Research Article Stability and Time Delay Tolerance ...downloads.hindawi.com/journals/mpe/2015/812070.pdf · Stability and Time Delay Tolerance Analysis Approach for Networked Control

Page 1

Research ArticleStability and Time Delay Tolerance Analysis Approach forNetworked Control Systems

Ashraf F Khalil1 and Jihong Wang2

1Electrical and Electronic Engineering Department University of Benghazi Benghazi Libya2School of Engineering University of Warwick Coventry CV4 7AL UK

Correspondence should be addressed to Ashraf F Khalil ashrafkhaliluobeduly

Received 9 August 2014 Revised 27 October 2014 Accepted 29 October 2014

Academic Editor Yun-Bo Zhao

Copyright copy 2015 A F Khalil and J Wang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Networked control system is a research area where the theory is behind practice Closing the feedback loop through shared networkinduces time delay and some of the data could be lost So the network induced time delay and data loss are inevitable in networkedcontrol SystemsThe time delay may degrade the performance of control systems or even worse lead to system instability Once thestructure of a networked control system is confirmed it is essential to identify the maximum time delay allowed for maintainingthe system stability which in turn is also associated with the process of controller design Some studies reported methods forestimating the maximum time delay allowed for maintaining system stability however most of the reported methods are normallyovercomplicated for practical applications A method based on the finite difference approximation is proposed in this paper forestimating the maximum time delay tolerance which has a simple structure and is easy to apply

1 Introduction

The key feature of networked control systems (NCSs) is thatthe information is exchanged through a network amongcontrol system components So the network induced timedelay is inevitable in NCSs The time delay either constant(up to jitter) or random may degrade the performance ofcontrol systems and even destabilize the systems NCSs canbe defined as a control system where the control loop isclosed through a real-time communication network [1] Theterm networked control systems first appeared in Walshrsquosarticle in 1999 [2] A typical organization of an NCS is shownin Figure 1 The reference input plant output and controlinput are exchanged through a real-time communicationnetwork The main advantages of NCSs are modularitysimplified wiring low cost reduced weight decentralizationof control integrated diagnosis simple installation quickand easy maintenance [3] and flexible expandability (easyto addremove sensors actuators or controllers with lowcost) NCSs are able to easily fuse global information tomake intelligent decisions over large physical spaces whichis important for many engineering systems such as the powersystem

As the control loop is closed through a communicationnetwork the time delay and data dropout are unavoidableTherefore networked control system can be regarded as aspecial case time delay system and many authors appliedthe time delay theorems to study NCSs [4] Time delay nodoubt increases complexity in analysis and design of NCSsConventional control theories built on a number of standingassumptions including synchronized control and nondelayedsensing and actuationmust be reevaluated before they can beapplied for NCSs [5]

The main goal of the most recent work is to reducethe conservativeness of the maximum time delay by usingLyapunov-Krasovskii functional with improved algorithmsfor solving the linear matrix inequalities (LMIs) set butwith the expense of increasing complexity and computationtime Analytical and graphical methods have been studiedin the literature (see eg [6]) The stability criteria forNCSs based on Lyapunov-Krasovskii functional approachhave been reported in [7ndash9] In [7] a Lyapunov-Krasovskiifunction is used to derive a set of LMIs and the stabilityproblem is generalized to a feasibility problem for the LMIsset Inmany of the previously reportedworks the controller is

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 812070 9 pageshttpdxdoiorg1011552015812070

Page 2

2 Mathematical Problems in Engineering

PlantActuator Sensor

Controller

Network

Figure 1 A typical networked control system

designed in the absence of the time delay In [10] an improvedLyapunov-Krasovskii function is used with triple integraltermsThe LMImethods require the closed-loop system to beHurwitz [8 11 12] In [13] a modified cone complementarylinearization algorithm based on the Lyapunov-Krasovskiiapproach is implemented And the method reported in [14]is claimed to be less conservative and the computationalcomplexity is reduced

The authors in [15] derived an LMI-based method inthe frequency domain and then the LMI is transformedonto an equivalent nonfrequency domain LMI by applyingKalman-Yakubovich-Popov lemma It has been reported in[16] that the ordinary Lyapunov stability analysis is linked bya suggested variable to state vectors through convolution andthe stability analysis is simplified to only require solving anonlinear algebraic matrix equation

In [11] the hybrid system technique is used to derive astability region An upper bound is derived for time delay inan inequality form and the results are rather conservativeThehybrid system stability analysis technique has also been usedin [17] A simple analytical relation is derived between thesampling period the time delay and the controller gainsThesame approach is used in [18] withmore conservative stabilityregion resultsThemodel-based approach for deriving neces-sary and sufficient conditions for stability is presented in [19]The stability criteria are derived in terms of the update timeand the parameters of the model The model-based approachis then extended to multiunits NCS in [20] The optimalstochastic control was studied in [21] with a discrete-timesystem model where the random time delays are modeledusing Markov chains and the controller uses the knowledgeof the past state time delays by time stamping

Most of the previously developed approaches requireexcessive load of computations and also for higher ordersystems the load of computations will increase dramaticallyIn practice engineers may find it difficult to apply thoseavailable methods in control system design because of thecomplexity of the methods and lack of guideline in linkingbetween the design parameters and the system performanceAlmost all the design procedures highly depend on thepostdesign simulation to determine the design parametersSo there is a demand for a simple design approach withclear guidance for practical applications In this paper a newstability analysis and control design method is proposedin which the design approach is simple and a clear designprocedure is given

The paper starts from the mathematical model of NCSand then the proposed method for estimating the maximumallowable delay bound is briefly describedA few examples areillustrated and the results are compared with those previouslypublished in the literature The cart and inverted pendulumproblem is used to study the effect of the parameters on themaximum allowable delay bound

2 Mathematical Analysis

Although the issues involved with time delays in controlsystems have been studied for a long time it is difficultto find a method simple enough to be accepted by controlsystem design engineers It is found that the most previouslyreported methods rely on LMI techniques and they aregenerally too complicated for practical engineers to useand also involve heavy load of numerical calculations andcomputing timeThepaper proposes a newmethodwhich hasa simple structure and is used for estimating the maximumtime delay allowed while the system stability can still bemaintained In most control systems the sampling time ispreferred to be small [22] The maximum allowable delaybound (MADB) can be defined as the maximum samplingperiod that guarantees the stability even with poor systemperformance A continuous time-invariant linear system isshown in Figure 2 and given by

x (119905) = Ax (119905) + Bu (119905)

y (119905) = Cx (119905) +Du (119905)

(1)

where x(119905) isin R119899 is the system state vector u(119905) isin R119898 is thesystem control input y(119905) isin R119901 is the system output andA isin R119899times119899 B isin R119899times119898 C isin R119901times119899 andD isin R119901times119898 are constantmatrices with appropriate sizes

Suppose that the control signals are connected to thecontrol plant through a kind of network so the time delayis inevitable to be involved in the feedback loop The statefeedback is therefore can be written as

u (119905) = Kx (119905 minus 120591

119904119888minus 120591

119888minus 120591

119888119886) (2)

where 120591

119904119888is the time delay between the sensor and the

controller 120591119888is the time delay in the controller and 120591

119888119886is the

time delay from the controller to the actuator K representsthe feedback control gains with appropriate size From (2) thenetworked control system can be modeled where the timedelay is lumped between the sensor and the controller asshown in Figure 3

The time delaymay be constant variable or even randomIn NCSs the time delay is composed of the time delayfrom sensors to controllers time delay in the controller andcontrollers to actuators time delay which is given by

120591 = 120591

119904119888+ 120591

119888+ 120591

119888119886 (3)

For a general formulation the packet dropouts can be incor-porated in (3) as follows

120591 = 120591

119904119888+ 120591

119888+ 120591

119888119886+ 119889ℎ (4)

Page 3

Mathematical Problems in Engineering 3

Actuator Plant Sensor

Controller

120591ca 120591sc

x(t)

Kx(t minus 120591sc minus 120591c minus 120591ca)

Kx(t minus 120591sc minus 120591c) x(t minus 120591sc)

Figure 2 Anetworked control systemwith the time delay both fromthe sensor to the controller and from the controller to the actuator

Actuator Plant Sensor

Controller

120591ca + 120591sc

x(t)

Kx(t minus 120591sc minus 120591c minus 120591ca)

x(t minus 120591sc minus 120591ca)

Figure 3 A simplified model of the networked control system

where119889 is the number of dropouts and ℎ the sampling periodAnd by (4) the data dropouts can be considered as a specialcase of time delay [23 24] It is supposed that the followinghypotheses hold

Hypothesis 1 (H1) (i) The sensors are clock driven (ii) Thecontrollers and the actuators are event driven (iii) The dataare transmitted as a single packet (iv) The old packets arediscarded (v) All the states are available for measurementsand hence for transmission

Hypothesis 2 (H2) The time delay 120591 is small to be less thanone unit of its measurement

Definition 1 (D1) For a function119891(119905) the 119899th order reminderfor its Taylorrsquos series expansion is defined by

119877

119899(119891 (119905) 120591) =

infin

sum

119899

119891

(119899)

(120585)

119899

120591

119899

(5)

Applying the state feedback proposed in (2) to the system (1)we have

x (119905) = Αx (119905) + BKx (119905 minus 120591) (6)

From (6) the following can be derived

x (119905) = (A + BK) x (119905) + BK [x (119905 minus 120591) minus x (119905)] (7)

Theorem 2 Suppose that (H1) and (H2) hold For system (1)with the feedback control of (2) the closed-loop system is glob-ally asymptotically stable if 120582

119894(Ψ) isin 119862

minus for 119894 = 1 2 119899 andall the state variablesrsquo 2nd order reminders are small enough forthe given value of 120591 whereΨ is given by

Ψ = [(119868 + 120591BK)

minus1

(A + BK)] (8)

Proof The expression for x(119905 minus 120591) can be obtained by Taylorexpansion as

x (119905 minus 120591) =

infin

sum

119899=0

(minus1)

119899120591

119899

119899

x(119899) (119905) (9)

where x(119899)(119905) is the 119899th order derivative The first orderapproximation of the delay term is given by

x (119905 minus 120591) = x (119905) minus 120591x (119905) + (

120591

2

2

) x (119905) + R3(x 120591)

x (119905 minus 120591) asymp x (119905) minus 120591x (119905) + (

120591

2

2

) x (119905)

x (119905 minus 120591) = x (119905) minus 120591x (119905) + R2(x 120591)

(10)

From (10) it can be seen that R2(119909 120591) depends on the time

delay 120591 and the higher order derivatives of x(119905)which can beneglected if the time delay and the norm ofR

2(119909 120591) are small

Then we have

x (119905 minus 120591) asymp x (119905) minus 120591x (119905) (11)

The assumption in (11) can be used without significanterror and this can be true for the following reasons Firstlythe time delay in a computer network is very small in orderof milli- or microseconds and at the worst few tenths ofthe second Secondly in most of the real control systemapplications the linearizedmodel is used and the higher orderterms are already neglected Additionally the higher orderderivatives will be multiplied by 120591

119899

119899 which is much moresmaller than 120591 because 120591 ≪ 1 Substituting (11) into (7) thefollowing can be derived

x (119905) asymp (A + BK) x (119905) minus 120591BKx (119905) (12)

x (119905) asymp [(I + 120591BK)

minus1

(A + BK)] x (119905) (13)

Ψ = [(I + 120591BK)

minus1

(A + BK)] (14)

The system (13) will be globally asymptotically stable if

120582

119894(Ψ) isin Cminus for 119894 = 1 2 119899 (15)

Corollary 3 Suppose (H1) and (H2) hold For the controlsystem (1) with the control law (2) the closed-loop system isglobally asymptotically stable if

120591 lt

1

BK

(16)

Proof For system (1) suppose that the state feedback hasbeen designed to ensure 120582(A + BK) isin Cminus Therefore for achosen positive definite matrix P = PT it will find a positivedefinite matrixQ = QT to have

P (A + BK) + (A + BK)

T P = minusQ (17)

Page 4

4 Mathematical Problems in Engineering

Choose a Lyapunov functional candidate as

V (119909) = xTPx gt 0 forallx = 0 (18)

The objective for the next step is to find the range of 120591 thatwill ensure ( V(119909) lt 0 forallx = 0) [25ndash27] Taking the derivativeof (18)

V (119909) = xTPx + xTPx

asymp xT [(A + BK)

T PPminus1 (I + 120591BK)

minusT P

+P (I + 120591BK)

minus1 Pminus1P (A + BK)] x

minus xT [P (A + BK) + (A + BK)

T P] x

+ xT [P (A + BK) + (A + BK)

T P] x

asymp xT [(A + BK)

T PPminus1 (I + 120591BK)

minusT P

minus (A + BK)

T P + P (I + 120591BK)

minus1 Pminus1P (A + BK)

minus P (A + BK) ] x minus xTQx(19)

Rearranging the terms in the above equation then

V (x) asymp xT (A + BK)

T P [Pminus1 (I + 120591BK)

minusT P minus I]

+ [P (I + 120591BK)

minus1 Pminus1 minus I]P (A + BK) x

minus xTQx

(20)

If P(I + 120591BK)

minus1Pminus1 minus I = I then (20) will become

xT [P (A + BK) + (A + BK)

T P] x minus xTQx = 0 (21)

Move the last term to the right hand side the following willbe derived

xT [P (A + BK) + (A + BK)

T P] x = xTQx (22)

So P(A + BK) + (A + BK)

TP sdot x2 = Q sdot x2Assuming that we can find a positive number to make the

following hold1003817

1003817

1003817

1003817

1003817

P (A + BK) + (A + BK)

T P1003817100381710038171003817

1003817

= 2120574

1003817

1003817

1003817

1003817

1003817

(A + BK)

T P1003817100381710038171003817

1003817

= Q

(23)

then 120574 can be considered as the norm of Pminus1(I+120591BK)

minus1Pminus ITherefore we have

xT [(A + BK)

T P [Pminus1 (I + 120591BK)

minusT P minus I]

+ [P (I + 120591BK)

minus1 Pminus1 minus I]P (A + BK)] x

le 2

1003817

1003817

1003817

1003817

1003817

(Pminus1 (I + 120591BK)

minusT P minus I)P (A + BK)

1003817

1003817

1003817

1003817

1003817

sdot x2

(24)

Choose1003817

1003817

1003817

1003817

1003817

Pminus1 (I + 120591BK)

minus1 P minus I1003817100381710038171003817

1003817

le 1 (25)

UseNeumann series formula for the inverse of the sumof twomatrices

(I + 120591BK)

minus1

= I minus 120591BK + 120591

2

(BK)

2

minus 120591

3

(BK)

3

+ sdot sdot sdot minus

(26)

For small time delays 120591 ≪ 1 (26) can be given as

(I + 120591BK)

minus1

asymp I minus 120591BK (27)

Applying (27) into (25) then we have1003817

1003817

1003817

1003817

1003817

Pminus1 (I + 120591BK)

minus1 P minus I1003817100381710038171003817

1003817

asymp

1003817

1003817

1003817

1003817

1003817

Pminus1 (I minus 120591BK)P minus I1003817100381710038171003817

1003817

= 120591BK lt 1

(28)

And finally we get

120591 lt

1

BK

(29)

That is for any 120591 lt 1BK V(119909) lt 0 the system will beglobally asymptotically stable

Theorem 2 and Corollary 3 give us a simple tool inestimating the maximum allowable time delay for NCSsFurther analysis in the frequency domain is described belowTaking Laplace transform of (12) we have

119904X (119904) = (A + BK)X (119904) minus 120591119904BKX (119904)

[119904I minus (A + BK) + 120591119904BK]X (119904) = 0

(30)

The characteristics equation is defined as

[119904I minus (A + BK) + 120591119904BK] = 0 (31)

For a stable system the roots of the characteristics equation(31) must lie in the left hand side of the 119904-plane Fromthe characteristics equation it is clear that the term 120591119904BKinfluences the system performance and the stability as theterm of 120591119904BKmay push the closed-loop system poles towardthe right hand side of the 119904-plane

As we have seen the system characteristic is determinedby the term 120591BKx(119905) in a certain level This term can beregarded as a differentiator in the feedback loop so it willintroduce extra zeros to the closed-loop system and thetime delay can be considered to have resulted in a variablegain to the feedback path For more accurate estimation thesecond or third-order difference approximation can be usedas follows

[119904I minus (A + BK) + 120591119904BK minus

120591

2

119904

2

2

BK] = 0

[119904I minus (A + BK) + 120591119904BK minus

120591

2

119904

2

2

BK +

120591

3

119904

3

6

BK] = 0

(32)

In the following a simple corollary for estimating the MADBin single-input-single-output NCS will be derived

Page 5

Mathematical Problems in Engineering 5

Corollary 4 Suppose that (H1) and (H2) holdThe system (2)with the controller (3) is asymptotically stable if

120591 lt

1

1003816

1003816

1003816

1003816

120582min (BK)

1003816

1003816

1003816

1003816

(33)

Proof The main assumption is that the eigenvalues of thecompensator BK are all negative 119904

1lt 0 119904

119899lt 0 and

are given by

BK minus 119904I119899times119899

=

[

[

[

[

[

119886

11minus 119904 119886

12sdot sdot sdot 119886

1119899

119886

21119886

22minus 119904 sdot sdot sdot 119886

2119899

d

119886

1198991119886

1198992sdot sdot sdot 119886

119899119899minus 119904

]

]

]

]

]

(34)

The characteristic equation is the determinant of (34) As-sume that the eigenvalues are given by

119904

1= 120572

1 119904

119899= 120572

119899

120572

1lt 0 120572

119899lt 0

(35)

Preliminary 1 (inverse eigenvalues theorem [28]) Given amatrix X that is nonsingular with eigenvalues 120582

1 120582

119899gt

0 120582

1 120582

119899are eigenvalues of X if and only if 120582

1

minus1

120582

119899

minus1

are eigenvalues of Xminus1The eigenvalues of (I

119899times119899+ 120591BK) are given by

120591 sdot BK + I119899times119899

minus 120582I119899times119899

=

[

[

[

[

[

120591119886

11+ 1 minus 120582 120591119886

12sdot sdot sdot 120591119886

1119899

120591119886

21120591119886

22+ 1 minus 120582 sdot sdot sdot 120591119886

2119899

d

120591119886

1198991120591119886

1198992sdot sdot sdot 120591119886

119899119899+ 1 minus 120582

]

]

]

]

]

(36)

Δ (120591 sdot BK + I119899times119899

minus 120582I119899times119899

)

= det([

[

[

[

[

120591119886

11+ 1 minus 120582 120591119886

12sdot sdot sdot 120591119886

1119899

120591119886

21120591119886

22+ 1 minus 120582 sdot sdot sdot 120591119886

2119899

d

120591119886

1198991120591119886

1198992sdot sdot sdot 120591119886

119899119899+ 1 minus 120582

]

]

]

]

]

)

Δ (120591 sdot BK + I119899times119899

minus 120582I119899times119899

)

= 120591

119899 det(

(

(

(

[

[

[

[

[

[

[

[

[

[

119886

11+

1 minus 120582

120591

119886

12sdot sdot sdot 119886

1119899

119886

21119886

22+

1 minus 120582

120591

sdot sdot sdot 119886

2119899

d

119886

1198991119886

1198992sdot sdot sdot 119886

119899119899+

1 minus 120582

120591

]

]

]

]

]

]

]

]

]

]

)

)

)

)

(37)

Replacing (1 minus 120582)120591 by minus119904 in (37) we get

= 120591

119899 det([

[

[

[

[

119886

11minus 119904 119886

12sdot sdot sdot 119886

1119899

119886

21119886

22minus 119904 sdot sdot sdot 119886

2119899

d

119886

1198991119886

1198992sdot sdot sdot 119886

119899119899minus 119904

]

]

]

]

]

) = 0 (38)

Solving (38) the eigenvalues are given as

(120582

1minus 1)

120591

= 120572

1

(120582

119899minus 1)

120591

= 120572

119899

120572

1lt 0 120572

119899lt 0

120582

1= 1 + 120591120572

1 120582

119899= 1 + 120591120572

119899

120572

1lt 0 120572

119899lt 0

(39)

If 120591 lt 1|120572max| then all the eigenvalues are positive and thesystem is asymptotically stable and if 120591 gt 1|120572max| at leastone of the eigenvalues will be negative then

If 120591 lt 1|120582min(BK)| and (H1) and (H2) hold then thesystem is asymptotically stable

Corollary 5 Suppose that (H1) and (H2) hold For system(1) with the control law (2) the closed-loop system is globallyasymptotically stable if

120591 lt

1

119886119887119904 (KB)(119908ℎ119890119903119890 119886119887119904 119894119904 119905ℎ119890 119886119887119904119900119897119906119905119890 V119886119897119906119890) (40)

From Preliminary 1 the signs of the eigenvalues of (I119899times119899

+

120591BK)

minus1 and (I119899times119899

+ 120591BK) are the same For a single-input-single-output control system the matrix BK can be written as

BK =

[

[

[

[

[

119887

1

119887

2

119887

119899

]

]

]

]

]

[119896

1119896

2sdot sdot sdot 119896

119899] =

[

[

[

[

[

119887

1119896

1119887

1119896

2sdot sdot sdot 119887

1119896

119899

119887

2119896

1119887

2119896

2sdot sdot sdot 119887

2119896

119899

d

119887

119899119896

1119887

119899119896

2sdot sdot sdot 119887

119899119896

119899

]

]

]

]

]

(41)

The interesting property of BK is that it is singular The eigen-values of BK are given by

BK minus 120582I119899times119899

=

[

[

[

[

[

[

[

119887

1119896

1minus 120582 119887

1119896

2sdot sdot sdot 119887

1119896

119899

119887

2119896

1119887

2119896

2minus 120582 sdot sdot sdot 119887

2119896

119899

d

119887

119899119896

1119887

119899119896

2sdot sdot sdot 119887

119899119896

119899minus 120582

]

]

]

]

]

]

]

(42)

The characteristics equation of BK is the determinant of (42)and is given by

120582

2

minus Tr (BK) 120582 +

1

2

[Tr (BK2) minus Tr (BK)

2

]

120582

119899

minus Tr (BK) 120582

119899minus1

+

1

2

[Tr (BK2) minus Tr (BK)

2

] 120582

119899minus2

+ sdot sdot sdot +

1

2

[Tr (BK2) minus Tr (BK)

2

]

(43)

Because BK is singular det(BK) = 0 and hence

det (BK) =

1

2

[Tr (BK2) minus Tr (BK)

2

] = 0

Tr (BK2) = Tr (BK)

2

(44)

Page 6

6 Mathematical Problems in Engineering

Substituting (44) into (43) then (43) becomes

120582

2

minus Tr (BK) 120582 997888rarr 120582 (120582 minus Tr (BK))

(minus1)

119899

120582

119899

minus Tr (BK) 120582

119899minus1

997888rarr (minus1)

119899

120582

119899minus1

(120582 minus Tr (BK))

(45)

Finally the eigenvalues of BK are

120582

1 120582

119899minus1= 0 120582

119899= Tr (BK) lt 0 (46)

Equation (46) shows that the minimum eigenvalue of BKequals Tr(BK) If the eigenvalues of (I

119899times119899+120591BK) are 119904

1 119904

119899

then the eigenvalues of (I119899times119899

+ 120591BK)

minus1 are 11199041 1119904

119899 The

eigenvalues of (I119899times119899

+ 120591BK) are given by

120591 sdot BK + I119899times119899

minus 119904I119899times119899

=

[

[

[

[

[

[

[

120591119887

1119896

1+ 1 minus 119904 120591119887

1119896

2sdot sdot sdot 120591119887

1119896

119899

120591119887

2119896

1120591119887

2119896

2+ 1 minus 119904 sdot sdot sdot 120591119887

2119896

119899

d

120591119887

119899119896

1120591119887

119899119896

2sdot sdot sdot 120591119887

119899119896

119899+ 1 minus 119904

]

]

]

]

]

]

]

(47)

By solving (47) it can be found that

119904

1 119904

119899minus1= 1

119904

119899= 1 + 120591 sdot Tr (BK) = 1 + 120591 sdot 120582max (BK)

(48)

if 120591 lt 1|Tr(BK)| rarr 119904

119899gt 0 rarr 119904

1 119904

119899gt 0

For single-input-single-output NCS we have

119886119887119904 (KB) = Tr (BK) 119905ℎ119890119899 (49)

if 120591 lt 1|KB| and both (H1) and (H2) hold then the system isasymptotically stable

This inequality can be used as a simple and fast toolfor estimating the MADB in NCS and involves only singlecalculation

3 Stability Analysis Case Studies

In general two approaches are applied to controller designfor NCSs The first approach is to design a controller withoutconsidering time delay and then to design a communicationprotocol that minimizes the effects caused by time delaysThe second approach is to design the controller while takingthe time delay and data dropouts into account [11 29] Theproposedmethod in this paper is used to estimate theMADBfor predesigned control system In this section a number ofexamples are studied to demonstrate the proposed methodand compare its results with the previously published casesin the literature In particular the results derived using themethod proposed in this paper have been compared withthe results using the LMI method given in [7] and withthe fourth-order Pade approximationThe fourth-order Padeapproximation [6] is used for the delay term in the 119904-domainand is defined as

119890

minus120591119904

asymp 119875

119889(119904) =

119873

119889(119904)

119863

119889(119904)

=

(sum

119899

119896=0(minus1)

119896

119888

119896120591

119896

119904

119896

)

(sum

119899

119896=0119888

119896120591

119896119904

119896)

(50)

The coefficients are given by

119888

119896=

((2119899 minus 119896)119899)

(2119899119896 (119899 minus 119896))

119896 = 0 1 119899 (119899 = 4) (51)

With the fourth-order Pade approximation the truncationerror in the time delay calculation is less than 00001 TheLMI-basedmethod which has been used for the comparisonsis based on using Lyapunov-Krasovskii functional and can besummarized as follows

Corollary 6 (see [7]) For a given scalar 120591 and a matrix K ifthere exist matrices P gt 0 T gt 0 N

119894 and M

119894(119894 = 1 2 3) of

appropriate dimension such that

[

[

[

[

[

[

[

[

[

[

M1+M1198791minus N1A minus A119879N119879

1M1198792minusM1minus A119879N119879

2minus N1BK M119879

3minus A119879N119879

3+ N1+ P 120591M

1

lowast minusM2minusM1198792minus N2BK minus (BK)

119879N1198792

minusM1198793+ N2minus (BK)

119879N1198793

120591M2

lowast lowast N3+ N1198793+ 120591T 120591M

3

lowast lowast lowast minus120591T

]

]

]

]

]

]

]

]

]

]

lt 0 (52)

then the system (1)-(2) is exponentially asymptotically stableWith a given controller gain K solving the LMI in Corollary 6using the LMIMatlab Toolbox the maximum time delay can becomputed

Example 7 The system in this example is the most widelyused example in the literature and is described by the

following equation

(119905) = [

0 1

0 minus01

] 119909 (119905) + [

0

01

] 119906 (119905) (53)

In previous reports [1 7] the feedback control is chosen to be

119906 (119905) = [minus375 minus115] 119909 (119905) (54)

Page 7

Mathematical Problems in Engineering 7

From Corollary 3 1BK = 08695 so the MADB is esti-mated to be 08695 s Using Theorem 2 and Corollary 5 theMADB is 08695 s The same result can be obtained using theLMImethod as reported in [7 23 24 30] In [11 17] the valuereported for MADB is 45 times 10

minus4 s and in [22] it is 00538 sIn [29] the MADB is 0785 s It has been reported in [10]where an improved Lyapunov-Krasovskii approach has beenused that the MADB is 10551 s and also 105 s reported in[23] with improved algorithm for solving the LMI In [1] theMADB is 10081 s Using the proposed method with secondorder finite difference approximation we can obtain 113 s astheMADBThe system response with 08695 s time delay andx(0) = [01 0]

T is shown in Figure 4which proves the systemis stable with the estimated MADB

Example 8 (see [31]) Consider

(119905) =

[

[

0 1 0

0 0 1

0 minus2 minus3

]

]

119909 (119905) +

[

[

0

0

1

]

]

119906 (119905)

119906 (119905) = [minus160 minus54 minus11] 119909 (119905)

(55)

For this third-order system both the LMI and ourmethodgive 00909 s as theMADB Also with Corollary 5 theMADBis 00909 s

Example 9 (see [31]) The last example is the fourth-ordermodel of the inverted pendulum shown in Figure 5 which isin many papers reduced to a second order system in order toverify the stability of NCSs The pendulum mass is denotedby119898 and the cart mass is119872 the length of the pendulum rodis 119871 The open loop system is unstable The states are definedas 1199091= 119909 119909

2= 119909

3= 120579 and 119909

4=

120579 The model is given by

x (119905) =[

[

[

[

[

[

[

0 1 0 0

0 0

minus119898119892

119872

0

0 0 0 1

0 0

(119872 + 119898) 119892

119872119871

0

]

]

]

]

]

]

]

x (119905) +[

[

[

[

[

[

[

0

1

119872

0

minus1

119872119871

]

]

]

]

]

]

]

119906 (119905)

y (119905) = [

119909

120579

] = [

1 0 0 0

0 0 1 0

] x (119905)

(56)

The parameters used are 119872 = 2 kg 119898 = 01 kg and 119871 =

05mThen the linear model becomes

(119905) =

[

[

[

[

0 1000 0 0

20601 0 0 0

0 0 0 1

minus04905 0 0 0

]

]

]

]

119909 (119905) +

[

[

[

[

0

minus1

0

05

]

]

]

]

119906 (119905) (57)

Using the LQR Matlab function with Q = I and R = 1 thecontroller is given by

KLQR = [521238 115850 1000 27252] (58)

Using the LMI method the MADB is 00978 s and ourmethod gives 00978 s using Theorem 2 and Corollary 5 Wenoted that there is a good agreement between our method

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus006

minus004

minus002

0

002

004

006

008

01

The s

yste

m re

spon

se

Figure 4 The response of the system in Example 7 with 08695 sdelay

u

x

M

L

120579

m

Figure 5 The inverted pendulum on a cart

and the LMI method because 120591 is small enough to make thefinite difference approximation hold The system responsewith 00978 s time delay and with 119909 = 0 and 120579 = 01

is shown in Figure 6 which shows the system is stableMany examples have been studied to compare the resultsobtained using the method proposed in this paper with theresults obtained using the LMI method [7] and the fourth-order Pade approximation method The calculation resultsare summarized in Table 1 along with the simulation basedMADB

Remarks From Table 1 it can be seen that the proposednew method can give values of MADB similar to the valuesobtained using the LMI method and the other methodshowever the method proposed in this paper has a much

Page 8

8 Mathematical Problems in Engineering

Table 1 The MADB (seconds) using the proposed method with 1st 2nd and 3rd order finite difference approximation for the delay termthe LMI method the fourth-order Pade approximation method and the simulation based method

The finite difference method The LMI Pade approximation Simulation based1st order 2nd order 3rd order

1 08695 08427 11321 08696 11672 11802 01000 00995 01421 01000 01475 01493 00100 00099 00149 00100 00156 001574 01428 01385 01808 01429 01855 018605 08217 08489 09085 08217 09091 091406 05000 04816 06303 05000 06474 065107 09940 09940 09960 09940 09960 099708 00856 00854 01192 00856 01230 012309 00906 00919 01251 00909 01284 0128510 00416 00400 00496 00416 00505 00505

0 1 2 3 4 5 6 7 8 9 10Time (s)

The s

yste

m re

spon

se

minus01

minus005

0

005

01

015

02

025

03

035

120579 (rad)x (m)

Figure 6 The response of the system in Example 9 with 00978 sdelay

simpler procedure and it should have no difficulties for prac-tical design engineers to accept this approach Clearly theMADB with the first-order finite difference approximation iscomparable with the LMI method Furthermore we foundgood agreement between the third-order finite differenceapproximation and the fourth-order Pade approximationThe simulation based results for the MADB show that theestimated MADB through the proposed method sufficientlyachieves the system stability A simple controller designmethod has been developed by the authors based on themethod presented in this paper In the controller designmethod a stabilizing controller can be derived for a givennetwork time delay In all the case studies or examplesonly linear system examples are given The method is lim-ited to linear systems only The authors are now workingon extending the methods to nonlinear systems such asmulticonverter and inverter system and engine and electricalpower generation systems [32 33]

The application of the finite difference approximation forrepresenting the time delay is not new but we found in thispaper that using higher order approximations can sufficientlyrepresent the time delay linear system From Table 1 it canbe concluded that using the first order approximation theestimated MADB is comparable with the other two methodsThis is because the derivation of the linear model from thenonlinear model is based on neglecting the higher orderderivative terms In some cases we need to use the higherderivative terms for the time delay in order to achieve moreaccurate results for the MADB The current research is toderive sufficient conditions for applying the method in orderto find the tolerance of the estimated MADB

4 Concluding Remarks

The main contribution of the paper is to have derived anew method for estimating the maximum time delay inNCSs The most attractive feature of the new method isthat it is a simple approach and easy to be applied whichcan be easily interpreted to design engineers in industrialsectors The results obtained in this method are comparedwith those obtained through the methods introduced in theliterature The method has demonstrated its merits in usingless computation time due to its simple structure and givingless conservative results while showing good agreement withother methods The method is limited to linear systemsonly and the work for extending the method for a class ofnonlinear systems is on-going

Conflict of Interests

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

References

[1] X Jiang Q L Han S Liu and A Xue ldquoA new119867

infinstabilization

criterion for networked control systemsrdquo IEEE Transactions onAutomatic Control vol 53 no 4 pp 1025ndash1032 2008

Page 9

Mathematical Problems in Engineering 9

[2] G C Walsh H Ye and L Bushnell ldquoStability analysis ofnetworked control systemsrdquo in Proceedings of the AmericanControl Conference (ACC rsquo99) vol 4 pp 2876ndash2880 San DiegoCalif USA June 1999

[3] G C Walsh H Ye and L G Bushnell ldquoStability analysisof networked control systemsrdquo IEEE Transactions on ControlSystems Technology vol 10 no 3 pp 438ndash446 2002

[4] M S Mahmoud Robust Control and Filtering For Time Delaysystems Marcel Dekker New York NY USA 2000

[5] M S Mahmoud and A Ismail ldquoRole of delays in networkedcontrol systemsrdquo in Proceedings of the 10th IEEE InternationalConference on Electronics Circuits and Systems (ICECS rsquo03) vol1 pp 40ndash43 December 2003

[6] J E Marshall H Gorecki A Korytowski and K WaltonTime-Delay Systems Stability and Performance Criteria withApplications Ellis Horwood 1992

[7] D Yue Q-L Han and C Peng ldquoState feedback controllerdesign of networked control systemsrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 51 no 11 pp 640ndash644 2004

[8] P-L Liu ldquoExponential stability for linear time-delay systemswith delay dependencerdquo Journal of the Franklin Institute vol340 no 6-7 pp 481ndash488 2003

[9] B Tang G P Liu and W H Gui ldquoImprovement of statefeedback controller design for networked control systemsrdquoIEEE Transactions on Circuits and Systems vol 55 no 5 pp464ndash468 2008

[10] J Sun G Liu and J Chen ldquoState feedback stabilization ofnetworked control systemsrdquo in Proceedings of the 27th ChineseControl Conference (CCC rsquo08) pp 457ndash461 Kunming ChinaJuly 2008

[11] W Zhang M S Branicky and S M Phillips ldquoStability ofnetworked control systemsrdquo IEEE Control Systems Magazinevol 21 no 1 pp 84ndash97 2001

[12] X Li and C E de Souza ldquoDelay-dependent stability of lineartime-delay systems an LMI approachrdquo IEEE Transactions onAutomatic Control vol 42 no 8 pp 1144ndash1148 1997

[13] W Min and H Yong ldquoImproved stabilization method fornetworked control systemsrdquo in Proceedings of the 26th ChineseControl Conference (CCC rsquo07) pp 544ndash548 Hunan China July2007

[14] X-L Zhu and G-H Yang ldquoNew results on stability analysisof networked control systemsrdquo in Proceedings of the AmericanControl Conference pp 3792ndash3797 Seattle Wash USA June2008

[15] M Jun and M G Safonov ldquoStability analysis of a systemwith time-delay statesrdquo in Proceeding of the American ControlConference Chicago Ill USA June 2000

[16] K Kim ldquoA delay-dependent stability criterion in time delaysystemrdquo Journal of the Korean Society for Industrial and AppliedMathematics vol 9 no 2 pp 1ndash11 2005

[17] M S Branicky S M Phillips and W Zhang ldquoStability ofnetworked control systems explicit analysis of delayrdquo in Pro-ceedings of the American Control Conference pp 2352ndash2357Chicago Ill USA June 2000

[18] L Xie J-M Zhang and S-Q Wang ldquoStability analysis ofnetworked control systemrdquo in Proceedings of 1st InternationalConference on Machine Learning and Cybernetics pp 757ndash759Beijing China November 2002

[19] A Luis and P J Montestruque ldquoModel-based networked con-trol systems- necessary and sufficient conditions for stabilityrdquo

in Proceedings of the 10th Mediterranean Conference on Controland Automation pp 1ndash58 Lisbon Portugal July 2002

[20] Y Sun and N H El-Farra ldquoQuasi-decentralized model-basednetworked control of process systemsrdquo Computers amp ChemicalEngineering vol 32 no 9 pp 2016ndash2029 2008

[21] J Nilsson Real-time control systems with delays [PhD thesis]Institute of Technology Lund Sweden 1998

[22] H S Park Y H KimD-S Kim andWH Kwon ldquoA schedulingmethod for network-based control systemsrdquo IEEE Transactionson Control Systems Technology vol 10 no 3 pp 318ndash330 2002

[23] Y Zhang Q Zhong and LWei ldquoStability analysis of networkedcontrol systems with communication constraintsrdquo in Proceed-ings of the Chinese Control andDecisionConference pp 335ndash339July 2008

[24] Y Zhang Q Zhong and LWei ldquoStability analysis of networkedcontrol systems with transmission delaysrdquo in Proceedings ofthe Chinese Control and Decision Conference pp 340ndash343 July2008

[25] J Wang U Kotta and J Ke ldquoTracking control of nonlinearpneumatic actuator systems using static state feedback lin-earisation of inputoutput maprdquo Proceedings of the EstonianAcademy of Sciences Physics Mathematics vol 56 no 1 pp 47ndash66 2007

[26] D P Goodall and J Wang ldquoStabilization of a class of uncertainnonlinear affine systems subject to control constraintsrdquo Interna-tional Journal of Robust and Nonlinear Control vol 11 no 9 pp797ndash818 2001

[27] J Wang J Pu P R Moore and Z Zhang ldquoModelling studyand servo-control of air motor systemsrdquo International Journalof Control vol 71 no 3 pp 459ndash476 1998

[28] C DMeyerMatrix Analysis and Applied Linear Algebra SIAM2000

[29] F Yang and H Fang ldquoControl strategy design of networkedcontrol systems based on maximum allowable delay boundsrdquoin Proceedings of the IEEE International Conference on Controland Automation (ICCA rsquo07) pp 794ndash797 Guangzhou ChinaJune 2007

[30] P Naghshtabrizi Delay impulsive systems a framework formodeling networked control systems [PhD thesis] University ofCalifornia Los Angeles Calif USA 2007

[31] K OgataModern Control Engineering PrenticeHall NewYorkNY USA 3rd edition 1997

[32] J L Wei J Wang and Q H Wu ldquoDevelopment of a multi-segment coal mill model using an evolutionary computationtechniquerdquo IEEE Transactions on Energy Conversion vol 22 no3 pp 718ndash727 2007

[33] N Jia JWangKNuttall et al ldquoHCCI enginemodeling for real-time implementation and control developmentrdquo IEEEASMETransactions on Mechatronics vol 12 no 6 pp 581ndash589 2007

Page 10

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Mathematical Problems in Engineering

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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