the integral part of

119909 119864init is the given initial power of the sensor 119905(119894) 119894 isin

1 2 sum is the time that the sensor node sends the 119894thdata packet and 119905(0) is the initial time

Survival time is dependent on transfer intervals It canbe prolonged by increasing the transfer intervals Based onthis premise as well as on the knowledge of the relationshipbetween sampling period and control performance we cancooperatively design the control and the network perfor-mances by adaptively adjusting the sampling period with aproper rule

3 Adaptive Sampling PowerEfficiency Algorithm

31 Update Rule of Sampling Period For the consideration ofsimplicity and generality we choose one of the control loopsin the WNCS to describe the power control algorithm Let119879min be the lower bound of the sampling period to guarantee

the schedulability of the network and let 119879max be the upperbound of the sampling period to ensure system stabilitySupposing that there are 119872 candidate sampling periods forchoosing in the allowable range between the maximum andminimum bounds 119879

119897isin [119879

min 119879

max] 119897 = 1 2 119872

Furthermore an update period 119879119872 119879119872

gt 119879max is designed

for the sampling period renewal The sampling period isrenewed at each update instant as follows

119879 (119895 + 1) = 119879 (119895) + Δ119879 (119895 + 1)

Δ119879 (119895 + 1) = minus sgn ( (119895) minus 119871119890) sdot

119878 (119879max

minus 119879min

)

119872 minus 1

(3)

4 Mathematical Problems in Engineering

where

119878 = min

[[[

[

2 (119872 minus 1)10038161003816100381610038161003816 (119895) minus 119871

119890

10038161003816100381610038161003816

times ((119864init

119864

minus

119872

sum

119897=1

119896119897(119895)

119879119872

119879119897

minus119879119872

119879119895

)

times (119879max

minus 119879min

))

minus1]]]

]

119872 minus 1

(4)

119871119890is the demand of sensor survival time 119879(119895) is the value

of the sampling period in the 119895th updating interval sgn(119909)is the signal of scalar 119909 lfloor119909rfloor is the integral part of scalar119909 119896119897(119895) is the number of updating intervals in which the

sampling period is 119879119897during the previous 119895 updates (119895) is

the current predicted survival time of sensor node calculatedby the following formula

(119895) = 119895119879119872+119864rem (119895)

119864

times 119879 (119895) (5)

119864rem(119895) is the current remaining power at the updating instant119895119879119872

32 Convergence of Power Efficiency Algorithm

Theorem 1 For the WNCS described in Figure 1 consideringthe update rule of adaptive sampling period (3) if theminimumsampling period satisfies

119879min

gt119871119890

119904119906119898 (6)

then the actual survival time of sensor node will reach itsexpected value 119871

119890through the proposed rule of sampling period

update

Proof According to Formula (1) we have

(119895 + 1) = (119895 + 1) 119879119872+119864rem (119895 + 1)

119864

times 119879 (119895 + 1) (7)

The remaining power relationship at the two adjacentupdating instants is given by

119864rem (119895 + 1) = 119864rem (119895) minus119879119872

119879 (119895)times 119864 (8)

We assume that the sensor node has sampled the plantwith the sampling period 119879

119897for 119896119897(119895) times from the initial

to the current time instant The remaining power can becalculated based on the initial power and consumed power

119864rem (119895) = 119864init minus119872

sum

119897=1

119896119897(119895)

119879119872119864

119879119897

(9)

Power control error is defined as 119890(119895) = (119895) minus 119871119890

According to formulas (7) (8) and (9) the dynamics of theerror can be described as

119890 (119895 + 1) = 119890 (119895) + 119870 (119895) Δ119879 (119895 + 1) (10)

where 119870(119895) is defined as 119870(119895) = sum minus sum119872

119897=1119896119897(119895)(119879119872119879119897) minus

(119879119872119879(119895)) At the current time instant 119870(119895) is a known

variableThe following Lyapunov function is introduced to prove

convergence of the adaptive sampling power efficiency algo-rithm

119881 (119895) =1

21198902(119895) (11)

Considering Formula (10) it follows that

Δ119881 (119895) = 119881 (119895 + 1) minus 119881 (119895)

= 119870 (119895) Δ119879 (119895 + 1) 119890 (119895) +1

21198702(119895) Δ119879

2(119895 + 1)

(12)

With formula (3) we obtain

119890 (119895) gt 0 Δ119879 (119895 + 1) lt 0

119890 (119895) lt 0 Δ119879 (119895 + 1) gt 0

997904rArr Δ119879 (119895 + 1) 119890 (119895) lt 0

1003816100381610038161003816Δ119879 (119895 + 1)1003816100381610038161003816

1003816100381610038161003816119890 (119895)1003816100381610038161003816 gt

1

2119870 (119895) Δ119879

2(119895 + 1)

(13)

From inequalities (13) it can be concluded that Δ119881(119895) lt0 Furthermore to guarantee that the minimum survivaltime can reach the expected value the minimum samplingperiod is bounded by 119879

minge 119871119890sum Consequently the

error system is stable and the survival time can convergeto the expected value if the conditions in Theorem 1 aresatisfied

Remark 2 (prediction of survival time) The actual survivaltime is unavailable at the current instant because the powerconsumption is time varying However it can be predictedby the known information of the remaining power andsampling period at the current instant Formula (5) providesthe prediction and indicates that the survival time of thenode will be (119895) if the sensor node maintains the samplingperiod 119879(119895) as unchanged from the current instant 119905 = 119895119879

119872

The prediction of the survival time serves as a substitute forreal survival time and is used to calculate the new samplingperiod

Remark 3 (lower bound of sampling period) Taking IEEE80211b as an example the lower bound of the samplingperiod of sensor 119879min can be determined by the followingformula

119879min

= max119871119890

sum119878mc times 2 times 119873

119876 (14)

Mathematical Problems in Engineering 5

where (119878mc times 2 times 119873)119876 is the allowable minimum samplingperiod when the wireless network can be schedulable (see in[6]) and

119876 =119878mc

119879DIFS + (119862119882min times 119879SIFS2) + 119879frame + 119879SIFS + 119879ACK(15)

119879frame =119878PHY119877119897

+119878MAC + 119878mc

119877119905

119879frame =119878PHY119877119897

+119878MAC + 119878mc

119877119905

(16)

119877119905is the transmission rate 119877

119897is the legacy transmission rate

119878mc is the measurement-control data size 119878PHY is the sizeof control frame in physical layer and 119878MAC is the data sizeof ACK and is a confirmed sign in the header of TCP datapacket that confirms the received TCP message 119879SIFS is theshortest time period of the 80211b protocol for the intervalof frames requiring immediate response 119879DIFS is the timesegment for the interval of the time frame of the distributedcoordination function for sending in IEEE 80211b 119879PIFS isthe time segment for the interval of the time frame of thecentralized coordination function for sending which satisfies

119879PIFS = 119879DIFS minus 119879slot 119879slot = 119879PIFS minus 119879SIFS (17)

CW is the contention window Wireless network parametersunder the 80211b direct sequence spread spectrum are shownin Table 2

Remark 4 (upper bound of sampling period) For a SISOsystem the maximum sampling period can be obtainedusing Shannon sampling theorem For a MIMO system thefollowing method can be used to obtain the upper bound Ifthe system feedback control law is given ahead then119879max canbe obtained by solving the following optimal problem

max 119879

st100381710038171003817100381710038171003817100381710038171003817

eig(119890A119879 + int

119879

0

119890A119879

119889119905BK0)

100381710038171003817100381710038171003817100381710038171003817

lt 1

(18)

where A is the system matrix and B is the control inputmatrix K

0is the Kalman gain which satisfies

Re (eig (A + BK0)) lt 0 (19)

where Re(120592) denotes the real part of 120592 and eig(X) denotes theeigenvalues of matrix X Gain K

0can be determined by the

pole assignment in the continuous time domain

However the above optimal problem is difficult to solvedirectlyThe following iterationmethod can be used to obtainthe approximate optimal value of 119879max

Step 1 Let 119902 = 1 and the initial value of 119879max is 119879max(119902) =

2119879min If the condition can satisfy the constraints of the

optimal problem go to Step 2 or else go to Step 3

Table 2 Wireless network parameters under 80211b directsequence spread spectrum

Parameter Value119877119905

11Mbps119877119897

1Mbps119879SIFS 119879DIFS 119879slot 10 50 20 us119878MAC 119878PHY 34 24 bytes119878mc 80 bytes119878ACK 14 bytes + PHY headerCWmin CWmax 32 1024

Step 2 Let 119902 = 119902 + 1 and let 119879max(119902) = 2119879

max(119902 minus 1) If the

condition can fulfill the constraints of the optimal problemcycle Step 2 or else end the iteration and then let 119879max

=

119879max

(119902 minus 1)

Step 3 Let 119879max(119902) = (1 + (34)

119902)119879

min If the condition stilldoes not satisfy the constraints of the optimal problem let119902 = 119902 + 1 and cycle Step 3 or else end the iteration and let119879max

= 119879max

(119902)

4 Modeling and Stability Analysis of AdaptiveSampling Period WNCS

41 Modeling of Adaptive Sampling Period WNCS Consid-ering the generality one of the control loops in the WNCSis chosen as an example to illustrate the modeling approachFor theWNCS power consumptionmanaged by algorithm inSection 2 the dynamics of the control system is time-varyingSince the sampling period varies among the 119872 candidatesthe system can be considered a switched system with 119872

modes from the perspective of the switched control systemscheme Each switching mode is corresponded to one ofthe candidates According to Assumption 1 the sensor andactuator nodes are assumed to be clock-driven It results thatthe switching occurs at some of the sampling instants Addi-tionally according to Assumption 3 the inevitable existenceof network-induced delay is taken into account and it is lessthan one sampling period

We consider a plant of control loop 119894 in the WNCS withthe following dynamics

119894(119905) = A119894119909119894 (119905) + B119894119906119894 (119905)

119910119894(119905) = C119894119909119894 (119905)

(20)

where 119909119894(119905) isin R119899 is the plant state 119906119894(119905) isin R119898 is the controlinput and 119910

119894(119905) isin R119901 is the plant output A119894 isin R119899times119899 B119894 isin

R119899times119898 and C119894 isin R119901times119899 are the matrices of state control inputand output matrices respectively Due to the generality of 119894we omit the superscript 119894 in the following model descriptionand deduction

Discretizing system (20) with sampling rate 119879119897and

considering network-induced delay less than one sampling

6 Mathematical Problems in Engineering

Mixed mode Mixed model(k minus 1) ne l(k) l(k minus 1) = l(k) l(k + 3) ne l(k + 2)

u(k minus 1)

u(k + 1)120591l(k)(k)

k + 1 k + 2 k + 3

Sensor

Controller

node

node

nodeu(k minus 1)= Kl(kminus1)x(k minus 1)

Actuator

k minus 1 k

u(k)

Nominal behavior

One monitor interval

Tl(kminus1) Tl(k) Tl(k +3)

TM

Figure 2 Evolution over one period of switched WNCS with two types of behavior

period the discrete dynamics of the open control loop canbe described as

119909 (119896 + 1) = Φ119897(119896)119909 (119896) + Γ119897(119896)119906 (119896 minus 1)

119910 (119896) = C119909 (119896) (21)

whereΦ119897(119896)

= 119890A119879119897(119896) Γ

119897= int119879119897(119896)

0119890A119905119889119905B and 119897(119896) is the identi-

fication of sampling period at the 119896th sampling instant 119897(119896) isinZ rarr ℓ = 0 1 119872 minus 1

For the discrete switched system (21) a switched statefeedback controller is introduced in the following form

119906 (119896) = K120574(119896)

119909 (119896) (22)

where 120574(119896) isin Z rarr ℓ = 0 1 119872 minus 1 denotes the switch-ing signal used in the control

Consequently the closed-loop WNCS can be written as

[119909 (119896 + 1)

119909 (119896)] = [Φ119897(119896)Γ119897(119896)

K120574(119896)

I 0] [

119909 (119896)

119909 (119896 minus 1)] (23)

42 Stability Analysis of Adaptive Sampling Period WNCSThe control gains K

120574(119896)are assumed to be designed in such

a way that the closed-loop system is asymptotically stablewhen 120574(119896) = 119897(119896) Ideally the switching signal used in control120574(119896) is the same as the real signal 119897(119896) However this viewis unrealistic in WNCS with network-induced delay where120574(119896) = 119897(119896 minus 1) The evolution over one sampling period canbe described by two distinct types of behavior the nominaland mixed mode sampling periods as shown in Figure 2

(1)Nominal sampling period is when the system evolutionuses the right switching information

120574 (119896) = 119897 (119896 minus 1) 119897 (119896) = 119897 (119896 minus 1) (24)

(2) Mixed mode sampling period is when the systemcommand uses a wrong feedback gain

120574 (119896) = 119897 (119896 minus 1) 119897 (119896) = 119897 (119896 minus 1) (25)

Denoting Δ119898 as the samples spent in the mixed mode

since the delay is less than one sampling period it follows that

Δ119898 has a range of 0 le Δ

119898le 1 Moreover we assume that

the system is controlled using the right gains for at least Δ119899samples before another switching occurs

The next thing we should do is to guarantee the closed-loop system remaining stable with the designed state feed-back control gains when the switching signal is temporarilyuncertain Consider the scalars 120579

119899gt 0 120579

119898gt 0 and the

symmetric positive definite matrices P119899119897 P119898(119897120574)

with (119897 120574) isin

ℓ times ℓ which satisfy the following matrix inequalities

Φ119879

(119897120574)(P119898(119897120574)

)minus1

Φ(119897120574)

minus 120579119898P119898(119897120574)

lt 0 forall (119897 120574) isin ℓ times ℓ

Φ119879

(119897119897)P119899119897Φ(119897119897)

minus 120579119899P119899119897lt 0 forall119897 isin ℓ

(26)

Moreover consider the following two scalars

120573119899=max119897isinℓeigmax (P

119899

119897)

min119897isinℓeigmin (P119899119897 )

120573119898=

max(119897120574)isinℓtimesℓ

eigmax (P119898

(119897120574))

min(119897120574)isinℓtimesℓ

eigmin (P119898(119897120574))

(27)

where eigmax(X) and eigmin(X) denote the maximum and theminimum eigenvalues of matrix X respectively

Then the stability of the closed-loop WNCS (23) can beguaranteed by the following theorem

Theorem 5 Let 120579lowast119899 120579lowast119898be the solutions of the optimization

problems 120579lowast119899

= min 120579119899and 120579

lowast

119898= min 120579

119898subject to matrix

inequalities (26) Closed-loop system (23) is asymptoticallystable if

120573119899sdot 120573119898sdot (120579lowast

119899)Δ119899

sdot 120579lowast

119898lt 1 (28)

Proof We consider the following Lyapunov functions

119881119899(119896) = 119909

119879(119896)P119899119897119909 (119896) 119881

119898(119896) = 119909

119879(119896)P119898(119897120574)

119909 (119896)

(29)

Inequalities (13) show that

119881119899(1198961) lt (120579

lowast

119899)1198961minus1198960

119881119899(1198960) (30)

Mathematical Problems in Engineering 7

forall119896 isin [1198960 1198961) if the right switching signal is used in the

control 119897(119896) = 120574(119896) = 119897(1198960) then

119881119898(1198961) lt (120579

lowast

119898)1198961minus1198960

119881119898(1198960) (31)

forall119896 isin [1198960 1198961) if the switching signal in the control is not

necessarily the same as the real signal the pair 119897(119896) 120574(119896) takesan arbitrary value 119897(119896

0) 120574(119896

0) in ℓ times ℓ

Since the controller gains K119897are designed to make the

matricesΦ(119897119897)

stable the scalar 120579119899is smaller than one 120579

119899lt 1

The scalar 120579119898may be greater than one 120579

119898gt 1 since the

gains K120574are not designed to stabilize combinations other

thanΦ(120574120574)

Combining inequalities (26) yield

min119897isinℓ

eigmin (P119899

119897)1003817100381710038171003817119909(1198961)

1003817100381710038171003817

2

lt (120579lowast

119899)1198961minus1198960max119897isinℓ

eigmax (P119899

119897)1003817100381710038171003817119909(1198960)

1003817100381710038171003817

2forall119897 isin ℓ

min(119897120574)isinℓtimesℓ

eigmin (P119898

(119897120574))1003817100381710038171003817119909(1198961)

1003817100381710038171003817

2

lt max(119897120574)isinℓtimesℓ

eigmax (P119898

(119897120574))1003817100381710038171003817119909(1198960)

1003817100381710038171003817

2forall (119897 120574) isin ℓ times ℓ

(32)

With the definitions of 120573119899and 120573

119898 we can obtain the state

vector norm decay or growth rate in a nominal regime and inan uncertain switching signal regime as follows

1003817100381710038171003817119909(1198961)1003817100381710038171003817

2lt 120573119899sdot (120579lowast

119899)1198961minus11989601003817100381710038171003817119909(1198960)

1003817100381710038171003817

2 forall119897 isin ℓ

1003817100381710038171003817119909(1198961)1003817100381710038171003817

2lt 120573119898sdot (120579lowast

119898)1198961minus11989601003817100381710038171003817119909(1198960)

1003817100381710038171003817

2 forall (119897 120574) isin ℓ times ℓ

(33)

Let 119896119898119904describe the instants when the closed-loop system

jumps to a mixed mode with uncertain switching signal andlet 119896119899119904be the instance when the system enters into a normal

regime With definitions and bounds of Δ119898 and Δ119899 it followsthat

0 le Δ119898= 119896119899

119904minus 119896119898

119904le 1 119896

119898

119904+1minus 119896119899

119904le Δ119899 (34)

Without loss of generality we assume that the systemstarts with a mixed-mode behavior 119896119898

119904lt 119896119899

119904 The system

behavior in time interval 119896 isin [119896119898

119904 119896119898

119904+1) is then analyzed

Given that 119897(119896) = 120574(119896) forall119896 isin [119896119898

119904 119896119899

119904) and 119897(119896) =

120574(119896) forall119896 isin [119896119899

119904 119896119898

119904+1) using inequalities (26) the norm of the

state at the end of the sequence can be upper bounded asd

1003817100381710038171003817119909(119896119898

119904+1)1003817100381710038171003817

2lt 120573119898sdot 120573119899sdot (120579lowast

119898)119896119899

119904minus119896119906

119904sdot (120579lowast

119899)119896119898

119904+1minus119896119899

119904 1003817100381710038171003817119909(119896119898

119904)1003817100381710038171003817

2

lt1003817100381710038171003817119909(119896119898

119904+1)1003817100381710038171003817

2lt 120573119898sdot 120573119899sdot 120579lowast

119898sdot (120579lowast

119899)Δ1198991003817100381710038171003817119909(119896119898

119904)1003817100381710038171003817

2

(35)

It indicates that closed loop (16) will be asymptotically stableif condition (28) in Theorem 5 is satisfied

43 Choosing Rule of Update Period 119879119872

Theorem 6 Consider the WNCS with adaptive samplingperiod rule (3) The WNCS can be stabilized by the switched

state feedback controllers (22) whereas the survival time canmeet its expected value 119871

119890 if the update period 119879

119872satisfies the

following conditions

(1) 119879119872

ge max (Δ119899119894+ 1) [119879

max1

119879max2

119879max119899

]

(2) lfloor119871119890

119879119872

rfloor sdot 119879119872

le119864119894119899119894119905

119864

min 119879min1

119879min2

119879min119899

(36)

where Δ119899119894is the least nominal sampling period of control loop

119894 and the solution of Theorem 5 [119879max1

119879max2

119879max119899

] is theleast commonmultiple of the maximum sampling period for allthe 119899 control loops in the WNCS

Proof According to Theorem 5 control loop 119894 in the WNCSshould stay at least Δ119899

119894in a nominal sampling period in one

updating interval for stability With the consideration of 0 le

Δ119898

119894le 1 if the update period satisfies condition (1) all control

loops in the WNCS will meet the demands of their leastnominal sampling periods and will be stable As a completesystem WNCS is composed of 119899 control loops that will bestable when condition (1) is satisfied

Condition (2) inTheorem 6 provides the upper bound ofthe updating period For the power efficiency algorithm inTheorem 1 with the definition of 119896

119897(119895) there exists

119872

sum

119897=1

119896119897(119895)

119879119872

119879119897

+119879119872

119879119895

le lfloor119871119890

119879119872

rfloor sdot119879119872

min 119879min1

119879min2

119879min119899

(37)

If condition (2) inTheorem 6 is satisfied the item ((119864init119864)minus

sum119872

119897=1119896119897(119895)(119879119872119879119897) minus (119879

119872119879119895)) in the denominator of formula

(3) will be greater than zero which guarantees that formula(3) has physical meaning and is solvable The survival timewill thus reach the expected value by applying the powerefficiency algorithm

5 Numerical Example

Simulation studies are performed on a WNCS closed by anIEEE 80211b wireless network with two control loops sharingthe network resources The two control loops are assumed tohave the same dynamics but with different initial conditions

119894(119905) = [

minus1 minus01

0 095] 119909119894(119905) + [

minus015

minus043] 119906119894(119905)

119910119894(119905) = [1 1] 119909

119894(119905)

119894 = 1 2

1199091(0) = [minus5 5]

119879 119909

2(0) = [minus5 10]

119879

(38)

The wireless network parameters are set as in Table 2The time delays in both loops are less than one samplingperiod Some other necessary parameters are given as shownin Table 2

In Table 3 the minimum and maximum sampling peri-ods are computed by the methods in Section 3 Solving

8 Mathematical Problems in Engineering

Table 3 Simulation parameters

Parameter ValueInitial energy of both sensors 119864init 015 JSensor 1 expected survival time 119871

119890175 s

Sensor 2 expected survival time 1198711198902

70 sUnit transmission energy 119864 25 dbmNumber of sampling period candidates119872 10The minimum sampling period of loop1 119879min

11ms

The maximum sampling period of loop1 119879max1

256msThe minimum sampling period of loop2 119879min

21ms

The maximum sampling period of loop2 119879max2

256ms

Table 4 Controller gains of ten sampling modes

Sampling period (ms) Controller gain1198790= 1 119870

0= [11176 11188]

1198791= 29 119870

1= [394837 405512]

1198792= 58 119870

2= [194678 204319]

1198793= 86 119870

3= [125668 134279]

1198794= 114 119870

4= [90353 97947]

1198795= 143 119870

5= [68663 75265]

1198796= 171 119870

6= [5383 5947]

1198797= 199 119870

7= [4295 4769]

1198798= 227 119870

8= [4225 3852]

1198799= 256 119870

9= [3707 295]

inequality (28) in Theorem 5 it yields Δ1198991= Δ119899

2= 8 Δ119898 = 1

Also the update period can be chosen byTheorem 6 as 119879119872

=

150msSolving matrix inequalities (18) and (19) the controller

gains of ten switching modes can be obtained as in Table 4With the above simulation parameters and controller

gains the curves of survival time prediction power consump-tion and control output of both control loops are shown inFigures 3 to 8

Analyzing the simulation curves we have the followingresults

(1) Figures 3 6 4 and 7 imply that both sensor 1and sensor 2 can meet their expected survival timerequirements

(2) The power consumption in three cases of minimumsampling maximum sampling and the proposedadaptive sampling is compared in Figures 4 and7 It is obvious that the power is consumed muchfaster than the other two cases In the case of theadaptive sampling the power consumption variesaccording to both the requirements of the controlperformance and survival time At the beginning thepower consumption curves vary quickly and morepower are consumed because of the control systemsnot reaching stable yetThen after the control systems

0 20 40 60 800

50

100

150

200

Sens

or 1

life

span

(s)

250

300

350

400

Time (s)

Figure 3 Sensor 1 survival time prediction

0 15 30 45 60 750

005

01

015

Time (s)Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Ener

gy o

f sen

sor1

(J)

T 1T 1

Figure 4 Sensor 1 power consumption comparison in three cases

are settled it tends to theminimum consumption ratewhich is corresponding to the maximum samplingperiod

(3) Figures 5 and 8 are the control outputs of loop1and loop2 in the three cases mentioned above Thefigures show that the control systems can be stabilizedthrough the proposed joint design methods In threecases the adaptive sampling can get the control per-formances closed to the case of minimum sampling

(4) Combining Figures 4 5 7 and 8 it can be concludedthat the proposed joint design method achieves atradeoff between the performances of control andpower efficiency

Mathematical Problems in Engineering 9

Time (s)

The proposed adaptive sampling

0 15 30 45 60 75

0

2

4

6

8

Out

puty

of lo

op1

minus2

minus4

minus6

minus8

Constant sampling with minimum T1 = 1 msConstant sampling with maximum T1 = 256 ms

Figure 5 Control loop1 output comparison in three cases

0 20 40 60 800

50

100

150

Sens

or 2

life

span

(s)

200

250

300

350

Time (s)

Figure 6 Sensor 2 survival time prediction

6 Conclusion

This paper presents a joint design method for wirelessnetworked control systems with limited power constraint Apower efficiency algorithm based on the adaptive samplingperiod is put forward to satisfy the demands of sensorsurvival time and system stability Then the time-varyingcontrol system with transmission delay is modeled as aswitched system with uncertain switching signals A dwell-time-dependent controlmethod is discussed to guarantee thestability of WNCS Simulation results show the effectivenessof the proposed method and indicate that it can achievegood tradeoff performance Methods by which to reducepower consumption from the aspect of a single node as wellas balancing power consumption from the global networkperspective are worthy of further exploration

0 15 30 45 60 750

005

01

015

Time (s)

Ener

gy o

f sen

sor2

(J)

The proposed adaptive sampling

Constant sampling with minimum T2 = 1msConstant sampling with maximum T2 = 256ms

Figure 7 Sensor 2 power consumption comparison in three cases

0 15 30 45 60 75

0

5

10

15

Out

put y

of l

oop

2

minus15

minus10

minus5

Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Time (s)

T2T2

Figure 8 Control loop2 output comparison in three cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The working is sponsored by The NSFC (no 61202473) andThe Natural Science Foundation of Jiangsu Province (noBK2012551)

10 Mathematical Problems in Engineering

References

[1] W Dieterle H D Kochs and E Dittmar ldquoCommunicationarchitectures for distributed computer control systemsrdquoControlEngineering Practice vol 3 no 8 pp 1171ndash1176 1995

[2] H A Thompson ldquoWireless and Internet communicationstechnologies for monitoring and controlrdquo Control EngineeringPractice vol 12 no 6 pp 781ndash791 2004

[3] P R Kumar ldquoNew technological vistas for systems and controlrdquoIEEE Control Systems Magazine vol 21 no 1 pp 24ndash37 2001

[4] X Liu and A Goldsmith ldquoWireless medium access controlin networked control systemsrdquo in Proceedings of the AmericanControl Conference (AAC rsquo04) pp 3605ndash3610 Boston MassUSA July 2004

[5] J Colandairaj G W Irwin and W G Scanlon ldquoWirelessnetworked control systems with QoS-based samplingrdquo IETControl Theory and Applications vol 1 no 1 pp 430ndash438 2007

[6] N Israr W G Scanlon and G W Irwin ldquoA cross-layer com-munication framework forwireless networked control systemsrdquoin Proceedings of the 1st International Conference on WirelessCommunication Vehicular Technology Information Theory andAerospace and Electronic Systems Technology (VITAE rsquo09) pp577ndash581 Aalborg Denmark May 2009

[7] S Dai H Lin and S S Ge ldquoScheduling-and-control codesignfor a collection of networked control systems with uncertaindelaysrdquo IEEE Transactions on Control Systems Technology vol18 no 1 pp 66ndash78 2010

[8] A V Savkin ldquoAnalysis and synthesis of networked control sys-tems topological entropy observability robustness and optimalcontrolrdquo Automatica vol 42 no 1 pp 51ndash62 2006

[9] J Z Luo F Dong and J X Cao ldquoA novel task schedul-ing algorithm based on dynamic critical path and effectiveduplication for pervasive computing environmentrdquo WirelessCommunications amp Mobile Computing vol 10 no 10 pp 1283ndash1302 2010

[10] X Yin X Zhou Z Li and S Li ldquoJint rate control and powercontrol for lifetime maximization in Wreless Sensor NtworksrdquoJournal of Internet Technology vol 12 no 1 pp 69ndash78 2011

[11] M Pajic S Sundaram G J Pappas and R Mangharam ldquoThewireless control network a new approach for control overnetworksrdquo IEEE Transactions on Automatic Control vol 56 no10 pp 2305ndash2318 2011

[12] J Colandairaj G W Irwin and W G Scanlon ldquoA co-designsolution for wireless feedback controlrdquo in Proceeding of theInternational Conference on Networking Sensing and Control(ICNSC 07) pp 404ndash409 London UK April 2007

[13] W Liang and Y Yang ldquoMaximizing battery life routing in wire-less ad hoc networksrdquo in Proceedings of the 37th InternationalConference on System Sciences pp 4739ndash4746 IEEE HonoluluHawaii USA January 2004

[14] K Brian J Haberman and W Sheppard Overlapping ParticleSwarms for Energy-Efficient Routing in Sensor Networks Wire-less Network Springer 2011

[15] D Kim K Dantu andM Pedram ldquoLifetime prediction routinginmobile AdHoc networksrdquo in Proceedings of the IEEEWirelessCommunication and Networking Conference New York NYUSA 2003

[16] L Hetel J Daafouz and C Iung ldquoStability analysis for discretetime switched systems with temporary uncertain switchingsignalrdquo in Proceedings of the 46th IEEE Conference on Decisionand Control (CDC rsquo07) pp 5623ndash5628 New Orleans Fla USADecember 2007

[17] S Limin Wireless Sensor Network Tsinghua University PressBeijing China 2006

[18] Y He I Lee and L Guan ldquoDistributed algorithms for networklifetimemaximization in wireless visual sensor networksrdquo IEEETransactions on Circuits and Systems for Video Technology vol19 no 5 pp 704ndash718 2009

[19] B A Bakr and L Lilien ldquoA quantitative comparison of energyconsumption and WSN lifetime for LEACH and LEACH-SMrdquo in Proceedings of the 31st International Conference onDistributed Computing Systems Workshops (ICDCSW rsquo11) pp182ndash191 Minneapolis Minn USA June 2011

[20] C Fischione A Bonivento A Sangiovanni-Vincentelli F San-tucci and K H Johansson ldquoPerformance analysis of collabora-tive spatio-temporal processing for wireless sensor networksrdquoin Proceedings of the 3rd IEEE Consumer Communications andNetworking Conference (CCNC rsquo06) pp 325ndash329 Las VegasNev USA January 2006

[21] Y Iino ldquoSome considerations of wireless sensor network basedcontrol systemsrdquo in Proceedings of the SICE Conference onControl Systems (CCS rsquo07) Tokyo Japan 2007

[22] Y Iino T Hatanaka and M Fujita ldquoEvent-predictive controlfor energy saving of wireless networked control systemrdquo inProceedings of the American Control Conference (ACC rsquo09) pp2236ndash2242 St Louis Mo USA June 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

the purpose of adapting the sampling rate is to optimizebandwidth utilization not to save power A robust anddynamic cross-layer communication architecture for wirelessnetworked control system is presented in [6] by Israr et alThe protocol stack for WNCS comprises five layers Eachlayer contributes to the overall goal of reliable power-efficientcommunication However the control performance is nottaken into account in their work

Anumber of studies related to power efficiency inwirelesssensor networks (WSNs) and wireless networks have alsobeen conducted in [12ndash18] Current power efficiency researchalways falls into two categories one is reducing the powerconsumption of each single node in the network and theother is balancing the power consumption of all the nodes innetwork In [12] Colandairaj et al present a dynamic powercontrol strategy to minimize the communication powerconsumption of nodes by varying the transmission rate Aprotocol intending to balance power consumption from theremaining battery power of the node-based routing policy isproposed by Liang and Yang in [13] The nodes with greaterremaining power are allocated with more communicationtasks In [15] Kim et al propose a lifetime-based routingstrategy in which the survival time is estimated according tothe residual power and current ratio of power consumptionThe path with the longest node survival time is selected fordata transmission

Recently limited studies in [19ndash21] are conducted oneffective power saving strategies that specifically target atWNCS Fischione et al [20] propose a trade-off betweenwireless output power related to reliability and power con-sumption where a physical characteristic model revealedquantitative relations with communication outage probabil-ity They also focus on the lower layer optimal protocoldesign by considering the application layer requirementsLino [21] discusses the optimal sleepmode control of wirelessnetwork nodes and proposes a trade-off method betweencontrol performance and power consumption An optimalcontrol strategy is applied to optimize the control period In[22] event-predictive control for power saving of wirelessnetworked control system is discussed The key idea is tosave power by maximizing the control interval with con-strains of appropriate control performance The proposedcontrol method is rather complicated and requires onlineoptimizationmixed integer programming which reduces thepracticability Thus a simpler trade-off method for WNCS isrequired

Power consumption communication reliability and sys-tem stability exist simultaneously and react with one anotherin wireless networked control systems Supposed that thethree factors are interdependent most results achieved inwireless network power management and wired networkedcontrol systems cannot be directly applied to WNCS Thusthe motivation of this paper is to find a bridge which canlink the three factors and make a balance among thesefactors through the bridge parameter such that the overallsatisfactory performance can be achieved Fortunately thesampling period of sensor node is found to be the bridgeparameter From this point a joint designmethod of adaptivesampling power efficiency algorithm and coordinated control

method are discussed in this paper An updating rule ofsampling period is presented to satisfy the demands of wire-less life span under constrains of network schedulability andcontrol system stability Convergence of the power efficiencyalgorithm is further proved Subsequently the control systemis a varying-period system since the sampling periods ofsensors are time-varying It is then modeled as a class ofswitched control system with two types of behavior in eachupdate periodThe switched control law is applied to stabilizethe control system and stability conditions are discussedAlso the choosing rule of update period is given

The remaining sections are organized as follows Section 2is the problem formulation Section 3 presents the adaptivepower efficiency algorithm Section 4 discusses the coor-dinated wireless networked control system modeling anddesign method Numerical simulation is given in Section 5Section 6 is the conclusion of this paper

2 Problem Formulation

21 Description of WNCSs Consider the wireless networkedcontrol systems shown in Figure 1 There are three kindsof node in the system Power consumption varies for thedifferent kinds of node

Besides some necessary assumptions are made in thispaper as the follows

Assumption 1 The power of sensor and actuator nodes issupplied by battery while the power of controller is suppliedby base station

Assumption 2 The sensor and the actuator are clock drivenwhile the controller is event driven The sampling data ispacked in one packet for transmission with time stamp

Assumption 3 There exists transmission delay in the controlloop and it is assumed to be less than one sampling period

22 Analysis of Power Consumption inWNCSs Sensor poweris consumed by three processes data sampling sampledata reading by the ADC and data transfer The powerconsumption of the controller node is also consumed by threeprocesses receiving data calculating control variables andsending data packet The power of the actuator is consumedby two processes receiving data and DA conversion Thepower consumption of different tasks is shown in Table 1 (seein [6])

From the table we get the following two conclusions

(1) when the sensor transfers the same amount of data asthe actuator receives it will consume 25 times morepower than the actuator will consume

(2) data transfer consumes over 90 of the total sensorpower consumption

Given that the power required by control nodes can besupplied by the base station in most situations the powerrequired by the sensor nodes and actuators can be providedby batteries Thus sensors utilize the maximum amount

Mathematical Problems in Engineering 3

Actuator 1

Remote controller 1

Remote controller n

Actuator n

Plant 1

Plant n

Sensor 1

QoS data

Sensor node 1

Sensor node n

Adaptivesamplingalgorithm

modul

Adaptivesamplingalgorithm

modulQoS data

Sensor n

u1(j)

x1(j)

xn(j)

un(j) xn(j)

x1(j)

u1(j)

un(j)

T1(j + 1)

Tn(j + 1)

Figure 1 Structure of wireless networked control systems

Table 1 Power consumption of different tasks

Task Energy consumption (nAh)Receive data 8Transfer data 20Read data 0011Sample data 108

of power consumption in WNCS Managing the powerconsumption of sensors is the key to prolonging the survivaltime of the wireless network A direct and effective methodis to reduce as much of the transmission consumption aspossible by properly adjusting the amount of sample dataThis principle is the basis of our power control algorithm

In wireless networks the average power consumption ofsending a packet can be described as [3]

119864 = 119887 times packet size + 119888 (1)

where 119887 is the coefficient of power consumption and 119888 is thefixed power consumption of the node sending a data packetAccording to Assumptions 1 and 2 a sensor node sends apacket to the corresponding controller at every samplingtime Given the lack of packet retransmission the survivaltime of the sensor can be described as

119871 =

sumsum

119894=1

(119905 (119894) minus 119905 (119894 minus 1)) (2)

where 119871 is the survival time of the sensor node sum is themaximum number of packets sent by the sensor given aninitial power sum = lfloor119864init119864rfloor lfloor119909rfloor is the integral part of

119909 119864init is the given initial power of the sensor 119905(119894) 119894 isin

1 2 sum is the time that the sensor node sends the 119894thdata packet and 119905(0) is the initial time

Survival time is dependent on transfer intervals It canbe prolonged by increasing the transfer intervals Based onthis premise as well as on the knowledge of the relationshipbetween sampling period and control performance we cancooperatively design the control and the network perfor-mances by adaptively adjusting the sampling period with aproper rule

3 Adaptive Sampling PowerEfficiency Algorithm

31 Update Rule of Sampling Period For the consideration ofsimplicity and generality we choose one of the control loopsin the WNCS to describe the power control algorithm Let119879min be the lower bound of the sampling period to guarantee

the schedulability of the network and let 119879max be the upperbound of the sampling period to ensure system stabilitySupposing that there are 119872 candidate sampling periods forchoosing in the allowable range between the maximum andminimum bounds 119879

119897isin [119879

min 119879

max] 119897 = 1 2 119872

Furthermore an update period 119879119872 119879119872

gt 119879max is designed

for the sampling period renewal The sampling period isrenewed at each update instant as follows

119879 (119895 + 1) = 119879 (119895) + Δ119879 (119895 + 1)

Δ119879 (119895 + 1) = minus sgn ( (119895) minus 119871119890) sdot

119878 (119879max

minus 119879min

)

119872 minus 1

(3)

4 Mathematical Problems in Engineering

where

119878 = min

[[[

[

2 (119872 minus 1)10038161003816100381610038161003816 (119895) minus 119871

119890

10038161003816100381610038161003816

times ((119864init

119864

minus

119872

sum

119897=1

119896119897(119895)

119879119872

119879119897

minus119879119872

119879119895

)

times (119879max

minus 119879min

))

minus1]]]

]

119872 minus 1

(4)

119871119890is the demand of sensor survival time 119879(119895) is the value

of the sampling period in the 119895th updating interval sgn(119909)is the signal of scalar 119909 lfloor119909rfloor is the integral part of scalar119909 119896119897(119895) is the number of updating intervals in which the

sampling period is 119879119897during the previous 119895 updates (119895) is

the current predicted survival time of sensor node calculatedby the following formula

(119895) = 119895119879119872+119864rem (119895)

119864

times 119879 (119895) (5)

119864rem(119895) is the current remaining power at the updating instant119895119879119872

32 Convergence of Power Efficiency Algorithm

Theorem 1 For the WNCS described in Figure 1 consideringthe update rule of adaptive sampling period (3) if theminimumsampling period satisfies

119879min

gt119871119890

119904119906119898 (6)

then the actual survival time of sensor node will reach itsexpected value 119871

119890through the proposed rule of sampling period

update

Proof According to Formula (1) we have

(119895 + 1) = (119895 + 1) 119879119872+119864rem (119895 + 1)

119864

times 119879 (119895 + 1) (7)

The remaining power relationship at the two adjacentupdating instants is given by

119864rem (119895 + 1) = 119864rem (119895) minus119879119872

119879 (119895)times 119864 (8)

We assume that the sensor node has sampled the plantwith the sampling period 119879

119897for 119896119897(119895) times from the initial

to the current time instant The remaining power can becalculated based on the initial power and consumed power

119864rem (119895) = 119864init minus119872

sum

119897=1

119896119897(119895)

119879119872119864

119879119897

(9)

Power control error is defined as 119890(119895) = (119895) minus 119871119890

According to formulas (7) (8) and (9) the dynamics of theerror can be described as

119890 (119895 + 1) = 119890 (119895) + 119870 (119895) Δ119879 (119895 + 1) (10)

where 119870(119895) is defined as 119870(119895) = sum minus sum119872

119897=1119896119897(119895)(119879119872119879119897) minus

(119879119872119879(119895)) At the current time instant 119870(119895) is a known

variableThe following Lyapunov function is introduced to prove

convergence of the adaptive sampling power efficiency algo-rithm

119881 (119895) =1

21198902(119895) (11)

Considering Formula (10) it follows that

Δ119881 (119895) = 119881 (119895 + 1) minus 119881 (119895)

= 119870 (119895) Δ119879 (119895 + 1) 119890 (119895) +1

21198702(119895) Δ119879

2(119895 + 1)

(12)

With formula (3) we obtain

119890 (119895) gt 0 Δ119879 (119895 + 1) lt 0

119890 (119895) lt 0 Δ119879 (119895 + 1) gt 0

997904rArr Δ119879 (119895 + 1) 119890 (119895) lt 0

1003816100381610038161003816Δ119879 (119895 + 1)1003816100381610038161003816

1003816100381610038161003816119890 (119895)1003816100381610038161003816 gt

1

2119870 (119895) Δ119879

2(119895 + 1)

(13)

From inequalities (13) it can be concluded that Δ119881(119895) lt0 Furthermore to guarantee that the minimum survivaltime can reach the expected value the minimum samplingperiod is bounded by 119879

minge 119871119890sum Consequently the

error system is stable and the survival time can convergeto the expected value if the conditions in Theorem 1 aresatisfied

Remark 2 (prediction of survival time) The actual survivaltime is unavailable at the current instant because the powerconsumption is time varying However it can be predictedby the known information of the remaining power andsampling period at the current instant Formula (5) providesthe prediction and indicates that the survival time of thenode will be (119895) if the sensor node maintains the samplingperiod 119879(119895) as unchanged from the current instant 119905 = 119895119879

119872

The prediction of the survival time serves as a substitute forreal survival time and is used to calculate the new samplingperiod

Remark 3 (lower bound of sampling period) Taking IEEE80211b as an example the lower bound of the samplingperiod of sensor 119879min can be determined by the followingformula

119879min

= max119871119890

sum119878mc times 2 times 119873

119876 (14)

Mathematical Problems in Engineering 5

where (119878mc times 2 times 119873)119876 is the allowable minimum samplingperiod when the wireless network can be schedulable (see in[6]) and

119876 =119878mc

119879DIFS + (119862119882min times 119879SIFS2) + 119879frame + 119879SIFS + 119879ACK(15)

119879frame =119878PHY119877119897

+119878MAC + 119878mc

119877119905

119879frame =119878PHY119877119897

+119878MAC + 119878mc

119877119905

(16)

119877119905is the transmission rate 119877

119897is the legacy transmission rate

119878mc is the measurement-control data size 119878PHY is the sizeof control frame in physical layer and 119878MAC is the data sizeof ACK and is a confirmed sign in the header of TCP datapacket that confirms the received TCP message 119879SIFS is theshortest time period of the 80211b protocol for the intervalof frames requiring immediate response 119879DIFS is the timesegment for the interval of the time frame of the distributedcoordination function for sending in IEEE 80211b 119879PIFS isthe time segment for the interval of the time frame of thecentralized coordination function for sending which satisfies

119879PIFS = 119879DIFS minus 119879slot 119879slot = 119879PIFS minus 119879SIFS (17)

CW is the contention window Wireless network parametersunder the 80211b direct sequence spread spectrum are shownin Table 2

Remark 4 (upper bound of sampling period) For a SISOsystem the maximum sampling period can be obtainedusing Shannon sampling theorem For a MIMO system thefollowing method can be used to obtain the upper bound Ifthe system feedback control law is given ahead then119879max canbe obtained by solving the following optimal problem

max 119879

st100381710038171003817100381710038171003817100381710038171003817

eig(119890A119879 + int

119879

0

119890A119879

119889119905BK0)

100381710038171003817100381710038171003817100381710038171003817

lt 1

(18)

where A is the system matrix and B is the control inputmatrix K

0is the Kalman gain which satisfies

Re (eig (A + BK0)) lt 0 (19)

where Re(120592) denotes the real part of 120592 and eig(X) denotes theeigenvalues of matrix X Gain K

0can be determined by the

pole assignment in the continuous time domain

However the above optimal problem is difficult to solvedirectlyThe following iterationmethod can be used to obtainthe approximate optimal value of 119879max

Step 1 Let 119902 = 1 and the initial value of 119879max is 119879max(119902) =

2119879min If the condition can satisfy the constraints of the

optimal problem go to Step 2 or else go to Step 3

Table 2 Wireless network parameters under 80211b directsequence spread spectrum

Parameter Value119877119905

11Mbps119877119897

1Mbps119879SIFS 119879DIFS 119879slot 10 50 20 us119878MAC 119878PHY 34 24 bytes119878mc 80 bytes119878ACK 14 bytes + PHY headerCWmin CWmax 32 1024

Step 2 Let 119902 = 119902 + 1 and let 119879max(119902) = 2119879

max(119902 minus 1) If the

condition can fulfill the constraints of the optimal problemcycle Step 2 or else end the iteration and then let 119879max

=

119879max

(119902 minus 1)

Step 3 Let 119879max(119902) = (1 + (34)

119902)119879

min If the condition stilldoes not satisfy the constraints of the optimal problem let119902 = 119902 + 1 and cycle Step 3 or else end the iteration and let119879max

= 119879max

(119902)

4 Modeling and Stability Analysis of AdaptiveSampling Period WNCS

41 Modeling of Adaptive Sampling Period WNCS Consid-ering the generality one of the control loops in the WNCSis chosen as an example to illustrate the modeling approachFor theWNCS power consumptionmanaged by algorithm inSection 2 the dynamics of the control system is time-varyingSince the sampling period varies among the 119872 candidatesthe system can be considered a switched system with 119872

modes from the perspective of the switched control systemscheme Each switching mode is corresponded to one ofthe candidates According to Assumption 1 the sensor andactuator nodes are assumed to be clock-driven It results thatthe switching occurs at some of the sampling instants Addi-tionally according to Assumption 3 the inevitable existenceof network-induced delay is taken into account and it is lessthan one sampling period

We consider a plant of control loop 119894 in the WNCS withthe following dynamics

119894(119905) = A119894119909119894 (119905) + B119894119906119894 (119905)

119910119894(119905) = C119894119909119894 (119905)

(20)

where 119909119894(119905) isin R119899 is the plant state 119906119894(119905) isin R119898 is the controlinput and 119910

119894(119905) isin R119901 is the plant output A119894 isin R119899times119899 B119894 isin

R119899times119898 and C119894 isin R119901times119899 are the matrices of state control inputand output matrices respectively Due to the generality of 119894we omit the superscript 119894 in the following model descriptionand deduction

Discretizing system (20) with sampling rate 119879119897and

considering network-induced delay less than one sampling

6 Mathematical Problems in Engineering

Mixed mode Mixed model(k minus 1) ne l(k) l(k minus 1) = l(k) l(k + 3) ne l(k + 2)

u(k minus 1)

u(k + 1)120591l(k)(k)

k + 1 k + 2 k + 3

Sensor

Controller

node

node

nodeu(k minus 1)= Kl(kminus1)x(k minus 1)

Actuator

k minus 1 k

u(k)

Nominal behavior

One monitor interval

Tl(kminus1) Tl(k) Tl(k +3)

TM

Figure 2 Evolution over one period of switched WNCS with two types of behavior

period the discrete dynamics of the open control loop canbe described as

119909 (119896 + 1) = Φ119897(119896)119909 (119896) + Γ119897(119896)119906 (119896 minus 1)

119910 (119896) = C119909 (119896) (21)

whereΦ119897(119896)

= 119890A119879119897(119896) Γ

119897= int119879119897(119896)

0119890A119905119889119905B and 119897(119896) is the identi-

fication of sampling period at the 119896th sampling instant 119897(119896) isinZ rarr ℓ = 0 1 119872 minus 1

For the discrete switched system (21) a switched statefeedback controller is introduced in the following form

119906 (119896) = K120574(119896)

119909 (119896) (22)

where 120574(119896) isin Z rarr ℓ = 0 1 119872 minus 1 denotes the switch-ing signal used in the control

Consequently the closed-loop WNCS can be written as

[119909 (119896 + 1)

119909 (119896)] = [Φ119897(119896)Γ119897(119896)

K120574(119896)

I 0] [

119909 (119896)

119909 (119896 minus 1)] (23)

42 Stability Analysis of Adaptive Sampling Period WNCSThe control gains K

120574(119896)are assumed to be designed in such

a way that the closed-loop system is asymptotically stablewhen 120574(119896) = 119897(119896) Ideally the switching signal used in control120574(119896) is the same as the real signal 119897(119896) However this viewis unrealistic in WNCS with network-induced delay where120574(119896) = 119897(119896 minus 1) The evolution over one sampling period canbe described by two distinct types of behavior the nominaland mixed mode sampling periods as shown in Figure 2

(1)Nominal sampling period is when the system evolutionuses the right switching information

120574 (119896) = 119897 (119896 minus 1) 119897 (119896) = 119897 (119896 minus 1) (24)

(2) Mixed mode sampling period is when the systemcommand uses a wrong feedback gain

120574 (119896) = 119897 (119896 minus 1) 119897 (119896) = 119897 (119896 minus 1) (25)

Denoting Δ119898 as the samples spent in the mixed mode

since the delay is less than one sampling period it follows that

Δ119898 has a range of 0 le Δ

119898le 1 Moreover we assume that

the system is controlled using the right gains for at least Δ119899samples before another switching occurs

The next thing we should do is to guarantee the closed-loop system remaining stable with the designed state feed-back control gains when the switching signal is temporarilyuncertain Consider the scalars 120579

119899gt 0 120579

119898gt 0 and the

symmetric positive definite matrices P119899119897 P119898(119897120574)

with (119897 120574) isin

ℓ times ℓ which satisfy the following matrix inequalities

Φ119879

(119897120574)(P119898(119897120574)

)minus1

Φ(119897120574)

minus 120579119898P119898(119897120574)

lt 0 forall (119897 120574) isin ℓ times ℓ

Φ119879

(119897119897)P119899119897Φ(119897119897)

minus 120579119899P119899119897lt 0 forall119897 isin ℓ

(26)

Moreover consider the following two scalars

120573119899=max119897isinℓeigmax (P

119899

119897)

min119897isinℓeigmin (P119899119897 )

120573119898=

max(119897120574)isinℓtimesℓ

eigmax (P119898

(119897120574))

min(119897120574)isinℓtimesℓ

eigmin (P119898(119897120574))

(27)

where eigmax(X) and eigmin(X) denote the maximum and theminimum eigenvalues of matrix X respectively

Then the stability of the closed-loop WNCS (23) can beguaranteed by the following theorem

Theorem 5 Let 120579lowast119899 120579lowast119898be the solutions of the optimization

problems 120579lowast119899

= min 120579119899and 120579

lowast

119898= min 120579

119898subject to matrix

inequalities (26) Closed-loop system (23) is asymptoticallystable if

120573119899sdot 120573119898sdot (120579lowast

119899)Δ119899

sdot 120579lowast

119898lt 1 (28)

Proof We consider the following Lyapunov functions

119881119899(119896) = 119909

119879(119896)P119899119897119909 (119896) 119881

119898(119896) = 119909

119879(119896)P119898(119897120574)

119909 (119896)

(29)

Inequalities (13) show that

119881119899(1198961) lt (120579

lowast

119899)1198961minus1198960

119881119899(1198960) (30)

Mathematical Problems in Engineering 7

forall119896 isin [1198960 1198961) if the right switching signal is used in the

control 119897(119896) = 120574(119896) = 119897(1198960) then

119881119898(1198961) lt (120579

lowast

119898)1198961minus1198960

119881119898(1198960) (31)

forall119896 isin [1198960 1198961) if the switching signal in the control is not

necessarily the same as the real signal the pair 119897(119896) 120574(119896) takesan arbitrary value 119897(119896

0) 120574(119896

0) in ℓ times ℓ

Since the controller gains K119897are designed to make the

matricesΦ(119897119897)

stable the scalar 120579119899is smaller than one 120579

119899lt 1

The scalar 120579119898may be greater than one 120579

119898gt 1 since the

gains K120574are not designed to stabilize combinations other

thanΦ(120574120574)

Combining inequalities (26) yield

min119897isinℓ

eigmin (P119899

119897)1003817100381710038171003817119909(1198961)

1003817100381710038171003817

2

lt (120579lowast

119899)1198961minus1198960max119897isinℓ

eigmax (P119899

119897)1003817100381710038171003817119909(1198960)

1003817100381710038171003817

2forall119897 isin ℓ

min(119897120574)isinℓtimesℓ

eigmin (P119898

(119897120574))1003817100381710038171003817119909(1198961)

1003817100381710038171003817

2

lt max(119897120574)isinℓtimesℓ

eigmax (P119898

(119897120574))1003817100381710038171003817119909(1198960)

1003817100381710038171003817

2forall (119897 120574) isin ℓ times ℓ

(32)

With the definitions of 120573119899and 120573

119898 we can obtain the state

vector norm decay or growth rate in a nominal regime and inan uncertain switching signal regime as follows

1003817100381710038171003817119909(1198961)1003817100381710038171003817

2lt 120573119899sdot (120579lowast

119899)1198961minus11989601003817100381710038171003817119909(1198960)

1003817100381710038171003817

2 forall119897 isin ℓ

1003817100381710038171003817119909(1198961)1003817100381710038171003817

2lt 120573119898sdot (120579lowast

119898)1198961minus11989601003817100381710038171003817119909(1198960)

1003817100381710038171003817

2 forall (119897 120574) isin ℓ times ℓ

(33)

Let 119896119898119904describe the instants when the closed-loop system

jumps to a mixed mode with uncertain switching signal andlet 119896119899119904be the instance when the system enters into a normal

regime With definitions and bounds of Δ119898 and Δ119899 it followsthat

0 le Δ119898= 119896119899

119904minus 119896119898

119904le 1 119896

119898

119904+1minus 119896119899

119904le Δ119899 (34)

Without loss of generality we assume that the systemstarts with a mixed-mode behavior 119896119898

119904lt 119896119899

119904 The system

behavior in time interval 119896 isin [119896119898

119904 119896119898

119904+1) is then analyzed

Given that 119897(119896) = 120574(119896) forall119896 isin [119896119898

119904 119896119899

119904) and 119897(119896) =

120574(119896) forall119896 isin [119896119899

119904 119896119898

119904+1) using inequalities (26) the norm of the

state at the end of the sequence can be upper bounded asd

1003817100381710038171003817119909(119896119898

119904+1)1003817100381710038171003817

2lt 120573119898sdot 120573119899sdot (120579lowast

119898)119896119899

119904minus119896119906

119904sdot (120579lowast

119899)119896119898

119904+1minus119896119899

119904 1003817100381710038171003817119909(119896119898

119904)1003817100381710038171003817

2

lt1003817100381710038171003817119909(119896119898

119904+1)1003817100381710038171003817

2lt 120573119898sdot 120573119899sdot 120579lowast

119898sdot (120579lowast

119899)Δ1198991003817100381710038171003817119909(119896119898

119904)1003817100381710038171003817

2

(35)

It indicates that closed loop (16) will be asymptotically stableif condition (28) in Theorem 5 is satisfied

43 Choosing Rule of Update Period 119879119872

Theorem 6 Consider the WNCS with adaptive samplingperiod rule (3) The WNCS can be stabilized by the switched

state feedback controllers (22) whereas the survival time canmeet its expected value 119871

119890 if the update period 119879

119872satisfies the

following conditions

(1) 119879119872

ge max (Δ119899119894+ 1) [119879

max1

119879max2

119879max119899

]

(2) lfloor119871119890

119879119872

rfloor sdot 119879119872

le119864119894119899119894119905

119864

min 119879min1

119879min2

119879min119899

(36)

where Δ119899119894is the least nominal sampling period of control loop

119894 and the solution of Theorem 5 [119879max1

119879max2

119879max119899

] is theleast commonmultiple of the maximum sampling period for allthe 119899 control loops in the WNCS

Proof According to Theorem 5 control loop 119894 in the WNCSshould stay at least Δ119899

119894in a nominal sampling period in one

updating interval for stability With the consideration of 0 le

Δ119898

119894le 1 if the update period satisfies condition (1) all control

loops in the WNCS will meet the demands of their leastnominal sampling periods and will be stable As a completesystem WNCS is composed of 119899 control loops that will bestable when condition (1) is satisfied

Condition (2) inTheorem 6 provides the upper bound ofthe updating period For the power efficiency algorithm inTheorem 1 with the definition of 119896

119897(119895) there exists

119872

sum

119897=1

119896119897(119895)

119879119872

119879119897

+119879119872

119879119895

le lfloor119871119890

119879119872

rfloor sdot119879119872

min 119879min1

119879min2

119879min119899

(37)

If condition (2) inTheorem 6 is satisfied the item ((119864init119864)minus

sum119872

119897=1119896119897(119895)(119879119872119879119897) minus (119879

119872119879119895)) in the denominator of formula

(3) will be greater than zero which guarantees that formula(3) has physical meaning and is solvable The survival timewill thus reach the expected value by applying the powerefficiency algorithm

5 Numerical Example

Simulation studies are performed on a WNCS closed by anIEEE 80211b wireless network with two control loops sharingthe network resources The two control loops are assumed tohave the same dynamics but with different initial conditions

119894(119905) = [

minus1 minus01

0 095] 119909119894(119905) + [

minus015

minus043] 119906119894(119905)

119910119894(119905) = [1 1] 119909

119894(119905)

119894 = 1 2

1199091(0) = [minus5 5]

119879 119909

2(0) = [minus5 10]

119879

(38)

The wireless network parameters are set as in Table 2The time delays in both loops are less than one samplingperiod Some other necessary parameters are given as shownin Table 2

In Table 3 the minimum and maximum sampling peri-ods are computed by the methods in Section 3 Solving

8 Mathematical Problems in Engineering

Table 3 Simulation parameters

Parameter ValueInitial energy of both sensors 119864init 015 JSensor 1 expected survival time 119871

119890175 s

Sensor 2 expected survival time 1198711198902

70 sUnit transmission energy 119864 25 dbmNumber of sampling period candidates119872 10The minimum sampling period of loop1 119879min

11ms

The maximum sampling period of loop1 119879max1

256msThe minimum sampling period of loop2 119879min

21ms

The maximum sampling period of loop2 119879max2

256ms

Table 4 Controller gains of ten sampling modes

Sampling period (ms) Controller gain1198790= 1 119870

0= [11176 11188]

1198791= 29 119870

1= [394837 405512]

1198792= 58 119870

2= [194678 204319]

1198793= 86 119870

3= [125668 134279]

1198794= 114 119870

4= [90353 97947]

1198795= 143 119870

5= [68663 75265]

1198796= 171 119870

6= [5383 5947]

1198797= 199 119870

7= [4295 4769]

1198798= 227 119870

8= [4225 3852]

1198799= 256 119870

9= [3707 295]

inequality (28) in Theorem 5 it yields Δ1198991= Δ119899

2= 8 Δ119898 = 1

Also the update period can be chosen byTheorem 6 as 119879119872

=

150msSolving matrix inequalities (18) and (19) the controller

gains of ten switching modes can be obtained as in Table 4With the above simulation parameters and controller

gains the curves of survival time prediction power consump-tion and control output of both control loops are shown inFigures 3 to 8

Analyzing the simulation curves we have the followingresults

(1) Figures 3 6 4 and 7 imply that both sensor 1and sensor 2 can meet their expected survival timerequirements

(2) The power consumption in three cases of minimumsampling maximum sampling and the proposedadaptive sampling is compared in Figures 4 and7 It is obvious that the power is consumed muchfaster than the other two cases In the case of theadaptive sampling the power consumption variesaccording to both the requirements of the controlperformance and survival time At the beginning thepower consumption curves vary quickly and morepower are consumed because of the control systemsnot reaching stable yetThen after the control systems

0 20 40 60 800

50

100

150

200

Sens

or 1

life

span

(s)

250

300

350

400

Time (s)

Figure 3 Sensor 1 survival time prediction

0 15 30 45 60 750

005

01

015

Time (s)Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Ener

gy o

f sen

sor1

(J)

T 1T 1

Figure 4 Sensor 1 power consumption comparison in three cases

are settled it tends to theminimum consumption ratewhich is corresponding to the maximum samplingperiod

(3) Figures 5 and 8 are the control outputs of loop1and loop2 in the three cases mentioned above Thefigures show that the control systems can be stabilizedthrough the proposed joint design methods In threecases the adaptive sampling can get the control per-formances closed to the case of minimum sampling

(4) Combining Figures 4 5 7 and 8 it can be concludedthat the proposed joint design method achieves atradeoff between the performances of control andpower efficiency

Mathematical Problems in Engineering 9

Time (s)

The proposed adaptive sampling

0 15 30 45 60 75

0

2

4

6

8

Out

puty

of lo

op1

minus2

minus4

minus6

minus8

Constant sampling with minimum T1 = 1 msConstant sampling with maximum T1 = 256 ms

Figure 5 Control loop1 output comparison in three cases

0 20 40 60 800

50

100

150

Sens

or 2

life

span

(s)

200

250

300

350

Time (s)

Figure 6 Sensor 2 survival time prediction

6 Conclusion

This paper presents a joint design method for wirelessnetworked control systems with limited power constraint Apower efficiency algorithm based on the adaptive samplingperiod is put forward to satisfy the demands of sensorsurvival time and system stability Then the time-varyingcontrol system with transmission delay is modeled as aswitched system with uncertain switching signals A dwell-time-dependent controlmethod is discussed to guarantee thestability of WNCS Simulation results show the effectivenessof the proposed method and indicate that it can achievegood tradeoff performance Methods by which to reducepower consumption from the aspect of a single node as wellas balancing power consumption from the global networkperspective are worthy of further exploration

0 15 30 45 60 750

005

01

015

Time (s)

Ener

gy o

f sen

sor2

(J)

The proposed adaptive sampling

Constant sampling with minimum T2 = 1msConstant sampling with maximum T2 = 256ms

Figure 7 Sensor 2 power consumption comparison in three cases

0 15 30 45 60 75

0

5

10

15

Out

put y

of l

oop

2

minus15

minus10

minus5

Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Time (s)

T2T2

Figure 8 Control loop2 output comparison in three cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The working is sponsored by The NSFC (no 61202473) andThe Natural Science Foundation of Jiangsu Province (noBK2012551)

10 Mathematical Problems in Engineering

References

[1] W Dieterle H D Kochs and E Dittmar ldquoCommunicationarchitectures for distributed computer control systemsrdquoControlEngineering Practice vol 3 no 8 pp 1171ndash1176 1995

[2] H A Thompson ldquoWireless and Internet communicationstechnologies for monitoring and controlrdquo Control EngineeringPractice vol 12 no 6 pp 781ndash791 2004

[3] P R Kumar ldquoNew technological vistas for systems and controlrdquoIEEE Control Systems Magazine vol 21 no 1 pp 24ndash37 2001

[4] X Liu and A Goldsmith ldquoWireless medium access controlin networked control systemsrdquo in Proceedings of the AmericanControl Conference (AAC rsquo04) pp 3605ndash3610 Boston MassUSA July 2004

[5] J Colandairaj G W Irwin and W G Scanlon ldquoWirelessnetworked control systems with QoS-based samplingrdquo IETControl Theory and Applications vol 1 no 1 pp 430ndash438 2007

[6] N Israr W G Scanlon and G W Irwin ldquoA cross-layer com-munication framework forwireless networked control systemsrdquoin Proceedings of the 1st International Conference on WirelessCommunication Vehicular Technology Information Theory andAerospace and Electronic Systems Technology (VITAE rsquo09) pp577ndash581 Aalborg Denmark May 2009

[7] S Dai H Lin and S S Ge ldquoScheduling-and-control codesignfor a collection of networked control systems with uncertaindelaysrdquo IEEE Transactions on Control Systems Technology vol18 no 1 pp 66ndash78 2010

[8] A V Savkin ldquoAnalysis and synthesis of networked control sys-tems topological entropy observability robustness and optimalcontrolrdquo Automatica vol 42 no 1 pp 51ndash62 2006

[9] J Z Luo F Dong and J X Cao ldquoA novel task schedul-ing algorithm based on dynamic critical path and effectiveduplication for pervasive computing environmentrdquo WirelessCommunications amp Mobile Computing vol 10 no 10 pp 1283ndash1302 2010

[10] X Yin X Zhou Z Li and S Li ldquoJint rate control and powercontrol for lifetime maximization in Wreless Sensor NtworksrdquoJournal of Internet Technology vol 12 no 1 pp 69ndash78 2011

[11] M Pajic S Sundaram G J Pappas and R Mangharam ldquoThewireless control network a new approach for control overnetworksrdquo IEEE Transactions on Automatic Control vol 56 no10 pp 2305ndash2318 2011

[12] J Colandairaj G W Irwin and W G Scanlon ldquoA co-designsolution for wireless feedback controlrdquo in Proceeding of theInternational Conference on Networking Sensing and Control(ICNSC 07) pp 404ndash409 London UK April 2007

[13] W Liang and Y Yang ldquoMaximizing battery life routing in wire-less ad hoc networksrdquo in Proceedings of the 37th InternationalConference on System Sciences pp 4739ndash4746 IEEE HonoluluHawaii USA January 2004

[14] K Brian J Haberman and W Sheppard Overlapping ParticleSwarms for Energy-Efficient Routing in Sensor Networks Wire-less Network Springer 2011

[15] D Kim K Dantu andM Pedram ldquoLifetime prediction routinginmobile AdHoc networksrdquo in Proceedings of the IEEEWirelessCommunication and Networking Conference New York NYUSA 2003

[16] L Hetel J Daafouz and C Iung ldquoStability analysis for discretetime switched systems with temporary uncertain switchingsignalrdquo in Proceedings of the 46th IEEE Conference on Decisionand Control (CDC rsquo07) pp 5623ndash5628 New Orleans Fla USADecember 2007

[17] S Limin Wireless Sensor Network Tsinghua University PressBeijing China 2006

[18] Y He I Lee and L Guan ldquoDistributed algorithms for networklifetimemaximization in wireless visual sensor networksrdquo IEEETransactions on Circuits and Systems for Video Technology vol19 no 5 pp 704ndash718 2009

[19] B A Bakr and L Lilien ldquoA quantitative comparison of energyconsumption and WSN lifetime for LEACH and LEACH-SMrdquo in Proceedings of the 31st International Conference onDistributed Computing Systems Workshops (ICDCSW rsquo11) pp182ndash191 Minneapolis Minn USA June 2011

[20] C Fischione A Bonivento A Sangiovanni-Vincentelli F San-tucci and K H Johansson ldquoPerformance analysis of collabora-tive spatio-temporal processing for wireless sensor networksrdquoin Proceedings of the 3rd IEEE Consumer Communications andNetworking Conference (CCNC rsquo06) pp 325ndash329 Las VegasNev USA January 2006

[21] Y Iino ldquoSome considerations of wireless sensor network basedcontrol systemsrdquo in Proceedings of the SICE Conference onControl Systems (CCS rsquo07) Tokyo Japan 2007

[22] Y Iino T Hatanaka and M Fujita ldquoEvent-predictive controlfor energy saving of wireless networked control systemrdquo inProceedings of the American Control Conference (ACC rsquo09) pp2236ndash2242 St Louis Mo USA June 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering 3

Actuator 1

Remote controller 1

Remote controller n

Actuator n

Plant 1

Plant n

Sensor 1

QoS data

Sensor node 1

Sensor node n

Adaptivesamplingalgorithm

modul

Adaptivesamplingalgorithm

modulQoS data

Sensor n

u1(j)

x1(j)

xn(j)

un(j) xn(j)

x1(j)

u1(j)

un(j)

T1(j + 1)

Tn(j + 1)

Figure 1 Structure of wireless networked control systems

Table 1 Power consumption of different tasks

Task Energy consumption (nAh)Receive data 8Transfer data 20Read data 0011Sample data 108

of power consumption in WNCS Managing the powerconsumption of sensors is the key to prolonging the survivaltime of the wireless network A direct and effective methodis to reduce as much of the transmission consumption aspossible by properly adjusting the amount of sample dataThis principle is the basis of our power control algorithm

In wireless networks the average power consumption ofsending a packet can be described as [3]

119864 = 119887 times packet size + 119888 (1)

where 119887 is the coefficient of power consumption and 119888 is thefixed power consumption of the node sending a data packetAccording to Assumptions 1 and 2 a sensor node sends apacket to the corresponding controller at every samplingtime Given the lack of packet retransmission the survivaltime of the sensor can be described as

119871 =

sumsum

119894=1

(119905 (119894) minus 119905 (119894 minus 1)) (2)

where 119871 is the survival time of the sensor node sum is themaximum number of packets sent by the sensor given aninitial power sum = lfloor119864init119864rfloor lfloor119909rfloor is the integral part of

119909 119864init is the given initial power of the sensor 119905(119894) 119894 isin

1 2 sum is the time that the sensor node sends the 119894thdata packet and 119905(0) is the initial time

Survival time is dependent on transfer intervals It canbe prolonged by increasing the transfer intervals Based onthis premise as well as on the knowledge of the relationshipbetween sampling period and control performance we cancooperatively design the control and the network perfor-mances by adaptively adjusting the sampling period with aproper rule

3 Adaptive Sampling PowerEfficiency Algorithm

31 Update Rule of Sampling Period For the consideration ofsimplicity and generality we choose one of the control loopsin the WNCS to describe the power control algorithm Let119879min be the lower bound of the sampling period to guarantee

the schedulability of the network and let 119879max be the upperbound of the sampling period to ensure system stabilitySupposing that there are 119872 candidate sampling periods forchoosing in the allowable range between the maximum andminimum bounds 119879

119897isin [119879

min 119879

max] 119897 = 1 2 119872

Furthermore an update period 119879119872 119879119872

gt 119879max is designed

for the sampling period renewal The sampling period isrenewed at each update instant as follows

119879 (119895 + 1) = 119879 (119895) + Δ119879 (119895 + 1)

Δ119879 (119895 + 1) = minus sgn ( (119895) minus 119871119890) sdot

119878 (119879max

minus 119879min

)

119872 minus 1

(3)

4 Mathematical Problems in Engineering

where

119878 = min

[[[

[

2 (119872 minus 1)10038161003816100381610038161003816 (119895) minus 119871

119890

10038161003816100381610038161003816

times ((119864init

119864

minus

119872

sum

119897=1

119896119897(119895)

119879119872

119879119897

minus119879119872

119879119895

)

times (119879max

minus 119879min

))

minus1]]]

]

119872 minus 1

(4)

119871119890is the demand of sensor survival time 119879(119895) is the value

of the sampling period in the 119895th updating interval sgn(119909)is the signal of scalar 119909 lfloor119909rfloor is the integral part of scalar119909 119896119897(119895) is the number of updating intervals in which the

sampling period is 119879119897during the previous 119895 updates (119895) is

the current predicted survival time of sensor node calculatedby the following formula

(119895) = 119895119879119872+119864rem (119895)

119864

times 119879 (119895) (5)

119864rem(119895) is the current remaining power at the updating instant119895119879119872

32 Convergence of Power Efficiency Algorithm

Theorem 1 For the WNCS described in Figure 1 consideringthe update rule of adaptive sampling period (3) if theminimumsampling period satisfies

119879min

gt119871119890

119904119906119898 (6)

then the actual survival time of sensor node will reach itsexpected value 119871

119890through the proposed rule of sampling period

update

Proof According to Formula (1) we have

(119895 + 1) = (119895 + 1) 119879119872+119864rem (119895 + 1)

119864

times 119879 (119895 + 1) (7)

The remaining power relationship at the two adjacentupdating instants is given by

119864rem (119895 + 1) = 119864rem (119895) minus119879119872

119879 (119895)times 119864 (8)

We assume that the sensor node has sampled the plantwith the sampling period 119879

119897for 119896119897(119895) times from the initial

to the current time instant The remaining power can becalculated based on the initial power and consumed power

119864rem (119895) = 119864init minus119872

sum

119897=1

119896119897(119895)

119879119872119864

119879119897

(9)

Power control error is defined as 119890(119895) = (119895) minus 119871119890

According to formulas (7) (8) and (9) the dynamics of theerror can be described as

119890 (119895 + 1) = 119890 (119895) + 119870 (119895) Δ119879 (119895 + 1) (10)

where 119870(119895) is defined as 119870(119895) = sum minus sum119872

119897=1119896119897(119895)(119879119872119879119897) minus

(119879119872119879(119895)) At the current time instant 119870(119895) is a known

variableThe following Lyapunov function is introduced to prove

convergence of the adaptive sampling power efficiency algo-rithm

119881 (119895) =1

21198902(119895) (11)

Considering Formula (10) it follows that

Δ119881 (119895) = 119881 (119895 + 1) minus 119881 (119895)

= 119870 (119895) Δ119879 (119895 + 1) 119890 (119895) +1

21198702(119895) Δ119879

2(119895 + 1)

(12)

With formula (3) we obtain

119890 (119895) gt 0 Δ119879 (119895 + 1) lt 0

119890 (119895) lt 0 Δ119879 (119895 + 1) gt 0

997904rArr Δ119879 (119895 + 1) 119890 (119895) lt 0

1003816100381610038161003816Δ119879 (119895 + 1)1003816100381610038161003816

1003816100381610038161003816119890 (119895)1003816100381610038161003816 gt

1

2119870 (119895) Δ119879

2(119895 + 1)

(13)

From inequalities (13) it can be concluded that Δ119881(119895) lt0 Furthermore to guarantee that the minimum survivaltime can reach the expected value the minimum samplingperiod is bounded by 119879

minge 119871119890sum Consequently the

error system is stable and the survival time can convergeto the expected value if the conditions in Theorem 1 aresatisfied

Remark 2 (prediction of survival time) The actual survivaltime is unavailable at the current instant because the powerconsumption is time varying However it can be predictedby the known information of the remaining power andsampling period at the current instant Formula (5) providesthe prediction and indicates that the survival time of thenode will be (119895) if the sensor node maintains the samplingperiod 119879(119895) as unchanged from the current instant 119905 = 119895119879

119872

The prediction of the survival time serves as a substitute forreal survival time and is used to calculate the new samplingperiod

Remark 3 (lower bound of sampling period) Taking IEEE80211b as an example the lower bound of the samplingperiod of sensor 119879min can be determined by the followingformula

119879min

= max119871119890

sum119878mc times 2 times 119873

119876 (14)

Mathematical Problems in Engineering 5

where (119878mc times 2 times 119873)119876 is the allowable minimum samplingperiod when the wireless network can be schedulable (see in[6]) and

119876 =119878mc

119879DIFS + (119862119882min times 119879SIFS2) + 119879frame + 119879SIFS + 119879ACK(15)

119879frame =119878PHY119877119897

+119878MAC + 119878mc

119877119905

119879frame =119878PHY119877119897

+119878MAC + 119878mc

119877119905

(16)

119877119905is the transmission rate 119877

119897is the legacy transmission rate

119878mc is the measurement-control data size 119878PHY is the sizeof control frame in physical layer and 119878MAC is the data sizeof ACK and is a confirmed sign in the header of TCP datapacket that confirms the received TCP message 119879SIFS is theshortest time period of the 80211b protocol for the intervalof frames requiring immediate response 119879DIFS is the timesegment for the interval of the time frame of the distributedcoordination function for sending in IEEE 80211b 119879PIFS isthe time segment for the interval of the time frame of thecentralized coordination function for sending which satisfies

119879PIFS = 119879DIFS minus 119879slot 119879slot = 119879PIFS minus 119879SIFS (17)

CW is the contention window Wireless network parametersunder the 80211b direct sequence spread spectrum are shownin Table 2

Remark 4 (upper bound of sampling period) For a SISOsystem the maximum sampling period can be obtainedusing Shannon sampling theorem For a MIMO system thefollowing method can be used to obtain the upper bound Ifthe system feedback control law is given ahead then119879max canbe obtained by solving the following optimal problem

max 119879

st100381710038171003817100381710038171003817100381710038171003817

eig(119890A119879 + int

119879

0

119890A119879

119889119905BK0)

100381710038171003817100381710038171003817100381710038171003817

lt 1

(18)

where A is the system matrix and B is the control inputmatrix K

0is the Kalman gain which satisfies

Re (eig (A + BK0)) lt 0 (19)

where Re(120592) denotes the real part of 120592 and eig(X) denotes theeigenvalues of matrix X Gain K

0can be determined by the

pole assignment in the continuous time domain

However the above optimal problem is difficult to solvedirectlyThe following iterationmethod can be used to obtainthe approximate optimal value of 119879max

Step 1 Let 119902 = 1 and the initial value of 119879max is 119879max(119902) =

2119879min If the condition can satisfy the constraints of the

optimal problem go to Step 2 or else go to Step 3

Table 2 Wireless network parameters under 80211b directsequence spread spectrum

Parameter Value119877119905

11Mbps119877119897

1Mbps119879SIFS 119879DIFS 119879slot 10 50 20 us119878MAC 119878PHY 34 24 bytes119878mc 80 bytes119878ACK 14 bytes + PHY headerCWmin CWmax 32 1024

Step 2 Let 119902 = 119902 + 1 and let 119879max(119902) = 2119879

max(119902 minus 1) If the

condition can fulfill the constraints of the optimal problemcycle Step 2 or else end the iteration and then let 119879max

=

119879max

(119902 minus 1)

Step 3 Let 119879max(119902) = (1 + (34)

119902)119879

min If the condition stilldoes not satisfy the constraints of the optimal problem let119902 = 119902 + 1 and cycle Step 3 or else end the iteration and let119879max

= 119879max

(119902)

4 Modeling and Stability Analysis of AdaptiveSampling Period WNCS

41 Modeling of Adaptive Sampling Period WNCS Consid-ering the generality one of the control loops in the WNCSis chosen as an example to illustrate the modeling approachFor theWNCS power consumptionmanaged by algorithm inSection 2 the dynamics of the control system is time-varyingSince the sampling period varies among the 119872 candidatesthe system can be considered a switched system with 119872

modes from the perspective of the switched control systemscheme Each switching mode is corresponded to one ofthe candidates According to Assumption 1 the sensor andactuator nodes are assumed to be clock-driven It results thatthe switching occurs at some of the sampling instants Addi-tionally according to Assumption 3 the inevitable existenceof network-induced delay is taken into account and it is lessthan one sampling period

We consider a plant of control loop 119894 in the WNCS withthe following dynamics

119894(119905) = A119894119909119894 (119905) + B119894119906119894 (119905)

119910119894(119905) = C119894119909119894 (119905)

(20)

where 119909119894(119905) isin R119899 is the plant state 119906119894(119905) isin R119898 is the controlinput and 119910

119894(119905) isin R119901 is the plant output A119894 isin R119899times119899 B119894 isin

R119899times119898 and C119894 isin R119901times119899 are the matrices of state control inputand output matrices respectively Due to the generality of 119894we omit the superscript 119894 in the following model descriptionand deduction

Discretizing system (20) with sampling rate 119879119897and

considering network-induced delay less than one sampling

6 Mathematical Problems in Engineering

Mixed mode Mixed model(k minus 1) ne l(k) l(k minus 1) = l(k) l(k + 3) ne l(k + 2)

u(k minus 1)

u(k + 1)120591l(k)(k)

k + 1 k + 2 k + 3

Sensor

Controller

node

node

nodeu(k minus 1)= Kl(kminus1)x(k minus 1)

Actuator

k minus 1 k

u(k)

Nominal behavior

One monitor interval

Tl(kminus1) Tl(k) Tl(k +3)

TM

Figure 2 Evolution over one period of switched WNCS with two types of behavior

period the discrete dynamics of the open control loop canbe described as

119909 (119896 + 1) = Φ119897(119896)119909 (119896) + Γ119897(119896)119906 (119896 minus 1)

119910 (119896) = C119909 (119896) (21)

whereΦ119897(119896)

= 119890A119879119897(119896) Γ

119897= int119879119897(119896)

0119890A119905119889119905B and 119897(119896) is the identi-

fication of sampling period at the 119896th sampling instant 119897(119896) isinZ rarr ℓ = 0 1 119872 minus 1

For the discrete switched system (21) a switched statefeedback controller is introduced in the following form

119906 (119896) = K120574(119896)

119909 (119896) (22)

where 120574(119896) isin Z rarr ℓ = 0 1 119872 minus 1 denotes the switch-ing signal used in the control

Consequently the closed-loop WNCS can be written as

[119909 (119896 + 1)

119909 (119896)] = [Φ119897(119896)Γ119897(119896)

K120574(119896)

I 0] [

119909 (119896)

119909 (119896 minus 1)] (23)

42 Stability Analysis of Adaptive Sampling Period WNCSThe control gains K

120574(119896)are assumed to be designed in such

a way that the closed-loop system is asymptotically stablewhen 120574(119896) = 119897(119896) Ideally the switching signal used in control120574(119896) is the same as the real signal 119897(119896) However this viewis unrealistic in WNCS with network-induced delay where120574(119896) = 119897(119896 minus 1) The evolution over one sampling period canbe described by two distinct types of behavior the nominaland mixed mode sampling periods as shown in Figure 2

(1)Nominal sampling period is when the system evolutionuses the right switching information

120574 (119896) = 119897 (119896 minus 1) 119897 (119896) = 119897 (119896 minus 1) (24)

(2) Mixed mode sampling period is when the systemcommand uses a wrong feedback gain

120574 (119896) = 119897 (119896 minus 1) 119897 (119896) = 119897 (119896 minus 1) (25)

Denoting Δ119898 as the samples spent in the mixed mode

since the delay is less than one sampling period it follows that

Δ119898 has a range of 0 le Δ

119898le 1 Moreover we assume that

the system is controlled using the right gains for at least Δ119899samples before another switching occurs

The next thing we should do is to guarantee the closed-loop system remaining stable with the designed state feed-back control gains when the switching signal is temporarilyuncertain Consider the scalars 120579

119899gt 0 120579

119898gt 0 and the

symmetric positive definite matrices P119899119897 P119898(119897120574)

with (119897 120574) isin

ℓ times ℓ which satisfy the following matrix inequalities

Φ119879

(119897120574)(P119898(119897120574)

)minus1

Φ(119897120574)

minus 120579119898P119898(119897120574)

lt 0 forall (119897 120574) isin ℓ times ℓ

Φ119879

(119897119897)P119899119897Φ(119897119897)

minus 120579119899P119899119897lt 0 forall119897 isin ℓ

(26)

Moreover consider the following two scalars

120573119899=max119897isinℓeigmax (P

119899

119897)

min119897isinℓeigmin (P119899119897 )

120573119898=

max(119897120574)isinℓtimesℓ

eigmax (P119898

(119897120574))

min(119897120574)isinℓtimesℓ

eigmin (P119898(119897120574))

(27)

where eigmax(X) and eigmin(X) denote the maximum and theminimum eigenvalues of matrix X respectively

Then the stability of the closed-loop WNCS (23) can beguaranteed by the following theorem

Theorem 5 Let 120579lowast119899 120579lowast119898be the solutions of the optimization

problems 120579lowast119899

= min 120579119899and 120579

lowast

119898= min 120579

119898subject to matrix

inequalities (26) Closed-loop system (23) is asymptoticallystable if

120573119899sdot 120573119898sdot (120579lowast

119899)Δ119899

sdot 120579lowast

119898lt 1 (28)

Proof We consider the following Lyapunov functions

119881119899(119896) = 119909

119879(119896)P119899119897119909 (119896) 119881

119898(119896) = 119909

119879(119896)P119898(119897120574)

119909 (119896)

(29)

Inequalities (13) show that

119881119899(1198961) lt (120579

lowast

119899)1198961minus1198960

119881119899(1198960) (30)

Mathematical Problems in Engineering 7

forall119896 isin [1198960 1198961) if the right switching signal is used in the

control 119897(119896) = 120574(119896) = 119897(1198960) then

119881119898(1198961) lt (120579

lowast

119898)1198961minus1198960

119881119898(1198960) (31)

forall119896 isin [1198960 1198961) if the switching signal in the control is not

necessarily the same as the real signal the pair 119897(119896) 120574(119896) takesan arbitrary value 119897(119896

0) 120574(119896

0) in ℓ times ℓ

Since the controller gains K119897are designed to make the

matricesΦ(119897119897)

stable the scalar 120579119899is smaller than one 120579

119899lt 1

The scalar 120579119898may be greater than one 120579

119898gt 1 since the

gains K120574are not designed to stabilize combinations other

thanΦ(120574120574)

Combining inequalities (26) yield

min119897isinℓ

eigmin (P119899

119897)1003817100381710038171003817119909(1198961)

1003817100381710038171003817

2

lt (120579lowast

119899)1198961minus1198960max119897isinℓ

eigmax (P119899

119897)1003817100381710038171003817119909(1198960)

1003817100381710038171003817

2forall119897 isin ℓ

min(119897120574)isinℓtimesℓ

eigmin (P119898

(119897120574))1003817100381710038171003817119909(1198961)

1003817100381710038171003817

2

lt max(119897120574)isinℓtimesℓ

eigmax (P119898

(119897120574))1003817100381710038171003817119909(1198960)

1003817100381710038171003817

2forall (119897 120574) isin ℓ times ℓ

(32)

With the definitions of 120573119899and 120573

119898 we can obtain the state

vector norm decay or growth rate in a nominal regime and inan uncertain switching signal regime as follows

1003817100381710038171003817119909(1198961)1003817100381710038171003817

2lt 120573119899sdot (120579lowast

119899)1198961minus11989601003817100381710038171003817119909(1198960)

1003817100381710038171003817

2 forall119897 isin ℓ

1003817100381710038171003817119909(1198961)1003817100381710038171003817

2lt 120573119898sdot (120579lowast

119898)1198961minus11989601003817100381710038171003817119909(1198960)

1003817100381710038171003817

2 forall (119897 120574) isin ℓ times ℓ

(33)

Let 119896119898119904describe the instants when the closed-loop system

jumps to a mixed mode with uncertain switching signal andlet 119896119899119904be the instance when the system enters into a normal

regime With definitions and bounds of Δ119898 and Δ119899 it followsthat

0 le Δ119898= 119896119899

119904minus 119896119898

119904le 1 119896

119898

119904+1minus 119896119899

119904le Δ119899 (34)

Without loss of generality we assume that the systemstarts with a mixed-mode behavior 119896119898

119904lt 119896119899

119904 The system

behavior in time interval 119896 isin [119896119898

119904 119896119898

119904+1) is then analyzed

Given that 119897(119896) = 120574(119896) forall119896 isin [119896119898

119904 119896119899

119904) and 119897(119896) =

120574(119896) forall119896 isin [119896119899

119904 119896119898

119904+1) using inequalities (26) the norm of the

state at the end of the sequence can be upper bounded asd

1003817100381710038171003817119909(119896119898

119904+1)1003817100381710038171003817

2lt 120573119898sdot 120573119899sdot (120579lowast

119898)119896119899

119904minus119896119906

119904sdot (120579lowast

119899)119896119898

119904+1minus119896119899

119904 1003817100381710038171003817119909(119896119898

119904)1003817100381710038171003817

2

lt1003817100381710038171003817119909(119896119898

119904+1)1003817100381710038171003817

2lt 120573119898sdot 120573119899sdot 120579lowast

119898sdot (120579lowast

119899)Δ1198991003817100381710038171003817119909(119896119898

119904)1003817100381710038171003817

2

(35)

It indicates that closed loop (16) will be asymptotically stableif condition (28) in Theorem 5 is satisfied

43 Choosing Rule of Update Period 119879119872

Theorem 6 Consider the WNCS with adaptive samplingperiod rule (3) The WNCS can be stabilized by the switched

state feedback controllers (22) whereas the survival time canmeet its expected value 119871

119890 if the update period 119879

119872satisfies the

following conditions

(1) 119879119872

ge max (Δ119899119894+ 1) [119879

max1

119879max2

119879max119899

]

(2) lfloor119871119890

119879119872

rfloor sdot 119879119872

le119864119894119899119894119905

119864

min 119879min1

119879min2

119879min119899

(36)

where Δ119899119894is the least nominal sampling period of control loop

119894 and the solution of Theorem 5 [119879max1

119879max2

119879max119899

] is theleast commonmultiple of the maximum sampling period for allthe 119899 control loops in the WNCS

Proof According to Theorem 5 control loop 119894 in the WNCSshould stay at least Δ119899

119894in a nominal sampling period in one

updating interval for stability With the consideration of 0 le

Δ119898

119894le 1 if the update period satisfies condition (1) all control

loops in the WNCS will meet the demands of their leastnominal sampling periods and will be stable As a completesystem WNCS is composed of 119899 control loops that will bestable when condition (1) is satisfied

Condition (2) inTheorem 6 provides the upper bound ofthe updating period For the power efficiency algorithm inTheorem 1 with the definition of 119896

119897(119895) there exists

119872

sum

119897=1

119896119897(119895)

119879119872

119879119897

+119879119872

119879119895

le lfloor119871119890

119879119872

rfloor sdot119879119872

min 119879min1

119879min2

119879min119899

(37)

If condition (2) inTheorem 6 is satisfied the item ((119864init119864)minus

sum119872

119897=1119896119897(119895)(119879119872119879119897) minus (119879

119872119879119895)) in the denominator of formula

(3) will be greater than zero which guarantees that formula(3) has physical meaning and is solvable The survival timewill thus reach the expected value by applying the powerefficiency algorithm

5 Numerical Example

Simulation studies are performed on a WNCS closed by anIEEE 80211b wireless network with two control loops sharingthe network resources The two control loops are assumed tohave the same dynamics but with different initial conditions

119894(119905) = [

minus1 minus01

0 095] 119909119894(119905) + [

minus015

minus043] 119906119894(119905)

119910119894(119905) = [1 1] 119909

119894(119905)

119894 = 1 2

1199091(0) = [minus5 5]

119879 119909

2(0) = [minus5 10]

119879

(38)

The wireless network parameters are set as in Table 2The time delays in both loops are less than one samplingperiod Some other necessary parameters are given as shownin Table 2

In Table 3 the minimum and maximum sampling peri-ods are computed by the methods in Section 3 Solving

8 Mathematical Problems in Engineering

Table 3 Simulation parameters

Parameter ValueInitial energy of both sensors 119864init 015 JSensor 1 expected survival time 119871

119890175 s

Sensor 2 expected survival time 1198711198902

70 sUnit transmission energy 119864 25 dbmNumber of sampling period candidates119872 10The minimum sampling period of loop1 119879min

11ms

The maximum sampling period of loop1 119879max1

256msThe minimum sampling period of loop2 119879min

21ms

The maximum sampling period of loop2 119879max2

256ms

Table 4 Controller gains of ten sampling modes

Sampling period (ms) Controller gain1198790= 1 119870

0= [11176 11188]

1198791= 29 119870

1= [394837 405512]

1198792= 58 119870

2= [194678 204319]

1198793= 86 119870

3= [125668 134279]

1198794= 114 119870

4= [90353 97947]

1198795= 143 119870

5= [68663 75265]

1198796= 171 119870

6= [5383 5947]

1198797= 199 119870

7= [4295 4769]

1198798= 227 119870

8= [4225 3852]

1198799= 256 119870

9= [3707 295]

inequality (28) in Theorem 5 it yields Δ1198991= Δ119899

2= 8 Δ119898 = 1

Also the update period can be chosen byTheorem 6 as 119879119872

=

150msSolving matrix inequalities (18) and (19) the controller

gains of ten switching modes can be obtained as in Table 4With the above simulation parameters and controller

gains the curves of survival time prediction power consump-tion and control output of both control loops are shown inFigures 3 to 8

Analyzing the simulation curves we have the followingresults

(1) Figures 3 6 4 and 7 imply that both sensor 1and sensor 2 can meet their expected survival timerequirements

(2) The power consumption in three cases of minimumsampling maximum sampling and the proposedadaptive sampling is compared in Figures 4 and7 It is obvious that the power is consumed muchfaster than the other two cases In the case of theadaptive sampling the power consumption variesaccording to both the requirements of the controlperformance and survival time At the beginning thepower consumption curves vary quickly and morepower are consumed because of the control systemsnot reaching stable yetThen after the control systems

0 20 40 60 800

50

100

150

200

Sens

or 1

life

span

(s)

250

300

350

400

Time (s)

Figure 3 Sensor 1 survival time prediction

0 15 30 45 60 750

005

01

015

Time (s)Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Ener

gy o

f sen

sor1

(J)

T 1T 1

Figure 4 Sensor 1 power consumption comparison in three cases

are settled it tends to theminimum consumption ratewhich is corresponding to the maximum samplingperiod

(3) Figures 5 and 8 are the control outputs of loop1and loop2 in the three cases mentioned above Thefigures show that the control systems can be stabilizedthrough the proposed joint design methods In threecases the adaptive sampling can get the control per-formances closed to the case of minimum sampling

(4) Combining Figures 4 5 7 and 8 it can be concludedthat the proposed joint design method achieves atradeoff between the performances of control andpower efficiency

Mathematical Problems in Engineering 9

Time (s)

The proposed adaptive sampling

0 15 30 45 60 75

0

2

4

6

8

Out

puty

of lo

op1

minus2

minus4

minus6

minus8

Constant sampling with minimum T1 = 1 msConstant sampling with maximum T1 = 256 ms

Figure 5 Control loop1 output comparison in three cases

0 20 40 60 800

50

100

150

Sens

or 2

life

span

(s)

200

250

300

350

Time (s)

Figure 6 Sensor 2 survival time prediction

6 Conclusion

This paper presents a joint design method for wirelessnetworked control systems with limited power constraint Apower efficiency algorithm based on the adaptive samplingperiod is put forward to satisfy the demands of sensorsurvival time and system stability Then the time-varyingcontrol system with transmission delay is modeled as aswitched system with uncertain switching signals A dwell-time-dependent controlmethod is discussed to guarantee thestability of WNCS Simulation results show the effectivenessof the proposed method and indicate that it can achievegood tradeoff performance Methods by which to reducepower consumption from the aspect of a single node as wellas balancing power consumption from the global networkperspective are worthy of further exploration

0 15 30 45 60 750

005

01

015

Time (s)

Ener

gy o

f sen

sor2

(J)

The proposed adaptive sampling

Constant sampling with minimum T2 = 1msConstant sampling with maximum T2 = 256ms

Figure 7 Sensor 2 power consumption comparison in three cases

0 15 30 45 60 75

0

5

10

15

Out

put y

of l

oop

2

minus15

minus10

minus5

Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Time (s)

T2T2

Figure 8 Control loop2 output comparison in three cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The working is sponsored by The NSFC (no 61202473) andThe Natural Science Foundation of Jiangsu Province (noBK2012551)

10 Mathematical Problems in Engineering

References

[1] W Dieterle H D Kochs and E Dittmar ldquoCommunicationarchitectures for distributed computer control systemsrdquoControlEngineering Practice vol 3 no 8 pp 1171ndash1176 1995

[2] H A Thompson ldquoWireless and Internet communicationstechnologies for monitoring and controlrdquo Control EngineeringPractice vol 12 no 6 pp 781ndash791 2004

[3] P R Kumar ldquoNew technological vistas for systems and controlrdquoIEEE Control Systems Magazine vol 21 no 1 pp 24ndash37 2001

[4] X Liu and A Goldsmith ldquoWireless medium access controlin networked control systemsrdquo in Proceedings of the AmericanControl Conference (AAC rsquo04) pp 3605ndash3610 Boston MassUSA July 2004

[5] J Colandairaj G W Irwin and W G Scanlon ldquoWirelessnetworked control systems with QoS-based samplingrdquo IETControl Theory and Applications vol 1 no 1 pp 430ndash438 2007

[6] N Israr W G Scanlon and G W Irwin ldquoA cross-layer com-munication framework forwireless networked control systemsrdquoin Proceedings of the 1st International Conference on WirelessCommunication Vehicular Technology Information Theory andAerospace and Electronic Systems Technology (VITAE rsquo09) pp577ndash581 Aalborg Denmark May 2009

[7] S Dai H Lin and S S Ge ldquoScheduling-and-control codesignfor a collection of networked control systems with uncertaindelaysrdquo IEEE Transactions on Control Systems Technology vol18 no 1 pp 66ndash78 2010

[8] A V Savkin ldquoAnalysis and synthesis of networked control sys-tems topological entropy observability robustness and optimalcontrolrdquo Automatica vol 42 no 1 pp 51ndash62 2006

[9] J Z Luo F Dong and J X Cao ldquoA novel task schedul-ing algorithm based on dynamic critical path and effectiveduplication for pervasive computing environmentrdquo WirelessCommunications amp Mobile Computing vol 10 no 10 pp 1283ndash1302 2010

[10] X Yin X Zhou Z Li and S Li ldquoJint rate control and powercontrol for lifetime maximization in Wreless Sensor NtworksrdquoJournal of Internet Technology vol 12 no 1 pp 69ndash78 2011

[11] M Pajic S Sundaram G J Pappas and R Mangharam ldquoThewireless control network a new approach for control overnetworksrdquo IEEE Transactions on Automatic Control vol 56 no10 pp 2305ndash2318 2011

[12] J Colandairaj G W Irwin and W G Scanlon ldquoA co-designsolution for wireless feedback controlrdquo in Proceeding of theInternational Conference on Networking Sensing and Control(ICNSC 07) pp 404ndash409 London UK April 2007

[13] W Liang and Y Yang ldquoMaximizing battery life routing in wire-less ad hoc networksrdquo in Proceedings of the 37th InternationalConference on System Sciences pp 4739ndash4746 IEEE HonoluluHawaii USA January 2004

[14] K Brian J Haberman and W Sheppard Overlapping ParticleSwarms for Energy-Efficient Routing in Sensor Networks Wire-less Network Springer 2011

[15] D Kim K Dantu andM Pedram ldquoLifetime prediction routinginmobile AdHoc networksrdquo in Proceedings of the IEEEWirelessCommunication and Networking Conference New York NYUSA 2003

[16] L Hetel J Daafouz and C Iung ldquoStability analysis for discretetime switched systems with temporary uncertain switchingsignalrdquo in Proceedings of the 46th IEEE Conference on Decisionand Control (CDC rsquo07) pp 5623ndash5628 New Orleans Fla USADecember 2007

[17] S Limin Wireless Sensor Network Tsinghua University PressBeijing China 2006

[18] Y He I Lee and L Guan ldquoDistributed algorithms for networklifetimemaximization in wireless visual sensor networksrdquo IEEETransactions on Circuits and Systems for Video Technology vol19 no 5 pp 704ndash718 2009

[19] B A Bakr and L Lilien ldquoA quantitative comparison of energyconsumption and WSN lifetime for LEACH and LEACH-SMrdquo in Proceedings of the 31st International Conference onDistributed Computing Systems Workshops (ICDCSW rsquo11) pp182ndash191 Minneapolis Minn USA June 2011

[20] C Fischione A Bonivento A Sangiovanni-Vincentelli F San-tucci and K H Johansson ldquoPerformance analysis of collabora-tive spatio-temporal processing for wireless sensor networksrdquoin Proceedings of the 3rd IEEE Consumer Communications andNetworking Conference (CCNC rsquo06) pp 325ndash329 Las VegasNev USA January 2006

[21] Y Iino ldquoSome considerations of wireless sensor network basedcontrol systemsrdquo in Proceedings of the SICE Conference onControl Systems (CCS rsquo07) Tokyo Japan 2007

[22] Y Iino T Hatanaka and M Fujita ldquoEvent-predictive controlfor energy saving of wireless networked control systemrdquo inProceedings of the American Control Conference (ACC rsquo09) pp2236ndash2242 St Louis Mo USA June 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

where

119878 = min

[[[

[

2 (119872 minus 1)10038161003816100381610038161003816 (119895) minus 119871

119890

10038161003816100381610038161003816

times ((119864init

119864

minus

119872

sum

119897=1

119896119897(119895)

119879119872

119879119897

minus119879119872

119879119895

)

times (119879max

minus 119879min

))

minus1]]]

]

119872 minus 1

(4)

119871119890is the demand of sensor survival time 119879(119895) is the value

of the sampling period in the 119895th updating interval sgn(119909)is the signal of scalar 119909 lfloor119909rfloor is the integral part of scalar119909 119896119897(119895) is the number of updating intervals in which the

sampling period is 119879119897during the previous 119895 updates (119895) is

the current predicted survival time of sensor node calculatedby the following formula

(119895) = 119895119879119872+119864rem (119895)

119864

times 119879 (119895) (5)

119864rem(119895) is the current remaining power at the updating instant119895119879119872

32 Convergence of Power Efficiency Algorithm

Theorem 1 For the WNCS described in Figure 1 consideringthe update rule of adaptive sampling period (3) if theminimumsampling period satisfies

119879min

gt119871119890

119904119906119898 (6)

then the actual survival time of sensor node will reach itsexpected value 119871

119890through the proposed rule of sampling period

update

Proof According to Formula (1) we have

(119895 + 1) = (119895 + 1) 119879119872+119864rem (119895 + 1)

119864

times 119879 (119895 + 1) (7)

The remaining power relationship at the two adjacentupdating instants is given by

119864rem (119895 + 1) = 119864rem (119895) minus119879119872

119879 (119895)times 119864 (8)

We assume that the sensor node has sampled the plantwith the sampling period 119879

119897for 119896119897(119895) times from the initial

to the current time instant The remaining power can becalculated based on the initial power and consumed power

119864rem (119895) = 119864init minus119872

sum

119897=1

119896119897(119895)

119879119872119864

119879119897

(9)

Power control error is defined as 119890(119895) = (119895) minus 119871119890

According to formulas (7) (8) and (9) the dynamics of theerror can be described as

119890 (119895 + 1) = 119890 (119895) + 119870 (119895) Δ119879 (119895 + 1) (10)

where 119870(119895) is defined as 119870(119895) = sum minus sum119872

119897=1119896119897(119895)(119879119872119879119897) minus

(119879119872119879(119895)) At the current time instant 119870(119895) is a known

variableThe following Lyapunov function is introduced to prove

convergence of the adaptive sampling power efficiency algo-rithm

119881 (119895) =1

21198902(119895) (11)

Considering Formula (10) it follows that

Δ119881 (119895) = 119881 (119895 + 1) minus 119881 (119895)

= 119870 (119895) Δ119879 (119895 + 1) 119890 (119895) +1

21198702(119895) Δ119879

2(119895 + 1)

(12)

With formula (3) we obtain

119890 (119895) gt 0 Δ119879 (119895 + 1) lt 0

119890 (119895) lt 0 Δ119879 (119895 + 1) gt 0

997904rArr Δ119879 (119895 + 1) 119890 (119895) lt 0

1003816100381610038161003816Δ119879 (119895 + 1)1003816100381610038161003816

1003816100381610038161003816119890 (119895)1003816100381610038161003816 gt

1

2119870 (119895) Δ119879

2(119895 + 1)

(13)

From inequalities (13) it can be concluded that Δ119881(119895) lt0 Furthermore to guarantee that the minimum survivaltime can reach the expected value the minimum samplingperiod is bounded by 119879

minge 119871119890sum Consequently the

error system is stable and the survival time can convergeto the expected value if the conditions in Theorem 1 aresatisfied

Remark 2 (prediction of survival time) The actual survivaltime is unavailable at the current instant because the powerconsumption is time varying However it can be predictedby the known information of the remaining power andsampling period at the current instant Formula (5) providesthe prediction and indicates that the survival time of thenode will be (119895) if the sensor node maintains the samplingperiod 119879(119895) as unchanged from the current instant 119905 = 119895119879

119872

The prediction of the survival time serves as a substitute forreal survival time and is used to calculate the new samplingperiod

Remark 3 (lower bound of sampling period) Taking IEEE80211b as an example the lower bound of the samplingperiod of sensor 119879min can be determined by the followingformula

119879min

= max119871119890

sum119878mc times 2 times 119873

119876 (14)

Mathematical Problems in Engineering 5

where (119878mc times 2 times 119873)119876 is the allowable minimum samplingperiod when the wireless network can be schedulable (see in[6]) and

119876 =119878mc

119879DIFS + (119862119882min times 119879SIFS2) + 119879frame + 119879SIFS + 119879ACK(15)

119879frame =119878PHY119877119897

+119878MAC + 119878mc

119877119905

119879frame =119878PHY119877119897

+119878MAC + 119878mc

119877119905

(16)

119877119905is the transmission rate 119877

119897is the legacy transmission rate

119878mc is the measurement-control data size 119878PHY is the sizeof control frame in physical layer and 119878MAC is the data sizeof ACK and is a confirmed sign in the header of TCP datapacket that confirms the received TCP message 119879SIFS is theshortest time period of the 80211b protocol for the intervalof frames requiring immediate response 119879DIFS is the timesegment for the interval of the time frame of the distributedcoordination function for sending in IEEE 80211b 119879PIFS isthe time segment for the interval of the time frame of thecentralized coordination function for sending which satisfies

119879PIFS = 119879DIFS minus 119879slot 119879slot = 119879PIFS minus 119879SIFS (17)

CW is the contention window Wireless network parametersunder the 80211b direct sequence spread spectrum are shownin Table 2

Remark 4 (upper bound of sampling period) For a SISOsystem the maximum sampling period can be obtainedusing Shannon sampling theorem For a MIMO system thefollowing method can be used to obtain the upper bound Ifthe system feedback control law is given ahead then119879max canbe obtained by solving the following optimal problem

max 119879

st100381710038171003817100381710038171003817100381710038171003817

eig(119890A119879 + int

119879

0

119890A119879

119889119905BK0)

100381710038171003817100381710038171003817100381710038171003817

lt 1

(18)

where A is the system matrix and B is the control inputmatrix K

0is the Kalman gain which satisfies

Re (eig (A + BK0)) lt 0 (19)

where Re(120592) denotes the real part of 120592 and eig(X) denotes theeigenvalues of matrix X Gain K

0can be determined by the

pole assignment in the continuous time domain

However the above optimal problem is difficult to solvedirectlyThe following iterationmethod can be used to obtainthe approximate optimal value of 119879max

Step 1 Let 119902 = 1 and the initial value of 119879max is 119879max(119902) =

2119879min If the condition can satisfy the constraints of the

optimal problem go to Step 2 or else go to Step 3

Table 2 Wireless network parameters under 80211b directsequence spread spectrum

Parameter Value119877119905

11Mbps119877119897

1Mbps119879SIFS 119879DIFS 119879slot 10 50 20 us119878MAC 119878PHY 34 24 bytes119878mc 80 bytes119878ACK 14 bytes + PHY headerCWmin CWmax 32 1024

Step 2 Let 119902 = 119902 + 1 and let 119879max(119902) = 2119879

max(119902 minus 1) If the

condition can fulfill the constraints of the optimal problemcycle Step 2 or else end the iteration and then let 119879max

=

119879max

(119902 minus 1)

Step 3 Let 119879max(119902) = (1 + (34)

119902)119879

min If the condition stilldoes not satisfy the constraints of the optimal problem let119902 = 119902 + 1 and cycle Step 3 or else end the iteration and let119879max

= 119879max

(119902)

4 Modeling and Stability Analysis of AdaptiveSampling Period WNCS

41 Modeling of Adaptive Sampling Period WNCS Consid-ering the generality one of the control loops in the WNCSis chosen as an example to illustrate the modeling approachFor theWNCS power consumptionmanaged by algorithm inSection 2 the dynamics of the control system is time-varyingSince the sampling period varies among the 119872 candidatesthe system can be considered a switched system with 119872

modes from the perspective of the switched control systemscheme Each switching mode is corresponded to one ofthe candidates According to Assumption 1 the sensor andactuator nodes are assumed to be clock-driven It results thatthe switching occurs at some of the sampling instants Addi-tionally according to Assumption 3 the inevitable existenceof network-induced delay is taken into account and it is lessthan one sampling period

We consider a plant of control loop 119894 in the WNCS withthe following dynamics

119894(119905) = A119894119909119894 (119905) + B119894119906119894 (119905)

119910119894(119905) = C119894119909119894 (119905)

(20)

where 119909119894(119905) isin R119899 is the plant state 119906119894(119905) isin R119898 is the controlinput and 119910

119894(119905) isin R119901 is the plant output A119894 isin R119899times119899 B119894 isin

R119899times119898 and C119894 isin R119901times119899 are the matrices of state control inputand output matrices respectively Due to the generality of 119894we omit the superscript 119894 in the following model descriptionand deduction

Discretizing system (20) with sampling rate 119879119897and

considering network-induced delay less than one sampling

6 Mathematical Problems in Engineering

Mixed mode Mixed model(k minus 1) ne l(k) l(k minus 1) = l(k) l(k + 3) ne l(k + 2)

u(k minus 1)

u(k + 1)120591l(k)(k)

k + 1 k + 2 k + 3

Sensor

Controller

node

node

nodeu(k minus 1)= Kl(kminus1)x(k minus 1)

Actuator

k minus 1 k

u(k)

Nominal behavior

One monitor interval

Tl(kminus1) Tl(k) Tl(k +3)

TM

Figure 2 Evolution over one period of switched WNCS with two types of behavior

period the discrete dynamics of the open control loop canbe described as

119909 (119896 + 1) = Φ119897(119896)119909 (119896) + Γ119897(119896)119906 (119896 minus 1)

119910 (119896) = C119909 (119896) (21)

whereΦ119897(119896)

= 119890A119879119897(119896) Γ

119897= int119879119897(119896)

0119890A119905119889119905B and 119897(119896) is the identi-

fication of sampling period at the 119896th sampling instant 119897(119896) isinZ rarr ℓ = 0 1 119872 minus 1

For the discrete switched system (21) a switched statefeedback controller is introduced in the following form

119906 (119896) = K120574(119896)

119909 (119896) (22)

where 120574(119896) isin Z rarr ℓ = 0 1 119872 minus 1 denotes the switch-ing signal used in the control

Consequently the closed-loop WNCS can be written as

[119909 (119896 + 1)

119909 (119896)] = [Φ119897(119896)Γ119897(119896)

K120574(119896)

I 0] [

119909 (119896)

119909 (119896 minus 1)] (23)

42 Stability Analysis of Adaptive Sampling Period WNCSThe control gains K

120574(119896)are assumed to be designed in such

a way that the closed-loop system is asymptotically stablewhen 120574(119896) = 119897(119896) Ideally the switching signal used in control120574(119896) is the same as the real signal 119897(119896) However this viewis unrealistic in WNCS with network-induced delay where120574(119896) = 119897(119896 minus 1) The evolution over one sampling period canbe described by two distinct types of behavior the nominaland mixed mode sampling periods as shown in Figure 2

(1)Nominal sampling period is when the system evolutionuses the right switching information

120574 (119896) = 119897 (119896 minus 1) 119897 (119896) = 119897 (119896 minus 1) (24)

(2) Mixed mode sampling period is when the systemcommand uses a wrong feedback gain

120574 (119896) = 119897 (119896 minus 1) 119897 (119896) = 119897 (119896 minus 1) (25)

Denoting Δ119898 as the samples spent in the mixed mode

since the delay is less than one sampling period it follows that

Δ119898 has a range of 0 le Δ

119898le 1 Moreover we assume that

the system is controlled using the right gains for at least Δ119899samples before another switching occurs

The next thing we should do is to guarantee the closed-loop system remaining stable with the designed state feed-back control gains when the switching signal is temporarilyuncertain Consider the scalars 120579

119899gt 0 120579

119898gt 0 and the

symmetric positive definite matrices P119899119897 P119898(119897120574)

with (119897 120574) isin

ℓ times ℓ which satisfy the following matrix inequalities

Φ119879

(119897120574)(P119898(119897120574)

)minus1

Φ(119897120574)

minus 120579119898P119898(119897120574)

lt 0 forall (119897 120574) isin ℓ times ℓ

Φ119879

(119897119897)P119899119897Φ(119897119897)

minus 120579119899P119899119897lt 0 forall119897 isin ℓ

(26)

Moreover consider the following two scalars

120573119899=max119897isinℓeigmax (P

119899

119897)

min119897isinℓeigmin (P119899119897 )

120573119898=

max(119897120574)isinℓtimesℓ

eigmax (P119898

(119897120574))

min(119897120574)isinℓtimesℓ

eigmin (P119898(119897120574))

(27)

where eigmax(X) and eigmin(X) denote the maximum and theminimum eigenvalues of matrix X respectively

Then the stability of the closed-loop WNCS (23) can beguaranteed by the following theorem

Theorem 5 Let 120579lowast119899 120579lowast119898be the solutions of the optimization

problems 120579lowast119899

= min 120579119899and 120579

lowast

119898= min 120579

119898subject to matrix

inequalities (26) Closed-loop system (23) is asymptoticallystable if

120573119899sdot 120573119898sdot (120579lowast

119899)Δ119899

sdot 120579lowast

119898lt 1 (28)

Proof We consider the following Lyapunov functions

119881119899(119896) = 119909

119879(119896)P119899119897119909 (119896) 119881

119898(119896) = 119909

119879(119896)P119898(119897120574)

119909 (119896)

(29)

Inequalities (13) show that

119881119899(1198961) lt (120579

lowast

119899)1198961minus1198960

119881119899(1198960) (30)

Mathematical Problems in Engineering 7

forall119896 isin [1198960 1198961) if the right switching signal is used in the

control 119897(119896) = 120574(119896) = 119897(1198960) then

119881119898(1198961) lt (120579

lowast

119898)1198961minus1198960

119881119898(1198960) (31)

forall119896 isin [1198960 1198961) if the switching signal in the control is not

necessarily the same as the real signal the pair 119897(119896) 120574(119896) takesan arbitrary value 119897(119896

0) 120574(119896

0) in ℓ times ℓ

Since the controller gains K119897are designed to make the

matricesΦ(119897119897)

stable the scalar 120579119899is smaller than one 120579

119899lt 1

The scalar 120579119898may be greater than one 120579

119898gt 1 since the

gains K120574are not designed to stabilize combinations other

thanΦ(120574120574)

Combining inequalities (26) yield

min119897isinℓ

eigmin (P119899

119897)1003817100381710038171003817119909(1198961)

1003817100381710038171003817

2

lt (120579lowast

119899)1198961minus1198960max119897isinℓ

eigmax (P119899

119897)1003817100381710038171003817119909(1198960)

1003817100381710038171003817

2forall119897 isin ℓ

min(119897120574)isinℓtimesℓ

eigmin (P119898

(119897120574))1003817100381710038171003817119909(1198961)

1003817100381710038171003817

2

lt max(119897120574)isinℓtimesℓ

eigmax (P119898

(119897120574))1003817100381710038171003817119909(1198960)

1003817100381710038171003817

2forall (119897 120574) isin ℓ times ℓ

(32)

With the definitions of 120573119899and 120573

119898 we can obtain the state

vector norm decay or growth rate in a nominal regime and inan uncertain switching signal regime as follows

1003817100381710038171003817119909(1198961)1003817100381710038171003817

2lt 120573119899sdot (120579lowast

119899)1198961minus11989601003817100381710038171003817119909(1198960)

1003817100381710038171003817

2 forall119897 isin ℓ

1003817100381710038171003817119909(1198961)1003817100381710038171003817

2lt 120573119898sdot (120579lowast

119898)1198961minus11989601003817100381710038171003817119909(1198960)

1003817100381710038171003817

2 forall (119897 120574) isin ℓ times ℓ

(33)

Let 119896119898119904describe the instants when the closed-loop system

jumps to a mixed mode with uncertain switching signal andlet 119896119899119904be the instance when the system enters into a normal

regime With definitions and bounds of Δ119898 and Δ119899 it followsthat

0 le Δ119898= 119896119899

119904minus 119896119898

119904le 1 119896

119898

119904+1minus 119896119899

119904le Δ119899 (34)

Without loss of generality we assume that the systemstarts with a mixed-mode behavior 119896119898

119904lt 119896119899

119904 The system

behavior in time interval 119896 isin [119896119898

119904 119896119898

119904+1) is then analyzed

Given that 119897(119896) = 120574(119896) forall119896 isin [119896119898

119904 119896119899

119904) and 119897(119896) =

120574(119896) forall119896 isin [119896119899

119904 119896119898

119904+1) using inequalities (26) the norm of the

state at the end of the sequence can be upper bounded asd

1003817100381710038171003817119909(119896119898

119904+1)1003817100381710038171003817

2lt 120573119898sdot 120573119899sdot (120579lowast

119898)119896119899

119904minus119896119906

119904sdot (120579lowast

119899)119896119898

119904+1minus119896119899

119904 1003817100381710038171003817119909(119896119898

119904)1003817100381710038171003817

2

lt1003817100381710038171003817119909(119896119898

119904+1)1003817100381710038171003817

2lt 120573119898sdot 120573119899sdot 120579lowast

119898sdot (120579lowast

119899)Δ1198991003817100381710038171003817119909(119896119898

119904)1003817100381710038171003817

2

(35)

It indicates that closed loop (16) will be asymptotically stableif condition (28) in Theorem 5 is satisfied

43 Choosing Rule of Update Period 119879119872

Theorem 6 Consider the WNCS with adaptive samplingperiod rule (3) The WNCS can be stabilized by the switched

state feedback controllers (22) whereas the survival time canmeet its expected value 119871

119890 if the update period 119879

119872satisfies the

following conditions

(1) 119879119872

ge max (Δ119899119894+ 1) [119879

max1

119879max2

119879max119899

]

(2) lfloor119871119890

119879119872

rfloor sdot 119879119872

le119864119894119899119894119905

119864

min 119879min1

119879min2

119879min119899

(36)

where Δ119899119894is the least nominal sampling period of control loop

119894 and the solution of Theorem 5 [119879max1

119879max2

119879max119899

] is theleast commonmultiple of the maximum sampling period for allthe 119899 control loops in the WNCS

Proof According to Theorem 5 control loop 119894 in the WNCSshould stay at least Δ119899

119894in a nominal sampling period in one

updating interval for stability With the consideration of 0 le

Δ119898

119894le 1 if the update period satisfies condition (1) all control

loops in the WNCS will meet the demands of their leastnominal sampling periods and will be stable As a completesystem WNCS is composed of 119899 control loops that will bestable when condition (1) is satisfied

Condition (2) inTheorem 6 provides the upper bound ofthe updating period For the power efficiency algorithm inTheorem 1 with the definition of 119896

119897(119895) there exists

119872

sum

119897=1

119896119897(119895)

119879119872

119879119897

+119879119872

119879119895

le lfloor119871119890

119879119872

rfloor sdot119879119872

min 119879min1

119879min2

119879min119899

(37)

If condition (2) inTheorem 6 is satisfied the item ((119864init119864)minus

sum119872

119897=1119896119897(119895)(119879119872119879119897) minus (119879

119872119879119895)) in the denominator of formula

(3) will be greater than zero which guarantees that formula(3) has physical meaning and is solvable The survival timewill thus reach the expected value by applying the powerefficiency algorithm

5 Numerical Example

Simulation studies are performed on a WNCS closed by anIEEE 80211b wireless network with two control loops sharingthe network resources The two control loops are assumed tohave the same dynamics but with different initial conditions

119894(119905) = [

minus1 minus01

0 095] 119909119894(119905) + [

minus015

minus043] 119906119894(119905)

119910119894(119905) = [1 1] 119909

119894(119905)

119894 = 1 2

1199091(0) = [minus5 5]

119879 119909

2(0) = [minus5 10]

119879

(38)

The wireless network parameters are set as in Table 2The time delays in both loops are less than one samplingperiod Some other necessary parameters are given as shownin Table 2

In Table 3 the minimum and maximum sampling peri-ods are computed by the methods in Section 3 Solving

8 Mathematical Problems in Engineering

Table 3 Simulation parameters

Parameter ValueInitial energy of both sensors 119864init 015 JSensor 1 expected survival time 119871

119890175 s

Sensor 2 expected survival time 1198711198902

70 sUnit transmission energy 119864 25 dbmNumber of sampling period candidates119872 10The minimum sampling period of loop1 119879min

11ms

The maximum sampling period of loop1 119879max1

256msThe minimum sampling period of loop2 119879min

21ms

The maximum sampling period of loop2 119879max2

256ms

Table 4 Controller gains of ten sampling modes

Sampling period (ms) Controller gain1198790= 1 119870

0= [11176 11188]

1198791= 29 119870

1= [394837 405512]

1198792= 58 119870

2= [194678 204319]

1198793= 86 119870

3= [125668 134279]

1198794= 114 119870

4= [90353 97947]

1198795= 143 119870

5= [68663 75265]

1198796= 171 119870

6= [5383 5947]

1198797= 199 119870

7= [4295 4769]

1198798= 227 119870

8= [4225 3852]

1198799= 256 119870

9= [3707 295]

inequality (28) in Theorem 5 it yields Δ1198991= Δ119899

2= 8 Δ119898 = 1

Also the update period can be chosen byTheorem 6 as 119879119872

=

150msSolving matrix inequalities (18) and (19) the controller

gains of ten switching modes can be obtained as in Table 4With the above simulation parameters and controller

gains the curves of survival time prediction power consump-tion and control output of both control loops are shown inFigures 3 to 8

Analyzing the simulation curves we have the followingresults

(1) Figures 3 6 4 and 7 imply that both sensor 1and sensor 2 can meet their expected survival timerequirements

(2) The power consumption in three cases of minimumsampling maximum sampling and the proposedadaptive sampling is compared in Figures 4 and7 It is obvious that the power is consumed muchfaster than the other two cases In the case of theadaptive sampling the power consumption variesaccording to both the requirements of the controlperformance and survival time At the beginning thepower consumption curves vary quickly and morepower are consumed because of the control systemsnot reaching stable yetThen after the control systems

0 20 40 60 800

50

100

150

200

Sens

or 1

life

span

(s)

250

300

350

400

Time (s)

Figure 3 Sensor 1 survival time prediction

0 15 30 45 60 750

005

01

015

Time (s)Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Ener

gy o

f sen

sor1

(J)

T 1T 1

Figure 4 Sensor 1 power consumption comparison in three cases

are settled it tends to theminimum consumption ratewhich is corresponding to the maximum samplingperiod

(3) Figures 5 and 8 are the control outputs of loop1and loop2 in the three cases mentioned above Thefigures show that the control systems can be stabilizedthrough the proposed joint design methods In threecases the adaptive sampling can get the control per-formances closed to the case of minimum sampling

(4) Combining Figures 4 5 7 and 8 it can be concludedthat the proposed joint design method achieves atradeoff between the performances of control andpower efficiency

Mathematical Problems in Engineering 9

Time (s)

The proposed adaptive sampling

0 15 30 45 60 75

0

2

4

6

8

Out

puty

of lo

op1

minus2

minus4

minus6

minus8

Constant sampling with minimum T1 = 1 msConstant sampling with maximum T1 = 256 ms

Figure 5 Control loop1 output comparison in three cases

0 20 40 60 800

50

100

150

Sens

or 2

life

span

(s)

200

250

300

350

Time (s)

Figure 6 Sensor 2 survival time prediction

6 Conclusion

This paper presents a joint design method for wirelessnetworked control systems with limited power constraint Apower efficiency algorithm based on the adaptive samplingperiod is put forward to satisfy the demands of sensorsurvival time and system stability Then the time-varyingcontrol system with transmission delay is modeled as aswitched system with uncertain switching signals A dwell-time-dependent controlmethod is discussed to guarantee thestability of WNCS Simulation results show the effectivenessof the proposed method and indicate that it can achievegood tradeoff performance Methods by which to reducepower consumption from the aspect of a single node as wellas balancing power consumption from the global networkperspective are worthy of further exploration

0 15 30 45 60 750

005

01

015

Time (s)

Ener

gy o

f sen

sor2

(J)

The proposed adaptive sampling

Constant sampling with minimum T2 = 1msConstant sampling with maximum T2 = 256ms

Figure 7 Sensor 2 power consumption comparison in three cases

0 15 30 45 60 75

0

5

10

15

Out

put y

of l

oop

2

minus15

minus10

minus5

Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Time (s)

T2T2

Figure 8 Control loop2 output comparison in three cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The working is sponsored by The NSFC (no 61202473) andThe Natural Science Foundation of Jiangsu Province (noBK2012551)

10 Mathematical Problems in Engineering

References

[1] W Dieterle H D Kochs and E Dittmar ldquoCommunicationarchitectures for distributed computer control systemsrdquoControlEngineering Practice vol 3 no 8 pp 1171ndash1176 1995

[2] H A Thompson ldquoWireless and Internet communicationstechnologies for monitoring and controlrdquo Control EngineeringPractice vol 12 no 6 pp 781ndash791 2004

[3] P R Kumar ldquoNew technological vistas for systems and controlrdquoIEEE Control Systems Magazine vol 21 no 1 pp 24ndash37 2001

[4] X Liu and A Goldsmith ldquoWireless medium access controlin networked control systemsrdquo in Proceedings of the AmericanControl Conference (AAC rsquo04) pp 3605ndash3610 Boston MassUSA July 2004

[5] J Colandairaj G W Irwin and W G Scanlon ldquoWirelessnetworked control systems with QoS-based samplingrdquo IETControl Theory and Applications vol 1 no 1 pp 430ndash438 2007

[6] N Israr W G Scanlon and G W Irwin ldquoA cross-layer com-munication framework forwireless networked control systemsrdquoin Proceedings of the 1st International Conference on WirelessCommunication Vehicular Technology Information Theory andAerospace and Electronic Systems Technology (VITAE rsquo09) pp577ndash581 Aalborg Denmark May 2009

[7] S Dai H Lin and S S Ge ldquoScheduling-and-control codesignfor a collection of networked control systems with uncertaindelaysrdquo IEEE Transactions on Control Systems Technology vol18 no 1 pp 66ndash78 2010

[8] A V Savkin ldquoAnalysis and synthesis of networked control sys-tems topological entropy observability robustness and optimalcontrolrdquo Automatica vol 42 no 1 pp 51ndash62 2006

[9] J Z Luo F Dong and J X Cao ldquoA novel task schedul-ing algorithm based on dynamic critical path and effectiveduplication for pervasive computing environmentrdquo WirelessCommunications amp Mobile Computing vol 10 no 10 pp 1283ndash1302 2010

[10] X Yin X Zhou Z Li and S Li ldquoJint rate control and powercontrol for lifetime maximization in Wreless Sensor NtworksrdquoJournal of Internet Technology vol 12 no 1 pp 69ndash78 2011

[11] M Pajic S Sundaram G J Pappas and R Mangharam ldquoThewireless control network a new approach for control overnetworksrdquo IEEE Transactions on Automatic Control vol 56 no10 pp 2305ndash2318 2011

[12] J Colandairaj G W Irwin and W G Scanlon ldquoA co-designsolution for wireless feedback controlrdquo in Proceeding of theInternational Conference on Networking Sensing and Control(ICNSC 07) pp 404ndash409 London UK April 2007

[13] W Liang and Y Yang ldquoMaximizing battery life routing in wire-less ad hoc networksrdquo in Proceedings of the 37th InternationalConference on System Sciences pp 4739ndash4746 IEEE HonoluluHawaii USA January 2004

[14] K Brian J Haberman and W Sheppard Overlapping ParticleSwarms for Energy-Efficient Routing in Sensor Networks Wire-less Network Springer 2011

[15] D Kim K Dantu andM Pedram ldquoLifetime prediction routinginmobile AdHoc networksrdquo in Proceedings of the IEEEWirelessCommunication and Networking Conference New York NYUSA 2003

[16] L Hetel J Daafouz and C Iung ldquoStability analysis for discretetime switched systems with temporary uncertain switchingsignalrdquo in Proceedings of the 46th IEEE Conference on Decisionand Control (CDC rsquo07) pp 5623ndash5628 New Orleans Fla USADecember 2007

[17] S Limin Wireless Sensor Network Tsinghua University PressBeijing China 2006

[18] Y He I Lee and L Guan ldquoDistributed algorithms for networklifetimemaximization in wireless visual sensor networksrdquo IEEETransactions on Circuits and Systems for Video Technology vol19 no 5 pp 704ndash718 2009

[19] B A Bakr and L Lilien ldquoA quantitative comparison of energyconsumption and WSN lifetime for LEACH and LEACH-SMrdquo in Proceedings of the 31st International Conference onDistributed Computing Systems Workshops (ICDCSW rsquo11) pp182ndash191 Minneapolis Minn USA June 2011

[20] C Fischione A Bonivento A Sangiovanni-Vincentelli F San-tucci and K H Johansson ldquoPerformance analysis of collabora-tive spatio-temporal processing for wireless sensor networksrdquoin Proceedings of the 3rd IEEE Consumer Communications andNetworking Conference (CCNC rsquo06) pp 325ndash329 Las VegasNev USA January 2006

[21] Y Iino ldquoSome considerations of wireless sensor network basedcontrol systemsrdquo in Proceedings of the SICE Conference onControl Systems (CCS rsquo07) Tokyo Japan 2007

[22] Y Iino T Hatanaka and M Fujita ldquoEvent-predictive controlfor energy saving of wireless networked control systemrdquo inProceedings of the American Control Conference (ACC rsquo09) pp2236ndash2242 St Louis Mo USA June 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

where (119878mc times 2 times 119873)119876 is the allowable minimum samplingperiod when the wireless network can be schedulable (see in[6]) and

119876 =119878mc

119879DIFS + (119862119882min times 119879SIFS2) + 119879frame + 119879SIFS + 119879ACK(15)

119879frame =119878PHY119877119897

+119878MAC + 119878mc

119877119905

119879frame =119878PHY119877119897

+119878MAC + 119878mc

119877119905

(16)

119877119905is the transmission rate 119877

119897is the legacy transmission rate

119878mc is the measurement-control data size 119878PHY is the sizeof control frame in physical layer and 119878MAC is the data sizeof ACK and is a confirmed sign in the header of TCP datapacket that confirms the received TCP message 119879SIFS is theshortest time period of the 80211b protocol for the intervalof frames requiring immediate response 119879DIFS is the timesegment for the interval of the time frame of the distributedcoordination function for sending in IEEE 80211b 119879PIFS isthe time segment for the interval of the time frame of thecentralized coordination function for sending which satisfies

119879PIFS = 119879DIFS minus 119879slot 119879slot = 119879PIFS minus 119879SIFS (17)

CW is the contention window Wireless network parametersunder the 80211b direct sequence spread spectrum are shownin Table 2

Remark 4 (upper bound of sampling period) For a SISOsystem the maximum sampling period can be obtainedusing Shannon sampling theorem For a MIMO system thefollowing method can be used to obtain the upper bound Ifthe system feedback control law is given ahead then119879max canbe obtained by solving the following optimal problem

max 119879

st100381710038171003817100381710038171003817100381710038171003817

eig(119890A119879 + int

119879

0

119890A119879

119889119905BK0)

100381710038171003817100381710038171003817100381710038171003817

lt 1

(18)

where A is the system matrix and B is the control inputmatrix K

0is the Kalman gain which satisfies

Re (eig (A + BK0)) lt 0 (19)

where Re(120592) denotes the real part of 120592 and eig(X) denotes theeigenvalues of matrix X Gain K

0can be determined by the

pole assignment in the continuous time domain

However the above optimal problem is difficult to solvedirectlyThe following iterationmethod can be used to obtainthe approximate optimal value of 119879max

Step 1 Let 119902 = 1 and the initial value of 119879max is 119879max(119902) =

2119879min If the condition can satisfy the constraints of the

optimal problem go to Step 2 or else go to Step 3

Table 2 Wireless network parameters under 80211b directsequence spread spectrum

Parameter Value119877119905

11Mbps119877119897

1Mbps119879SIFS 119879DIFS 119879slot 10 50 20 us119878MAC 119878PHY 34 24 bytes119878mc 80 bytes119878ACK 14 bytes + PHY headerCWmin CWmax 32 1024

Step 2 Let 119902 = 119902 + 1 and let 119879max(119902) = 2119879

max(119902 minus 1) If the

condition can fulfill the constraints of the optimal problemcycle Step 2 or else end the iteration and then let 119879max

=

119879max

(119902 minus 1)

Step 3 Let 119879max(119902) = (1 + (34)

119902)119879

min If the condition stilldoes not satisfy the constraints of the optimal problem let119902 = 119902 + 1 and cycle Step 3 or else end the iteration and let119879max

= 119879max

(119902)

4 Modeling and Stability Analysis of AdaptiveSampling Period WNCS

41 Modeling of Adaptive Sampling Period WNCS Consid-ering the generality one of the control loops in the WNCSis chosen as an example to illustrate the modeling approachFor theWNCS power consumptionmanaged by algorithm inSection 2 the dynamics of the control system is time-varyingSince the sampling period varies among the 119872 candidatesthe system can be considered a switched system with 119872

modes from the perspective of the switched control systemscheme Each switching mode is corresponded to one ofthe candidates According to Assumption 1 the sensor andactuator nodes are assumed to be clock-driven It results thatthe switching occurs at some of the sampling instants Addi-tionally according to Assumption 3 the inevitable existenceof network-induced delay is taken into account and it is lessthan one sampling period

We consider a plant of control loop 119894 in the WNCS withthe following dynamics

119894(119905) = A119894119909119894 (119905) + B119894119906119894 (119905)

119910119894(119905) = C119894119909119894 (119905)

(20)

where 119909119894(119905) isin R119899 is the plant state 119906119894(119905) isin R119898 is the controlinput and 119910

119894(119905) isin R119901 is the plant output A119894 isin R119899times119899 B119894 isin

R119899times119898 and C119894 isin R119901times119899 are the matrices of state control inputand output matrices respectively Due to the generality of 119894we omit the superscript 119894 in the following model descriptionand deduction

Discretizing system (20) with sampling rate 119879119897and

considering network-induced delay less than one sampling

6 Mathematical Problems in Engineering

Mixed mode Mixed model(k minus 1) ne l(k) l(k minus 1) = l(k) l(k + 3) ne l(k + 2)

u(k minus 1)

u(k + 1)120591l(k)(k)

k + 1 k + 2 k + 3

Sensor

Controller

node

node

nodeu(k minus 1)= Kl(kminus1)x(k minus 1)

Actuator

k minus 1 k

u(k)

Nominal behavior

One monitor interval

Tl(kminus1) Tl(k) Tl(k +3)

TM

Figure 2 Evolution over one period of switched WNCS with two types of behavior

period the discrete dynamics of the open control loop canbe described as

119909 (119896 + 1) = Φ119897(119896)119909 (119896) + Γ119897(119896)119906 (119896 minus 1)

119910 (119896) = C119909 (119896) (21)

whereΦ119897(119896)

= 119890A119879119897(119896) Γ

119897= int119879119897(119896)

0119890A119905119889119905B and 119897(119896) is the identi-

fication of sampling period at the 119896th sampling instant 119897(119896) isinZ rarr ℓ = 0 1 119872 minus 1

For the discrete switched system (21) a switched statefeedback controller is introduced in the following form

119906 (119896) = K120574(119896)

119909 (119896) (22)

where 120574(119896) isin Z rarr ℓ = 0 1 119872 minus 1 denotes the switch-ing signal used in the control

Consequently the closed-loop WNCS can be written as

[119909 (119896 + 1)

119909 (119896)] = [Φ119897(119896)Γ119897(119896)

K120574(119896)

I 0] [

119909 (119896)

119909 (119896 minus 1)] (23)

42 Stability Analysis of Adaptive Sampling Period WNCSThe control gains K

120574(119896)are assumed to be designed in such

a way that the closed-loop system is asymptotically stablewhen 120574(119896) = 119897(119896) Ideally the switching signal used in control120574(119896) is the same as the real signal 119897(119896) However this viewis unrealistic in WNCS with network-induced delay where120574(119896) = 119897(119896 minus 1) The evolution over one sampling period canbe described by two distinct types of behavior the nominaland mixed mode sampling periods as shown in Figure 2

(1)Nominal sampling period is when the system evolutionuses the right switching information

120574 (119896) = 119897 (119896 minus 1) 119897 (119896) = 119897 (119896 minus 1) (24)

(2) Mixed mode sampling period is when the systemcommand uses a wrong feedback gain

120574 (119896) = 119897 (119896 minus 1) 119897 (119896) = 119897 (119896 minus 1) (25)

Denoting Δ119898 as the samples spent in the mixed mode

since the delay is less than one sampling period it follows that

Δ119898 has a range of 0 le Δ

119898le 1 Moreover we assume that

the system is controlled using the right gains for at least Δ119899samples before another switching occurs

The next thing we should do is to guarantee the closed-loop system remaining stable with the designed state feed-back control gains when the switching signal is temporarilyuncertain Consider the scalars 120579

119899gt 0 120579

119898gt 0 and the

symmetric positive definite matrices P119899119897 P119898(119897120574)

with (119897 120574) isin

ℓ times ℓ which satisfy the following matrix inequalities

Φ119879

(119897120574)(P119898(119897120574)

)minus1

Φ(119897120574)

minus 120579119898P119898(119897120574)

lt 0 forall (119897 120574) isin ℓ times ℓ

Φ119879

(119897119897)P119899119897Φ(119897119897)

minus 120579119899P119899119897lt 0 forall119897 isin ℓ

(26)

Moreover consider the following two scalars

120573119899=max119897isinℓeigmax (P

119899

119897)

min119897isinℓeigmin (P119899119897 )

120573119898=

max(119897120574)isinℓtimesℓ

eigmax (P119898

(119897120574))

min(119897120574)isinℓtimesℓ

eigmin (P119898(119897120574))

(27)

where eigmax(X) and eigmin(X) denote the maximum and theminimum eigenvalues of matrix X respectively

Then the stability of the closed-loop WNCS (23) can beguaranteed by the following theorem

Theorem 5 Let 120579lowast119899 120579lowast119898be the solutions of the optimization

problems 120579lowast119899

= min 120579119899and 120579

lowast

119898= min 120579

119898subject to matrix

inequalities (26) Closed-loop system (23) is asymptoticallystable if

120573119899sdot 120573119898sdot (120579lowast

119899)Δ119899

sdot 120579lowast

119898lt 1 (28)

Proof We consider the following Lyapunov functions

119881119899(119896) = 119909

119879(119896)P119899119897119909 (119896) 119881

119898(119896) = 119909

119879(119896)P119898(119897120574)

119909 (119896)

(29)

Inequalities (13) show that

119881119899(1198961) lt (120579

lowast

119899)1198961minus1198960

119881119899(1198960) (30)

Mathematical Problems in Engineering 7

forall119896 isin [1198960 1198961) if the right switching signal is used in the

control 119897(119896) = 120574(119896) = 119897(1198960) then

119881119898(1198961) lt (120579

lowast

119898)1198961minus1198960

119881119898(1198960) (31)

forall119896 isin [1198960 1198961) if the switching signal in the control is not

necessarily the same as the real signal the pair 119897(119896) 120574(119896) takesan arbitrary value 119897(119896

0) 120574(119896

0) in ℓ times ℓ

Since the controller gains K119897are designed to make the

matricesΦ(119897119897)

stable the scalar 120579119899is smaller than one 120579

119899lt 1

The scalar 120579119898may be greater than one 120579

119898gt 1 since the

gains K120574are not designed to stabilize combinations other

thanΦ(120574120574)

Combining inequalities (26) yield

min119897isinℓ

eigmin (P119899

119897)1003817100381710038171003817119909(1198961)

1003817100381710038171003817

2

lt (120579lowast

119899)1198961minus1198960max119897isinℓ

eigmax (P119899

119897)1003817100381710038171003817119909(1198960)

1003817100381710038171003817

2forall119897 isin ℓ

min(119897120574)isinℓtimesℓ

eigmin (P119898

(119897120574))1003817100381710038171003817119909(1198961)

1003817100381710038171003817

2

lt max(119897120574)isinℓtimesℓ

eigmax (P119898

(119897120574))1003817100381710038171003817119909(1198960)

1003817100381710038171003817

2forall (119897 120574) isin ℓ times ℓ

(32)

With the definitions of 120573119899and 120573

119898 we can obtain the state

vector norm decay or growth rate in a nominal regime and inan uncertain switching signal regime as follows

1003817100381710038171003817119909(1198961)1003817100381710038171003817

2lt 120573119899sdot (120579lowast

119899)1198961minus11989601003817100381710038171003817119909(1198960)

1003817100381710038171003817

2 forall119897 isin ℓ

1003817100381710038171003817119909(1198961)1003817100381710038171003817

2lt 120573119898sdot (120579lowast

119898)1198961minus11989601003817100381710038171003817119909(1198960)

1003817100381710038171003817

2 forall (119897 120574) isin ℓ times ℓ

(33)

Let 119896119898119904describe the instants when the closed-loop system

jumps to a mixed mode with uncertain switching signal andlet 119896119899119904be the instance when the system enters into a normal

regime With definitions and bounds of Δ119898 and Δ119899 it followsthat

0 le Δ119898= 119896119899

119904minus 119896119898

119904le 1 119896

119898

119904+1minus 119896119899

119904le Δ119899 (34)

Without loss of generality we assume that the systemstarts with a mixed-mode behavior 119896119898

119904lt 119896119899

119904 The system

behavior in time interval 119896 isin [119896119898

119904 119896119898

119904+1) is then analyzed

Given that 119897(119896) = 120574(119896) forall119896 isin [119896119898

119904 119896119899

119904) and 119897(119896) =

120574(119896) forall119896 isin [119896119899

119904 119896119898

119904+1) using inequalities (26) the norm of the

state at the end of the sequence can be upper bounded asd

1003817100381710038171003817119909(119896119898

119904+1)1003817100381710038171003817

2lt 120573119898sdot 120573119899sdot (120579lowast

119898)119896119899

119904minus119896119906

119904sdot (120579lowast

119899)119896119898

119904+1minus119896119899

119904 1003817100381710038171003817119909(119896119898

119904)1003817100381710038171003817

2

lt1003817100381710038171003817119909(119896119898

119904+1)1003817100381710038171003817

2lt 120573119898sdot 120573119899sdot 120579lowast

119898sdot (120579lowast

119899)Δ1198991003817100381710038171003817119909(119896119898

119904)1003817100381710038171003817

2

(35)

It indicates that closed loop (16) will be asymptotically stableif condition (28) in Theorem 5 is satisfied

43 Choosing Rule of Update Period 119879119872

Theorem 6 Consider the WNCS with adaptive samplingperiod rule (3) The WNCS can be stabilized by the switched

state feedback controllers (22) whereas the survival time canmeet its expected value 119871

119890 if the update period 119879

119872satisfies the

following conditions

(1) 119879119872

ge max (Δ119899119894+ 1) [119879

max1

119879max2

119879max119899

]

(2) lfloor119871119890

119879119872

rfloor sdot 119879119872

le119864119894119899119894119905

119864

min 119879min1

119879min2

119879min119899

(36)

where Δ119899119894is the least nominal sampling period of control loop

119894 and the solution of Theorem 5 [119879max1

119879max2

119879max119899

] is theleast commonmultiple of the maximum sampling period for allthe 119899 control loops in the WNCS

Proof According to Theorem 5 control loop 119894 in the WNCSshould stay at least Δ119899

119894in a nominal sampling period in one

updating interval for stability With the consideration of 0 le

Δ119898

119894le 1 if the update period satisfies condition (1) all control

loops in the WNCS will meet the demands of their leastnominal sampling periods and will be stable As a completesystem WNCS is composed of 119899 control loops that will bestable when condition (1) is satisfied

Condition (2) inTheorem 6 provides the upper bound ofthe updating period For the power efficiency algorithm inTheorem 1 with the definition of 119896

119897(119895) there exists

119872

sum

119897=1

119896119897(119895)

119879119872

119879119897

+119879119872

119879119895

le lfloor119871119890

119879119872

rfloor sdot119879119872

min 119879min1

119879min2

119879min119899

(37)

If condition (2) inTheorem 6 is satisfied the item ((119864init119864)minus

sum119872

119897=1119896119897(119895)(119879119872119879119897) minus (119879

119872119879119895)) in the denominator of formula

(3) will be greater than zero which guarantees that formula(3) has physical meaning and is solvable The survival timewill thus reach the expected value by applying the powerefficiency algorithm

5 Numerical Example

Simulation studies are performed on a WNCS closed by anIEEE 80211b wireless network with two control loops sharingthe network resources The two control loops are assumed tohave the same dynamics but with different initial conditions

119894(119905) = [

minus1 minus01

0 095] 119909119894(119905) + [

minus015

minus043] 119906119894(119905)

119910119894(119905) = [1 1] 119909

119894(119905)

119894 = 1 2

1199091(0) = [minus5 5]

119879 119909

2(0) = [minus5 10]

119879

(38)

The wireless network parameters are set as in Table 2The time delays in both loops are less than one samplingperiod Some other necessary parameters are given as shownin Table 2

In Table 3 the minimum and maximum sampling peri-ods are computed by the methods in Section 3 Solving

8 Mathematical Problems in Engineering

Table 3 Simulation parameters

Parameter ValueInitial energy of both sensors 119864init 015 JSensor 1 expected survival time 119871

119890175 s

Sensor 2 expected survival time 1198711198902

70 sUnit transmission energy 119864 25 dbmNumber of sampling period candidates119872 10The minimum sampling period of loop1 119879min

11ms

The maximum sampling period of loop1 119879max1

256msThe minimum sampling period of loop2 119879min

21ms

The maximum sampling period of loop2 119879max2

256ms

Table 4 Controller gains of ten sampling modes

Sampling period (ms) Controller gain1198790= 1 119870

0= [11176 11188]

1198791= 29 119870

1= [394837 405512]

1198792= 58 119870

2= [194678 204319]

1198793= 86 119870

3= [125668 134279]

1198794= 114 119870

4= [90353 97947]

1198795= 143 119870

5= [68663 75265]

1198796= 171 119870

6= [5383 5947]

1198797= 199 119870

7= [4295 4769]

1198798= 227 119870

8= [4225 3852]

1198799= 256 119870

9= [3707 295]

inequality (28) in Theorem 5 it yields Δ1198991= Δ119899

2= 8 Δ119898 = 1

Also the update period can be chosen byTheorem 6 as 119879119872

=

150msSolving matrix inequalities (18) and (19) the controller

gains of ten switching modes can be obtained as in Table 4With the above simulation parameters and controller

gains the curves of survival time prediction power consump-tion and control output of both control loops are shown inFigures 3 to 8

Analyzing the simulation curves we have the followingresults

(1) Figures 3 6 4 and 7 imply that both sensor 1and sensor 2 can meet their expected survival timerequirements

(2) The power consumption in three cases of minimumsampling maximum sampling and the proposedadaptive sampling is compared in Figures 4 and7 It is obvious that the power is consumed muchfaster than the other two cases In the case of theadaptive sampling the power consumption variesaccording to both the requirements of the controlperformance and survival time At the beginning thepower consumption curves vary quickly and morepower are consumed because of the control systemsnot reaching stable yetThen after the control systems

0 20 40 60 800

50

100

150

200

Sens

or 1

life

span

(s)

250

300

350

400

Time (s)

Figure 3 Sensor 1 survival time prediction

0 15 30 45 60 750

005

01

015

Time (s)Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Ener

gy o

f sen

sor1

(J)

T 1T 1

Figure 4 Sensor 1 power consumption comparison in three cases

are settled it tends to theminimum consumption ratewhich is corresponding to the maximum samplingperiod

(3) Figures 5 and 8 are the control outputs of loop1and loop2 in the three cases mentioned above Thefigures show that the control systems can be stabilizedthrough the proposed joint design methods In threecases the adaptive sampling can get the control per-formances closed to the case of minimum sampling

(4) Combining Figures 4 5 7 and 8 it can be concludedthat the proposed joint design method achieves atradeoff between the performances of control andpower efficiency

Mathematical Problems in Engineering 9

Time (s)

The proposed adaptive sampling

0 15 30 45 60 75

0

2

4

6

8

Out

puty

of lo

op1

minus2

minus4

minus6

minus8

Constant sampling with minimum T1 = 1 msConstant sampling with maximum T1 = 256 ms

Figure 5 Control loop1 output comparison in three cases

0 20 40 60 800

50

100

150

Sens

or 2

life

span

(s)

200

250

300

350

Time (s)

Figure 6 Sensor 2 survival time prediction

6 Conclusion

This paper presents a joint design method for wirelessnetworked control systems with limited power constraint Apower efficiency algorithm based on the adaptive samplingperiod is put forward to satisfy the demands of sensorsurvival time and system stability Then the time-varyingcontrol system with transmission delay is modeled as aswitched system with uncertain switching signals A dwell-time-dependent controlmethod is discussed to guarantee thestability of WNCS Simulation results show the effectivenessof the proposed method and indicate that it can achievegood tradeoff performance Methods by which to reducepower consumption from the aspect of a single node as wellas balancing power consumption from the global networkperspective are worthy of further exploration

0 15 30 45 60 750

005

01

015

Time (s)

Ener

gy o

f sen

sor2

(J)

The proposed adaptive sampling

Constant sampling with minimum T2 = 1msConstant sampling with maximum T2 = 256ms

Figure 7 Sensor 2 power consumption comparison in three cases

0 15 30 45 60 75

0

5

10

15

Out

put y

of l

oop

2

minus15

minus10

minus5

Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Time (s)

T2T2

Figure 8 Control loop2 output comparison in three cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The working is sponsored by The NSFC (no 61202473) andThe Natural Science Foundation of Jiangsu Province (noBK2012551)

10 Mathematical Problems in Engineering

References

[1] W Dieterle H D Kochs and E Dittmar ldquoCommunicationarchitectures for distributed computer control systemsrdquoControlEngineering Practice vol 3 no 8 pp 1171ndash1176 1995

[2] H A Thompson ldquoWireless and Internet communicationstechnologies for monitoring and controlrdquo Control EngineeringPractice vol 12 no 6 pp 781ndash791 2004

[3] P R Kumar ldquoNew technological vistas for systems and controlrdquoIEEE Control Systems Magazine vol 21 no 1 pp 24ndash37 2001

[4] X Liu and A Goldsmith ldquoWireless medium access controlin networked control systemsrdquo in Proceedings of the AmericanControl Conference (AAC rsquo04) pp 3605ndash3610 Boston MassUSA July 2004

[5] J Colandairaj G W Irwin and W G Scanlon ldquoWirelessnetworked control systems with QoS-based samplingrdquo IETControl Theory and Applications vol 1 no 1 pp 430ndash438 2007

[6] N Israr W G Scanlon and G W Irwin ldquoA cross-layer com-munication framework forwireless networked control systemsrdquoin Proceedings of the 1st International Conference on WirelessCommunication Vehicular Technology Information Theory andAerospace and Electronic Systems Technology (VITAE rsquo09) pp577ndash581 Aalborg Denmark May 2009

[7] S Dai H Lin and S S Ge ldquoScheduling-and-control codesignfor a collection of networked control systems with uncertaindelaysrdquo IEEE Transactions on Control Systems Technology vol18 no 1 pp 66ndash78 2010

[8] A V Savkin ldquoAnalysis and synthesis of networked control sys-tems topological entropy observability robustness and optimalcontrolrdquo Automatica vol 42 no 1 pp 51ndash62 2006

[9] J Z Luo F Dong and J X Cao ldquoA novel task schedul-ing algorithm based on dynamic critical path and effectiveduplication for pervasive computing environmentrdquo WirelessCommunications amp Mobile Computing vol 10 no 10 pp 1283ndash1302 2010

[10] X Yin X Zhou Z Li and S Li ldquoJint rate control and powercontrol for lifetime maximization in Wreless Sensor NtworksrdquoJournal of Internet Technology vol 12 no 1 pp 69ndash78 2011

[11] M Pajic S Sundaram G J Pappas and R Mangharam ldquoThewireless control network a new approach for control overnetworksrdquo IEEE Transactions on Automatic Control vol 56 no10 pp 2305ndash2318 2011

[12] J Colandairaj G W Irwin and W G Scanlon ldquoA co-designsolution for wireless feedback controlrdquo in Proceeding of theInternational Conference on Networking Sensing and Control(ICNSC 07) pp 404ndash409 London UK April 2007

[13] W Liang and Y Yang ldquoMaximizing battery life routing in wire-less ad hoc networksrdquo in Proceedings of the 37th InternationalConference on System Sciences pp 4739ndash4746 IEEE HonoluluHawaii USA January 2004

[14] K Brian J Haberman and W Sheppard Overlapping ParticleSwarms for Energy-Efficient Routing in Sensor Networks Wire-less Network Springer 2011

[15] D Kim K Dantu andM Pedram ldquoLifetime prediction routinginmobile AdHoc networksrdquo in Proceedings of the IEEEWirelessCommunication and Networking Conference New York NYUSA 2003

[16] L Hetel J Daafouz and C Iung ldquoStability analysis for discretetime switched systems with temporary uncertain switchingsignalrdquo in Proceedings of the 46th IEEE Conference on Decisionand Control (CDC rsquo07) pp 5623ndash5628 New Orleans Fla USADecember 2007

[17] S Limin Wireless Sensor Network Tsinghua University PressBeijing China 2006

[18] Y He I Lee and L Guan ldquoDistributed algorithms for networklifetimemaximization in wireless visual sensor networksrdquo IEEETransactions on Circuits and Systems for Video Technology vol19 no 5 pp 704ndash718 2009

[19] B A Bakr and L Lilien ldquoA quantitative comparison of energyconsumption and WSN lifetime for LEACH and LEACH-SMrdquo in Proceedings of the 31st International Conference onDistributed Computing Systems Workshops (ICDCSW rsquo11) pp182ndash191 Minneapolis Minn USA June 2011

[20] C Fischione A Bonivento A Sangiovanni-Vincentelli F San-tucci and K H Johansson ldquoPerformance analysis of collabora-tive spatio-temporal processing for wireless sensor networksrdquoin Proceedings of the 3rd IEEE Consumer Communications andNetworking Conference (CCNC rsquo06) pp 325ndash329 Las VegasNev USA January 2006

[21] Y Iino ldquoSome considerations of wireless sensor network basedcontrol systemsrdquo in Proceedings of the SICE Conference onControl Systems (CCS rsquo07) Tokyo Japan 2007

[22] Y Iino T Hatanaka and M Fujita ldquoEvent-predictive controlfor energy saving of wireless networked control systemrdquo inProceedings of the American Control Conference (ACC rsquo09) pp2236ndash2242 St Louis Mo USA June 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Mixed mode Mixed model(k minus 1) ne l(k) l(k minus 1) = l(k) l(k + 3) ne l(k + 2)

u(k minus 1)

u(k + 1)120591l(k)(k)

k + 1 k + 2 k + 3

Sensor

Controller

node

node

nodeu(k minus 1)= Kl(kminus1)x(k minus 1)

Actuator

k minus 1 k

u(k)

Nominal behavior

One monitor interval

Tl(kminus1) Tl(k) Tl(k +3)

TM

Figure 2 Evolution over one period of switched WNCS with two types of behavior

period the discrete dynamics of the open control loop canbe described as

119909 (119896 + 1) = Φ119897(119896)119909 (119896) + Γ119897(119896)119906 (119896 minus 1)

119910 (119896) = C119909 (119896) (21)

whereΦ119897(119896)

= 119890A119879119897(119896) Γ

119897= int119879119897(119896)

0119890A119905119889119905B and 119897(119896) is the identi-

fication of sampling period at the 119896th sampling instant 119897(119896) isinZ rarr ℓ = 0 1 119872 minus 1

For the discrete switched system (21) a switched statefeedback controller is introduced in the following form

119906 (119896) = K120574(119896)

119909 (119896) (22)

where 120574(119896) isin Z rarr ℓ = 0 1 119872 minus 1 denotes the switch-ing signal used in the control

Consequently the closed-loop WNCS can be written as

[119909 (119896 + 1)

119909 (119896)] = [Φ119897(119896)Γ119897(119896)

K120574(119896)

I 0] [

119909 (119896)

119909 (119896 minus 1)] (23)

42 Stability Analysis of Adaptive Sampling Period WNCSThe control gains K

120574(119896)are assumed to be designed in such

a way that the closed-loop system is asymptotically stablewhen 120574(119896) = 119897(119896) Ideally the switching signal used in control120574(119896) is the same as the real signal 119897(119896) However this viewis unrealistic in WNCS with network-induced delay where120574(119896) = 119897(119896 minus 1) The evolution over one sampling period canbe described by two distinct types of behavior the nominaland mixed mode sampling periods as shown in Figure 2

(1)Nominal sampling period is when the system evolutionuses the right switching information

120574 (119896) = 119897 (119896 minus 1) 119897 (119896) = 119897 (119896 minus 1) (24)

(2) Mixed mode sampling period is when the systemcommand uses a wrong feedback gain

120574 (119896) = 119897 (119896 minus 1) 119897 (119896) = 119897 (119896 minus 1) (25)

Denoting Δ119898 as the samples spent in the mixed mode

since the delay is less than one sampling period it follows that

Δ119898 has a range of 0 le Δ

119898le 1 Moreover we assume that

the system is controlled using the right gains for at least Δ119899samples before another switching occurs

The next thing we should do is to guarantee the closed-loop system remaining stable with the designed state feed-back control gains when the switching signal is temporarilyuncertain Consider the scalars 120579

119899gt 0 120579

119898gt 0 and the

symmetric positive definite matrices P119899119897 P119898(119897120574)

with (119897 120574) isin

ℓ times ℓ which satisfy the following matrix inequalities

Φ119879

(119897120574)(P119898(119897120574)

)minus1

Φ(119897120574)

minus 120579119898P119898(119897120574)

lt 0 forall (119897 120574) isin ℓ times ℓ

Φ119879

(119897119897)P119899119897Φ(119897119897)

minus 120579119899P119899119897lt 0 forall119897 isin ℓ

(26)

Moreover consider the following two scalars

120573119899=max119897isinℓeigmax (P

119899

119897)

min119897isinℓeigmin (P119899119897 )

120573119898=

max(119897120574)isinℓtimesℓ

eigmax (P119898

(119897120574))

min(119897120574)isinℓtimesℓ

eigmin (P119898(119897120574))

(27)

where eigmax(X) and eigmin(X) denote the maximum and theminimum eigenvalues of matrix X respectively

Then the stability of the closed-loop WNCS (23) can beguaranteed by the following theorem

Theorem 5 Let 120579lowast119899 120579lowast119898be the solutions of the optimization

problems 120579lowast119899

= min 120579119899and 120579

lowast

119898= min 120579

119898subject to matrix

inequalities (26) Closed-loop system (23) is asymptoticallystable if

120573119899sdot 120573119898sdot (120579lowast

119899)Δ119899

sdot 120579lowast

119898lt 1 (28)

Proof We consider the following Lyapunov functions

119881119899(119896) = 119909

119879(119896)P119899119897119909 (119896) 119881

119898(119896) = 119909

119879(119896)P119898(119897120574)

119909 (119896)

(29)

Inequalities (13) show that

119881119899(1198961) lt (120579

lowast

119899)1198961minus1198960

119881119899(1198960) (30)

Mathematical Problems in Engineering 7

forall119896 isin [1198960 1198961) if the right switching signal is used in the

control 119897(119896) = 120574(119896) = 119897(1198960) then

119881119898(1198961) lt (120579

lowast

119898)1198961minus1198960

119881119898(1198960) (31)

forall119896 isin [1198960 1198961) if the switching signal in the control is not

necessarily the same as the real signal the pair 119897(119896) 120574(119896) takesan arbitrary value 119897(119896

0) 120574(119896

0) in ℓ times ℓ

Since the controller gains K119897are designed to make the

matricesΦ(119897119897)

stable the scalar 120579119899is smaller than one 120579

119899lt 1

The scalar 120579119898may be greater than one 120579

119898gt 1 since the

gains K120574are not designed to stabilize combinations other

thanΦ(120574120574)

Combining inequalities (26) yield

min119897isinℓ

eigmin (P119899

119897)1003817100381710038171003817119909(1198961)

1003817100381710038171003817

2

lt (120579lowast

119899)1198961minus1198960max119897isinℓ

eigmax (P119899

119897)1003817100381710038171003817119909(1198960)

1003817100381710038171003817

2forall119897 isin ℓ

min(119897120574)isinℓtimesℓ

eigmin (P119898

(119897120574))1003817100381710038171003817119909(1198961)

1003817100381710038171003817

2

lt max(119897120574)isinℓtimesℓ

eigmax (P119898

(119897120574))1003817100381710038171003817119909(1198960)

1003817100381710038171003817

2forall (119897 120574) isin ℓ times ℓ

(32)

With the definitions of 120573119899and 120573

119898 we can obtain the state

vector norm decay or growth rate in a nominal regime and inan uncertain switching signal regime as follows

1003817100381710038171003817119909(1198961)1003817100381710038171003817

2lt 120573119899sdot (120579lowast

119899)1198961minus11989601003817100381710038171003817119909(1198960)

1003817100381710038171003817

2 forall119897 isin ℓ

1003817100381710038171003817119909(1198961)1003817100381710038171003817

2lt 120573119898sdot (120579lowast

119898)1198961minus11989601003817100381710038171003817119909(1198960)

1003817100381710038171003817

2 forall (119897 120574) isin ℓ times ℓ

(33)

Let 119896119898119904describe the instants when the closed-loop system

jumps to a mixed mode with uncertain switching signal andlet 119896119899119904be the instance when the system enters into a normal

regime With definitions and bounds of Δ119898 and Δ119899 it followsthat

0 le Δ119898= 119896119899

119904minus 119896119898

119904le 1 119896

119898

119904+1minus 119896119899

119904le Δ119899 (34)

Without loss of generality we assume that the systemstarts with a mixed-mode behavior 119896119898

119904lt 119896119899

119904 The system

behavior in time interval 119896 isin [119896119898

119904 119896119898

119904+1) is then analyzed

Given that 119897(119896) = 120574(119896) forall119896 isin [119896119898

119904 119896119899

119904) and 119897(119896) =

120574(119896) forall119896 isin [119896119899

119904 119896119898

119904+1) using inequalities (26) the norm of the

state at the end of the sequence can be upper bounded asd

1003817100381710038171003817119909(119896119898

119904+1)1003817100381710038171003817

2lt 120573119898sdot 120573119899sdot (120579lowast

119898)119896119899

119904minus119896119906

119904sdot (120579lowast

119899)119896119898

119904+1minus119896119899

119904 1003817100381710038171003817119909(119896119898

119904)1003817100381710038171003817

2

lt1003817100381710038171003817119909(119896119898

119904+1)1003817100381710038171003817

2lt 120573119898sdot 120573119899sdot 120579lowast

119898sdot (120579lowast

119899)Δ1198991003817100381710038171003817119909(119896119898

119904)1003817100381710038171003817

2

(35)

It indicates that closed loop (16) will be asymptotically stableif condition (28) in Theorem 5 is satisfied

43 Choosing Rule of Update Period 119879119872

Theorem 6 Consider the WNCS with adaptive samplingperiod rule (3) The WNCS can be stabilized by the switched

state feedback controllers (22) whereas the survival time canmeet its expected value 119871

119890 if the update period 119879

119872satisfies the

following conditions

(1) 119879119872

ge max (Δ119899119894+ 1) [119879

max1

119879max2

119879max119899

]

(2) lfloor119871119890

119879119872

rfloor sdot 119879119872

le119864119894119899119894119905

119864

min 119879min1

119879min2

119879min119899

(36)

where Δ119899119894is the least nominal sampling period of control loop

119894 and the solution of Theorem 5 [119879max1

119879max2

119879max119899

] is theleast commonmultiple of the maximum sampling period for allthe 119899 control loops in the WNCS

Proof According to Theorem 5 control loop 119894 in the WNCSshould stay at least Δ119899

119894in a nominal sampling period in one

updating interval for stability With the consideration of 0 le

Δ119898

119894le 1 if the update period satisfies condition (1) all control

loops in the WNCS will meet the demands of their leastnominal sampling periods and will be stable As a completesystem WNCS is composed of 119899 control loops that will bestable when condition (1) is satisfied

Condition (2) inTheorem 6 provides the upper bound ofthe updating period For the power efficiency algorithm inTheorem 1 with the definition of 119896

119897(119895) there exists

119872

sum

119897=1

119896119897(119895)

119879119872

119879119897

+119879119872

119879119895

le lfloor119871119890

119879119872

rfloor sdot119879119872

min 119879min1

119879min2

119879min119899

(37)

If condition (2) inTheorem 6 is satisfied the item ((119864init119864)minus

sum119872

119897=1119896119897(119895)(119879119872119879119897) minus (119879

119872119879119895)) in the denominator of formula

(3) will be greater than zero which guarantees that formula(3) has physical meaning and is solvable The survival timewill thus reach the expected value by applying the powerefficiency algorithm

5 Numerical Example

Simulation studies are performed on a WNCS closed by anIEEE 80211b wireless network with two control loops sharingthe network resources The two control loops are assumed tohave the same dynamics but with different initial conditions

119894(119905) = [

minus1 minus01

0 095] 119909119894(119905) + [

minus015

minus043] 119906119894(119905)

119910119894(119905) = [1 1] 119909

119894(119905)

119894 = 1 2

1199091(0) = [minus5 5]

119879 119909

2(0) = [minus5 10]

119879

(38)

The wireless network parameters are set as in Table 2The time delays in both loops are less than one samplingperiod Some other necessary parameters are given as shownin Table 2

In Table 3 the minimum and maximum sampling peri-ods are computed by the methods in Section 3 Solving

8 Mathematical Problems in Engineering

Table 3 Simulation parameters

Parameter ValueInitial energy of both sensors 119864init 015 JSensor 1 expected survival time 119871

119890175 s

Sensor 2 expected survival time 1198711198902

70 sUnit transmission energy 119864 25 dbmNumber of sampling period candidates119872 10The minimum sampling period of loop1 119879min

11ms

The maximum sampling period of loop1 119879max1

256msThe minimum sampling period of loop2 119879min

21ms

The maximum sampling period of loop2 119879max2

256ms

Table 4 Controller gains of ten sampling modes

Sampling period (ms) Controller gain1198790= 1 119870

0= [11176 11188]

1198791= 29 119870

1= [394837 405512]

1198792= 58 119870

2= [194678 204319]

1198793= 86 119870

3= [125668 134279]

1198794= 114 119870

4= [90353 97947]

1198795= 143 119870

5= [68663 75265]

1198796= 171 119870

6= [5383 5947]

1198797= 199 119870

7= [4295 4769]

1198798= 227 119870

8= [4225 3852]

1198799= 256 119870

9= [3707 295]

inequality (28) in Theorem 5 it yields Δ1198991= Δ119899

2= 8 Δ119898 = 1

Also the update period can be chosen byTheorem 6 as 119879119872

=

150msSolving matrix inequalities (18) and (19) the controller

gains of ten switching modes can be obtained as in Table 4With the above simulation parameters and controller

gains the curves of survival time prediction power consump-tion and control output of both control loops are shown inFigures 3 to 8

Analyzing the simulation curves we have the followingresults

(1) Figures 3 6 4 and 7 imply that both sensor 1and sensor 2 can meet their expected survival timerequirements

(2) The power consumption in three cases of minimumsampling maximum sampling and the proposedadaptive sampling is compared in Figures 4 and7 It is obvious that the power is consumed muchfaster than the other two cases In the case of theadaptive sampling the power consumption variesaccording to both the requirements of the controlperformance and survival time At the beginning thepower consumption curves vary quickly and morepower are consumed because of the control systemsnot reaching stable yetThen after the control systems

0 20 40 60 800

50

100

150

200

Sens

or 1

life

span

(s)

250

300

350

400

Time (s)

Figure 3 Sensor 1 survival time prediction

0 15 30 45 60 750

005

01

015

Time (s)Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Ener

gy o

f sen

sor1

(J)

T 1T 1

Figure 4 Sensor 1 power consumption comparison in three cases

are settled it tends to theminimum consumption ratewhich is corresponding to the maximum samplingperiod

(3) Figures 5 and 8 are the control outputs of loop1and loop2 in the three cases mentioned above Thefigures show that the control systems can be stabilizedthrough the proposed joint design methods In threecases the adaptive sampling can get the control per-formances closed to the case of minimum sampling

(4) Combining Figures 4 5 7 and 8 it can be concludedthat the proposed joint design method achieves atradeoff between the performances of control andpower efficiency

Mathematical Problems in Engineering 9

Time (s)

The proposed adaptive sampling

0 15 30 45 60 75

0

2

4

6

8

Out

puty

of lo

op1

minus2

minus4

minus6

minus8

Constant sampling with minimum T1 = 1 msConstant sampling with maximum T1 = 256 ms

Figure 5 Control loop1 output comparison in three cases

0 20 40 60 800

50

100

150

Sens

or 2

life

span

(s)

200

250

300

350

Time (s)

Figure 6 Sensor 2 survival time prediction

6 Conclusion

This paper presents a joint design method for wirelessnetworked control systems with limited power constraint Apower efficiency algorithm based on the adaptive samplingperiod is put forward to satisfy the demands of sensorsurvival time and system stability Then the time-varyingcontrol system with transmission delay is modeled as aswitched system with uncertain switching signals A dwell-time-dependent controlmethod is discussed to guarantee thestability of WNCS Simulation results show the effectivenessof the proposed method and indicate that it can achievegood tradeoff performance Methods by which to reducepower consumption from the aspect of a single node as wellas balancing power consumption from the global networkperspective are worthy of further exploration

0 15 30 45 60 750

005

01

015

Time (s)

Ener

gy o

f sen

sor2

(J)

The proposed adaptive sampling

Constant sampling with minimum T2 = 1msConstant sampling with maximum T2 = 256ms

Figure 7 Sensor 2 power consumption comparison in three cases

0 15 30 45 60 75

0

5

10

15

Out

put y

of l

oop

2

minus15

minus10

minus5

Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Time (s)

T2T2

Figure 8 Control loop2 output comparison in three cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The working is sponsored by The NSFC (no 61202473) andThe Natural Science Foundation of Jiangsu Province (noBK2012551)

10 Mathematical Problems in Engineering

References

[1] W Dieterle H D Kochs and E Dittmar ldquoCommunicationarchitectures for distributed computer control systemsrdquoControlEngineering Practice vol 3 no 8 pp 1171ndash1176 1995

[2] H A Thompson ldquoWireless and Internet communicationstechnologies for monitoring and controlrdquo Control EngineeringPractice vol 12 no 6 pp 781ndash791 2004

[3] P R Kumar ldquoNew technological vistas for systems and controlrdquoIEEE Control Systems Magazine vol 21 no 1 pp 24ndash37 2001

[4] X Liu and A Goldsmith ldquoWireless medium access controlin networked control systemsrdquo in Proceedings of the AmericanControl Conference (AAC rsquo04) pp 3605ndash3610 Boston MassUSA July 2004

[5] J Colandairaj G W Irwin and W G Scanlon ldquoWirelessnetworked control systems with QoS-based samplingrdquo IETControl Theory and Applications vol 1 no 1 pp 430ndash438 2007

[6] N Israr W G Scanlon and G W Irwin ldquoA cross-layer com-munication framework forwireless networked control systemsrdquoin Proceedings of the 1st International Conference on WirelessCommunication Vehicular Technology Information Theory andAerospace and Electronic Systems Technology (VITAE rsquo09) pp577ndash581 Aalborg Denmark May 2009

[7] S Dai H Lin and S S Ge ldquoScheduling-and-control codesignfor a collection of networked control systems with uncertaindelaysrdquo IEEE Transactions on Control Systems Technology vol18 no 1 pp 66ndash78 2010

[8] A V Savkin ldquoAnalysis and synthesis of networked control sys-tems topological entropy observability robustness and optimalcontrolrdquo Automatica vol 42 no 1 pp 51ndash62 2006

[9] J Z Luo F Dong and J X Cao ldquoA novel task schedul-ing algorithm based on dynamic critical path and effectiveduplication for pervasive computing environmentrdquo WirelessCommunications amp Mobile Computing vol 10 no 10 pp 1283ndash1302 2010

[10] X Yin X Zhou Z Li and S Li ldquoJint rate control and powercontrol for lifetime maximization in Wreless Sensor NtworksrdquoJournal of Internet Technology vol 12 no 1 pp 69ndash78 2011

[11] M Pajic S Sundaram G J Pappas and R Mangharam ldquoThewireless control network a new approach for control overnetworksrdquo IEEE Transactions on Automatic Control vol 56 no10 pp 2305ndash2318 2011

[12] J Colandairaj G W Irwin and W G Scanlon ldquoA co-designsolution for wireless feedback controlrdquo in Proceeding of theInternational Conference on Networking Sensing and Control(ICNSC 07) pp 404ndash409 London UK April 2007

[13] W Liang and Y Yang ldquoMaximizing battery life routing in wire-less ad hoc networksrdquo in Proceedings of the 37th InternationalConference on System Sciences pp 4739ndash4746 IEEE HonoluluHawaii USA January 2004

[14] K Brian J Haberman and W Sheppard Overlapping ParticleSwarms for Energy-Efficient Routing in Sensor Networks Wire-less Network Springer 2011

[15] D Kim K Dantu andM Pedram ldquoLifetime prediction routinginmobile AdHoc networksrdquo in Proceedings of the IEEEWirelessCommunication and Networking Conference New York NYUSA 2003

[16] L Hetel J Daafouz and C Iung ldquoStability analysis for discretetime switched systems with temporary uncertain switchingsignalrdquo in Proceedings of the 46th IEEE Conference on Decisionand Control (CDC rsquo07) pp 5623ndash5628 New Orleans Fla USADecember 2007

[17] S Limin Wireless Sensor Network Tsinghua University PressBeijing China 2006

[18] Y He I Lee and L Guan ldquoDistributed algorithms for networklifetimemaximization in wireless visual sensor networksrdquo IEEETransactions on Circuits and Systems for Video Technology vol19 no 5 pp 704ndash718 2009

[19] B A Bakr and L Lilien ldquoA quantitative comparison of energyconsumption and WSN lifetime for LEACH and LEACH-SMrdquo in Proceedings of the 31st International Conference onDistributed Computing Systems Workshops (ICDCSW rsquo11) pp182ndash191 Minneapolis Minn USA June 2011

[20] C Fischione A Bonivento A Sangiovanni-Vincentelli F San-tucci and K H Johansson ldquoPerformance analysis of collabora-tive spatio-temporal processing for wireless sensor networksrdquoin Proceedings of the 3rd IEEE Consumer Communications andNetworking Conference (CCNC rsquo06) pp 325ndash329 Las VegasNev USA January 2006

[21] Y Iino ldquoSome considerations of wireless sensor network basedcontrol systemsrdquo in Proceedings of the SICE Conference onControl Systems (CCS rsquo07) Tokyo Japan 2007

[22] Y Iino T Hatanaka and M Fujita ldquoEvent-predictive controlfor energy saving of wireless networked control systemrdquo inProceedings of the American Control Conference (ACC rsquo09) pp2236ndash2242 St Louis Mo USA June 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

forall119896 isin [1198960 1198961) if the right switching signal is used in the

control 119897(119896) = 120574(119896) = 119897(1198960) then

119881119898(1198961) lt (120579

lowast

119898)1198961minus1198960

119881119898(1198960) (31)

forall119896 isin [1198960 1198961) if the switching signal in the control is not

necessarily the same as the real signal the pair 119897(119896) 120574(119896) takesan arbitrary value 119897(119896

0) 120574(119896

0) in ℓ times ℓ

Since the controller gains K119897are designed to make the

matricesΦ(119897119897)

stable the scalar 120579119899is smaller than one 120579

119899lt 1

The scalar 120579119898may be greater than one 120579

119898gt 1 since the

gains K120574are not designed to stabilize combinations other

thanΦ(120574120574)

Combining inequalities (26) yield

min119897isinℓ

eigmin (P119899

119897)1003817100381710038171003817119909(1198961)

1003817100381710038171003817

2

lt (120579lowast

119899)1198961minus1198960max119897isinℓ

eigmax (P119899

119897)1003817100381710038171003817119909(1198960)

1003817100381710038171003817

2forall119897 isin ℓ

min(119897120574)isinℓtimesℓ

eigmin (P119898

(119897120574))1003817100381710038171003817119909(1198961)

1003817100381710038171003817

2

lt max(119897120574)isinℓtimesℓ

eigmax (P119898

(119897120574))1003817100381710038171003817119909(1198960)

1003817100381710038171003817

2forall (119897 120574) isin ℓ times ℓ

(32)

With the definitions of 120573119899and 120573

119898 we can obtain the state

vector norm decay or growth rate in a nominal regime and inan uncertain switching signal regime as follows

1003817100381710038171003817119909(1198961)1003817100381710038171003817

2lt 120573119899sdot (120579lowast

119899)1198961minus11989601003817100381710038171003817119909(1198960)

1003817100381710038171003817

2 forall119897 isin ℓ

1003817100381710038171003817119909(1198961)1003817100381710038171003817

2lt 120573119898sdot (120579lowast

119898)1198961minus11989601003817100381710038171003817119909(1198960)

1003817100381710038171003817

2 forall (119897 120574) isin ℓ times ℓ

(33)

Let 119896119898119904describe the instants when the closed-loop system

jumps to a mixed mode with uncertain switching signal andlet 119896119899119904be the instance when the system enters into a normal

regime With definitions and bounds of Δ119898 and Δ119899 it followsthat

0 le Δ119898= 119896119899

119904minus 119896119898

119904le 1 119896

119898

119904+1minus 119896119899

119904le Δ119899 (34)

Without loss of generality we assume that the systemstarts with a mixed-mode behavior 119896119898

119904lt 119896119899

119904 The system

behavior in time interval 119896 isin [119896119898

119904 119896119898

119904+1) is then analyzed

Given that 119897(119896) = 120574(119896) forall119896 isin [119896119898

119904 119896119899

119904) and 119897(119896) =

120574(119896) forall119896 isin [119896119899

119904 119896119898

119904+1) using inequalities (26) the norm of the

state at the end of the sequence can be upper bounded asd

1003817100381710038171003817119909(119896119898

119904+1)1003817100381710038171003817

2lt 120573119898sdot 120573119899sdot (120579lowast

119898)119896119899

119904minus119896119906

119904sdot (120579lowast

119899)119896119898

119904+1minus119896119899

119904 1003817100381710038171003817119909(119896119898

119904)1003817100381710038171003817

2

lt1003817100381710038171003817119909(119896119898

119904+1)1003817100381710038171003817

2lt 120573119898sdot 120573119899sdot 120579lowast

119898sdot (120579lowast

119899)Δ1198991003817100381710038171003817119909(119896119898

119904)1003817100381710038171003817

2

(35)

It indicates that closed loop (16) will be asymptotically stableif condition (28) in Theorem 5 is satisfied

43 Choosing Rule of Update Period 119879119872

Theorem 6 Consider the WNCS with adaptive samplingperiod rule (3) The WNCS can be stabilized by the switched

state feedback controllers (22) whereas the survival time canmeet its expected value 119871

119890 if the update period 119879

119872satisfies the

following conditions

(1) 119879119872

ge max (Δ119899119894+ 1) [119879

max1

119879max2

119879max119899

]

(2) lfloor119871119890

119879119872

rfloor sdot 119879119872

le119864119894119899119894119905

119864

min 119879min1

119879min2

119879min119899

(36)

where Δ119899119894is the least nominal sampling period of control loop

119894 and the solution of Theorem 5 [119879max1

119879max2

119879max119899

] is theleast commonmultiple of the maximum sampling period for allthe 119899 control loops in the WNCS

Proof According to Theorem 5 control loop 119894 in the WNCSshould stay at least Δ119899

119894in a nominal sampling period in one

updating interval for stability With the consideration of 0 le

Δ119898

119894le 1 if the update period satisfies condition (1) all control

loops in the WNCS will meet the demands of their leastnominal sampling periods and will be stable As a completesystem WNCS is composed of 119899 control loops that will bestable when condition (1) is satisfied

Condition (2) inTheorem 6 provides the upper bound ofthe updating period For the power efficiency algorithm inTheorem 1 with the definition of 119896

119897(119895) there exists

119872

sum

119897=1

119896119897(119895)

119879119872

119879119897

+119879119872

119879119895

le lfloor119871119890

119879119872

rfloor sdot119879119872

min 119879min1

119879min2

119879min119899

(37)

If condition (2) inTheorem 6 is satisfied the item ((119864init119864)minus

sum119872

119897=1119896119897(119895)(119879119872119879119897) minus (119879

119872119879119895)) in the denominator of formula

(3) will be greater than zero which guarantees that formula(3) has physical meaning and is solvable The survival timewill thus reach the expected value by applying the powerefficiency algorithm

5 Numerical Example

Simulation studies are performed on a WNCS closed by anIEEE 80211b wireless network with two control loops sharingthe network resources The two control loops are assumed tohave the same dynamics but with different initial conditions

119894(119905) = [

minus1 minus01

0 095] 119909119894(119905) + [

minus015

minus043] 119906119894(119905)

119910119894(119905) = [1 1] 119909

119894(119905)

119894 = 1 2

1199091(0) = [minus5 5]

119879 119909

2(0) = [minus5 10]

119879

(38)

The wireless network parameters are set as in Table 2The time delays in both loops are less than one samplingperiod Some other necessary parameters are given as shownin Table 2

In Table 3 the minimum and maximum sampling peri-ods are computed by the methods in Section 3 Solving

8 Mathematical Problems in Engineering

Table 3 Simulation parameters

Parameter ValueInitial energy of both sensors 119864init 015 JSensor 1 expected survival time 119871

119890175 s

Sensor 2 expected survival time 1198711198902

70 sUnit transmission energy 119864 25 dbmNumber of sampling period candidates119872 10The minimum sampling period of loop1 119879min

11ms

The maximum sampling period of loop1 119879max1

256msThe minimum sampling period of loop2 119879min

21ms

The maximum sampling period of loop2 119879max2

256ms

Table 4 Controller gains of ten sampling modes

Sampling period (ms) Controller gain1198790= 1 119870

0= [11176 11188]

1198791= 29 119870

1= [394837 405512]

1198792= 58 119870

2= [194678 204319]

1198793= 86 119870

3= [125668 134279]

1198794= 114 119870

4= [90353 97947]

1198795= 143 119870

5= [68663 75265]

1198796= 171 119870

6= [5383 5947]

1198797= 199 119870

7= [4295 4769]

1198798= 227 119870

8= [4225 3852]

1198799= 256 119870

9= [3707 295]

inequality (28) in Theorem 5 it yields Δ1198991= Δ119899

2= 8 Δ119898 = 1

Also the update period can be chosen byTheorem 6 as 119879119872

=

150msSolving matrix inequalities (18) and (19) the controller

gains of ten switching modes can be obtained as in Table 4With the above simulation parameters and controller

gains the curves of survival time prediction power consump-tion and control output of both control loops are shown inFigures 3 to 8

Analyzing the simulation curves we have the followingresults

(1) Figures 3 6 4 and 7 imply that both sensor 1and sensor 2 can meet their expected survival timerequirements

(2) The power consumption in three cases of minimumsampling maximum sampling and the proposedadaptive sampling is compared in Figures 4 and7 It is obvious that the power is consumed muchfaster than the other two cases In the case of theadaptive sampling the power consumption variesaccording to both the requirements of the controlperformance and survival time At the beginning thepower consumption curves vary quickly and morepower are consumed because of the control systemsnot reaching stable yetThen after the control systems

0 20 40 60 800

50

100

150

200

Sens

or 1

life

span

(s)

250

300

350

400

Time (s)

Figure 3 Sensor 1 survival time prediction

0 15 30 45 60 750

005

01

015

Time (s)Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Ener

gy o

f sen

sor1

(J)

T 1T 1

Figure 4 Sensor 1 power consumption comparison in three cases

are settled it tends to theminimum consumption ratewhich is corresponding to the maximum samplingperiod

(3) Figures 5 and 8 are the control outputs of loop1and loop2 in the three cases mentioned above Thefigures show that the control systems can be stabilizedthrough the proposed joint design methods In threecases the adaptive sampling can get the control per-formances closed to the case of minimum sampling

(4) Combining Figures 4 5 7 and 8 it can be concludedthat the proposed joint design method achieves atradeoff between the performances of control andpower efficiency

Mathematical Problems in Engineering 9

Time (s)

The proposed adaptive sampling

0 15 30 45 60 75

0

2

4

6

8

Out

puty

of lo

op1

minus2

minus4

minus6

minus8

Constant sampling with minimum T1 = 1 msConstant sampling with maximum T1 = 256 ms

Figure 5 Control loop1 output comparison in three cases

0 20 40 60 800

50

100

150

Sens

or 2

life

span

(s)

200

250

300

350

Time (s)

Figure 6 Sensor 2 survival time prediction

6 Conclusion

This paper presents a joint design method for wirelessnetworked control systems with limited power constraint Apower efficiency algorithm based on the adaptive samplingperiod is put forward to satisfy the demands of sensorsurvival time and system stability Then the time-varyingcontrol system with transmission delay is modeled as aswitched system with uncertain switching signals A dwell-time-dependent controlmethod is discussed to guarantee thestability of WNCS Simulation results show the effectivenessof the proposed method and indicate that it can achievegood tradeoff performance Methods by which to reducepower consumption from the aspect of a single node as wellas balancing power consumption from the global networkperspective are worthy of further exploration

0 15 30 45 60 750

005

01

015

Time (s)

Ener

gy o

f sen

sor2

(J)

The proposed adaptive sampling

Constant sampling with minimum T2 = 1msConstant sampling with maximum T2 = 256ms

Figure 7 Sensor 2 power consumption comparison in three cases

0 15 30 45 60 75

0

5

10

15

Out

put y

of l

oop

2

minus15

minus10

minus5

Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Time (s)

T2T2

Figure 8 Control loop2 output comparison in three cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The working is sponsored by The NSFC (no 61202473) andThe Natural Science Foundation of Jiangsu Province (noBK2012551)

10 Mathematical Problems in Engineering

References

[1] W Dieterle H D Kochs and E Dittmar ldquoCommunicationarchitectures for distributed computer control systemsrdquoControlEngineering Practice vol 3 no 8 pp 1171ndash1176 1995

[2] H A Thompson ldquoWireless and Internet communicationstechnologies for monitoring and controlrdquo Control EngineeringPractice vol 12 no 6 pp 781ndash791 2004

[3] P R Kumar ldquoNew technological vistas for systems and controlrdquoIEEE Control Systems Magazine vol 21 no 1 pp 24ndash37 2001

[4] X Liu and A Goldsmith ldquoWireless medium access controlin networked control systemsrdquo in Proceedings of the AmericanControl Conference (AAC rsquo04) pp 3605ndash3610 Boston MassUSA July 2004

[5] J Colandairaj G W Irwin and W G Scanlon ldquoWirelessnetworked control systems with QoS-based samplingrdquo IETControl Theory and Applications vol 1 no 1 pp 430ndash438 2007

[6] N Israr W G Scanlon and G W Irwin ldquoA cross-layer com-munication framework forwireless networked control systemsrdquoin Proceedings of the 1st International Conference on WirelessCommunication Vehicular Technology Information Theory andAerospace and Electronic Systems Technology (VITAE rsquo09) pp577ndash581 Aalborg Denmark May 2009

[7] S Dai H Lin and S S Ge ldquoScheduling-and-control codesignfor a collection of networked control systems with uncertaindelaysrdquo IEEE Transactions on Control Systems Technology vol18 no 1 pp 66ndash78 2010

[8] A V Savkin ldquoAnalysis and synthesis of networked control sys-tems topological entropy observability robustness and optimalcontrolrdquo Automatica vol 42 no 1 pp 51ndash62 2006

[9] J Z Luo F Dong and J X Cao ldquoA novel task schedul-ing algorithm based on dynamic critical path and effectiveduplication for pervasive computing environmentrdquo WirelessCommunications amp Mobile Computing vol 10 no 10 pp 1283ndash1302 2010

[10] X Yin X Zhou Z Li and S Li ldquoJint rate control and powercontrol for lifetime maximization in Wreless Sensor NtworksrdquoJournal of Internet Technology vol 12 no 1 pp 69ndash78 2011

[11] M Pajic S Sundaram G J Pappas and R Mangharam ldquoThewireless control network a new approach for control overnetworksrdquo IEEE Transactions on Automatic Control vol 56 no10 pp 2305ndash2318 2011

[12] J Colandairaj G W Irwin and W G Scanlon ldquoA co-designsolution for wireless feedback controlrdquo in Proceeding of theInternational Conference on Networking Sensing and Control(ICNSC 07) pp 404ndash409 London UK April 2007

[13] W Liang and Y Yang ldquoMaximizing battery life routing in wire-less ad hoc networksrdquo in Proceedings of the 37th InternationalConference on System Sciences pp 4739ndash4746 IEEE HonoluluHawaii USA January 2004

[14] K Brian J Haberman and W Sheppard Overlapping ParticleSwarms for Energy-Efficient Routing in Sensor Networks Wire-less Network Springer 2011

[15] D Kim K Dantu andM Pedram ldquoLifetime prediction routinginmobile AdHoc networksrdquo in Proceedings of the IEEEWirelessCommunication and Networking Conference New York NYUSA 2003

[16] L Hetel J Daafouz and C Iung ldquoStability analysis for discretetime switched systems with temporary uncertain switchingsignalrdquo in Proceedings of the 46th IEEE Conference on Decisionand Control (CDC rsquo07) pp 5623ndash5628 New Orleans Fla USADecember 2007

[17] S Limin Wireless Sensor Network Tsinghua University PressBeijing China 2006

[18] Y He I Lee and L Guan ldquoDistributed algorithms for networklifetimemaximization in wireless visual sensor networksrdquo IEEETransactions on Circuits and Systems for Video Technology vol19 no 5 pp 704ndash718 2009

[19] B A Bakr and L Lilien ldquoA quantitative comparison of energyconsumption and WSN lifetime for LEACH and LEACH-SMrdquo in Proceedings of the 31st International Conference onDistributed Computing Systems Workshops (ICDCSW rsquo11) pp182ndash191 Minneapolis Minn USA June 2011

[20] C Fischione A Bonivento A Sangiovanni-Vincentelli F San-tucci and K H Johansson ldquoPerformance analysis of collabora-tive spatio-temporal processing for wireless sensor networksrdquoin Proceedings of the 3rd IEEE Consumer Communications andNetworking Conference (CCNC rsquo06) pp 325ndash329 Las VegasNev USA January 2006

[21] Y Iino ldquoSome considerations of wireless sensor network basedcontrol systemsrdquo in Proceedings of the SICE Conference onControl Systems (CCS rsquo07) Tokyo Japan 2007

[22] Y Iino T Hatanaka and M Fujita ldquoEvent-predictive controlfor energy saving of wireless networked control systemrdquo inProceedings of the American Control Conference (ACC rsquo09) pp2236ndash2242 St Louis Mo USA June 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 3 Simulation parameters

Parameter ValueInitial energy of both sensors 119864init 015 JSensor 1 expected survival time 119871

119890175 s

Sensor 2 expected survival time 1198711198902

70 sUnit transmission energy 119864 25 dbmNumber of sampling period candidates119872 10The minimum sampling period of loop1 119879min

11ms

The maximum sampling period of loop1 119879max1

256msThe minimum sampling period of loop2 119879min

21ms

The maximum sampling period of loop2 119879max2

256ms

Table 4 Controller gains of ten sampling modes

Sampling period (ms) Controller gain1198790= 1 119870

0= [11176 11188]

1198791= 29 119870

1= [394837 405512]

1198792= 58 119870

2= [194678 204319]

1198793= 86 119870

3= [125668 134279]

1198794= 114 119870

4= [90353 97947]

1198795= 143 119870

5= [68663 75265]

1198796= 171 119870

6= [5383 5947]

1198797= 199 119870

7= [4295 4769]

1198798= 227 119870

8= [4225 3852]

1198799= 256 119870

9= [3707 295]

inequality (28) in Theorem 5 it yields Δ1198991= Δ119899

2= 8 Δ119898 = 1

Also the update period can be chosen byTheorem 6 as 119879119872

=

150msSolving matrix inequalities (18) and (19) the controller

gains of ten switching modes can be obtained as in Table 4With the above simulation parameters and controller

gains the curves of survival time prediction power consump-tion and control output of both control loops are shown inFigures 3 to 8

Analyzing the simulation curves we have the followingresults

(1) Figures 3 6 4 and 7 imply that both sensor 1and sensor 2 can meet their expected survival timerequirements

(2) The power consumption in three cases of minimumsampling maximum sampling and the proposedadaptive sampling is compared in Figures 4 and7 It is obvious that the power is consumed muchfaster than the other two cases In the case of theadaptive sampling the power consumption variesaccording to both the requirements of the controlperformance and survival time At the beginning thepower consumption curves vary quickly and morepower are consumed because of the control systemsnot reaching stable yetThen after the control systems

0 20 40 60 800

50

100

150

200

Sens

or 1

life

span

(s)

250

300

350

400

Time (s)

Figure 3 Sensor 1 survival time prediction

0 15 30 45 60 750

005

01

015

Time (s)Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Ener

gy o

f sen

sor1

(J)

T 1T 1

Figure 4 Sensor 1 power consumption comparison in three cases

are settled it tends to theminimum consumption ratewhich is corresponding to the maximum samplingperiod

(3) Figures 5 and 8 are the control outputs of loop1and loop2 in the three cases mentioned above Thefigures show that the control systems can be stabilizedthrough the proposed joint design methods In threecases the adaptive sampling can get the control per-formances closed to the case of minimum sampling

(4) Combining Figures 4 5 7 and 8 it can be concludedthat the proposed joint design method achieves atradeoff between the performances of control andpower efficiency

Mathematical Problems in Engineering 9

Time (s)

The proposed adaptive sampling

0 15 30 45 60 75

0

2

4

6

8

Out

puty

of lo

op1

minus2

minus4

minus6

minus8

Constant sampling with minimum T1 = 1 msConstant sampling with maximum T1 = 256 ms

Figure 5 Control loop1 output comparison in three cases

0 20 40 60 800

50

100

150

Sens

or 2

life

span

(s)

200

250

300

350

Time (s)

Figure 6 Sensor 2 survival time prediction

6 Conclusion

This paper presents a joint design method for wirelessnetworked control systems with limited power constraint Apower efficiency algorithm based on the adaptive samplingperiod is put forward to satisfy the demands of sensorsurvival time and system stability Then the time-varyingcontrol system with transmission delay is modeled as aswitched system with uncertain switching signals A dwell-time-dependent controlmethod is discussed to guarantee thestability of WNCS Simulation results show the effectivenessof the proposed method and indicate that it can achievegood tradeoff performance Methods by which to reducepower consumption from the aspect of a single node as wellas balancing power consumption from the global networkperspective are worthy of further exploration

0 15 30 45 60 750

005

01

015

Time (s)

Ener

gy o

f sen

sor2

(J)

The proposed adaptive sampling

Constant sampling with minimum T2 = 1msConstant sampling with maximum T2 = 256ms

Figure 7 Sensor 2 power consumption comparison in three cases

0 15 30 45 60 75

0

5

10

15

Out

put y

of l

oop

2

minus15

minus10

minus5

Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Time (s)

T2T2

Figure 8 Control loop2 output comparison in three cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The working is sponsored by The NSFC (no 61202473) andThe Natural Science Foundation of Jiangsu Province (noBK2012551)

10 Mathematical Problems in Engineering

References

[1] W Dieterle H D Kochs and E Dittmar ldquoCommunicationarchitectures for distributed computer control systemsrdquoControlEngineering Practice vol 3 no 8 pp 1171ndash1176 1995

[2] H A Thompson ldquoWireless and Internet communicationstechnologies for monitoring and controlrdquo Control EngineeringPractice vol 12 no 6 pp 781ndash791 2004

[3] P R Kumar ldquoNew technological vistas for systems and controlrdquoIEEE Control Systems Magazine vol 21 no 1 pp 24ndash37 2001

[4] X Liu and A Goldsmith ldquoWireless medium access controlin networked control systemsrdquo in Proceedings of the AmericanControl Conference (AAC rsquo04) pp 3605ndash3610 Boston MassUSA July 2004

[5] J Colandairaj G W Irwin and W G Scanlon ldquoWirelessnetworked control systems with QoS-based samplingrdquo IETControl Theory and Applications vol 1 no 1 pp 430ndash438 2007

[6] N Israr W G Scanlon and G W Irwin ldquoA cross-layer com-munication framework forwireless networked control systemsrdquoin Proceedings of the 1st International Conference on WirelessCommunication Vehicular Technology Information Theory andAerospace and Electronic Systems Technology (VITAE rsquo09) pp577ndash581 Aalborg Denmark May 2009

[7] S Dai H Lin and S S Ge ldquoScheduling-and-control codesignfor a collection of networked control systems with uncertaindelaysrdquo IEEE Transactions on Control Systems Technology vol18 no 1 pp 66ndash78 2010

[8] A V Savkin ldquoAnalysis and synthesis of networked control sys-tems topological entropy observability robustness and optimalcontrolrdquo Automatica vol 42 no 1 pp 51ndash62 2006

[9] J Z Luo F Dong and J X Cao ldquoA novel task schedul-ing algorithm based on dynamic critical path and effectiveduplication for pervasive computing environmentrdquo WirelessCommunications amp Mobile Computing vol 10 no 10 pp 1283ndash1302 2010

[10] X Yin X Zhou Z Li and S Li ldquoJint rate control and powercontrol for lifetime maximization in Wreless Sensor NtworksrdquoJournal of Internet Technology vol 12 no 1 pp 69ndash78 2011

[11] M Pajic S Sundaram G J Pappas and R Mangharam ldquoThewireless control network a new approach for control overnetworksrdquo IEEE Transactions on Automatic Control vol 56 no10 pp 2305ndash2318 2011

[12] J Colandairaj G W Irwin and W G Scanlon ldquoA co-designsolution for wireless feedback controlrdquo in Proceeding of theInternational Conference on Networking Sensing and Control(ICNSC 07) pp 404ndash409 London UK April 2007

[13] W Liang and Y Yang ldquoMaximizing battery life routing in wire-less ad hoc networksrdquo in Proceedings of the 37th InternationalConference on System Sciences pp 4739ndash4746 IEEE HonoluluHawaii USA January 2004

[14] K Brian J Haberman and W Sheppard Overlapping ParticleSwarms for Energy-Efficient Routing in Sensor Networks Wire-less Network Springer 2011

[15] D Kim K Dantu andM Pedram ldquoLifetime prediction routinginmobile AdHoc networksrdquo in Proceedings of the IEEEWirelessCommunication and Networking Conference New York NYUSA 2003

[16] L Hetel J Daafouz and C Iung ldquoStability analysis for discretetime switched systems with temporary uncertain switchingsignalrdquo in Proceedings of the 46th IEEE Conference on Decisionand Control (CDC rsquo07) pp 5623ndash5628 New Orleans Fla USADecember 2007

[17] S Limin Wireless Sensor Network Tsinghua University PressBeijing China 2006

[18] Y He I Lee and L Guan ldquoDistributed algorithms for networklifetimemaximization in wireless visual sensor networksrdquo IEEETransactions on Circuits and Systems for Video Technology vol19 no 5 pp 704ndash718 2009

[19] B A Bakr and L Lilien ldquoA quantitative comparison of energyconsumption and WSN lifetime for LEACH and LEACH-SMrdquo in Proceedings of the 31st International Conference onDistributed Computing Systems Workshops (ICDCSW rsquo11) pp182ndash191 Minneapolis Minn USA June 2011

[20] C Fischione A Bonivento A Sangiovanni-Vincentelli F San-tucci and K H Johansson ldquoPerformance analysis of collabora-tive spatio-temporal processing for wireless sensor networksrdquoin Proceedings of the 3rd IEEE Consumer Communications andNetworking Conference (CCNC rsquo06) pp 325ndash329 Las VegasNev USA January 2006

[21] Y Iino ldquoSome considerations of wireless sensor network basedcontrol systemsrdquo in Proceedings of the SICE Conference onControl Systems (CCS rsquo07) Tokyo Japan 2007

[22] Y Iino T Hatanaka and M Fujita ldquoEvent-predictive controlfor energy saving of wireless networked control systemrdquo inProceedings of the American Control Conference (ACC rsquo09) pp2236ndash2242 St Louis Mo USA June 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Time (s)

The proposed adaptive sampling

0 15 30 45 60 75

0

2

4

6

8

Out

puty

of lo

op1

minus2

minus4

minus6

minus8

Constant sampling with minimum T1 = 1 msConstant sampling with maximum T1 = 256 ms

Figure 5 Control loop1 output comparison in three cases

0 20 40 60 800

50

100

150

Sens

or 2

life

span

(s)

200

250

300

350

Time (s)

Figure 6 Sensor 2 survival time prediction

6 Conclusion

This paper presents a joint design method for wirelessnetworked control systems with limited power constraint Apower efficiency algorithm based on the adaptive samplingperiod is put forward to satisfy the demands of sensorsurvival time and system stability Then the time-varyingcontrol system with transmission delay is modeled as aswitched system with uncertain switching signals A dwell-time-dependent controlmethod is discussed to guarantee thestability of WNCS Simulation results show the effectivenessof the proposed method and indicate that it can achievegood tradeoff performance Methods by which to reducepower consumption from the aspect of a single node as wellas balancing power consumption from the global networkperspective are worthy of further exploration

0 15 30 45 60 750

005

01

015

Time (s)

Ener

gy o

f sen

sor2

(J)

The proposed adaptive sampling

Constant sampling with minimum T2 = 1msConstant sampling with maximum T2 = 256ms

Figure 7 Sensor 2 power consumption comparison in three cases

0 15 30 45 60 75

0

5

10

15

Out

put y

of l

oop

2

minus15

minus10

minus5

Constant sampling with minimum = 1 ms Constant sampling with maximum = 256 msThe proposed adaptive sampling

Time (s)

T2T2

Figure 8 Control loop2 output comparison in three cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The working is sponsored by The NSFC (no 61202473) andThe Natural Science Foundation of Jiangsu Province (noBK2012551)

10 Mathematical Problems in Engineering

References

[1] W Dieterle H D Kochs and E Dittmar ldquoCommunicationarchitectures for distributed computer control systemsrdquoControlEngineering Practice vol 3 no 8 pp 1171ndash1176 1995

[2] H A Thompson ldquoWireless and Internet communicationstechnologies for monitoring and controlrdquo Control EngineeringPractice vol 12 no 6 pp 781ndash791 2004

[3] P R Kumar ldquoNew technological vistas for systems and controlrdquoIEEE Control Systems Magazine vol 21 no 1 pp 24ndash37 2001

[4] X Liu and A Goldsmith ldquoWireless medium access controlin networked control systemsrdquo in Proceedings of the AmericanControl Conference (AAC rsquo04) pp 3605ndash3610 Boston MassUSA July 2004

[5] J Colandairaj G W Irwin and W G Scanlon ldquoWirelessnetworked control systems with QoS-based samplingrdquo IETControl Theory and Applications vol 1 no 1 pp 430ndash438 2007

[6] N Israr W G Scanlon and G W Irwin ldquoA cross-layer com-munication framework forwireless networked control systemsrdquoin Proceedings of the 1st International Conference on WirelessCommunication Vehicular Technology Information Theory andAerospace and Electronic Systems Technology (VITAE rsquo09) pp577ndash581 Aalborg Denmark May 2009

[7] S Dai H Lin and S S Ge ldquoScheduling-and-control codesignfor a collection of networked control systems with uncertaindelaysrdquo IEEE Transactions on Control Systems Technology vol18 no 1 pp 66ndash78 2010

[8] A V Savkin ldquoAnalysis and synthesis of networked control sys-tems topological entropy observability robustness and optimalcontrolrdquo Automatica vol 42 no 1 pp 51ndash62 2006

[9] J Z Luo F Dong and J X Cao ldquoA novel task schedul-ing algorithm based on dynamic critical path and effectiveduplication for pervasive computing environmentrdquo WirelessCommunications amp Mobile Computing vol 10 no 10 pp 1283ndash1302 2010

[10] X Yin X Zhou Z Li and S Li ldquoJint rate control and powercontrol for lifetime maximization in Wreless Sensor NtworksrdquoJournal of Internet Technology vol 12 no 1 pp 69ndash78 2011

[11] M Pajic S Sundaram G J Pappas and R Mangharam ldquoThewireless control network a new approach for control overnetworksrdquo IEEE Transactions on Automatic Control vol 56 no10 pp 2305ndash2318 2011

[12] J Colandairaj G W Irwin and W G Scanlon ldquoA co-designsolution for wireless feedback controlrdquo in Proceeding of theInternational Conference on Networking Sensing and Control(ICNSC 07) pp 404ndash409 London UK April 2007

[13] W Liang and Y Yang ldquoMaximizing battery life routing in wire-less ad hoc networksrdquo in Proceedings of the 37th InternationalConference on System Sciences pp 4739ndash4746 IEEE HonoluluHawaii USA January 2004

[14] K Brian J Haberman and W Sheppard Overlapping ParticleSwarms for Energy-Efficient Routing in Sensor Networks Wire-less Network Springer 2011

[15] D Kim K Dantu andM Pedram ldquoLifetime prediction routinginmobile AdHoc networksrdquo in Proceedings of the IEEEWirelessCommunication and Networking Conference New York NYUSA 2003

[16] L Hetel J Daafouz and C Iung ldquoStability analysis for discretetime switched systems with temporary uncertain switchingsignalrdquo in Proceedings of the 46th IEEE Conference on Decisionand Control (CDC rsquo07) pp 5623ndash5628 New Orleans Fla USADecember 2007

[17] S Limin Wireless Sensor Network Tsinghua University PressBeijing China 2006

[18] Y He I Lee and L Guan ldquoDistributed algorithms for networklifetimemaximization in wireless visual sensor networksrdquo IEEETransactions on Circuits and Systems for Video Technology vol19 no 5 pp 704ndash718 2009

[19] B A Bakr and L Lilien ldquoA quantitative comparison of energyconsumption and WSN lifetime for LEACH and LEACH-SMrdquo in Proceedings of the 31st International Conference onDistributed Computing Systems Workshops (ICDCSW rsquo11) pp182ndash191 Minneapolis Minn USA June 2011

[20] C Fischione A Bonivento A Sangiovanni-Vincentelli F San-tucci and K H Johansson ldquoPerformance analysis of collabora-tive spatio-temporal processing for wireless sensor networksrdquoin Proceedings of the 3rd IEEE Consumer Communications andNetworking Conference (CCNC rsquo06) pp 325ndash329 Las VegasNev USA January 2006

[21] Y Iino ldquoSome considerations of wireless sensor network basedcontrol systemsrdquo in Proceedings of the SICE Conference onControl Systems (CCS rsquo07) Tokyo Japan 2007

[22] Y Iino T Hatanaka and M Fujita ldquoEvent-predictive controlfor energy saving of wireless networked control systemrdquo inProceedings of the American Control Conference (ACC rsquo09) pp2236ndash2242 St Louis Mo USA June 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

References

[1] W Dieterle H D Kochs and E Dittmar ldquoCommunicationarchitectures for distributed computer control systemsrdquoControlEngineering Practice vol 3 no 8 pp 1171ndash1176 1995

[2] H A Thompson ldquoWireless and Internet communicationstechnologies for monitoring and controlrdquo Control EngineeringPractice vol 12 no 6 pp 781ndash791 2004

[3] P R Kumar ldquoNew technological vistas for systems and controlrdquoIEEE Control Systems Magazine vol 21 no 1 pp 24ndash37 2001

[4] X Liu and A Goldsmith ldquoWireless medium access controlin networked control systemsrdquo in Proceedings of the AmericanControl Conference (AAC rsquo04) pp 3605ndash3610 Boston MassUSA July 2004

[5] J Colandairaj G W Irwin and W G Scanlon ldquoWirelessnetworked control systems with QoS-based samplingrdquo IETControl Theory and Applications vol 1 no 1 pp 430ndash438 2007

[6] N Israr W G Scanlon and G W Irwin ldquoA cross-layer com-munication framework forwireless networked control systemsrdquoin Proceedings of the 1st International Conference on WirelessCommunication Vehicular Technology Information Theory andAerospace and Electronic Systems Technology (VITAE rsquo09) pp577ndash581 Aalborg Denmark May 2009

[7] S Dai H Lin and S S Ge ldquoScheduling-and-control codesignfor a collection of networked control systems with uncertaindelaysrdquo IEEE Transactions on Control Systems Technology vol18 no 1 pp 66ndash78 2010

[8] A V Savkin ldquoAnalysis and synthesis of networked control sys-tems topological entropy observability robustness and optimalcontrolrdquo Automatica vol 42 no 1 pp 51ndash62 2006

[9] J Z Luo F Dong and J X Cao ldquoA novel task schedul-ing algorithm based on dynamic critical path and effectiveduplication for pervasive computing environmentrdquo WirelessCommunications amp Mobile Computing vol 10 no 10 pp 1283ndash1302 2010

[10] X Yin X Zhou Z Li and S Li ldquoJint rate control and powercontrol for lifetime maximization in Wreless Sensor NtworksrdquoJournal of Internet Technology vol 12 no 1 pp 69ndash78 2011

[11] M Pajic S Sundaram G J Pappas and R Mangharam ldquoThewireless control network a new approach for control overnetworksrdquo IEEE Transactions on Automatic Control vol 56 no10 pp 2305ndash2318 2011

[12] J Colandairaj G W Irwin and W G Scanlon ldquoA co-designsolution for wireless feedback controlrdquo in Proceeding of theInternational Conference on Networking Sensing and Control(ICNSC 07) pp 404ndash409 London UK April 2007

[13] W Liang and Y Yang ldquoMaximizing battery life routing in wire-less ad hoc networksrdquo in Proceedings of the 37th InternationalConference on System Sciences pp 4739ndash4746 IEEE HonoluluHawaii USA January 2004

[14] K Brian J Haberman and W Sheppard Overlapping ParticleSwarms for Energy-Efficient Routing in Sensor Networks Wire-less Network Springer 2011

[15] D Kim K Dantu andM Pedram ldquoLifetime prediction routinginmobile AdHoc networksrdquo in Proceedings of the IEEEWirelessCommunication and Networking Conference New York NYUSA 2003

[16] L Hetel J Daafouz and C Iung ldquoStability analysis for discretetime switched systems with temporary uncertain switchingsignalrdquo in Proceedings of the 46th IEEE Conference on Decisionand Control (CDC rsquo07) pp 5623ndash5628 New Orleans Fla USADecember 2007

[17] S Limin Wireless Sensor Network Tsinghua University PressBeijing China 2006

[18] Y He I Lee and L Guan ldquoDistributed algorithms for networklifetimemaximization in wireless visual sensor networksrdquo IEEETransactions on Circuits and Systems for Video Technology vol19 no 5 pp 704ndash718 2009

[19] B A Bakr and L Lilien ldquoA quantitative comparison of energyconsumption and WSN lifetime for LEACH and LEACH-SMrdquo in Proceedings of the 31st International Conference onDistributed Computing Systems Workshops (ICDCSW rsquo11) pp182ndash191 Minneapolis Minn USA June 2011

[20] C Fischione A Bonivento A Sangiovanni-Vincentelli F San-tucci and K H Johansson ldquoPerformance analysis of collabora-tive spatio-temporal processing for wireless sensor networksrdquoin Proceedings of the 3rd IEEE Consumer Communications andNetworking Conference (CCNC rsquo06) pp 325ndash329 Las VegasNev USA January 2006

[21] Y Iino ldquoSome considerations of wireless sensor network basedcontrol systemsrdquo in Proceedings of the SICE Conference onControl Systems (CCS rsquo07) Tokyo Japan 2007

[22] Y Iino T Hatanaka and M Fujita ldquoEvent-predictive controlfor energy saving of wireless networked control systemrdquo inProceedings of the American Control Conference (ACC rsquo09) pp2236ndash2242 St Louis Mo USA June 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of