Research ArticleIdentification of LTI Time-Delay Systems with Missing OutputData Using GEM Algorithm

Xianqiang Yang1 and Hamid Reza Karimi2

1 Research Institute of Intelligent Control and Systems Harbin Institute of Technology Harbin Heilongjiang 150080 China2Department of Engineering Faculty of Engineering and Science University of Agder 4898 Grimstad Norway

Correspondence should be addressed to Xianqiang Yang xianqiangyanggmailcom

Received 28 December 2013 Accepted 13 February 2014 Published 17 March 2014

Academic Editor Xudong Zhao

Copyright copy 2014 X Yang and H R Karimi 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

This paper considers the parameter estimation for linear time-invariant (LTI) systems in an input-output setting with output error(OE) time-delay model structure The problem of missing data is commonly experienced in industry due to irregular samplingsensor failure data deletion in data preprocessing network transmission fault and so forth to deal with the identification ofLTI systems with time-delay in incomplete-data problem the generalized expectation-maximization (GEM) algorithm is adoptedto estimate the model parameters and the time-delay simultaneously Numerical examples are provided to demonstrate theeffectiveness of the proposed method

1 Introduction

The advanced process control theories have enjoyed rapiddevelopment in the past several decades to meet the grow-ing demands of closed-loop system performances such asimproved process safety and efficiency of plant operationconsistent product quality and economic optimization [1]These control strategies have improved the process automa-tion and stability through providing control solutions for theprocess operated under the abnormal working conditionssuch as process fault [2ndash5] network transmission delay [6]data packet dropouts [7] and modeling error Generallythe implementation of these control strategies relies on theunderstanding of the process dynamics and the availability ofan accuratemathematical model of the process In view of thedifficulties and complexities imposed by modeling using firstprinciple method the data-driven modeling method inwhich the process model is retrieved from the process datahas become a main modeling method

Typically the process data used in process modelingare generated by performing an identification experimentin which a testing signal is designed and utilized to excitethe process Most of the conventional parameter estima-tion methods such as prediction error method (PEM)

instrumental variable (IV) method and subspace methodassume that the identification data are sampled regularlyand recorded properly However this is not always true inpractical industry For example in the development of aninferential model for the sulfur content in the gas oil productthe sulfur concentration cannot be measured directly andthe lab analysis is required which takes a long time Theprocess variable can be sampled in every minute but thesulfur concentration is only available in every twelve hoursAnother example is the industrial process with data trans-mission through the network The recorded process dataare corrupted by many network-induced problems such astransmission delay and packet dropout ormissingThereforeparameter estimation with irregular data has not been exten-sively investigated in the literature

Time-delays are commonly encountered in various engi-neering systems such as chemical processes mechanicalsystems network control systems transmission line andeconomic systems [8] Since the existence of time-delayusually causes performance degradation of the inferentialmodel and is frequently a source of instability of the closed-loop system it should be handled carefully in the modelingprocess Common methods to estimate the time-delays are

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 242145 8 pageshttpdxdoiorg1011552014242145

2 Mathematical Problems in Engineering

nonparametric methods (eg step test or correlation anal-ysis) and grid searching method For example Wang andZhang [9] considered the robust identification problems oflinear continuous time-delay systems from step responses Alinear regression equation was derived from the solution ofthe output time response and its various-order integrals andsolved by using IV-least squares (LS)methodThe parametersof the transfer function were then recovered from the LSsolution Weyer [10] considered to build a model for theopen water channel The model parameters were estimatedusing the grid searching method in which one model wasestablished for each time-delay in a range The final modelwas selected as the model which gave the best prediction per-formance for a validation data set The time-delay estimationmethods mentioned above are to estimate the time-delay andmodel parameters in a separate way

Missing data problem is very common in process indus-try A special example is the irregularly sampling systemMany critical parameters such as the product concentrationsteam quality and 90 boiling point cannot be measureddirectly by using the sensorsThese parameters are measuredthrough lab analysis so only the slow rate data are availableHowever the process variables such as the temperaturepressure and flow rate can be measured on-line in fast rateby using the sensorsTherefore we can treat the data samplesbetween the slow rate data as missing data Another exampleis the network control system in which data transmit via thewireless network or internet Data missing occurs due to datapacket dropout or missing Other reasons for missing dataare sensor fault data recording system malfunction and soforth Several methods have been reported in the literatureto handle missing data problem in system identificationFor example F Ding and J Ding [11] proposed an auxiliarymodel-based approach to cope with the problems of parame-ter estimation and output estimation with irregularly missingoutput data using the PEM method The outputs of theauxiliary model were used in the identification process Zhuet al [12] considered the identification of systems with slowlyand irregularly sampled output data The output errormethod was employed to estimate the fast rate model basedon the fast input and slow output data However themethodsmentioned above just used part of the process data whichmay lead to information missing Moreover the statisticalproperties of the model parameters and the process noisecannot be given in these methods

The work introduced in this paper aims at handlingthe identification problem of the LTI systems with missingoutput data in the presence of time-delay The identificationproblem is formulated under the scheme of the generalizedexpectation-maximization (GEM) algorithm and the time-delay and missing output data are handled simultaneouslyThe GEM algorithm consists of expectation step (E-step) andmaximization step (M-step) In theM-step themaximizationproblem is transformed into an equivalent minimizationproblem and this problem is solved by using a generalnumerical optimization algorithm

The rest of this paper is organized as follows The prob-lem statement is presented in Section 2 A brief revisit ofthe GEM algorithm and the mathematical formulation of

the identification of LTI time-delay systems with incompletedata set are given in Section 3 Numerical examples are pre-sented in Section 4 to show the effectiveness of the proposedmethod The conclusions are given in Section 5

2 Problem Statement

Consider the LTI system described by the following outputerror (OE) time-delay model

119910119905= 119866 (119911

minus1) 119906119905minus120591+ 119890119905 (1)

where 120591 is the time-delay which is assumed to be integermultiples of the sampling period 119890

119905is the Gaussian white

noise with zero mean and variance 1205902 and 119910119905and 119906

119905are the

output and input respectively The transfer function 119866(119911minus1)has the following form

119866(119911minus1) =

sum119899119887

119894=1119887119894119911minus119894

1 + sum119899119886

119894=1119886119894119911minus119894 (2)

Here we assume that the model orders 119899119886and 119899

119887are known

a priori and the time-delay 120591 is uniformly distributed in aknown range of [119889

1 1198892]

The identification data 119910119905 119906119905119905=12119873

are collected Wedenote 119910

119905119905=12119873

as119884 and 119906119905119905=12119873

as119880 Since part of theoutput data are missing completely at random (MCAR) theoutput data set 119884 can be divided into 119884obs = 119910119905119905=1199051 119905120572 and119884mis = 119910119905119905=1198981 119898120573 Therefore the identification problem isto estimate the parameters 120579 = 119886

1 119886

119899119886 1198871 119887

119899119887 the

noise variance 1205902 and the time-delay 120591 based on the identifi-cation data 119884obs and 119880

3 Parameter Estimation Usingthe GEM Algorithm

31 GEMAlgorithmRevisit TheGEMalgorithm is a general-purpose iterative optimization algorithm to derive the max-imum likelihood (ML) estimate and it has attracted greatattentions of the researcher due to its flexibility in handlingthemissing data or hidden state [13] Denote themissing dataset by 119862mis and the observed data set by 119862obs The main ideaof the GEM algorithm is that instead of optimizing the likeli-hood of the observed 119884obs the conditional expectation of thecomplete data likelihood function with respect to themissingdata set is calculated in the E-step and the maximizationproblem is solved in the M-step The procedures of the GEMalgorithm to calculate the ML estimate can be described asfollows [13]

E-step given the 119862obs and the parameter estimate Θ119904 inprevious iteration the 119876-function can be calculated by

119876 (Θ | Θ119904) = 119864119862mis|119862obs Θ

119904 log 119901 (119862mis 119862obs | Θ) (3)

M-step find theΘ119904+1 to increase119876(Θ | Θ119904) over its valueat Θ119904 that is

119876(Θ119904+1| Θ119904) ⩾ 119876 (Θ

119904| Θ119904) (4)

Mathematical Problems in Engineering 3

The E-step and M-step alternate until the relative changeof the parameter estimate between neighboring iterations issmaller than a prespecified arbitrary small constant or themaximal iteration number is achieved

32 LTI Time-Delay System Identification withMissing OutputUsing GEM Algorithm Here we treat the time-delay 120591

as a hidden state variable The observed data set 119862obs isconstructed as 119862obs = 119884obs 119880 and the missing data set 119862misis constructed as 119862mis = 119884mis 120591 The parameter vector Θ isconstructed as Θ = 120579 1205902

Based on the Bayesian property the likelihood functionof the complete data set can be decomposed into

119901 (119862mis 119862obs | Θ) = 119901 (119884119880 120591 | Θ)

= 119901 (119884 | 119880 120591 Θ) 119901 (120591 | 119880Θ) 119901 (119880 | Θ)

(5)

The term 119901(119884 | 119880 120591 Θ) can be further decomposed into

119901 (119884 | 119880 120591 Θ)

= 119901 (119910119873 119910

1| 119906119873 119906

1 120591 Θ)

= 119901 (119910119873| 119910119873minus1 119910

1 119906119873 119906

1 120591 Θ)

times 119901 (119910119873minus1 119910

1| 119906119873 119906

1 120591 Θ)

=

119873

prod

119905=1

119901 (119910119905| 119910119905minus1 119910

1 119906119873 119906

1 120591 Θ)

(6)

Based on (1) and (2) 119910119905depends only on the previous input

sequence 119906119905minus1 1

= 119906119905minus1 119906

1 the time-delay 120591 and the

parameter vector Θ Therefore (6) can be rewritten as

119901 (119884 | 119880 120591 Θ) =

119873

prod

119905=1

119901 (119910119905| 119906119905minus1 1

120591 Θ) (7)

Since the time-delay 120591 is uniformly distributed in therange [119889

1 1198892] the probability of 120591 taking any value in this

range is a constant Since the input119880 ismeasurable data and itis independent of the parameter vector Θ the term 119901(119880 | Θ)

is a constant Therefore the last two terms of (5) will notplay a role in the following derivations The complete datalikelihood function can be further written as

119901 (119884119880 120591 | Θ) =

119873

prod

119905=1

119901 (119910119905| 119906119905minus1 1

120591 Θ)1198621 (8)

where 1198621= 119901(120591 | 119880Θ)119901(119880 | Θ)

Therefore the conditional expectation of the log completedata density 119876(Θ | Θ119904) in (3) can be written as

119876 (Θ | Θ119904)

= 119864119884mis 120591|119862obs Θ

119904 log 119901 (119884119880 120591 | Θ)

= 119864119884mis 120591|119862obs Θ

119904

119873

sum

119905=1

log119901 (119910119905| 119906119905minus1 1

120591 Θ) + log1198621

= 119864119884mis 120591|119862obs Θ

119904

119898120573

sum

119905=1198981

log119901 (119910119905| 119906119905minus1 1

120591 Θ)

+

119905120572

sum

119905=1199051

log119901 (119910119905| 119906119905minus1 1

120591 Θ) + log1198621

(9)

The expectation is firstly takenwith respect to the discretevariable 120591 then we have119876 (Θ | Θ

119904)

= 119864119884mis|119862obs 120591Θ

119904

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119898120573

sum

119905=1198981

log 119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ)

+

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119905120572

sum

119905=1199051

log119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ)

+ log 1198621

(10)

The expectation is then taken with respect to the contin-uous variable 119884miss so we have

119876 (Θ | Θ119904) =

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119898120573

sum

119905=1198981

int119901 (119910119905| 119862obs 120591 = 119889 Θ

119904)

times log 119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ) 119889119910119905

+

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119905120572

sum

119905=1199051

log 119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ) + log 1198621

(11)

4 Mathematical Problems in Engineering

In order to calculate 119876(Θ | Θ119904) the unknown terms

should be calculated firstly Consider

119901 (120591 = 119889 | 119862obs Θ119904)

= 119901 (120591 = 119889 | 119884obs 119880 Θ119904)

=119901 (119884obs | 119880 120591 = 119889Θ

119904) 119901 (120591 = 119889 | 119880Θ

119904)

sum1198892

119889=1198891119901 (119884obs | 119880 120591 = 119889 Θ

119904) 119901 (120591 = 119889 | 119880Θ119904)

=

prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904) 119901 (120591 = 119889)

sum1198892

119889=1198891prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904) 119901 (120591 = 119889)

=

prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904)

sum1198892

119889=1198891prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904)

int 119901 (119910119905| 119862obs 120591 = 119889 Θ

119904) log119901 (119910

119905| 119906119905minus1 1

120591 = 119889 Θ) 119889119910119905

= int119901 (119910119905| 119862obs 120591 = 119889 Θ

119904) log 1

radic21205871205902

times exp

minus

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

21205902

119889119910119905

= minus1

2log (21205871205902) minus 1

21205902int119901 (119910

119905| 119862obs 120591 = 119889 Θ

119904)

times (119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

119889119910119905

= minus1

2log (21205871205902) minus 1

21205902((120590119904)2

+ (119866119904(119911minus1) 119906119905minus119889)2

)

+1

1205902(119866 (119911minus1) 119906119905minus119889) (119866119904(119911minus1) 119906119905minus119889)

minus1

21205902(119866 (119911minus1) 119906119905minus119889)2

= minus1

2log (21205871205902)

minus1

21205902((120590119904)2

+ (119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

)

119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ)

=1

radic21205871205902exp

minus

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

21205902

(12)

Therefore the 119876-function can be rewritten as119876 (Θ | Θ

119904)

=

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119898120573

sum

119905=1198981

minus1

2log (21205871205902) minus 1

21205902

times ((120590119904)2

+ (119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

)

+

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119905120572

sum

119905=1199051

minus1

2log (21205871205902) minus 1

21205902(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

+ log1198621

(13)

In the M-step of the GEM algorithm the unknownparameters should be estimated to increase the119876-function bysolving an optimization problem Taking the gradient of the119876-function (13) with respect to the 1205902 and setting it to zeroswe have

2=1

119873

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119898120573

sum

119905=1198981

((119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

+ (120590119904)2

)

+

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119905120572

sum

119905=1199051

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

(14)

Substituting 2 into the 119876-function (13) we get

119876 (Θ | Θ119904)

= minus119873

2log (2120587) minus 119873

2

minus119873

2log

1

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

((119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

+(120590119904)2

)

+

119905120572

sum

119905=1199051

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

)]

]

(15)

Mathematical Problems in Engineering 5

Based on the monotonicity of the log function the problemis transformed into optimizing the following cost function

119869 (120579 | 120579119904)

=1

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

((119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

+ (120590119904)2

)

+

119905120572

sum

119905=1199051

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

)]

]

(16)

Here we introduce the variable119909120591119905denoting the noise-free

output with time-delay 120591 Based on (1) and (2) we have

119909120591

119905=

sum119899119887

119894=1119887119894119911minus119894

1 + sum119899119886

119894=1119886119894119911minus119894119906119905minus120591= (120601120591

119905minus120591)119879

120579 (17)

where 120601120591119905minus120591

= [minus119909120591

119905minus1 minus119909

120591

119905minus119899119886 119906119905minus120591minus1

119906119905minus120591minus119899119887

]119879 There-

fore the cost function can be rewritten as

119869 (120579 | 120579119904)

=1

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

(((120601119889

119905minus119889)119879

120579 minus (119909119889

119905)119904

)

2

+ (120590119904)2

)

+

119905120572

sum

119905=1199051

(119910119905minus (120601119889

119905minus119889)119879

120579)

2

)]

]

(18)

where (119909119889119905)119904

= (120601119889

119905minus119889)119879

120579119904 However the cost function

(18) cannot be optimized directly due to the unmeasurable119909119889

119905119905=1119873 119889=1198891 1198892

Here we adopt the auxiliary model prin-ciple and the auxiliarymodel can be constructed based on theestimates obtained in the previous iteration That is

119909120591

119905=

sum119899119887

119894=1119894119911minus119894

1 + sum119899119886

119894=1119886119894119911minus119894119906119905minus120591= (120601120591

119905minus120591)119879

120579 (19)

where 120601120591119905minus120591

= [minus119909120591

119905minus1 minus119909

120591

119905minus119899119886 119906119905minus120591minus1

119906119905minus120591minus119899119887

]119879 There-

fore the cost function (18) with 120601120591119905minus120591

substituted by 120601120591119905minus120591

canbe optimized by using the damped Newton algorithm

120579119904+1= 120579119904minus [119877119904]minus1

1198691015840(120579119904| 120579119904) (20)

where

1198691015840(120579119904| 120579119904)

=2

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

120601119889

119905minus119889((120601119889

119905minus119889)119879

120579119904minus (119909119889

119905)119904

)

+

119905120572

sum

119905=1199051

120601119889

119905minus119889((120601119889

119905minus119889)119879

120579119904minus 119910119905))]

119877119904=2

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

119873

sum

119905=1

120601119889

119905minus119889(120601119889

119905minus119889)119879]

]

+ 120582119868

(21)

where 120582 is a constantThe time-delay 120591 can be selected as the delay in the range

with maximal posterior probability That is

120591 = argmax119889

119901 (120591 = 119889 | 119862obs Θ119904) (22)

The E-step andM-step alternate until the convergence condi-tion of the GEM algorithm is met

4 Simulation Examples

41 A Numerical Simulation Example Consider the follow-ing LTI time-delay system described by the OE time-delaymodel

119910119905=

03119911minus1

1 minus 07119911minus1119906119905minus3+ 119890119905 (23)

The input data and output data are generated by simulationand the noise 119890

119905with zero mean and variance 001 is added

to the output The input and output data are shown inFigure 1 In the simulation 125 output data are randomlymissing The parameter range of the time-delay is set to[1 5] The method proposed in this paper is used to estimatethe parameters and the time-delay The parameter estimatetrajectories of the model parameters and the noise varianceare shown in Figures 2 and 3 respectively The estimatedtime-delay is 3 which is consistent with the true time-delay To further verify the effectiveness of the proposedmethod the simulations are also performed with 25 outputdata missing and 50 output data missing The estimatedparameters after 13 iterations are summarized in Table 1 Itcan be seen from these figures and the table that the proposedGEM algorithm has a good identification performance

42 The Continuous Stirred Tank Reactor The ContinuousStirred Tank Reactor (CSTR) is a benchmark example usedto test the performances of different modeling and control

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300

0

1

2

Time

y1

minus1

minus2

(a)

0 50 100 150 200 250 300

0

05

1

Time

u1

minus05

minus1

(b)

Figure 1 The input and output data

Table 1 Estimated parameters after 13 iterations

True value 119886 = minus07 119887 = 03 120591 = 3 1205902= 001

Proportion of missing output 119886 119887 120591 1205902

Full data set minus0695 03026 3 00114125 minus0694 03042 3 0011625 minus07012 02994 3 0012150 minus06979 03017 3 00124

0 2 4 6 8 10 12 14

0

02

04

06

a

b

minus02

minus04

minus06

minus08

s

Figure 2 The estimated model parameters in each iteration

algorithms and the first principle model of the CSTR isdescribed as [14]

119889119862119860

119889119905=119902

119881(1198621198600minus 119862119860) minus 1198960119862119860exp(minus119864

119877119879)

119889119879

119889119905=119902

119881(1198790minus 119879) minus

(Δ119867) 1198960119862119860

120588119862119901

exp(minus119864119877119879)

+

120588119888119862119901119888

1205881198621199011198811199021198881 minus exp( minusℎ119860

119902119888120588119862119901

) (1198791198880minus 119879)

(24)

where the product concentration 119862119860and the temperature 119879

are output variables and the coolant flow rate 119902119888is the input

0 2 4 6 8 10 12 140

002

004

006

008

01

012

s

Figure 3 The estimated noise variance in each iteration

variable The steady state values of the process variables canbe found in Gopaluni [14] In this simulation the CSTR isoperated at a steady state working point which is at 119902

119888=

97 Lmin 119862119860= 00795molL and 119879 = 4434566K The

task here is to build a first-order model between 119862119860and 119902

119888

The input and output data are generated through simulationand the noise with zero mean and variance 1 times 10

minus7 isadded to the output data Since time is needed tomeasure theconcentration119862

119860 so themeasurement delaywith 15minutes

is also added to the output data The input and output data isshown in Figure 4 In this simulation 25 output data arerandomly missing and the parameter range of the time-delayis set to [09 21] The proposed GEM algorithm is used toestimate the unknown parametersThe estimated parametersare 119886 = minus0468 = 00015 2 = 15 times 10

minus7 and 120591 = 15The self-validation and the cross-validation results are shownin Figures 5 and 6 It can be seen from these results that theproposed method has a good identification performance andthe estimated model can capture the dynamic behavior of theCSTR

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 4000075

008

0085

Time

y1

(a)

0 50 100 150 200 250 300 350 40095

96

97

98

99

Time

u1

(b)

Figure 4 The input and output data

0 50 100 150 200 250 300 350 4000075

0076

0077

0078

0079

008

0081

0082

0083

Figure 5The self-validation resultsThe blue line is the real processdata and the red line is the simulated output of the estimated model

0 50 100 150 200 250 300 350 4000075

0076

0077

0078

0079

008

0081

0082

0083

Figure 6 The cross-validation results The blue line is the realprocess data and the red line is the simulated output of the estimatedmodel

5 Conclusion

This paper considers the identification problem of LTI sys-tems with irregular data set The time-delay and the missing

data are commonly encountered problems in process indus-try and the existence of these problems makes the processmodeling a challenging taskThe identification problem withincomplete data set in the presence of time-delay is formu-lated under the scheme of the GEM algorithm and the modelparameters and the time-delay are estimated simultaneouslyin this algorithm Numerical examples are presented todemonstrate the efficacy of the proposed method

Conflict of Interests

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

References

[1] S Yin H Luo and S X Ding ldquoReal-Time implementation offault-tolerant control systems with performance optimizationrdquoIEEE Transactions on Industrial Electronics vol 61 no 5 pp2402ndash2411 2014

[2] S Yin S X Ding A Haghani and P Zhang ldquoA comparisonstudy of basic data-driven fault diagnosis and process monitor-ing methods on the benchmark Tennessee Eastman processrdquoJournal of Process Control vol 22 no 9 pp 1567ndash1581 2012

[3] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[4] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[5] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[6] H Dong Z Wang and H Gao ldquoDistributed 119867infin

filteringfor a class of Markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013

[7] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[8] J P Richard ldquoTime-delay systems an overview of some recentadvances and open problemsrdquo Automatica vol 39 no 10 pp1667ndash1694 2003

8 Mathematical Problems in Engineering

[9] Q G Wang and Y Zhang ldquoRobust identification of continuoussystems with dead-time from step responsesrdquo Automatica vol37 no 3 pp 377ndash390 2001

[10] E Weyer ldquoSystem identification of an open water channelrdquoControl Engineering Practice vol 9 no 12 pp 1289ndash1299 2001

[11] F Ding and J Ding ldquoLeast-squares parameter estimation forsystems with irregularly missing datardquo International Journal ofAdaptive Control and Signal Processing vol 24 no 7 pp 540ndash553 2010

[12] Y C Zhu H Telkamp J H Wang and Q L Fu ldquoSystemidentification using slow and irregular output samplesrdquo Journalof Process Control vol 19 no 1 pp 58ndash67 2009

[13] G J McLachlan and T Krishnan The EM Algorithm andExtensions John Wiley amp Sons New York NY USA 2007

[14] R B Gopaluni ldquoA particle filter approach to identification ofnonlinear processes undermissing observationsrdquoTheCanadianJournal of Chemical Engineering vol 86 no 6 pp 1081ndash10922008

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

nonparametric methods (eg step test or correlation anal-ysis) and grid searching method For example Wang andZhang [9] considered the robust identification problems oflinear continuous time-delay systems from step responses Alinear regression equation was derived from the solution ofthe output time response and its various-order integrals andsolved by using IV-least squares (LS)methodThe parametersof the transfer function were then recovered from the LSsolution Weyer [10] considered to build a model for theopen water channel The model parameters were estimatedusing the grid searching method in which one model wasestablished for each time-delay in a range The final modelwas selected as the model which gave the best prediction per-formance for a validation data set The time-delay estimationmethods mentioned above are to estimate the time-delay andmodel parameters in a separate way

Missing data problem is very common in process indus-try A special example is the irregularly sampling systemMany critical parameters such as the product concentrationsteam quality and 90 boiling point cannot be measureddirectly by using the sensorsThese parameters are measuredthrough lab analysis so only the slow rate data are availableHowever the process variables such as the temperaturepressure and flow rate can be measured on-line in fast rateby using the sensorsTherefore we can treat the data samplesbetween the slow rate data as missing data Another exampleis the network control system in which data transmit via thewireless network or internet Data missing occurs due to datapacket dropout or missing Other reasons for missing dataare sensor fault data recording system malfunction and soforth Several methods have been reported in the literatureto handle missing data problem in system identificationFor example F Ding and J Ding [11] proposed an auxiliarymodel-based approach to cope with the problems of parame-ter estimation and output estimation with irregularly missingoutput data using the PEM method The outputs of theauxiliary model were used in the identification process Zhuet al [12] considered the identification of systems with slowlyand irregularly sampled output data The output errormethod was employed to estimate the fast rate model basedon the fast input and slow output data However themethodsmentioned above just used part of the process data whichmay lead to information missing Moreover the statisticalproperties of the model parameters and the process noisecannot be given in these methods

The work introduced in this paper aims at handlingthe identification problem of the LTI systems with missingoutput data in the presence of time-delay The identificationproblem is formulated under the scheme of the generalizedexpectation-maximization (GEM) algorithm and the time-delay and missing output data are handled simultaneouslyThe GEM algorithm consists of expectation step (E-step) andmaximization step (M-step) In theM-step themaximizationproblem is transformed into an equivalent minimizationproblem and this problem is solved by using a generalnumerical optimization algorithm

The rest of this paper is organized as follows The prob-lem statement is presented in Section 2 A brief revisit ofthe GEM algorithm and the mathematical formulation of

the identification of LTI time-delay systems with incompletedata set are given in Section 3 Numerical examples are pre-sented in Section 4 to show the effectiveness of the proposedmethod The conclusions are given in Section 5

2 Problem Statement

Consider the LTI system described by the following outputerror (OE) time-delay model

119910119905= 119866 (119911

minus1) 119906119905minus120591+ 119890119905 (1)

where 120591 is the time-delay which is assumed to be integermultiples of the sampling period 119890

119905is the Gaussian white

noise with zero mean and variance 1205902 and 119910119905and 119906

119905are the

output and input respectively The transfer function 119866(119911minus1)has the following form

119866(119911minus1) =

sum119899119887

119894=1119887119894119911minus119894

1 + sum119899119886

119894=1119886119894119911minus119894 (2)

Here we assume that the model orders 119899119886and 119899

119887are known

a priori and the time-delay 120591 is uniformly distributed in aknown range of [119889

1 1198892]

The identification data 119910119905 119906119905119905=12119873

are collected Wedenote 119910

119905119905=12119873

as119884 and 119906119905119905=12119873

as119880 Since part of theoutput data are missing completely at random (MCAR) theoutput data set 119884 can be divided into 119884obs = 119910119905119905=1199051 119905120572 and119884mis = 119910119905119905=1198981 119898120573 Therefore the identification problem isto estimate the parameters 120579 = 119886

1 119886

119899119886 1198871 119887

119899119887 the

noise variance 1205902 and the time-delay 120591 based on the identifi-cation data 119884obs and 119880

3 Parameter Estimation Usingthe GEM Algorithm

31 GEMAlgorithmRevisit TheGEMalgorithm is a general-purpose iterative optimization algorithm to derive the max-imum likelihood (ML) estimate and it has attracted greatattentions of the researcher due to its flexibility in handlingthemissing data or hidden state [13] Denote themissing dataset by 119862mis and the observed data set by 119862obs The main ideaof the GEM algorithm is that instead of optimizing the likeli-hood of the observed 119884obs the conditional expectation of thecomplete data likelihood function with respect to themissingdata set is calculated in the E-step and the maximizationproblem is solved in the M-step The procedures of the GEMalgorithm to calculate the ML estimate can be described asfollows [13]

E-step given the 119862obs and the parameter estimate Θ119904 inprevious iteration the 119876-function can be calculated by

119876 (Θ | Θ119904) = 119864119862mis|119862obs Θ

119904 log 119901 (119862mis 119862obs | Θ) (3)

M-step find theΘ119904+1 to increase119876(Θ | Θ119904) over its valueat Θ119904 that is

119876(Θ119904+1| Θ119904) ⩾ 119876 (Θ

119904| Θ119904) (4)

Mathematical Problems in Engineering 3

The E-step and M-step alternate until the relative changeof the parameter estimate between neighboring iterations issmaller than a prespecified arbitrary small constant or themaximal iteration number is achieved

32 LTI Time-Delay System Identification withMissing OutputUsing GEM Algorithm Here we treat the time-delay 120591

as a hidden state variable The observed data set 119862obs isconstructed as 119862obs = 119884obs 119880 and the missing data set 119862misis constructed as 119862mis = 119884mis 120591 The parameter vector Θ isconstructed as Θ = 120579 1205902

Based on the Bayesian property the likelihood functionof the complete data set can be decomposed into

119901 (119862mis 119862obs | Θ) = 119901 (119884119880 120591 | Θ)

= 119901 (119884 | 119880 120591 Θ) 119901 (120591 | 119880Θ) 119901 (119880 | Θ)

(5)

The term 119901(119884 | 119880 120591 Θ) can be further decomposed into

119901 (119884 | 119880 120591 Θ)

= 119901 (119910119873 119910

1| 119906119873 119906

1 120591 Θ)

= 119901 (119910119873| 119910119873minus1 119910

1 119906119873 119906

1 120591 Θ)

times 119901 (119910119873minus1 119910

1| 119906119873 119906

1 120591 Θ)

=

119873

prod

119905=1

119901 (119910119905| 119910119905minus1 119910

1 119906119873 119906

1 120591 Θ)

(6)

Based on (1) and (2) 119910119905depends only on the previous input

sequence 119906119905minus1 1

= 119906119905minus1 119906

1 the time-delay 120591 and the

parameter vector Θ Therefore (6) can be rewritten as

119901 (119884 | 119880 120591 Θ) =

119873

prod

119905=1

119901 (119910119905| 119906119905minus1 1

120591 Θ) (7)

Since the time-delay 120591 is uniformly distributed in therange [119889

1 1198892] the probability of 120591 taking any value in this

range is a constant Since the input119880 ismeasurable data and itis independent of the parameter vector Θ the term 119901(119880 | Θ)

is a constant Therefore the last two terms of (5) will notplay a role in the following derivations The complete datalikelihood function can be further written as

119901 (119884119880 120591 | Θ) =

119873

prod

119905=1

119901 (119910119905| 119906119905minus1 1

120591 Θ)1198621 (8)

where 1198621= 119901(120591 | 119880Θ)119901(119880 | Θ)

Therefore the conditional expectation of the log completedata density 119876(Θ | Θ119904) in (3) can be written as

119876 (Θ | Θ119904)

= 119864119884mis 120591|119862obs Θ

119904 log 119901 (119884119880 120591 | Θ)

= 119864119884mis 120591|119862obs Θ

119904

119873

sum

119905=1

log119901 (119910119905| 119906119905minus1 1

120591 Θ) + log1198621

= 119864119884mis 120591|119862obs Θ

119904

119898120573

sum

119905=1198981

log119901 (119910119905| 119906119905minus1 1

120591 Θ)

+

119905120572

sum

119905=1199051

log119901 (119910119905| 119906119905minus1 1

120591 Θ) + log1198621

(9)

The expectation is firstly takenwith respect to the discretevariable 120591 then we have119876 (Θ | Θ

119904)

= 119864119884mis|119862obs 120591Θ

119904

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119898120573

sum

119905=1198981

log 119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ)

+

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119905120572

sum

119905=1199051

log119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ)

+ log 1198621

(10)

The expectation is then taken with respect to the contin-uous variable 119884miss so we have

119876 (Θ | Θ119904) =

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119898120573

sum

119905=1198981

int119901 (119910119905| 119862obs 120591 = 119889 Θ

119904)

times log 119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ) 119889119910119905

+

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119905120572

sum

119905=1199051

log 119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ) + log 1198621

(11)

4 Mathematical Problems in Engineering

In order to calculate 119876(Θ | Θ119904) the unknown terms

should be calculated firstly Consider

119901 (120591 = 119889 | 119862obs Θ119904)

= 119901 (120591 = 119889 | 119884obs 119880 Θ119904)

=119901 (119884obs | 119880 120591 = 119889Θ

119904) 119901 (120591 = 119889 | 119880Θ

119904)

sum1198892

119889=1198891119901 (119884obs | 119880 120591 = 119889 Θ

119904) 119901 (120591 = 119889 | 119880Θ119904)

=

prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904) 119901 (120591 = 119889)

sum1198892

119889=1198891prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904) 119901 (120591 = 119889)

=

prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904)

sum1198892

119889=1198891prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904)

int 119901 (119910119905| 119862obs 120591 = 119889 Θ

119904) log119901 (119910

119905| 119906119905minus1 1

120591 = 119889 Θ) 119889119910119905

= int119901 (119910119905| 119862obs 120591 = 119889 Θ

119904) log 1

radic21205871205902

times exp

minus

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

21205902

119889119910119905

= minus1

2log (21205871205902) minus 1

21205902int119901 (119910

119905| 119862obs 120591 = 119889 Θ

119904)

times (119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

119889119910119905

= minus1

2log (21205871205902) minus 1

21205902((120590119904)2

+ (119866119904(119911minus1) 119906119905minus119889)2

)

+1

1205902(119866 (119911minus1) 119906119905minus119889) (119866119904(119911minus1) 119906119905minus119889)

minus1

21205902(119866 (119911minus1) 119906119905minus119889)2

= minus1

2log (21205871205902)

minus1

21205902((120590119904)2

+ (119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

)

119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ)

=1

radic21205871205902exp

minus

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

21205902

(12)

Therefore the 119876-function can be rewritten as119876 (Θ | Θ

119904)

=

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119898120573

sum

119905=1198981

minus1

2log (21205871205902) minus 1

21205902

times ((120590119904)2

+ (119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

)

+

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119905120572

sum

119905=1199051

minus1

2log (21205871205902) minus 1

21205902(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

+ log1198621

(13)

In the M-step of the GEM algorithm the unknownparameters should be estimated to increase the119876-function bysolving an optimization problem Taking the gradient of the119876-function (13) with respect to the 1205902 and setting it to zeroswe have

2=1

119873

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119898120573

sum

119905=1198981

((119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

+ (120590119904)2

)

+

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119905120572

sum

119905=1199051

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

(14)

Substituting 2 into the 119876-function (13) we get

119876 (Θ | Θ119904)

= minus119873

2log (2120587) minus 119873

2

minus119873

2log

1

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

((119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

+(120590119904)2

)

+

119905120572

sum

119905=1199051

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

)]

]

(15)

Mathematical Problems in Engineering 5

Based on the monotonicity of the log function the problemis transformed into optimizing the following cost function

119869 (120579 | 120579119904)

=1

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

((119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

+ (120590119904)2

)

+

119905120572

sum

119905=1199051

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

)]

]

(16)

Here we introduce the variable119909120591119905denoting the noise-free

output with time-delay 120591 Based on (1) and (2) we have

119909120591

119905=

sum119899119887

119894=1119887119894119911minus119894

1 + sum119899119886

119894=1119886119894119911minus119894119906119905minus120591= (120601120591

119905minus120591)119879

120579 (17)

where 120601120591119905minus120591

= [minus119909120591

119905minus1 minus119909

120591

119905minus119899119886 119906119905minus120591minus1

119906119905minus120591minus119899119887

]119879 There-

fore the cost function can be rewritten as

119869 (120579 | 120579119904)

=1

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

(((120601119889

119905minus119889)119879

120579 minus (119909119889

119905)119904

)

2

+ (120590119904)2

)

+

119905120572

sum

119905=1199051

(119910119905minus (120601119889

119905minus119889)119879

120579)

2

)]

]

(18)

where (119909119889119905)119904

= (120601119889

119905minus119889)119879

120579119904 However the cost function

(18) cannot be optimized directly due to the unmeasurable119909119889

119905119905=1119873 119889=1198891 1198892

Here we adopt the auxiliary model prin-ciple and the auxiliarymodel can be constructed based on theestimates obtained in the previous iteration That is

119909120591

119905=

sum119899119887

119894=1119894119911minus119894

1 + sum119899119886

119894=1119886119894119911minus119894119906119905minus120591= (120601120591

119905minus120591)119879

120579 (19)

where 120601120591119905minus120591

= [minus119909120591

119905minus1 minus119909

120591

119905minus119899119886 119906119905minus120591minus1

119906119905minus120591minus119899119887

]119879 There-

fore the cost function (18) with 120601120591119905minus120591

substituted by 120601120591119905minus120591

canbe optimized by using the damped Newton algorithm

120579119904+1= 120579119904minus [119877119904]minus1

1198691015840(120579119904| 120579119904) (20)

where

1198691015840(120579119904| 120579119904)

=2

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

120601119889

119905minus119889((120601119889

119905minus119889)119879

120579119904minus (119909119889

119905)119904

)

+

119905120572

sum

119905=1199051

120601119889

119905minus119889((120601119889

119905minus119889)119879

120579119904minus 119910119905))]

119877119904=2

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

119873

sum

119905=1

120601119889

119905minus119889(120601119889

119905minus119889)119879]

]

+ 120582119868

(21)

where 120582 is a constantThe time-delay 120591 can be selected as the delay in the range

with maximal posterior probability That is

120591 = argmax119889

119901 (120591 = 119889 | 119862obs Θ119904) (22)

The E-step andM-step alternate until the convergence condi-tion of the GEM algorithm is met

4 Simulation Examples

41 A Numerical Simulation Example Consider the follow-ing LTI time-delay system described by the OE time-delaymodel

119910119905=

03119911minus1

1 minus 07119911minus1119906119905minus3+ 119890119905 (23)

The input data and output data are generated by simulationand the noise 119890

119905with zero mean and variance 001 is added

to the output The input and output data are shown inFigure 1 In the simulation 125 output data are randomlymissing The parameter range of the time-delay is set to[1 5] The method proposed in this paper is used to estimatethe parameters and the time-delay The parameter estimatetrajectories of the model parameters and the noise varianceare shown in Figures 2 and 3 respectively The estimatedtime-delay is 3 which is consistent with the true time-delay To further verify the effectiveness of the proposedmethod the simulations are also performed with 25 outputdata missing and 50 output data missing The estimatedparameters after 13 iterations are summarized in Table 1 Itcan be seen from these figures and the table that the proposedGEM algorithm has a good identification performance

42 The Continuous Stirred Tank Reactor The ContinuousStirred Tank Reactor (CSTR) is a benchmark example usedto test the performances of different modeling and control

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300

0

1

2

Time

y1

minus1

minus2

(a)

0 50 100 150 200 250 300

0

05

1

Time

u1

minus05

minus1

(b)

Figure 1 The input and output data

Table 1 Estimated parameters after 13 iterations

True value 119886 = minus07 119887 = 03 120591 = 3 1205902= 001

Proportion of missing output 119886 119887 120591 1205902

Full data set minus0695 03026 3 00114125 minus0694 03042 3 0011625 minus07012 02994 3 0012150 minus06979 03017 3 00124

0 2 4 6 8 10 12 14

0

02

04

06

a

b

minus02

minus04

minus06

minus08

s

Figure 2 The estimated model parameters in each iteration

algorithms and the first principle model of the CSTR isdescribed as [14]

119889119862119860

119889119905=119902

119881(1198621198600minus 119862119860) minus 1198960119862119860exp(minus119864

119877119879)

119889119879

119889119905=119902

119881(1198790minus 119879) minus

(Δ119867) 1198960119862119860

120588119862119901

exp(minus119864119877119879)

+

120588119888119862119901119888

1205881198621199011198811199021198881 minus exp( minusℎ119860

119902119888120588119862119901

) (1198791198880minus 119879)

(24)

where the product concentration 119862119860and the temperature 119879

are output variables and the coolant flow rate 119902119888is the input

0 2 4 6 8 10 12 140

002

004

006

008

01

012

s

Figure 3 The estimated noise variance in each iteration

variable The steady state values of the process variables canbe found in Gopaluni [14] In this simulation the CSTR isoperated at a steady state working point which is at 119902

119888=

97 Lmin 119862119860= 00795molL and 119879 = 4434566K The

task here is to build a first-order model between 119862119860and 119902

119888

The input and output data are generated through simulationand the noise with zero mean and variance 1 times 10

minus7 isadded to the output data Since time is needed tomeasure theconcentration119862

119860 so themeasurement delaywith 15minutes

is also added to the output data The input and output data isshown in Figure 4 In this simulation 25 output data arerandomly missing and the parameter range of the time-delayis set to [09 21] The proposed GEM algorithm is used toestimate the unknown parametersThe estimated parametersare 119886 = minus0468 = 00015 2 = 15 times 10

minus7 and 120591 = 15The self-validation and the cross-validation results are shownin Figures 5 and 6 It can be seen from these results that theproposed method has a good identification performance andthe estimated model can capture the dynamic behavior of theCSTR

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 4000075

008

0085

Time

y1

(a)

0 50 100 150 200 250 300 350 40095

96

97

98

99

Time

u1

(b)

Figure 4 The input and output data

0 50 100 150 200 250 300 350 4000075

0076

0077

0078

0079

008

0081

0082

0083

Figure 5The self-validation resultsThe blue line is the real processdata and the red line is the simulated output of the estimated model

0 50 100 150 200 250 300 350 4000075

0076

0077

0078

0079

008

0081

0082

0083

Figure 6 The cross-validation results The blue line is the realprocess data and the red line is the simulated output of the estimatedmodel

5 Conclusion

This paper considers the identification problem of LTI sys-tems with irregular data set The time-delay and the missing

data are commonly encountered problems in process indus-try and the existence of these problems makes the processmodeling a challenging taskThe identification problem withincomplete data set in the presence of time-delay is formu-lated under the scheme of the GEM algorithm and the modelparameters and the time-delay are estimated simultaneouslyin this algorithm Numerical examples are presented todemonstrate the efficacy of the proposed method

Conflict of Interests

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

References

[1] S Yin H Luo and S X Ding ldquoReal-Time implementation offault-tolerant control systems with performance optimizationrdquoIEEE Transactions on Industrial Electronics vol 61 no 5 pp2402ndash2411 2014

[2] S Yin S X Ding A Haghani and P Zhang ldquoA comparisonstudy of basic data-driven fault diagnosis and process monitor-ing methods on the benchmark Tennessee Eastman processrdquoJournal of Process Control vol 22 no 9 pp 1567ndash1581 2012

[3] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[4] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[5] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[6] H Dong Z Wang and H Gao ldquoDistributed 119867infin

filteringfor a class of Markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013

[7] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[8] J P Richard ldquoTime-delay systems an overview of some recentadvances and open problemsrdquo Automatica vol 39 no 10 pp1667ndash1694 2003

8 Mathematical Problems in Engineering

[9] Q G Wang and Y Zhang ldquoRobust identification of continuoussystems with dead-time from step responsesrdquo Automatica vol37 no 3 pp 377ndash390 2001

[10] E Weyer ldquoSystem identification of an open water channelrdquoControl Engineering Practice vol 9 no 12 pp 1289ndash1299 2001

[11] F Ding and J Ding ldquoLeast-squares parameter estimation forsystems with irregularly missing datardquo International Journal ofAdaptive Control and Signal Processing vol 24 no 7 pp 540ndash553 2010

[12] Y C Zhu H Telkamp J H Wang and Q L Fu ldquoSystemidentification using slow and irregular output samplesrdquo Journalof Process Control vol 19 no 1 pp 58ndash67 2009

[13] G J McLachlan and T Krishnan The EM Algorithm andExtensions John Wiley amp Sons New York NY USA 2007

[14] R B Gopaluni ldquoA particle filter approach to identification ofnonlinear processes undermissing observationsrdquoTheCanadianJournal of Chemical Engineering vol 86 no 6 pp 1081ndash10922008

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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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

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

The E-step and M-step alternate until the relative changeof the parameter estimate between neighboring iterations issmaller than a prespecified arbitrary small constant or themaximal iteration number is achieved

32 LTI Time-Delay System Identification withMissing OutputUsing GEM Algorithm Here we treat the time-delay 120591

as a hidden state variable The observed data set 119862obs isconstructed as 119862obs = 119884obs 119880 and the missing data set 119862misis constructed as 119862mis = 119884mis 120591 The parameter vector Θ isconstructed as Θ = 120579 1205902

Based on the Bayesian property the likelihood functionof the complete data set can be decomposed into

119901 (119862mis 119862obs | Θ) = 119901 (119884119880 120591 | Θ)

= 119901 (119884 | 119880 120591 Θ) 119901 (120591 | 119880Θ) 119901 (119880 | Θ)

(5)

The term 119901(119884 | 119880 120591 Θ) can be further decomposed into

119901 (119884 | 119880 120591 Θ)

= 119901 (119910119873 119910

1| 119906119873 119906

1 120591 Θ)

= 119901 (119910119873| 119910119873minus1 119910

1 119906119873 119906

1 120591 Θ)

times 119901 (119910119873minus1 119910

1| 119906119873 119906

1 120591 Θ)

=

119873

prod

119905=1

119901 (119910119905| 119910119905minus1 119910

1 119906119873 119906

1 120591 Θ)

(6)

Based on (1) and (2) 119910119905depends only on the previous input

sequence 119906119905minus1 1

= 119906119905minus1 119906

1 the time-delay 120591 and the

parameter vector Θ Therefore (6) can be rewritten as

119901 (119884 | 119880 120591 Θ) =

119873

prod

119905=1

119901 (119910119905| 119906119905minus1 1

120591 Θ) (7)

Since the time-delay 120591 is uniformly distributed in therange [119889

1 1198892] the probability of 120591 taking any value in this

range is a constant Since the input119880 ismeasurable data and itis independent of the parameter vector Θ the term 119901(119880 | Θ)

is a constant Therefore the last two terms of (5) will notplay a role in the following derivations The complete datalikelihood function can be further written as

119901 (119884119880 120591 | Θ) =

119873

prod

119905=1

119901 (119910119905| 119906119905minus1 1

120591 Θ)1198621 (8)

where 1198621= 119901(120591 | 119880Θ)119901(119880 | Θ)

Therefore the conditional expectation of the log completedata density 119876(Θ | Θ119904) in (3) can be written as

119876 (Θ | Θ119904)

= 119864119884mis 120591|119862obs Θ

119904 log 119901 (119884119880 120591 | Θ)

= 119864119884mis 120591|119862obs Θ

119904

119873

sum

119905=1

log119901 (119910119905| 119906119905minus1 1

120591 Θ) + log1198621

= 119864119884mis 120591|119862obs Θ

119904

119898120573

sum

119905=1198981

log119901 (119910119905| 119906119905minus1 1

120591 Θ)

+

119905120572

sum

119905=1199051

log119901 (119910119905| 119906119905minus1 1

120591 Θ) + log1198621

(9)

The expectation is firstly takenwith respect to the discretevariable 120591 then we have119876 (Θ | Θ

119904)

= 119864119884mis|119862obs 120591Θ

119904

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119898120573

sum

119905=1198981

log 119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ)

+

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119905120572

sum

119905=1199051

log119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ)

+ log 1198621

(10)

The expectation is then taken with respect to the contin-uous variable 119884miss so we have

119876 (Θ | Θ119904) =

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119898120573

sum

119905=1198981

int119901 (119910119905| 119862obs 120591 = 119889 Θ

119904)

times log 119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ) 119889119910119905

+

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119905120572

sum

119905=1199051

log 119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ) + log 1198621

(11)

4 Mathematical Problems in Engineering

In order to calculate 119876(Θ | Θ119904) the unknown terms

should be calculated firstly Consider

119901 (120591 = 119889 | 119862obs Θ119904)

= 119901 (120591 = 119889 | 119884obs 119880 Θ119904)

=119901 (119884obs | 119880 120591 = 119889Θ

119904) 119901 (120591 = 119889 | 119880Θ

119904)

sum1198892

119889=1198891119901 (119884obs | 119880 120591 = 119889 Θ

119904) 119901 (120591 = 119889 | 119880Θ119904)

=

prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904) 119901 (120591 = 119889)

sum1198892

119889=1198891prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904) 119901 (120591 = 119889)

=

prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904)

sum1198892

119889=1198891prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904)

int 119901 (119910119905| 119862obs 120591 = 119889 Θ

119904) log119901 (119910

119905| 119906119905minus1 1

120591 = 119889 Θ) 119889119910119905

= int119901 (119910119905| 119862obs 120591 = 119889 Θ

119904) log 1

radic21205871205902

times exp

minus

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

21205902

119889119910119905

= minus1

2log (21205871205902) minus 1

21205902int119901 (119910

119905| 119862obs 120591 = 119889 Θ

119904)

times (119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

119889119910119905

= minus1

2log (21205871205902) minus 1

21205902((120590119904)2

+ (119866119904(119911minus1) 119906119905minus119889)2

)

+1

1205902(119866 (119911minus1) 119906119905minus119889) (119866119904(119911minus1) 119906119905minus119889)

minus1

21205902(119866 (119911minus1) 119906119905minus119889)2

= minus1

2log (21205871205902)

minus1

21205902((120590119904)2

+ (119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

)

119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ)

=1

radic21205871205902exp

minus

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

21205902

(12)

Therefore the 119876-function can be rewritten as119876 (Θ | Θ

119904)

=

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119898120573

sum

119905=1198981

minus1

2log (21205871205902) minus 1

21205902

times ((120590119904)2

+ (119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

)

+

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119905120572

sum

119905=1199051

minus1

2log (21205871205902) minus 1

21205902(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

+ log1198621

(13)

In the M-step of the GEM algorithm the unknownparameters should be estimated to increase the119876-function bysolving an optimization problem Taking the gradient of the119876-function (13) with respect to the 1205902 and setting it to zeroswe have

2=1

119873

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119898120573

sum

119905=1198981

((119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

+ (120590119904)2

)

+

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119905120572

sum

119905=1199051

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

(14)

Substituting 2 into the 119876-function (13) we get

119876 (Θ | Θ119904)

= minus119873

2log (2120587) minus 119873

2

minus119873

2log

1

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

((119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

+(120590119904)2

)

+

119905120572

sum

119905=1199051

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

)]

]

(15)

Mathematical Problems in Engineering 5

Based on the monotonicity of the log function the problemis transformed into optimizing the following cost function

119869 (120579 | 120579119904)

=1

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

((119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

+ (120590119904)2

)

+

119905120572

sum

119905=1199051

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

)]

]

(16)

Here we introduce the variable119909120591119905denoting the noise-free

output with time-delay 120591 Based on (1) and (2) we have

119909120591

119905=

sum119899119887

119894=1119887119894119911minus119894

1 + sum119899119886

119894=1119886119894119911minus119894119906119905minus120591= (120601120591

119905minus120591)119879

120579 (17)

where 120601120591119905minus120591

= [minus119909120591

119905minus1 minus119909

120591

119905minus119899119886 119906119905minus120591minus1

119906119905minus120591minus119899119887

]119879 There-

fore the cost function can be rewritten as

119869 (120579 | 120579119904)

=1

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

(((120601119889

119905minus119889)119879

120579 minus (119909119889

119905)119904

)

2

+ (120590119904)2

)

+

119905120572

sum

119905=1199051

(119910119905minus (120601119889

119905minus119889)119879

120579)

2

)]

]

(18)

where (119909119889119905)119904

= (120601119889

119905minus119889)119879

120579119904 However the cost function

(18) cannot be optimized directly due to the unmeasurable119909119889

119905119905=1119873 119889=1198891 1198892

Here we adopt the auxiliary model prin-ciple and the auxiliarymodel can be constructed based on theestimates obtained in the previous iteration That is

119909120591

119905=

sum119899119887

119894=1119894119911minus119894

1 + sum119899119886

119894=1119886119894119911minus119894119906119905minus120591= (120601120591

119905minus120591)119879

120579 (19)

where 120601120591119905minus120591

= [minus119909120591

119905minus1 minus119909

120591

119905minus119899119886 119906119905minus120591minus1

119906119905minus120591minus119899119887

]119879 There-

fore the cost function (18) with 120601120591119905minus120591

substituted by 120601120591119905minus120591

canbe optimized by using the damped Newton algorithm

120579119904+1= 120579119904minus [119877119904]minus1

1198691015840(120579119904| 120579119904) (20)

where

1198691015840(120579119904| 120579119904)

=2

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

120601119889

119905minus119889((120601119889

119905minus119889)119879

120579119904minus (119909119889

119905)119904

)

+

119905120572

sum

119905=1199051

120601119889

119905minus119889((120601119889

119905minus119889)119879

120579119904minus 119910119905))]

119877119904=2

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

119873

sum

119905=1

120601119889

119905minus119889(120601119889

119905minus119889)119879]

]

+ 120582119868

(21)

where 120582 is a constantThe time-delay 120591 can be selected as the delay in the range

with maximal posterior probability That is

120591 = argmax119889

119901 (120591 = 119889 | 119862obs Θ119904) (22)

The E-step andM-step alternate until the convergence condi-tion of the GEM algorithm is met

4 Simulation Examples

41 A Numerical Simulation Example Consider the follow-ing LTI time-delay system described by the OE time-delaymodel

119910119905=

03119911minus1

1 minus 07119911minus1119906119905minus3+ 119890119905 (23)

The input data and output data are generated by simulationand the noise 119890

119905with zero mean and variance 001 is added

to the output The input and output data are shown inFigure 1 In the simulation 125 output data are randomlymissing The parameter range of the time-delay is set to[1 5] The method proposed in this paper is used to estimatethe parameters and the time-delay The parameter estimatetrajectories of the model parameters and the noise varianceare shown in Figures 2 and 3 respectively The estimatedtime-delay is 3 which is consistent with the true time-delay To further verify the effectiveness of the proposedmethod the simulations are also performed with 25 outputdata missing and 50 output data missing The estimatedparameters after 13 iterations are summarized in Table 1 Itcan be seen from these figures and the table that the proposedGEM algorithm has a good identification performance

42 The Continuous Stirred Tank Reactor The ContinuousStirred Tank Reactor (CSTR) is a benchmark example usedto test the performances of different modeling and control

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300

0

1

2

Time

y1

minus1

minus2

(a)

0 50 100 150 200 250 300

0

05

1

Time

u1

minus05

minus1

(b)

Figure 1 The input and output data

Table 1 Estimated parameters after 13 iterations

True value 119886 = minus07 119887 = 03 120591 = 3 1205902= 001

Proportion of missing output 119886 119887 120591 1205902

Full data set minus0695 03026 3 00114125 minus0694 03042 3 0011625 minus07012 02994 3 0012150 minus06979 03017 3 00124

0 2 4 6 8 10 12 14

0

02

04

06

a

b

minus02

minus04

minus06

minus08

s

Figure 2 The estimated model parameters in each iteration

algorithms and the first principle model of the CSTR isdescribed as [14]

119889119862119860

119889119905=119902

119881(1198621198600minus 119862119860) minus 1198960119862119860exp(minus119864

119877119879)

119889119879

119889119905=119902

119881(1198790minus 119879) minus

(Δ119867) 1198960119862119860

120588119862119901

exp(minus119864119877119879)

+

120588119888119862119901119888

1205881198621199011198811199021198881 minus exp( minusℎ119860

119902119888120588119862119901

) (1198791198880minus 119879)

(24)

where the product concentration 119862119860and the temperature 119879

are output variables and the coolant flow rate 119902119888is the input

0 2 4 6 8 10 12 140

002

004

006

008

01

012

s

Figure 3 The estimated noise variance in each iteration

variable The steady state values of the process variables canbe found in Gopaluni [14] In this simulation the CSTR isoperated at a steady state working point which is at 119902

119888=

97 Lmin 119862119860= 00795molL and 119879 = 4434566K The

task here is to build a first-order model between 119862119860and 119902

119888

The input and output data are generated through simulationand the noise with zero mean and variance 1 times 10

minus7 isadded to the output data Since time is needed tomeasure theconcentration119862

119860 so themeasurement delaywith 15minutes

is also added to the output data The input and output data isshown in Figure 4 In this simulation 25 output data arerandomly missing and the parameter range of the time-delayis set to [09 21] The proposed GEM algorithm is used toestimate the unknown parametersThe estimated parametersare 119886 = minus0468 = 00015 2 = 15 times 10

minus7 and 120591 = 15The self-validation and the cross-validation results are shownin Figures 5 and 6 It can be seen from these results that theproposed method has a good identification performance andthe estimated model can capture the dynamic behavior of theCSTR

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 4000075

008

0085

Time

y1

(a)

0 50 100 150 200 250 300 350 40095

96

97

98

99

Time

u1

(b)

Figure 4 The input and output data

0 50 100 150 200 250 300 350 4000075

0076

0077

0078

0079

008

0081

0082

0083

Figure 5The self-validation resultsThe blue line is the real processdata and the red line is the simulated output of the estimated model

0 50 100 150 200 250 300 350 4000075

0076

0077

0078

0079

008

0081

0082

0083

Figure 6 The cross-validation results The blue line is the realprocess data and the red line is the simulated output of the estimatedmodel

5 Conclusion

This paper considers the identification problem of LTI sys-tems with irregular data set The time-delay and the missing

data are commonly encountered problems in process indus-try and the existence of these problems makes the processmodeling a challenging taskThe identification problem withincomplete data set in the presence of time-delay is formu-lated under the scheme of the GEM algorithm and the modelparameters and the time-delay are estimated simultaneouslyin this algorithm Numerical examples are presented todemonstrate the efficacy of the proposed method

Conflict of Interests

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

References

[1] S Yin H Luo and S X Ding ldquoReal-Time implementation offault-tolerant control systems with performance optimizationrdquoIEEE Transactions on Industrial Electronics vol 61 no 5 pp2402ndash2411 2014

[2] S Yin S X Ding A Haghani and P Zhang ldquoA comparisonstudy of basic data-driven fault diagnosis and process monitor-ing methods on the benchmark Tennessee Eastman processrdquoJournal of Process Control vol 22 no 9 pp 1567ndash1581 2012

[3] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[4] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[5] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[6] H Dong Z Wang and H Gao ldquoDistributed 119867infin

filteringfor a class of Markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013

[7] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[8] J P Richard ldquoTime-delay systems an overview of some recentadvances and open problemsrdquo Automatica vol 39 no 10 pp1667ndash1694 2003

8 Mathematical Problems in Engineering

[9] Q G Wang and Y Zhang ldquoRobust identification of continuoussystems with dead-time from step responsesrdquo Automatica vol37 no 3 pp 377ndash390 2001

[10] E Weyer ldquoSystem identification of an open water channelrdquoControl Engineering Practice vol 9 no 12 pp 1289ndash1299 2001

[11] F Ding and J Ding ldquoLeast-squares parameter estimation forsystems with irregularly missing datardquo International Journal ofAdaptive Control and Signal Processing vol 24 no 7 pp 540ndash553 2010

[12] Y C Zhu H Telkamp J H Wang and Q L Fu ldquoSystemidentification using slow and irregular output samplesrdquo Journalof Process Control vol 19 no 1 pp 58ndash67 2009

[13] G J McLachlan and T Krishnan The EM Algorithm andExtensions John Wiley amp Sons New York NY USA 2007

[14] R B Gopaluni ldquoA particle filter approach to identification ofnonlinear processes undermissing observationsrdquoTheCanadianJournal of Chemical Engineering vol 86 no 6 pp 1081ndash10922008

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

In order to calculate 119876(Θ | Θ119904) the unknown terms

should be calculated firstly Consider

119901 (120591 = 119889 | 119862obs Θ119904)

= 119901 (120591 = 119889 | 119884obs 119880 Θ119904)

=119901 (119884obs | 119880 120591 = 119889Θ

119904) 119901 (120591 = 119889 | 119880Θ

119904)

sum1198892

119889=1198891119901 (119884obs | 119880 120591 = 119889 Θ

119904) 119901 (120591 = 119889 | 119880Θ119904)

=

prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904) 119901 (120591 = 119889)

sum1198892

119889=1198891prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904) 119901 (120591 = 119889)

=

prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904)

sum1198892

119889=1198891prod119905120572

119905=1199051119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ119904)

int 119901 (119910119905| 119862obs 120591 = 119889 Θ

119904) log119901 (119910

119905| 119906119905minus1 1

120591 = 119889 Θ) 119889119910119905

= int119901 (119910119905| 119862obs 120591 = 119889 Θ

119904) log 1

radic21205871205902

times exp

minus

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

21205902

119889119910119905

= minus1

2log (21205871205902) minus 1

21205902int119901 (119910

119905| 119862obs 120591 = 119889 Θ

119904)

times (119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

119889119910119905

= minus1

2log (21205871205902) minus 1

21205902((120590119904)2

+ (119866119904(119911minus1) 119906119905minus119889)2

)

+1

1205902(119866 (119911minus1) 119906119905minus119889) (119866119904(119911minus1) 119906119905minus119889)

minus1

21205902(119866 (119911minus1) 119906119905minus119889)2

= minus1

2log (21205871205902)

minus1

21205902((120590119904)2

+ (119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

)

119901 (119910119905| 119906119905minus1 1

120591 = 119889 Θ)

=1

radic21205871205902exp

minus

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

21205902

(12)

Therefore the 119876-function can be rewritten as119876 (Θ | Θ

119904)

=

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119898120573

sum

119905=1198981

minus1

2log (21205871205902) minus 1

21205902

times ((120590119904)2

+ (119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

)

+

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119905120572

sum

119905=1199051

minus1

2log (21205871205902) minus 1

21205902(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

+ log1198621

(13)

In the M-step of the GEM algorithm the unknownparameters should be estimated to increase the119876-function bysolving an optimization problem Taking the gradient of the119876-function (13) with respect to the 1205902 and setting it to zeroswe have

2=1

119873

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119898120573

sum

119905=1198981

((119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

+ (120590119904)2

)

+

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times

119905120572

sum

119905=1199051

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

(14)

Substituting 2 into the 119876-function (13) we get

119876 (Θ | Θ119904)

= minus119873

2log (2120587) minus 119873

2

minus119873

2log

1

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

((119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

+(120590119904)2

)

+

119905120572

sum

119905=1199051

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

)]

]

(15)

Mathematical Problems in Engineering 5

Based on the monotonicity of the log function the problemis transformed into optimizing the following cost function

119869 (120579 | 120579119904)

=1

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

((119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

+ (120590119904)2

)

+

119905120572

sum

119905=1199051

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

)]

]

(16)

Here we introduce the variable119909120591119905denoting the noise-free

output with time-delay 120591 Based on (1) and (2) we have

119909120591

119905=

sum119899119887

119894=1119887119894119911minus119894

1 + sum119899119886

119894=1119886119894119911minus119894119906119905minus120591= (120601120591

119905minus120591)119879

120579 (17)

where 120601120591119905minus120591

= [minus119909120591

119905minus1 minus119909

120591

119905minus119899119886 119906119905minus120591minus1

119906119905minus120591minus119899119887

]119879 There-

fore the cost function can be rewritten as

119869 (120579 | 120579119904)

=1

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

(((120601119889

119905minus119889)119879

120579 minus (119909119889

119905)119904

)

2

+ (120590119904)2

)

+

119905120572

sum

119905=1199051

(119910119905minus (120601119889

119905minus119889)119879

120579)

2

)]

]

(18)

where (119909119889119905)119904

= (120601119889

119905minus119889)119879

120579119904 However the cost function

(18) cannot be optimized directly due to the unmeasurable119909119889

119905119905=1119873 119889=1198891 1198892

Here we adopt the auxiliary model prin-ciple and the auxiliarymodel can be constructed based on theestimates obtained in the previous iteration That is

119909120591

119905=

sum119899119887

119894=1119894119911minus119894

1 + sum119899119886

119894=1119886119894119911minus119894119906119905minus120591= (120601120591

119905minus120591)119879

120579 (19)

where 120601120591119905minus120591

= [minus119909120591

119905minus1 minus119909

120591

119905minus119899119886 119906119905minus120591minus1

119906119905minus120591minus119899119887

]119879 There-

fore the cost function (18) with 120601120591119905minus120591

substituted by 120601120591119905minus120591

canbe optimized by using the damped Newton algorithm

120579119904+1= 120579119904minus [119877119904]minus1

1198691015840(120579119904| 120579119904) (20)

where

1198691015840(120579119904| 120579119904)

=2

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

120601119889

119905minus119889((120601119889

119905minus119889)119879

120579119904minus (119909119889

119905)119904

)

+

119905120572

sum

119905=1199051

120601119889

119905minus119889((120601119889

119905minus119889)119879

120579119904minus 119910119905))]

119877119904=2

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

119873

sum

119905=1

120601119889

119905minus119889(120601119889

119905minus119889)119879]

]

+ 120582119868

(21)

where 120582 is a constantThe time-delay 120591 can be selected as the delay in the range

with maximal posterior probability That is

120591 = argmax119889

119901 (120591 = 119889 | 119862obs Θ119904) (22)

The E-step andM-step alternate until the convergence condi-tion of the GEM algorithm is met

4 Simulation Examples

41 A Numerical Simulation Example Consider the follow-ing LTI time-delay system described by the OE time-delaymodel

119910119905=

03119911minus1

1 minus 07119911minus1119906119905minus3+ 119890119905 (23)

The input data and output data are generated by simulationand the noise 119890

119905with zero mean and variance 001 is added

to the output The input and output data are shown inFigure 1 In the simulation 125 output data are randomlymissing The parameter range of the time-delay is set to[1 5] The method proposed in this paper is used to estimatethe parameters and the time-delay The parameter estimatetrajectories of the model parameters and the noise varianceare shown in Figures 2 and 3 respectively The estimatedtime-delay is 3 which is consistent with the true time-delay To further verify the effectiveness of the proposedmethod the simulations are also performed with 25 outputdata missing and 50 output data missing The estimatedparameters after 13 iterations are summarized in Table 1 Itcan be seen from these figures and the table that the proposedGEM algorithm has a good identification performance

42 The Continuous Stirred Tank Reactor The ContinuousStirred Tank Reactor (CSTR) is a benchmark example usedto test the performances of different modeling and control

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300

0

1

2

Time

y1

minus1

minus2

(a)

0 50 100 150 200 250 300

0

05

1

Time

u1

minus05

minus1

(b)

Figure 1 The input and output data

Table 1 Estimated parameters after 13 iterations

True value 119886 = minus07 119887 = 03 120591 = 3 1205902= 001

Proportion of missing output 119886 119887 120591 1205902

Full data set minus0695 03026 3 00114125 minus0694 03042 3 0011625 minus07012 02994 3 0012150 minus06979 03017 3 00124

0 2 4 6 8 10 12 14

0

02

04

06

a

b

minus02

minus04

minus06

minus08

s

Figure 2 The estimated model parameters in each iteration

algorithms and the first principle model of the CSTR isdescribed as [14]

119889119862119860

119889119905=119902

119881(1198621198600minus 119862119860) minus 1198960119862119860exp(minus119864

119877119879)

119889119879

119889119905=119902

119881(1198790minus 119879) minus

(Δ119867) 1198960119862119860

120588119862119901

exp(minus119864119877119879)

+

120588119888119862119901119888

1205881198621199011198811199021198881 minus exp( minusℎ119860

119902119888120588119862119901

) (1198791198880minus 119879)

(24)

where the product concentration 119862119860and the temperature 119879

are output variables and the coolant flow rate 119902119888is the input

0 2 4 6 8 10 12 140

002

004

006

008

01

012

s

Figure 3 The estimated noise variance in each iteration

variable The steady state values of the process variables canbe found in Gopaluni [14] In this simulation the CSTR isoperated at a steady state working point which is at 119902

119888=

97 Lmin 119862119860= 00795molL and 119879 = 4434566K The

task here is to build a first-order model between 119862119860and 119902

119888

The input and output data are generated through simulationand the noise with zero mean and variance 1 times 10

minus7 isadded to the output data Since time is needed tomeasure theconcentration119862

119860 so themeasurement delaywith 15minutes

is also added to the output data The input and output data isshown in Figure 4 In this simulation 25 output data arerandomly missing and the parameter range of the time-delayis set to [09 21] The proposed GEM algorithm is used toestimate the unknown parametersThe estimated parametersare 119886 = minus0468 = 00015 2 = 15 times 10

minus7 and 120591 = 15The self-validation and the cross-validation results are shownin Figures 5 and 6 It can be seen from these results that theproposed method has a good identification performance andthe estimated model can capture the dynamic behavior of theCSTR

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 4000075

008

0085

Time

y1

(a)

0 50 100 150 200 250 300 350 40095

96

97

98

99

Time

u1

(b)

Figure 4 The input and output data

0 50 100 150 200 250 300 350 4000075

0076

0077

0078

0079

008

0081

0082

0083

Figure 5The self-validation resultsThe blue line is the real processdata and the red line is the simulated output of the estimated model

0 50 100 150 200 250 300 350 4000075

0076

0077

0078

0079

008

0081

0082

0083

Figure 6 The cross-validation results The blue line is the realprocess data and the red line is the simulated output of the estimatedmodel

5 Conclusion

This paper considers the identification problem of LTI sys-tems with irregular data set The time-delay and the missing

data are commonly encountered problems in process indus-try and the existence of these problems makes the processmodeling a challenging taskThe identification problem withincomplete data set in the presence of time-delay is formu-lated under the scheme of the GEM algorithm and the modelparameters and the time-delay are estimated simultaneouslyin this algorithm Numerical examples are presented todemonstrate the efficacy of the proposed method

Conflict of Interests

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

References

[1] S Yin H Luo and S X Ding ldquoReal-Time implementation offault-tolerant control systems with performance optimizationrdquoIEEE Transactions on Industrial Electronics vol 61 no 5 pp2402ndash2411 2014

[2] S Yin S X Ding A Haghani and P Zhang ldquoA comparisonstudy of basic data-driven fault diagnosis and process monitor-ing methods on the benchmark Tennessee Eastman processrdquoJournal of Process Control vol 22 no 9 pp 1567ndash1581 2012

[3] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[4] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[5] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[6] H Dong Z Wang and H Gao ldquoDistributed 119867infin

filteringfor a class of Markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013

[7] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[8] J P Richard ldquoTime-delay systems an overview of some recentadvances and open problemsrdquo Automatica vol 39 no 10 pp1667ndash1694 2003

8 Mathematical Problems in Engineering

[9] Q G Wang and Y Zhang ldquoRobust identification of continuoussystems with dead-time from step responsesrdquo Automatica vol37 no 3 pp 377ndash390 2001

[10] E Weyer ldquoSystem identification of an open water channelrdquoControl Engineering Practice vol 9 no 12 pp 1289ndash1299 2001

[11] F Ding and J Ding ldquoLeast-squares parameter estimation forsystems with irregularly missing datardquo International Journal ofAdaptive Control and Signal Processing vol 24 no 7 pp 540ndash553 2010

[12] Y C Zhu H Telkamp J H Wang and Q L Fu ldquoSystemidentification using slow and irregular output samplesrdquo Journalof Process Control vol 19 no 1 pp 58ndash67 2009

[13] G J McLachlan and T Krishnan The EM Algorithm andExtensions John Wiley amp Sons New York NY USA 2007

[14] R B Gopaluni ldquoA particle filter approach to identification ofnonlinear processes undermissing observationsrdquoTheCanadianJournal of Chemical Engineering vol 86 no 6 pp 1081ndash10922008

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

Based on the monotonicity of the log function the problemis transformed into optimizing the following cost function

119869 (120579 | 120579119904)

=1

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

((119866 (119911minus1) 119906119905minus119889minus 119866119904(119911minus1) 119906119905minus119889)2

+ (120590119904)2

)

+

119905120572

sum

119905=1199051

(119910119905minus 119866 (119911

minus1) 119906119905minus119889)2

)]

]

(16)

Here we introduce the variable119909120591119905denoting the noise-free

output with time-delay 120591 Based on (1) and (2) we have

119909120591

119905=

sum119899119887

119894=1119887119894119911minus119894

1 + sum119899119886

119894=1119886119894119911minus119894119906119905minus120591= (120601120591

119905minus120591)119879

120579 (17)

where 120601120591119905minus120591

= [minus119909120591

119905minus1 minus119909

120591

119905minus119899119886 119906119905minus120591minus1

119906119905minus120591minus119899119887

]119879 There-

fore the cost function can be rewritten as

119869 (120579 | 120579119904)

=1

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

(((120601119889

119905minus119889)119879

120579 minus (119909119889

119905)119904

)

2

+ (120590119904)2

)

+

119905120572

sum

119905=1199051

(119910119905minus (120601119889

119905minus119889)119879

120579)

2

)]

]

(18)

where (119909119889119905)119904

= (120601119889

119905minus119889)119879

120579119904 However the cost function

(18) cannot be optimized directly due to the unmeasurable119909119889

119905119905=1119873 119889=1198891 1198892

Here we adopt the auxiliary model prin-ciple and the auxiliarymodel can be constructed based on theestimates obtained in the previous iteration That is

119909120591

119905=

sum119899119887

119894=1119894119911minus119894

1 + sum119899119886

119894=1119886119894119911minus119894119906119905minus120591= (120601120591

119905minus120591)119879

120579 (19)

where 120601120591119905minus120591

= [minus119909120591

119905minus1 minus119909

120591

119905minus119899119886 119906119905minus120591minus1

119906119905minus120591minus119899119887

]119879 There-

fore the cost function (18) with 120601120591119905minus120591

substituted by 120601120591119905minus120591

canbe optimized by using the damped Newton algorithm

120579119904+1= 120579119904minus [119877119904]minus1

1198691015840(120579119904| 120579119904) (20)

where

1198691015840(120579119904| 120579119904)

=2

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

times (

119898120573

sum

119905=1198981

120601119889

119905minus119889((120601119889

119905minus119889)119879

120579119904minus (119909119889

119905)119904

)

+

119905120572

sum

119905=1199051

120601119889

119905minus119889((120601119889

119905minus119889)119879

120579119904minus 119910119905))]

119877119904=2

119873

[

[

1198892

sum

119889=1198891

119901 (120591 = 119889 | 119862obs Θ119904)

119873

sum

119905=1

120601119889

119905minus119889(120601119889

119905minus119889)119879]

]

+ 120582119868

(21)

where 120582 is a constantThe time-delay 120591 can be selected as the delay in the range

with maximal posterior probability That is

120591 = argmax119889

119901 (120591 = 119889 | 119862obs Θ119904) (22)

The E-step andM-step alternate until the convergence condi-tion of the GEM algorithm is met

4 Simulation Examples

41 A Numerical Simulation Example Consider the follow-ing LTI time-delay system described by the OE time-delaymodel

119910119905=

03119911minus1

1 minus 07119911minus1119906119905minus3+ 119890119905 (23)

The input data and output data are generated by simulationand the noise 119890

119905with zero mean and variance 001 is added

to the output The input and output data are shown inFigure 1 In the simulation 125 output data are randomlymissing The parameter range of the time-delay is set to[1 5] The method proposed in this paper is used to estimatethe parameters and the time-delay The parameter estimatetrajectories of the model parameters and the noise varianceare shown in Figures 2 and 3 respectively The estimatedtime-delay is 3 which is consistent with the true time-delay To further verify the effectiveness of the proposedmethod the simulations are also performed with 25 outputdata missing and 50 output data missing The estimatedparameters after 13 iterations are summarized in Table 1 Itcan be seen from these figures and the table that the proposedGEM algorithm has a good identification performance

42 The Continuous Stirred Tank Reactor The ContinuousStirred Tank Reactor (CSTR) is a benchmark example usedto test the performances of different modeling and control

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300

0

1

2

Time

y1

minus1

minus2

(a)

0 50 100 150 200 250 300

0

05

1

Time

u1

minus05

minus1

(b)

Figure 1 The input and output data

Table 1 Estimated parameters after 13 iterations

True value 119886 = minus07 119887 = 03 120591 = 3 1205902= 001

Proportion of missing output 119886 119887 120591 1205902

Full data set minus0695 03026 3 00114125 minus0694 03042 3 0011625 minus07012 02994 3 0012150 minus06979 03017 3 00124

0 2 4 6 8 10 12 14

0

02

04

06

a

b

minus02

minus04

minus06

minus08

s

Figure 2 The estimated model parameters in each iteration

algorithms and the first principle model of the CSTR isdescribed as [14]

119889119862119860

119889119905=119902

119881(1198621198600minus 119862119860) minus 1198960119862119860exp(minus119864

119877119879)

119889119879

119889119905=119902

119881(1198790minus 119879) minus

(Δ119867) 1198960119862119860

120588119862119901

exp(minus119864119877119879)

+

120588119888119862119901119888

1205881198621199011198811199021198881 minus exp( minusℎ119860

119902119888120588119862119901

) (1198791198880minus 119879)

(24)

where the product concentration 119862119860and the temperature 119879

are output variables and the coolant flow rate 119902119888is the input

0 2 4 6 8 10 12 140

002

004

006

008

01

012

s

Figure 3 The estimated noise variance in each iteration

variable The steady state values of the process variables canbe found in Gopaluni [14] In this simulation the CSTR isoperated at a steady state working point which is at 119902

119888=

97 Lmin 119862119860= 00795molL and 119879 = 4434566K The

task here is to build a first-order model between 119862119860and 119902

119888

The input and output data are generated through simulationand the noise with zero mean and variance 1 times 10

minus7 isadded to the output data Since time is needed tomeasure theconcentration119862

119860 so themeasurement delaywith 15minutes

is also added to the output data The input and output data isshown in Figure 4 In this simulation 25 output data arerandomly missing and the parameter range of the time-delayis set to [09 21] The proposed GEM algorithm is used toestimate the unknown parametersThe estimated parametersare 119886 = minus0468 = 00015 2 = 15 times 10

minus7 and 120591 = 15The self-validation and the cross-validation results are shownin Figures 5 and 6 It can be seen from these results that theproposed method has a good identification performance andthe estimated model can capture the dynamic behavior of theCSTR

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 4000075

008

0085

Time

y1

(a)

0 50 100 150 200 250 300 350 40095

96

97

98

99

Time

u1

(b)

Figure 4 The input and output data

0 50 100 150 200 250 300 350 4000075

0076

0077

0078

0079

008

0081

0082

0083

Figure 5The self-validation resultsThe blue line is the real processdata and the red line is the simulated output of the estimated model

0 50 100 150 200 250 300 350 4000075

0076

0077

0078

0079

008

0081

0082

0083

Figure 6 The cross-validation results The blue line is the realprocess data and the red line is the simulated output of the estimatedmodel

5 Conclusion

This paper considers the identification problem of LTI sys-tems with irregular data set The time-delay and the missing

data are commonly encountered problems in process indus-try and the existence of these problems makes the processmodeling a challenging taskThe identification problem withincomplete data set in the presence of time-delay is formu-lated under the scheme of the GEM algorithm and the modelparameters and the time-delay are estimated simultaneouslyin this algorithm Numerical examples are presented todemonstrate the efficacy of the proposed method

Conflict of Interests

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

References

[1] S Yin H Luo and S X Ding ldquoReal-Time implementation offault-tolerant control systems with performance optimizationrdquoIEEE Transactions on Industrial Electronics vol 61 no 5 pp2402ndash2411 2014

[2] S Yin S X Ding A Haghani and P Zhang ldquoA comparisonstudy of basic data-driven fault diagnosis and process monitor-ing methods on the benchmark Tennessee Eastman processrdquoJournal of Process Control vol 22 no 9 pp 1567ndash1581 2012

[3] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[4] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[5] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[6] H Dong Z Wang and H Gao ldquoDistributed 119867infin

filteringfor a class of Markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013

[7] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[8] J P Richard ldquoTime-delay systems an overview of some recentadvances and open problemsrdquo Automatica vol 39 no 10 pp1667ndash1694 2003

8 Mathematical Problems in Engineering

[9] Q G Wang and Y Zhang ldquoRobust identification of continuoussystems with dead-time from step responsesrdquo Automatica vol37 no 3 pp 377ndash390 2001

[10] E Weyer ldquoSystem identification of an open water channelrdquoControl Engineering Practice vol 9 no 12 pp 1289ndash1299 2001

[11] F Ding and J Ding ldquoLeast-squares parameter estimation forsystems with irregularly missing datardquo International Journal ofAdaptive Control and Signal Processing vol 24 no 7 pp 540ndash553 2010

[12] Y C Zhu H Telkamp J H Wang and Q L Fu ldquoSystemidentification using slow and irregular output samplesrdquo Journalof Process Control vol 19 no 1 pp 58ndash67 2009

[13] G J McLachlan and T Krishnan The EM Algorithm andExtensions John Wiley amp Sons New York NY USA 2007

[14] R B Gopaluni ldquoA particle filter approach to identification ofnonlinear processes undermissing observationsrdquoTheCanadianJournal of Chemical Engineering vol 86 no 6 pp 1081ndash10922008

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

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300

0

1

2

Time

y1

minus1

minus2

(a)

0 50 100 150 200 250 300

0

05

1

Time

u1

minus05

minus1

(b)

Figure 1 The input and output data

Table 1 Estimated parameters after 13 iterations

True value 119886 = minus07 119887 = 03 120591 = 3 1205902= 001

Proportion of missing output 119886 119887 120591 1205902

Full data set minus0695 03026 3 00114125 minus0694 03042 3 0011625 minus07012 02994 3 0012150 minus06979 03017 3 00124

0 2 4 6 8 10 12 14

0

02

04

06

a

b

minus02

minus04

minus06

minus08

s

Figure 2 The estimated model parameters in each iteration

algorithms and the first principle model of the CSTR isdescribed as [14]

119889119862119860

119889119905=119902

119881(1198621198600minus 119862119860) minus 1198960119862119860exp(minus119864

119877119879)

119889119879

119889119905=119902

119881(1198790minus 119879) minus

(Δ119867) 1198960119862119860

120588119862119901

exp(minus119864119877119879)

+

120588119888119862119901119888

1205881198621199011198811199021198881 minus exp( minusℎ119860

119902119888120588119862119901

) (1198791198880minus 119879)

(24)

where the product concentration 119862119860and the temperature 119879

are output variables and the coolant flow rate 119902119888is the input

0 2 4 6 8 10 12 140

002

004

006

008

01

012

s

Figure 3 The estimated noise variance in each iteration

variable The steady state values of the process variables canbe found in Gopaluni [14] In this simulation the CSTR isoperated at a steady state working point which is at 119902

119888=

97 Lmin 119862119860= 00795molL and 119879 = 4434566K The

task here is to build a first-order model between 119862119860and 119902

119888

The input and output data are generated through simulationand the noise with zero mean and variance 1 times 10

minus7 isadded to the output data Since time is needed tomeasure theconcentration119862

119860 so themeasurement delaywith 15minutes

is also added to the output data The input and output data isshown in Figure 4 In this simulation 25 output data arerandomly missing and the parameter range of the time-delayis set to [09 21] The proposed GEM algorithm is used toestimate the unknown parametersThe estimated parametersare 119886 = minus0468 = 00015 2 = 15 times 10

minus7 and 120591 = 15The self-validation and the cross-validation results are shownin Figures 5 and 6 It can be seen from these results that theproposed method has a good identification performance andthe estimated model can capture the dynamic behavior of theCSTR

Mathematical Problems in Engineering 7

0 50 100 150 200 250 300 350 4000075

008

0085

Time

y1

(a)

0 50 100 150 200 250 300 350 40095

96

97

98

99

Time

u1

(b)

Figure 4 The input and output data

0 50 100 150 200 250 300 350 4000075

0076

0077

0078

0079

008

0081

0082

0083

Figure 5The self-validation resultsThe blue line is the real processdata and the red line is the simulated output of the estimated model

0 50 100 150 200 250 300 350 4000075

0076

0077

0078

0079

008

0081

0082

0083

Figure 6 The cross-validation results The blue line is the realprocess data and the red line is the simulated output of the estimatedmodel

5 Conclusion

This paper considers the identification problem of LTI sys-tems with irregular data set The time-delay and the missing

data are commonly encountered problems in process indus-try and the existence of these problems makes the processmodeling a challenging taskThe identification problem withincomplete data set in the presence of time-delay is formu-lated under the scheme of the GEM algorithm and the modelparameters and the time-delay are estimated simultaneouslyin this algorithm Numerical examples are presented todemonstrate the efficacy of the proposed method

Conflict of Interests

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

References

[1] S Yin H Luo and S X Ding ldquoReal-Time implementation offault-tolerant control systems with performance optimizationrdquoIEEE Transactions on Industrial Electronics vol 61 no 5 pp2402ndash2411 2014

[2] S Yin S X Ding A Haghani and P Zhang ldquoA comparisonstudy of basic data-driven fault diagnosis and process monitor-ing methods on the benchmark Tennessee Eastman processrdquoJournal of Process Control vol 22 no 9 pp 1567ndash1581 2012

[3] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[4] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[5] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[6] H Dong Z Wang and H Gao ldquoDistributed 119867infin

filteringfor a class of Markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013

[7] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[8] J P Richard ldquoTime-delay systems an overview of some recentadvances and open problemsrdquo Automatica vol 39 no 10 pp1667ndash1694 2003

8 Mathematical Problems in Engineering

[9] Q G Wang and Y Zhang ldquoRobust identification of continuoussystems with dead-time from step responsesrdquo Automatica vol37 no 3 pp 377ndash390 2001

[10] E Weyer ldquoSystem identification of an open water channelrdquoControl Engineering Practice vol 9 no 12 pp 1289ndash1299 2001

[11] F Ding and J Ding ldquoLeast-squares parameter estimation forsystems with irregularly missing datardquo International Journal ofAdaptive Control and Signal Processing vol 24 no 7 pp 540ndash553 2010

[12] Y C Zhu H Telkamp J H Wang and Q L Fu ldquoSystemidentification using slow and irregular output samplesrdquo Journalof Process Control vol 19 no 1 pp 58ndash67 2009

[13] G J McLachlan and T Krishnan The EM Algorithm andExtensions John Wiley amp Sons New York NY USA 2007

[14] R B Gopaluni ldquoA particle filter approach to identification ofnonlinear processes undermissing observationsrdquoTheCanadianJournal of Chemical Engineering vol 86 no 6 pp 1081ndash10922008

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

0 50 100 150 200 250 300 350 4000075

008

0085

Time

y1

(a)

0 50 100 150 200 250 300 350 40095

96

97

98

99

Time

u1

(b)

Figure 4 The input and output data

0 50 100 150 200 250 300 350 4000075

0076

0077

0078

0079

008

0081

0082

0083

Figure 5The self-validation resultsThe blue line is the real processdata and the red line is the simulated output of the estimated model

0 50 100 150 200 250 300 350 4000075

0076

0077

0078

0079

008

0081

0082

0083

Figure 6 The cross-validation results The blue line is the realprocess data and the red line is the simulated output of the estimatedmodel

5 Conclusion

This paper considers the identification problem of LTI sys-tems with irregular data set The time-delay and the missing

data are commonly encountered problems in process indus-try and the existence of these problems makes the processmodeling a challenging taskThe identification problem withincomplete data set in the presence of time-delay is formu-lated under the scheme of the GEM algorithm and the modelparameters and the time-delay are estimated simultaneouslyin this algorithm Numerical examples are presented todemonstrate the efficacy of the proposed method

Conflict of Interests

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

References

[1] S Yin H Luo and S X Ding ldquoReal-Time implementation offault-tolerant control systems with performance optimizationrdquoIEEE Transactions on Industrial Electronics vol 61 no 5 pp2402ndash2411 2014

[2] S Yin S X Ding A Haghani and P Zhang ldquoA comparisonstudy of basic data-driven fault diagnosis and process monitor-ing methods on the benchmark Tennessee Eastman processrdquoJournal of Process Control vol 22 no 9 pp 1567ndash1581 2012

[3] S Yin X Yang and H R Karimi ldquoData-driven adaptiveobserver for fault diagnosisrdquo Mathematical Problems in Engi-neering vol 2012 Article ID 832836 21 pages 2012

[4] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[5] S Yin G Wang and H Karimi ldquoData-driven design of robustfault detection system for wind turbinesrdquoMechatronics 2013

[6] H Dong Z Wang and H Gao ldquoDistributed 119867infin

filteringfor a class of Markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013

[7] H Dong Z Wang J Lam and H Gao ldquoFuzzy-model-basedrobust fault detection with stochastic mixed time delays andsuccessive packet dropoutsrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 42 no 2 pp 365ndash376 2012

[8] J P Richard ldquoTime-delay systems an overview of some recentadvances and open problemsrdquo Automatica vol 39 no 10 pp1667ndash1694 2003

8 Mathematical Problems in Engineering

[9] Q G Wang and Y Zhang ldquoRobust identification of continuoussystems with dead-time from step responsesrdquo Automatica vol37 no 3 pp 377ndash390 2001

[10] E Weyer ldquoSystem identification of an open water channelrdquoControl Engineering Practice vol 9 no 12 pp 1289ndash1299 2001

[11] F Ding and J Ding ldquoLeast-squares parameter estimation forsystems with irregularly missing datardquo International Journal ofAdaptive Control and Signal Processing vol 24 no 7 pp 540ndash553 2010

[12] Y C Zhu H Telkamp J H Wang and Q L Fu ldquoSystemidentification using slow and irregular output samplesrdquo Journalof Process Control vol 19 no 1 pp 58ndash67 2009

[13] G J McLachlan and T Krishnan The EM Algorithm andExtensions John Wiley amp Sons New York NY USA 2007

[14] R B Gopaluni ldquoA particle filter approach to identification ofnonlinear processes undermissing observationsrdquoTheCanadianJournal of Chemical Engineering vol 86 no 6 pp 1081ndash10922008

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

[9] Q G Wang and Y Zhang ldquoRobust identification of continuoussystems with dead-time from step responsesrdquo Automatica vol37 no 3 pp 377ndash390 2001

[10] E Weyer ldquoSystem identification of an open water channelrdquoControl Engineering Practice vol 9 no 12 pp 1289ndash1299 2001

[11] F Ding and J Ding ldquoLeast-squares parameter estimation forsystems with irregularly missing datardquo International Journal ofAdaptive Control and Signal Processing vol 24 no 7 pp 540ndash553 2010

[12] Y C Zhu H Telkamp J H Wang and Q L Fu ldquoSystemidentification using slow and irregular output samplesrdquo Journalof Process Control vol 19 no 1 pp 58ndash67 2009

[13] G J McLachlan and T Krishnan The EM Algorithm andExtensions John Wiley amp Sons New York NY USA 2007

[14] R B Gopaluni ldquoA particle filter approach to identification ofnonlinear processes undermissing observationsrdquoTheCanadianJournal of Chemical Engineering vol 86 no 6 pp 1081ndash10922008

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

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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