Research ArticleModelling of Lime Kiln Using Subspace Method withNew Order Selection Criterion

Li Zhang1 Chengjin Zhang1 Qingyang Xu1 and Chaoyang Wang2

1 School of Mechanical Electrical and Information Engineering Shandong University Weihai 264209 China2 College of Mathematics and System Science Shandong University of Science and Technology Qingdao 266590 China

Correspondence should be addressed to Li Zhang zhangliwhsdueducn

Received 12 February 2014 Revised 16 August 2014 Accepted 30 August 2014 Published 23 October 2014

Academic Editor Huaguang Zhang

Copyright copy 2014 Li Zhang et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is taking actual control demand of rotary kiln as background and builds a calcining belt state space model using PO-Moesp subspace method A novel order-delay double parameters error criterion (ODC) is presented to reduce the modeling orderThe proposed subspace order identification method takes into account the influence of order and delay on model error criterionsimultaneously For the introduction of the delay factors the order is reduced dramatically in the systemmodeling Also in the dataprocessing part sliding-window method is adopted for stripping delay factor from historical data For this the parameters can bechanged flexibly Some practical problems in industrial kiln process modeling are also solved Finally it is applied to an industrialkiln case

1 Introduction

Rotary kiln is the key equipment in metallurgy buildingmaterials and many other industries A lot of rotary kilncontrol theories andmethodologies have been studied for thereason that kiln belongs to the typical complex process whichhas the characteristics of multivariables time delay seriouscoupling and difficult modelling [1ndash5]

In the field of rotary kiln modeling researchers startedstudying internal status of calcinations process as earlyas 1960s The initial researchers established control modelbased on solid mechanics heat transfer mechanism anddry calcined kinetics Imber and Paschkis calculated theoptimum kiln length for heating process in 1961 [6 7]These early approaches are all based upon the rmodynamicequilibrium and analysis of the kiln structure and theestablishment of these equations harshly demands materialflow balance So it is not a good way to put in use widelyIn 1980s dynamic system identification technology was usedin rotary kiln modeling but the identification model canonly characterize part states of kiln So this technology wasrestricted in application and exploitation In the 1990s afterthe emergence of the new intelligent modeling and controlmethods a large number of intelligent modeling ways started

to use in building model of kiln successfully [8ndash10] Aftera long-term development many mature experiences havealready been obtained but there are still many problemssuch as imprecise model excess restrictions and difficultyin promotion Consequently building model has become abottleneck on the development road of kiln process control

Subspace identification method as an effective identifica-tion modelling way for multi-input and multi-output systemhas drawn much attention recently [11ndash13] In comparisonwith the conventional methods it has obvious advantageThis is because the model can be directly got from theinputoutput (119868119874) data without nonlinear optimization anditeration Furthermore the computations are based on robusttools such as QR-factorization and singular value decompo-sition (SVD) for which numerically reliable algorithms areavailable [14 15]

Order and delay are the most important constructionparameters of industrial process model There exists anextensive literature for order estimation algorithms of lineardynamical state space systems [16ndash20] Among all of thesemethods the most famous one is AIC (Akaike informationcriterion) which is introduced by Akaike [21] And alsogreat deal of subsequent researches were done about theproperties effects of the penalty term [22 23] And then

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 816831 11 pageshttpdxdoiorg1011552014816831

2 Mathematical Problems in Engineering

Baure introduces another criterion for the Larimore type ofprocedures which is similar to AIC [24 25] From aboveliterature an obvious problem can be found that the delayparameter is always neglected in systems order selectionThe information about the delay of any process is valuablefor industrial process model However there exist onlyfew references dealing with the estimation of the order indelay system [26ndash28] When the system has input delaymany methods attempt to increase the model order as thecost to improve model accuracy This enhances the systemcomplexity

In this paper the calcining belt state spacemodel of rotarykiln is built by using PO-Moesp subspace method Afteranalyzing the main components technology and calcinationsreaction mechanism in detail the input and output variablesof modeling are selected according to industrial field opera-tional procedures Andmore a novel order-selectingmethodis put forward The original error criterion is replaced by thenew one which includes the two parametersmdashthe order andthe delay This improvement effectively solves the problem ofmodel order too high whenmodelling object with time delayand it also increases the precision of model

The outline of this paper is as follows in Section 2 thebasic process of rotary kiln is described in Section 3 thegeneral subspace identification algorithm is introduced andthe order selection problem is addressed Section 4 presentsthe main results of this paper The method to identify theorder and the delay is deduced in detail In Section 5 wediscuss the performance of the proposed method by meansof some simulation studies on an industrial rotary kilnillustration

2 Lime Rotary Kiln Process Description

The rotary lime kiln is a long cylinder that has a 3ndash5∘dslope and it rotates about its axis at roughly 05ndash15 rpmLime mud is fed from the elevated cold end and movesdown the kiln due to rotation and gravity The hot endin which the burner operates is typically maintained atabout 1200∘C by burning fuel [1] Active lime rotary kilnsystem has many technological processes such as long kilntechnology system chain grate cooling technology systemand vertical preheater technology system The process ismainly divided intomaterial system and air flow system Rawmaterial becomes product after preheating high temperaturecalcinations and cooling Primary air secondary air andcoke oven gas constitute the air flow system The wholetechnological process is shown in Figure 1 In this paper thelime kiln under study is 600 td 40m times 60m

The main calcinations process parameters are listed inTable 1

3 Problem Formulation and Assumptions

A discrete-time model in state space form is described by theequations

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896 minus 119889) + 119908 (119896)

119910 (119896) = 119862119909 (119896) + 119863119906 (119896 minus 119889) + V (119896) (1)

Table 1 Process parameters of rotary kiln

Parameter The parameter of kilnRaw material granularity 10sim40mmPreheater temperature 900sim1100∘CKiln head temperature 800sim950∘CKiln tail temperature 1100 plusmn 20∘CCalcining zone temperature 1260sim1340∘CSecond air temperature 750sim900∘CKiln head cover temperature 800sim950∘CExhaust gas temperature 250∘CLime temperature after cooling le150∘CKiln head pressure range minus10simminus20 paGas pressure ge6KPaCalcinations process pressure Negative pressure 100ndash150 PaPulse jet cleaning pressure 035sim05MPa3920 dust removel pressure difference le15 kPaKiln body inclination 3sim5Kiln speed range 02sim128 rminCaO content gt88Activity degree gt350mLExhaust gas residual carbon lt=1Exhaust gas oxygen content 07sim25Dust collector air volume 225000m3hMaterial level in preheater 5sim88mMaterial level in cooler le055mproduction 600 Tsim650 TSiO2 lt15 or lt2MgO 5Power consumption 65 kwsdothtsim60 kwsdothtHeat consumption 1630sim1650 kJkg

where input 119906119896isin 119877119898 and output 119910

119896isin 119877119897 119898 and 119897 are

input and output dimensions respectively 119909119896isin 119877119899 is the

system state at time 119896 and 119899 is the system order to beidentified process noise vector 119908

119896isin 119877119899 and measurement

noise vector V119896

isin 119877119897 are zero mean value white noiseseries

We introduce the following hypotheses (van Overscheeand de Moor 1995) [29]

(a) The system is asymptotically stable

(b) The system is observable and reachable

(c) All the process noise 119908(119896) and measurement noiseV(119896) are statistically independent of the input 119906(119896)that is mean 119864119906(119894)119908(119895) = 119864119906(119894)V(119895) = 0 where119864sdot = lim

119873rarrinfin(1119873)sum119864(sdot)

The goal of subspace methods consist in the onlineestimating order 119899 delay 119889 and system matrices 119860 119861 119862 and119863

Mathematical Problems in Engineering 3

Preheater

Rotary kiln

Cooler

Figure 1 Technological process chart of rotary kiln

31 Matrices Construct Suppose at time 119896 the past input andoutput data samples are available then construct pastfutureoutput vectors and the Hankel output matrices as follows

119880119901= (

1199060

1199061sdot sdot sdot 119906

119895minus1

1199061

1199062sdot sdot sdot 119906

119895

d

119906119894minus1

119906119894sdot sdot sdot 119906119894+119895minus2

)

(2a)

119884119891= (

119910119894

119910119894+1

sdot sdot sdot 119910119894+119895minus1

119910119894+1

119910119894+2

sdot sdot sdot 119910119894+119895

d

1199102119894minus1

1199102119894

sdot sdot sdot 1199102119894+119895minus2

) (2b)

where 119906119894= [119906

1198941 1199061198942 119906

119894119897] 119910119894= [119910

1198941 1199101198942 119910

119894119898] 119901

represent the past information and 119891 represent the futureinformation 119894 ge 119899 119895 ≫ max(119894119897 119894119898)

The state matrix 119883119901and 119883

119891is defined as follows 119883

119901=

[11990901199091sdot sdot sdot 119909119895minus1]119883119891= [119909119894119909119894+1

sdot sdot sdot 119909119895+119894minus1

]The augmented observation matrix Γ

119894

Γ119894= [119862 119862119860 sdot sdot sdot 119862119860119894]

119879

(3)

Block-triangular matrix Toeplitz matrix119867119894and119867119904

119894

119867119894=

[[[[[[[

[

119863 0 sdot sdot sdot 0

119862119861 119863 d

d 0

119862119894minus2119861 119862119894minus1119861 sdot sdot sdot 119863

]]]]]]]

]

119867119904

119894

=

[[[[[[[

[

0 0 sdot sdot sdot 0

119862 0 d

d 0

119862119894minus2 119862119894minus1 sdot sdot sdot 0

]]]]]]]

]

(4)

The augmented observation inverse matrix of 119860 119861 isΔ119894= [119860119894minus1

119861 119860119894minus2

119861 sdot sdot sdot 119861] And the augmented observationinverse matrix of 119860119870 is Δ119904

119894

= [119860119894minus1

119870 119860119894minus2

119870 sdot sdot sdot 119870]Suppose the input and state matrices are all row full

rank matrices and they are row space orthogonal Then theaugmented inputoutput matrices can be written as follows

119883119891= 119860119894

119883119901+ Δ119894119880119901+ Δ119904

119894

119864119891 (5a)

119884119901= Γ119894119883119901+ 119867119894119880119901+ 119867119904

119894

119864119901 (5b)

119884119891= Γ119894119883119891+ 119867119894119880119891+ 119867119904

119894

119864119891 (5c)

The key step of Moesp method is to estimate the aug-mented observability matrix through the projection future119868119874 data onto past 119868119874 data The augmented observabilitymatrix and state space vector estimation can be got throughSVD decomposition

32 General Algorithm In this section we introduce thegeneral subspace method framework which is profferedby Favoreel et al [30] Subspace identification algorithmconsists of two main steps The first step always performsa weighted projection of the row space of the previouslydefined data Hankel matrices From this projection theaugmented observabilitymatrix Γ

119894and estimate119883

119894of the state

sequence119883119894can be retrievedThen the order is got from SVD

decomposition In the second step the system matrices 119860 119861119862119863 and119876 119878119877 are determined through least squaremethodThe concrete Moesp algorithm is as follows

4 Mathematical Problems in Engineering

(1) Project the 119884119891row space into the orthogonal comple-

ment of the 119880119891row space

119884119891119880perp

119891

= Γ119894119883119891119880perp

119891

+ 119867119889

119894

119880119891119880perp

119891

+ 119867119904

119894

119864119891119880perp

119891

(6)

Since it is assumed that the noise is uncorrelated withthe inputs so 119864

119891119880perp119891

= 119864119891and 119880

119891119880perp119891

= 0 therefore119884119891119880perp119891

= Γ119894119883119891119880perp119891

+ 119867119904119894

119864119891

(2) Select the weighting matrices1198821and119882

2

1198821119884119891

119880perp119891

1198822

=1198821Γ119894119883119891

119880perp119891

1198822

+1198821119867119904

119894

1198641198911198822 (7)

The weighting matrices can be chosen appropriatelyaccording to different subspace methods includingN4SID MOESP CVA basic-4SID and IV-4SID [31ndash34]

Then we can get

119900119894=1198821119884119891

119880perp119891

1198822

=1198821Γ119894119883119891

119880perp119891

1198822

(8)

(3) Carry SVD decomposition

119900119894= (11988011198802) (11987810

0 0)(

119881119879

1

1198811198792

) (9)

And then take the number of nonzero eigenvalue asthe system order rank(119900

119894) = 119899

(4) The augmented observability matrix Γ119894= 119882minus11

119880111987812

1

or119883119894= 119883119894119880perp119891

1198822is derived from the third step

(5) Extract estimate 119860 119861 119862119863 from Γ119894or119883119894

Remark By reference to [30] the weighting matrices1198821and

1198822should satisfy the following three conditions

(1) rank(1198821sdot Γ119894) = rank Γ

119894

(2) rank(119883119894119880perp119891

sdot 1198822) = rank119883

119894

(3) 1198821sdot (119867119904119894

119872119891+ 119873119891) sdot 1198822= 0

The first two conditions guarantee that the rank-119899 prop-erty of Γ

119894119883119894is preserved after projection onto 119880perp

119891

andweighting by119882

1and119882

2 The third condition expresses that

1198822should be uncorrelated with the noise sequences 119908(119896)

and V(119896) By choosing the appropriate weighting matrices1198821and 119882

2 all subspace algorithms for LTI systems can

be interpreted in the above framework including N4SIDMOESP CVA Basic-4SID and IV-4SID

4 The Proposed Method

In the classical system identification theory the actual modelstructure is usually assumed to be known However inpractical it is always not clear Subspace system identificationmethod determines the order of the system by the nonzeroeigenvalue of the augmented observability matrix Howeverthe system nonzero singular values may be very small Thismay lead to the wrong system order and large identificationerror

41 The Order-Delay Double Parameters Error Criterion Themost directly order-selection method is based on the errorperformance criterion This idea is to choose the smallestpossible order that keeps the error below a certain levelThenthe MRSE (mean relative squared error) index is introducedby model error as follows

119869MRSE (119899) =1

119871

119871

sum119896=1

radicsum119899119910

119895=1

(119910119896(119895) minus 119910

119896(119895))2

sum119899119910

119895=1

119910119896(119895)2

(10)

where 119910119896(119895) minus 119910

119896(119895) is the model prediction error and 119871 is

the sample number In [35] use the AIC which was originallydeveloped by Akaike and then adapted by Larimore for SMIGiven a set of samples for a sequence of system order 119899 forexample 119899 isin [0 sdot sdot sdot 20] the order of the model will be the onewhich makes the following AIC index minimum

AIC119899(119899) = 119873 (119898 (1 + ln 2120587) + ln 1003816100381610038161003816Σ119899

1003816100381610038161003816) + 2120575119899119872119899 (11)

where

Σ119899=

1

119873

119873

sum119894=1

119890 (119896) 119890(119896)119879

119890 (119896) = 119910119899(119896) minus 119910

119899(119896)

119872119899= 2119899119898 +

119898 (119898 + 1)

2+ 119899119897 + 119898119897

120575119899=

119873

119873 minus ((119872119899119898) + ((119898 + 1) 2))

(12)

For calculating the AIC(119899) criterion we first suppose theupper bound 119899max of the system order and then calculate theAIC119873(1)AIC

119873(2) AIC

119873(119899max) sequence the appropri-

ate system order 119899 is the one which decrees the AIC indexobviously and the order should be as small as possible TheMRSE and AIC index 119869MRSE(119899) can be analyzed in the samemanner

However the performance index based on a single orderparameter cannot provide an effective solution to the delaysystem which is shown in (1) This led to the problem thatthe original identificationmethod had to increase the order ascost to improvemodel accuracy Here we introduce an order-delay double parameters error criterion which identifies thetwo key structural parameters at the same time That meansthat the index 119869(119899) is changed into the 119869(119899 119889) form

For each given individual 119889 a state space model canbe identified using the Moesp algorithm described in

Mathematical Problems in Engineering 5

Initialization 119899 = 2 119889 = 1 the modelling data after pretreatment 119906(119896 minus 119889max) 119906(119896) 119906(119871) and 119910(119896) 119910(119871)(1) for 119899 = 2 to 119899 = 119899max(2) for 119889 = 1 to 119889 = 119889max(3) Rolling the modelling input data 119906 based on the hypothesis delay 119889 get the data set 119906(119896 minus 119889) 119906(119896) 119906(119871 minus 119889)

119910(119896) 119910(119871)(4) Construct input and output Hankel matrices 119880

119901

119884119901

119880119891

119884119891

(5) Calculate 119860 119861 119862119863 by Moesp method in Section 3 based on Hankel matrices(6) Substitute the 119860 119861 119862119863 and 119889 into the formula (1)(7) Calculate and store the model error and performance index 119869(119899 119889)(8) end for(9) end for(10) Search the inflection point of the 119869(119899 119889) surface

Algorithm 1

y

y

L

L

xk k + Lo

xo k + 1 k + L + 1

Figure 2 Sliding time window sketch map

Section 2 Then its model error can be deserved as Σ119899119889

=

(1119873)sum119873

119894=1

119890(119896)119890(119896)119879 then the AIC(119899 119889) with respect to

individual (119899 119889) as

AIC119873(119899 119889) = 119873 (119898 (1 + ln 2120587) + ln 1003816100381610038161003816Σ119899119889

1003816100381610038161003816) + 2120575119899119872119899 (13)

Also the MRSE criterion 119869MSE2 has the similar form

119869MRSE (119899 119889) =1

119871

119871

sum119896=1

radicsum119899119910

119895=1

(119910119899119889119896

(119895) minus 119910119899119889119896

(119895))2

sum119899119910

119895=1

119910119899119889119896

(119895)2

(14)

Other performance index 119869(sdot) such as SVC IVC NICcriteria mentioned in [25] can also be modified as thismethod

The original performance index just identifies the order119899 Suppose 119899 isin [1 119899max] then 119869(1) 119869(2) 119869(3) 119869(119899max) arecalculated respectively then the inflection point 119899lowast is the best

order and the corresponding systemmatrices119860lowast 119861lowast 119862lowast119863lowastare the best model After improvement we add the delay asother optimal parameters So the system order 119899 isin [1 119899max]and input delay 119889 isin [1 119889max] are all embedded in 119869(119899 119889)Then calculate 119869(1 1) 119869(119899max 1) 119869(1 2) 119869(119899max 2)119869(1 119889max) 119869(119899max 119889max) respectively By searching theminimum point 119899lowast 119889lowast of surface 119869(119899 119889) the best orderdelay and system matrices can all be got Thus taking thedelay as another parameter in the modelling methods it caneffectively avoid high order results in the delay system

42 The Delay Factor Stripping from Historical Data Theintroduction of delay parameters in performance criterionhas resulted to a notable problem The modelling historicaldata matrices have already included delay information It isdifficult to change 119889 in the performance criterion 119869(119899 119889) arti-ficially To solve this problem the sliding-window method isadopted here Sliding-window principle is shown in Figure 2When new samples are added to the window the oldest datainside the window will be discarded

We use sliding window to change delay which is shownin Figure 3 Suppose the output data length is 119871 select theinput data region 119906(119896 minus 119889max) 119906(119896) 119906(119871) Then theinput data can be moved according to different delay from119889 to 119889max

43 The Algorithm Description The detailed procedure ofsubspace identification based on ODC algorithm can beexpressed as Algorithm 1

Then the best combination 119899lowast 119889lowast and the optimum

matching model parameters are all obtained

5 Simulation Results

To demonstrate the superiority of the proposed order selec-tion method in this paper over the conventional methodtheir performance is evaluated through a numerical exampleand an industrial illustrate

6 Mathematical Problems in Engineering

u(k minus 1)

u(k)

u(L)

(d = 0)

y(k)

y(k + 1)

y(L)

u(k minus 1)

u(L minus 1)

u(L)

(d = 1)

y(k)

y(k + 1)

y(L)

u(L minus 1)

u(L)

y(k)

y(k + 1)

y(L)

u(k minus dmax) u(k minus dmax) u(k minus dmax)

u(L minus dmax)

(d = dmax)

middot middot middot

Figure 3 Modeling data sliding sketch map

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

minus1

minus2

minus3

minus4

u1

(a) Input signal 1199061

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

minus1

minus2

minus3

minus4

u2

(b) Input signal 1199062

0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

y1

minus5

minus10

minus15

minus20

(c) Output signal 1199101

0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

y2

minus5

minus10

minus15

(d) Output signal 1199102

Figure 4 PO-Moesp subspace system identification inputoutput signal

51 Example 1 MIMO Process with Input Delay The firstmodel (15) is a MIMO system

119909 (119896 + 1) =[[[

[

0603 0603 0 0

minus0603 0603 0 0

0 0 minus0603 minus0603

0 0 0603 minus0603

]]]

]

119909 (119896)

+[[[

[

11650 minus06965

06268 16961

00751 00591

03516 17971

]]]

]

119906 (119896 minus 5)

119910 (119896) = [02641 minus14462 12460 05774

08717 minus07012 minus06390 minus03600] 119909 (119896)

+ [minus01356 minus12704

minus13493 09846] 119906 (119896 minus 5)

(15)

The time delay is 119889 = 5 and the sample is 119871 = 1000 Weuse the zeromeanwhite noise as the input 119906

1and 1199062sequence

to excite the system The input and output curves are shownin Figure 4

Firstly we use conventional subspace model selectionmethod which is dependant on the number of eigenvaluesFigure 5(a) is the situation of delay = 0 and Figure 5(b) is the

Mathematical Problems in Engineering 7

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Order

Eige

nval

ue

(a) 119889 = 0 situation

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

Order

Eige

nval

ue

(b) 119889 = 5 situation

Figure 5The number of eigenvalues generated when 119889 = 0 and 119889 =5 respectively (a) 119889 = 0 and (b) 119889 = 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

Perfo

rman

ce in

dexJ

Order n

JMRSEJMSE2

JMSE1

Figure 6 Obtain the input delay system order by original errorcriterion methods

situation of delay = 5 According to this strategy the order ofmodel increases from 4 to 13 when the system has delay

Also theMSE (mean squared error)MRSE (mean relativesquared error) and AIC criteria are all tested as shown inFigures 6 and 7Thesemethods all have the distinct inflectionpoint at 13 which is for the 4-order system with input delay

2 4 6 8 10 12 14 16 18

0

1

2

AIC

inde

x

minus6

minus5

minus4

minus3

minus2

minus1

Order n

times104

Figure 7 Obtain the input delay system order by AIC index

0246810 123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

1

Figure 8 The corresponding mean square error surface 119869MSE1generated by ODC

5 Obviously for obtaining the ideal model of a delay systemthese conventional methods have to increase the order

Next the performances of the proposed ODCmethod inthis paper are presented As the two search parameters areavailable so the index performance is shown as a surfaceThe 119869MSE1 of output 1199101 is shown in Figure 8 and the 119869MSE2 ofoutput 119910

2is shown in Figure 9 Three axes are the order 119899

delay 119889 and 119869(sdot) respectivelyTheAIC index surface is shownin Figure 10 The minimum value of these indexes can be goteasily they are also the inflection pointsThe order and delayresults are 119899 = 4 119889 = 5 This is the same with the actualsystem model

The corresponding identified model matrices

119860 =[[[

[

05017 06047 02024 01514

minus06256 05164 03929 minus00828

02228 03763 minus04884 minus06090

01883 minus00936 06069 minus05297

]]]

]

119861 =[[[

[

minus00386 18602

minus10028 00809

minus05982 minus10787

minus00583 minus04779

]]]

]

8 Mathematical Problems in Engineering

02

46

810

123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

2

Figure 9 The corresponding mean square error surface 119869MSE2generated by ODC

02468

10 12

34

56

0

2

AIC

inde

x

minus6

minus4

minus2

Order nDelay d

n = 4 d = 5

times104

Figure 10 The corresponding AIC criterion surface generated byODC

119862 = [minus14276 10033 minus11161 03224

minus11801 minus05779 04116 minus04398]

119863 = [minus01356 minus12704

minus13493 09846]

(16)

For verifying the model the model output error is drawnas 1198901and 1198902in Figure 11

52 Example 2 The Kiln Industrial Illustration To demon-strate the superiority of the proposed order selection methodin this paper over the conventional method their perfor-mance is evaluated through an industrial illustration Thedata come from actual kiln production data of an enterpriseHere the gas flow and the second air flow are selected asthe control input and the calcination temperature and kilntail temperature are taken as output variables The samplingtime is 119879

119904= 1 119904 then use Moesp method modelling the

kiln based on the inputoutput data after preprocessing Herethree main practical problems are solved

521 The First Problem The problem is that these two inputvariables have different delay They need to be identified

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus12

minus10

minus8

minus6

minus4

minus2

times10minus15

e 1

(a) 1198901

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus6

minus4

minus2

times10minus15

e 2

(b) 1198902

Figure 11 Modeling error curves 1198901

(a) and 1198902

(b)

respectively that is to say a triple loop about 1198891 1198892 and 119899

should be carried for solving 119869(119899 1198891 1198892)

In order to obtain the inflection point information moredirectly we identify the order 119899 firstly According to theindustrial field situation choose the possibility maximumvalues which are 119889

1max = 150 1198892max = 150 At first

travel all the possible 1198891 1198892and compute the smallest 119869MSE1

119869MSE2 and 119869MRSE corresponding to the different order 119899Thesecurves are shown in Figure 12 As can be seen from thecurves all of these three error criteria achieve the inflectionpoint at 119899 = 5 so the order is got Then specify 119899 =

5 the triple loop is reduced to double loop which is tocompute 119869MSE1 and 119869MSE2 corresponding to 1198891 and 1198892 between[0 150]

522The Second Problem From Figure 13 we notice anotherproblem that the outputs 119910

1and 119910

2generate different inflec-

tion pointIn order to solve this problem the criterion should choose

MRSE and AIC which take into account 1199101and 119910

2both

together Then get Figures 14 and 15It can be conducted that 119899 = 5 119889

1= 30 119889

2= 100

Mathematical Problems in Engineering 9

Table 2 When 119879119904

= 10 s the order the minmum 119869MRSE and the delay

119899 1 2 3 4 5 6 7 81198891

4 3 3 3 3 3 3 31198892

5 10 10 10 10 10 10 10119869MRSE 6981251 595572 499145 384619 239931 239931 239931 239931

1 2 3 4 5 6 7 8 9 10 11 120

50

100

1 2 3 4 5 6 7 8 9 10 11 120

200

400

1 2 3 4 5 6 7 8 9 10 11 120

02

04

Order n

Order n

Order n

J MRS

EJ M

SE2

J MSE

1

Figure 12 The corresponding minimum 119869(119899 1198891

1198892

) curves of eachorder when traversal 119889

1

1198892

0 50 100 150 0 50 100 150050

100150200250300350

Delay d2 Delay d1

J MSE

i

JMSE1

JMSE2

50 100

JMJJ SE1

JMJJ SE2

Figure 13 119869MSE1 and 119869MSE2 surface generated by ODC when 119899 = 5

0 50 100150 0 50 100 150

384

42444648

5

AIC

inde

x

d2 d1

times104

d1 = 30 d2 = 100

AIC = 40159e + 04

d1 = 3330000 dddd222 = 1000

AICAIC 4 0150159e + 0444

Figure 14 AIC surface generated by ODC when 119899 = 5

0 50 100 150 0 50 100 150

0

50

100

150

200

250

d2 d1

J MRS

E

d1 = 30 d2 = 100 JMRSE = 16448d = 30 d = 100 JMJJ RSE = 161 4448

Figure 15 119869MRSE surface generated by ODC when 119899 = 5

0 50100

150

050

100150

050

100150200250300350400

(3 10 239931)

J MRS

E

d2 d150

10050100

Figure 16 119869MRSE surface generated by ODC when 119899 = 5 and 119879119904

=

10 s

0 100 200 300 400 500 600 700 800 900 10001220

1240

1260

1280

1300

1320

1340

1360

1380

Time (s)

Measured valuePredictive value

Kiln

tail

tem

pera

ture

y1

(∘C)

Figure 17 Comparison of calcination temperature measured curveand model predictive curve

10 Mathematical Problems in Engineering

Measured valuePredictive value

0 100 200 300 400 500 600 700 800 900 10001010

1020

1030

1040

1050

1060

1070

1080

1090

Time (s)

Kiln

tail

tem

pera

ture

y2

(∘C)

Figure 18 Comparison of kiln tail temperaturemeasured curve andmodel predictive curve

523 The Third Problem However there is still a problemwhich needs to be resolved In general the model with largetime delay will increase the difficulty of controller designingWe know that when the sampling frequency is higher thanthe actual needed frequency there will be lots of redundantdata And this will raise the model order and the delayTherefore the delay can be reduced by properly decreasingsampling frequency Considering the kiln is a slow time-varying process changing the sampling time from 1 s to 10 swill not affect the model accuracy

From Table 2 it can be seen that when set 119879119904= 10 s the

order and inflexion point is still 119899 = 5 We can also changethe delay to 119889

1= 3010 = 3 119889

2= 10010 = 10 The 119869MRSE

surface can be got as in Figure 16 the results show that 1198891=

3 1198892= 10 This is in agreement with the analysis before

The corresponding rotary kiln calcining zone tempera-ture model is

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896 minus 1198891)

119910 (119896) = 119862119909 (119896) + 119863119906 (119896 minus 1198892)

(17)

and the order 119899 = 5 delay 1198891= 3 119889

2= 10

119860 =

[[[[[

[

09936 00015 minus00062 minus00007 minus00001

00063 09906 00170 minus00017 minus00001

minus00009 00026 09793 minus00147 00209

00000 minus00001 00018 09985 minus00069

minus00000 00001 minus00015 00062 09848

]]]]]

]

119861 = 10minus3

times

[[[[[

[

minus00908 minus00129

minus01650 00957

00680 minus00329

minus00734 00191

minus00093 minus00012

]]]]]

]

119862 = [minus57995 29327 minus05841 minus00157 00260

minus67753 minus23024 minus00247 minus00596 minus00063]

119863 = 10minus15

times [01205 02699

minus01066 00143]

(18)

Compare the measured value and the predicted valuegenerated by the model in Figures 17 and 18

6 Conclusion

In this paper the calcining belt state space model of rotarykiln is built using PO-Moesp subspace method And a noveldouble parameters error performance criterion for the orderchoosing in subspace modelling is introduced Since thepresented method considering the order and delay simulta-neously it can reduce the model order of the delay systemeffectively And also a strategy for stripping the delay factorsfrom the historical data is also proposed The algorithm isverified in identifying an industrial lime kiln In this examplewe solve the practical problem of industrial process withmultidelay and also reduce the order by adjusting samplingtime Further research could shed more light on the issue ofapplying the model online The problem in industrial field ismore complex than simulation environment How to extractproblems from industrial practice and guide the direction ofmodeling research has become the study focus

Conflict of Interests

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

References

[1] M Georgallis P Nowak M Salcudean and I S GartshoreldquoModelling the rotary lime kilnrdquo The Canadian Journal ofChemical Engineering vol 83 no 2 pp 212ndash223 2005

[2] Z Sogut Z Oktay and H Karakoc ldquoMathematical modeling ofheat recovery from a rotary kilnrdquo Applied Thermal Engineeringvol 30 no 8-9 pp 817ndash825 2010

[3] Y H Kim ldquoDevelopment of process model of a rotary kilnfor volatile organic compound recovery from coconut shellrdquoKorean Journal of Chemical Engineering vol 29 no 12 pp 1674ndash1679 2012

[4] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[5] W Weijtjens G de Sitter C Devriendt and P GuillaumeldquoOperational modal parameter estimation of MIMO systemsusing transmissibility functionsrdquo Automatica vol 50 no 2 pp559ndash564 2014

[6] M Imber and V Paschkis ldquoA new theory for a rotary-kiln heatexchangerrdquo International Journal of Heat andMass Transfer vol5 no 7 pp 623ndash638 1962

[7] A Sass ldquoSimulation of heat-transfer phenomena in a rotarykilnrdquo Industrial amp Engineering Chemistry Process Design andDevelopment vol 6 no 4 pp 532ndash535 1967

Mathematical Problems in Engineering 11

[8] S D Shelukar H G K Sundar R Semiat J T Richardson andD Luss ldquoContinuous rotary kiln calcination of yttrium bariumcopper oxide precursor powdersrdquo Industrial and EngineeringChemistry Research vol 33 no 2 pp 421ndash427 1994

[9] Y Yang J Rakhorst M A Reuter and J H L Voncken ldquoAnal-ysis of gas flow and mixing in a rotary kiln waste incineratorrdquoin Proceedings of the 2nd International Conference on CFD inthe Minerals and Process Industries pp 443ndash448 MelbourneAustralia

[10] Y Wang X H Fan and X L Chen ldquoMathematical modelsand expert system for grate-kiln process of iron ore oxide pelletproduction (Part I) mathematical models of grate processrdquoJournal of Central South University of Technology vol 19 no 4pp 1092ndash1097 2012

[11] G Mercere L Bako and S Lecœuche ldquoPropagator-basedmethods for recursive subspace model identificationrdquo SignalProcessing vol 88 no 3 pp 468ndash491 2008

[12] P Misra and M Nikolaou ldquoInput design for model orderdetermination in subspace identificationrdquo AIChE Journal vol49 no 8 pp 2124ndash2132 2003

[13] B Liu B Fang X Liu J Chen Z Huang and X HeldquoLarge margin subspace learning for feature selectionrdquo PatternRecognition vol 46 no 10 pp 2798ndash2806 2013

[14] H Oku and H Kimura ldquoRecursive 4SID algorithms usinggradient type subspace trackingrdquo Automatica vol 38 no 6 pp1035ndash1043 2002

[15] Y Subasi and M Demirekler ldquoQuantitative measure of observ-ability for linear stochastic systemsrdquo Automatica vol 50 no 6pp 1669ndash1674 2014

[16] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[17] X Pan H Zhu F Yang and X Zeng ldquoSubspace trajectorypiecewise-linear model order reduction for nonlinear circuitsrdquoCommunications in Computational Physics vol 14 no 3 pp639ndash663 2013

[18] M Doumlhler and L Mevel ldquoFast multi-order computationof system matrices in subspace-based system identificationrdquoControl Engineering Practice vol 20 no 9 pp 882ndash894 2012

[19] T Breiten and T Damm ldquoKrylov subspace methods for modelorder reduction of bilinear control systemsrdquo Systems and Con-trol Letters vol 59 no 8 pp 443ndash450 2010

[20] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[21] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol 19 no 6 pp 716ndash723 1974

[22] E E Ioannidis ldquoAkaikersquos information criterion correction forthe least-squares autoregressive spectral estimatorrdquo Journal ofTime Series Analysis vol 32 no 6 pp 618ndash630 2011

[23] K Peternell W Scherrer and M Deistler ldquoStatistical analysisof subspace identification methodsrdquo in Proceedings of the 3rdEuropean Control Conference (ECC rsquo95) vol 2 p 1342 1995

[24] D Bauer ldquoOrder estimation in the context of MOESP subspaceidentification methodsrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[25] D Bauer ldquoOrder estimation for subspace methodsrdquo Automat-ica vol 37 no 10 pp 1561ndash1573 2001

[26] J Shalchian A Khaki-Sedigh and A Fatehi ldquoA subspacebased method for time delay estimationrdquo in Proceedings of the

4th International Symposium on Communications Control andSignal Processing (ISCCSP rsquo10) p 4 March 2010

[27] J Lee and T F Edgar ldquoSubspace identification method forsimulation of closed-loop systems with time delaysrdquo AIChEJournal vol 48 no 2 pp 417ndash420 2002

[28] H Zhang T Ma G-B Huang and Z Wang ldquoRobust globalexponential synchronization of uncertain chaotic delayed neu-ral networks via dual-stage impulsive controlrdquo IEEE Transac-tions on Systems Man and Cybernetics B Cybernetics vol 40no 3 pp 831ndash844 2010

[29] P van Overschee and B deMoor ldquoA unifying theorem for threesubspace system identification algorithmsrdquo Automatica vol 31no 12 pp 1853ndash1864 1995

[30] W Favoreel B de Moor and P van Overschee ldquoSubspace statespace system identification for industrial processesrdquo Journal ofProcess Control vol 10 no 2 pp 149ndash155 2000

[31] P van Overschee and B deMoor ldquoN4SID subspace algorithmsfor the identification of combined deterministic-stochasticsystemsrdquo Automatica vol 30 no 1 pp 75ndash93 1994

[32] M Verhaegen ldquoIdentification of the deterministic part ofMIMO state space models given in innovations form frominput-output datardquo Automatica vol 30 no 1 pp 61ndash74 1994

[33] W E Larimore ldquoCanonical variate analysis in identificationfiltering and adaptive controlrdquo in Proceedings of the 29th IEEEConference on Decision and Control pp 596ndash604 December1990

[34] M Viberg ldquoSubspace-based methods for the identification oflinear time-invariant systemsrdquo Automatica vol 31 no 12 pp1835ndash1851 1995

[35] J Wang and S J Qin ldquoA new subspace identification approachbased on principal component analysisrdquo Journal of ProcessControl vol 12 no 8 pp 841ndash855 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Baure introduces another criterion for the Larimore type ofprocedures which is similar to AIC [24 25] From aboveliterature an obvious problem can be found that the delayparameter is always neglected in systems order selectionThe information about the delay of any process is valuablefor industrial process model However there exist onlyfew references dealing with the estimation of the order indelay system [26ndash28] When the system has input delaymany methods attempt to increase the model order as thecost to improve model accuracy This enhances the systemcomplexity

In this paper the calcining belt state spacemodel of rotarykiln is built by using PO-Moesp subspace method Afteranalyzing the main components technology and calcinationsreaction mechanism in detail the input and output variablesof modeling are selected according to industrial field opera-tional procedures Andmore a novel order-selectingmethodis put forward The original error criterion is replaced by thenew one which includes the two parametersmdashthe order andthe delay This improvement effectively solves the problem ofmodel order too high whenmodelling object with time delayand it also increases the precision of model

The outline of this paper is as follows in Section 2 thebasic process of rotary kiln is described in Section 3 thegeneral subspace identification algorithm is introduced andthe order selection problem is addressed Section 4 presentsthe main results of this paper The method to identify theorder and the delay is deduced in detail In Section 5 wediscuss the performance of the proposed method by meansof some simulation studies on an industrial rotary kilnillustration

2 Lime Rotary Kiln Process Description

The rotary lime kiln is a long cylinder that has a 3ndash5∘dslope and it rotates about its axis at roughly 05ndash15 rpmLime mud is fed from the elevated cold end and movesdown the kiln due to rotation and gravity The hot endin which the burner operates is typically maintained atabout 1200∘C by burning fuel [1] Active lime rotary kilnsystem has many technological processes such as long kilntechnology system chain grate cooling technology systemand vertical preheater technology system The process ismainly divided intomaterial system and air flow system Rawmaterial becomes product after preheating high temperaturecalcinations and cooling Primary air secondary air andcoke oven gas constitute the air flow system The wholetechnological process is shown in Figure 1 In this paper thelime kiln under study is 600 td 40m times 60m

The main calcinations process parameters are listed inTable 1

3 Problem Formulation and Assumptions

A discrete-time model in state space form is described by theequations

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896 minus 119889) + 119908 (119896)

119910 (119896) = 119862119909 (119896) + 119863119906 (119896 minus 119889) + V (119896) (1)

Table 1 Process parameters of rotary kiln

Parameter The parameter of kilnRaw material granularity 10sim40mmPreheater temperature 900sim1100∘CKiln head temperature 800sim950∘CKiln tail temperature 1100 plusmn 20∘CCalcining zone temperature 1260sim1340∘CSecond air temperature 750sim900∘CKiln head cover temperature 800sim950∘CExhaust gas temperature 250∘CLime temperature after cooling le150∘CKiln head pressure range minus10simminus20 paGas pressure ge6KPaCalcinations process pressure Negative pressure 100ndash150 PaPulse jet cleaning pressure 035sim05MPa3920 dust removel pressure difference le15 kPaKiln body inclination 3sim5Kiln speed range 02sim128 rminCaO content gt88Activity degree gt350mLExhaust gas residual carbon lt=1Exhaust gas oxygen content 07sim25Dust collector air volume 225000m3hMaterial level in preheater 5sim88mMaterial level in cooler le055mproduction 600 Tsim650 TSiO2 lt15 or lt2MgO 5Power consumption 65 kwsdothtsim60 kwsdothtHeat consumption 1630sim1650 kJkg

where input 119906119896isin 119877119898 and output 119910

119896isin 119877119897 119898 and 119897 are

input and output dimensions respectively 119909119896isin 119877119899 is the

system state at time 119896 and 119899 is the system order to beidentified process noise vector 119908

119896isin 119877119899 and measurement

noise vector V119896

isin 119877119897 are zero mean value white noiseseries

We introduce the following hypotheses (van Overscheeand de Moor 1995) [29]

(a) The system is asymptotically stable

(b) The system is observable and reachable

(c) All the process noise 119908(119896) and measurement noiseV(119896) are statistically independent of the input 119906(119896)that is mean 119864119906(119894)119908(119895) = 119864119906(119894)V(119895) = 0 where119864sdot = lim

119873rarrinfin(1119873)sum119864(sdot)

The goal of subspace methods consist in the onlineestimating order 119899 delay 119889 and system matrices 119860 119861 119862 and119863

Mathematical Problems in Engineering 3

Preheater

Rotary kiln

Cooler

Figure 1 Technological process chart of rotary kiln

31 Matrices Construct Suppose at time 119896 the past input andoutput data samples are available then construct pastfutureoutput vectors and the Hankel output matrices as follows

119880119901= (

1199060

1199061sdot sdot sdot 119906

119895minus1

1199061

1199062sdot sdot sdot 119906

119895

d

119906119894minus1

119906119894sdot sdot sdot 119906119894+119895minus2

)

(2a)

119884119891= (

119910119894

119910119894+1

sdot sdot sdot 119910119894+119895minus1

119910119894+1

119910119894+2

sdot sdot sdot 119910119894+119895

d

1199102119894minus1

1199102119894

sdot sdot sdot 1199102119894+119895minus2

) (2b)

where 119906119894= [119906

1198941 1199061198942 119906

119894119897] 119910119894= [119910

1198941 1199101198942 119910

119894119898] 119901

represent the past information and 119891 represent the futureinformation 119894 ge 119899 119895 ≫ max(119894119897 119894119898)

The state matrix 119883119901and 119883

119891is defined as follows 119883

119901=

[11990901199091sdot sdot sdot 119909119895minus1]119883119891= [119909119894119909119894+1

sdot sdot sdot 119909119895+119894minus1

]The augmented observation matrix Γ

119894

Γ119894= [119862 119862119860 sdot sdot sdot 119862119860119894]

119879

(3)

Block-triangular matrix Toeplitz matrix119867119894and119867119904

119894

119867119894=

[[[[[[[

[

119863 0 sdot sdot sdot 0

119862119861 119863 d

d 0

119862119894minus2119861 119862119894minus1119861 sdot sdot sdot 119863

]]]]]]]

]

119867119904

119894

=

[[[[[[[

[

0 0 sdot sdot sdot 0

119862 0 d

d 0

119862119894minus2 119862119894minus1 sdot sdot sdot 0

]]]]]]]

]

(4)

The augmented observation inverse matrix of 119860 119861 isΔ119894= [119860119894minus1

119861 119860119894minus2

119861 sdot sdot sdot 119861] And the augmented observationinverse matrix of 119860119870 is Δ119904

119894

= [119860119894minus1

119870 119860119894minus2

119870 sdot sdot sdot 119870]Suppose the input and state matrices are all row full

rank matrices and they are row space orthogonal Then theaugmented inputoutput matrices can be written as follows

119883119891= 119860119894

119883119901+ Δ119894119880119901+ Δ119904

119894

119864119891 (5a)

119884119901= Γ119894119883119901+ 119867119894119880119901+ 119867119904

119894

119864119901 (5b)

119884119891= Γ119894119883119891+ 119867119894119880119891+ 119867119904

119894

119864119891 (5c)

The key step of Moesp method is to estimate the aug-mented observability matrix through the projection future119868119874 data onto past 119868119874 data The augmented observabilitymatrix and state space vector estimation can be got throughSVD decomposition

32 General Algorithm In this section we introduce thegeneral subspace method framework which is profferedby Favoreel et al [30] Subspace identification algorithmconsists of two main steps The first step always performsa weighted projection of the row space of the previouslydefined data Hankel matrices From this projection theaugmented observabilitymatrix Γ

119894and estimate119883

119894of the state

sequence119883119894can be retrievedThen the order is got from SVD

decomposition In the second step the system matrices 119860 119861119862119863 and119876 119878119877 are determined through least squaremethodThe concrete Moesp algorithm is as follows

4 Mathematical Problems in Engineering

(1) Project the 119884119891row space into the orthogonal comple-

ment of the 119880119891row space

119884119891119880perp

119891

= Γ119894119883119891119880perp

119891

+ 119867119889

119894

119880119891119880perp

119891

+ 119867119904

119894

119864119891119880perp

119891

(6)

Since it is assumed that the noise is uncorrelated withthe inputs so 119864

119891119880perp119891

= 119864119891and 119880

119891119880perp119891

= 0 therefore119884119891119880perp119891

= Γ119894119883119891119880perp119891

+ 119867119904119894

119864119891

(2) Select the weighting matrices1198821and119882

2

1198821119884119891

119880perp119891

1198822

=1198821Γ119894119883119891

119880perp119891

1198822

+1198821119867119904

119894

1198641198911198822 (7)

The weighting matrices can be chosen appropriatelyaccording to different subspace methods includingN4SID MOESP CVA basic-4SID and IV-4SID [31ndash34]

Then we can get

119900119894=1198821119884119891

119880perp119891

1198822

=1198821Γ119894119883119891

119880perp119891

1198822

(8)

(3) Carry SVD decomposition

119900119894= (11988011198802) (11987810

0 0)(

119881119879

1

1198811198792

) (9)

And then take the number of nonzero eigenvalue asthe system order rank(119900

119894) = 119899

(4) The augmented observability matrix Γ119894= 119882minus11

119880111987812

1

or119883119894= 119883119894119880perp119891

1198822is derived from the third step

(5) Extract estimate 119860 119861 119862119863 from Γ119894or119883119894

Remark By reference to [30] the weighting matrices1198821and

1198822should satisfy the following three conditions

(1) rank(1198821sdot Γ119894) = rank Γ

119894

(2) rank(119883119894119880perp119891

sdot 1198822) = rank119883

119894

(3) 1198821sdot (119867119904119894

119872119891+ 119873119891) sdot 1198822= 0

The first two conditions guarantee that the rank-119899 prop-erty of Γ

119894119883119894is preserved after projection onto 119880perp

119891

andweighting by119882

1and119882

2 The third condition expresses that

1198822should be uncorrelated with the noise sequences 119908(119896)

and V(119896) By choosing the appropriate weighting matrices1198821and 119882

2 all subspace algorithms for LTI systems can

be interpreted in the above framework including N4SIDMOESP CVA Basic-4SID and IV-4SID

4 The Proposed Method

In the classical system identification theory the actual modelstructure is usually assumed to be known However inpractical it is always not clear Subspace system identificationmethod determines the order of the system by the nonzeroeigenvalue of the augmented observability matrix Howeverthe system nonzero singular values may be very small Thismay lead to the wrong system order and large identificationerror

41 The Order-Delay Double Parameters Error Criterion Themost directly order-selection method is based on the errorperformance criterion This idea is to choose the smallestpossible order that keeps the error below a certain levelThenthe MRSE (mean relative squared error) index is introducedby model error as follows

119869MRSE (119899) =1

119871

119871

sum119896=1

radicsum119899119910

119895=1

(119910119896(119895) minus 119910

119896(119895))2

sum119899119910

119895=1

119910119896(119895)2

(10)

where 119910119896(119895) minus 119910

119896(119895) is the model prediction error and 119871 is

the sample number In [35] use the AIC which was originallydeveloped by Akaike and then adapted by Larimore for SMIGiven a set of samples for a sequence of system order 119899 forexample 119899 isin [0 sdot sdot sdot 20] the order of the model will be the onewhich makes the following AIC index minimum

AIC119899(119899) = 119873 (119898 (1 + ln 2120587) + ln 1003816100381610038161003816Σ119899

1003816100381610038161003816) + 2120575119899119872119899 (11)

where

Σ119899=

1

119873

119873

sum119894=1

119890 (119896) 119890(119896)119879

119890 (119896) = 119910119899(119896) minus 119910

119899(119896)

119872119899= 2119899119898 +

119898 (119898 + 1)

2+ 119899119897 + 119898119897

120575119899=

119873

119873 minus ((119872119899119898) + ((119898 + 1) 2))

(12)

For calculating the AIC(119899) criterion we first suppose theupper bound 119899max of the system order and then calculate theAIC119873(1)AIC

119873(2) AIC

119873(119899max) sequence the appropri-

ate system order 119899 is the one which decrees the AIC indexobviously and the order should be as small as possible TheMRSE and AIC index 119869MRSE(119899) can be analyzed in the samemanner

However the performance index based on a single orderparameter cannot provide an effective solution to the delaysystem which is shown in (1) This led to the problem thatthe original identificationmethod had to increase the order ascost to improvemodel accuracy Here we introduce an order-delay double parameters error criterion which identifies thetwo key structural parameters at the same time That meansthat the index 119869(119899) is changed into the 119869(119899 119889) form

For each given individual 119889 a state space model canbe identified using the Moesp algorithm described in

Mathematical Problems in Engineering 5

Initialization 119899 = 2 119889 = 1 the modelling data after pretreatment 119906(119896 minus 119889max) 119906(119896) 119906(119871) and 119910(119896) 119910(119871)(1) for 119899 = 2 to 119899 = 119899max(2) for 119889 = 1 to 119889 = 119889max(3) Rolling the modelling input data 119906 based on the hypothesis delay 119889 get the data set 119906(119896 minus 119889) 119906(119896) 119906(119871 minus 119889)

119910(119896) 119910(119871)(4) Construct input and output Hankel matrices 119880

119901

119884119901

119880119891

119884119891

(5) Calculate 119860 119861 119862119863 by Moesp method in Section 3 based on Hankel matrices(6) Substitute the 119860 119861 119862119863 and 119889 into the formula (1)(7) Calculate and store the model error and performance index 119869(119899 119889)(8) end for(9) end for(10) Search the inflection point of the 119869(119899 119889) surface

Algorithm 1

y

y

L

L

xk k + Lo

xo k + 1 k + L + 1

Figure 2 Sliding time window sketch map

Section 2 Then its model error can be deserved as Σ119899119889

=

(1119873)sum119873

119894=1

119890(119896)119890(119896)119879 then the AIC(119899 119889) with respect to

individual (119899 119889) as

AIC119873(119899 119889) = 119873 (119898 (1 + ln 2120587) + ln 1003816100381610038161003816Σ119899119889

1003816100381610038161003816) + 2120575119899119872119899 (13)

Also the MRSE criterion 119869MSE2 has the similar form

119869MRSE (119899 119889) =1

119871

119871

sum119896=1

radicsum119899119910

119895=1

(119910119899119889119896

(119895) minus 119910119899119889119896

(119895))2

sum119899119910

119895=1

119910119899119889119896

(119895)2

(14)

Other performance index 119869(sdot) such as SVC IVC NICcriteria mentioned in [25] can also be modified as thismethod

The original performance index just identifies the order119899 Suppose 119899 isin [1 119899max] then 119869(1) 119869(2) 119869(3) 119869(119899max) arecalculated respectively then the inflection point 119899lowast is the best

order and the corresponding systemmatrices119860lowast 119861lowast 119862lowast119863lowastare the best model After improvement we add the delay asother optimal parameters So the system order 119899 isin [1 119899max]and input delay 119889 isin [1 119889max] are all embedded in 119869(119899 119889)Then calculate 119869(1 1) 119869(119899max 1) 119869(1 2) 119869(119899max 2)119869(1 119889max) 119869(119899max 119889max) respectively By searching theminimum point 119899lowast 119889lowast of surface 119869(119899 119889) the best orderdelay and system matrices can all be got Thus taking thedelay as another parameter in the modelling methods it caneffectively avoid high order results in the delay system

42 The Delay Factor Stripping from Historical Data Theintroduction of delay parameters in performance criterionhas resulted to a notable problem The modelling historicaldata matrices have already included delay information It isdifficult to change 119889 in the performance criterion 119869(119899 119889) arti-ficially To solve this problem the sliding-window method isadopted here Sliding-window principle is shown in Figure 2When new samples are added to the window the oldest datainside the window will be discarded

We use sliding window to change delay which is shownin Figure 3 Suppose the output data length is 119871 select theinput data region 119906(119896 minus 119889max) 119906(119896) 119906(119871) Then theinput data can be moved according to different delay from119889 to 119889max

43 The Algorithm Description The detailed procedure ofsubspace identification based on ODC algorithm can beexpressed as Algorithm 1

Then the best combination 119899lowast 119889lowast and the optimum

matching model parameters are all obtained

5 Simulation Results

To demonstrate the superiority of the proposed order selec-tion method in this paper over the conventional methodtheir performance is evaluated through a numerical exampleand an industrial illustrate

6 Mathematical Problems in Engineering

u(k minus 1)

u(k)

u(L)

(d = 0)

y(k)

y(k + 1)

y(L)

u(k minus 1)

u(L minus 1)

u(L)

(d = 1)

y(k)

y(k + 1)

y(L)

u(L minus 1)

u(L)

y(k)

y(k + 1)

y(L)

u(k minus dmax) u(k minus dmax) u(k minus dmax)

u(L minus dmax)

(d = dmax)

middot middot middot

Figure 3 Modeling data sliding sketch map

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

minus1

minus2

minus3

minus4

u1

(a) Input signal 1199061

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

minus1

minus2

minus3

minus4

u2

(b) Input signal 1199062

0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

y1

minus5

minus10

minus15

minus20

(c) Output signal 1199101

0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

y2

minus5

minus10

minus15

(d) Output signal 1199102

Figure 4 PO-Moesp subspace system identification inputoutput signal

51 Example 1 MIMO Process with Input Delay The firstmodel (15) is a MIMO system

119909 (119896 + 1) =[[[

[

0603 0603 0 0

minus0603 0603 0 0

0 0 minus0603 minus0603

0 0 0603 minus0603

]]]

]

119909 (119896)

+[[[

[

11650 minus06965

06268 16961

00751 00591

03516 17971

]]]

]

119906 (119896 minus 5)

119910 (119896) = [02641 minus14462 12460 05774

08717 minus07012 minus06390 minus03600] 119909 (119896)

+ [minus01356 minus12704

minus13493 09846] 119906 (119896 minus 5)

(15)

The time delay is 119889 = 5 and the sample is 119871 = 1000 Weuse the zeromeanwhite noise as the input 119906

1and 1199062sequence

to excite the system The input and output curves are shownin Figure 4

Firstly we use conventional subspace model selectionmethod which is dependant on the number of eigenvaluesFigure 5(a) is the situation of delay = 0 and Figure 5(b) is the

Mathematical Problems in Engineering 7

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Order

Eige

nval

ue

(a) 119889 = 0 situation

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

Order

Eige

nval

ue

(b) 119889 = 5 situation

Figure 5The number of eigenvalues generated when 119889 = 0 and 119889 =5 respectively (a) 119889 = 0 and (b) 119889 = 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

Perfo

rman

ce in

dexJ

Order n

JMRSEJMSE2

JMSE1

Figure 6 Obtain the input delay system order by original errorcriterion methods

situation of delay = 5 According to this strategy the order ofmodel increases from 4 to 13 when the system has delay

Also theMSE (mean squared error)MRSE (mean relativesquared error) and AIC criteria are all tested as shown inFigures 6 and 7Thesemethods all have the distinct inflectionpoint at 13 which is for the 4-order system with input delay

2 4 6 8 10 12 14 16 18

0

1

2

AIC

inde

x

minus6

minus5

minus4

minus3

minus2

minus1

Order n

times104

Figure 7 Obtain the input delay system order by AIC index

0246810 123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

1

Figure 8 The corresponding mean square error surface 119869MSE1generated by ODC

5 Obviously for obtaining the ideal model of a delay systemthese conventional methods have to increase the order

Next the performances of the proposed ODCmethod inthis paper are presented As the two search parameters areavailable so the index performance is shown as a surfaceThe 119869MSE1 of output 1199101 is shown in Figure 8 and the 119869MSE2 ofoutput 119910

2is shown in Figure 9 Three axes are the order 119899

delay 119889 and 119869(sdot) respectivelyTheAIC index surface is shownin Figure 10 The minimum value of these indexes can be goteasily they are also the inflection pointsThe order and delayresults are 119899 = 4 119889 = 5 This is the same with the actualsystem model

The corresponding identified model matrices

119860 =[[[

[

05017 06047 02024 01514

minus06256 05164 03929 minus00828

02228 03763 minus04884 minus06090

01883 minus00936 06069 minus05297

]]]

]

119861 =[[[

[

minus00386 18602

minus10028 00809

minus05982 minus10787

minus00583 minus04779

]]]

]

8 Mathematical Problems in Engineering

02

46

810

123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

2

Figure 9 The corresponding mean square error surface 119869MSE2generated by ODC

02468

10 12

34

56

0

2

AIC

inde

x

minus6

minus4

minus2

Order nDelay d

n = 4 d = 5

times104

Figure 10 The corresponding AIC criterion surface generated byODC

119862 = [minus14276 10033 minus11161 03224

minus11801 minus05779 04116 minus04398]

119863 = [minus01356 minus12704

minus13493 09846]

(16)

For verifying the model the model output error is drawnas 1198901and 1198902in Figure 11

52 Example 2 The Kiln Industrial Illustration To demon-strate the superiority of the proposed order selection methodin this paper over the conventional method their perfor-mance is evaluated through an industrial illustration Thedata come from actual kiln production data of an enterpriseHere the gas flow and the second air flow are selected asthe control input and the calcination temperature and kilntail temperature are taken as output variables The samplingtime is 119879

119904= 1 119904 then use Moesp method modelling the

kiln based on the inputoutput data after preprocessing Herethree main practical problems are solved

521 The First Problem The problem is that these two inputvariables have different delay They need to be identified

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus12

minus10

minus8

minus6

minus4

minus2

times10minus15

e 1

(a) 1198901

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus6

minus4

minus2

times10minus15

e 2

(b) 1198902

Figure 11 Modeling error curves 1198901

(a) and 1198902

(b)

respectively that is to say a triple loop about 1198891 1198892 and 119899

should be carried for solving 119869(119899 1198891 1198892)

In order to obtain the inflection point information moredirectly we identify the order 119899 firstly According to theindustrial field situation choose the possibility maximumvalues which are 119889

1max = 150 1198892max = 150 At first

travel all the possible 1198891 1198892and compute the smallest 119869MSE1

119869MSE2 and 119869MRSE corresponding to the different order 119899Thesecurves are shown in Figure 12 As can be seen from thecurves all of these three error criteria achieve the inflectionpoint at 119899 = 5 so the order is got Then specify 119899 =

5 the triple loop is reduced to double loop which is tocompute 119869MSE1 and 119869MSE2 corresponding to 1198891 and 1198892 between[0 150]

522The Second Problem From Figure 13 we notice anotherproblem that the outputs 119910

1and 119910

2generate different inflec-

tion pointIn order to solve this problem the criterion should choose

MRSE and AIC which take into account 1199101and 119910

2both

together Then get Figures 14 and 15It can be conducted that 119899 = 5 119889

1= 30 119889

2= 100

Mathematical Problems in Engineering 9

Table 2 When 119879119904

= 10 s the order the minmum 119869MRSE and the delay

119899 1 2 3 4 5 6 7 81198891

4 3 3 3 3 3 3 31198892

5 10 10 10 10 10 10 10119869MRSE 6981251 595572 499145 384619 239931 239931 239931 239931

1 2 3 4 5 6 7 8 9 10 11 120

50

100

1 2 3 4 5 6 7 8 9 10 11 120

200

400

1 2 3 4 5 6 7 8 9 10 11 120

02

04

Order n

Order n

Order n

J MRS

EJ M

SE2

J MSE

1

Figure 12 The corresponding minimum 119869(119899 1198891

1198892

) curves of eachorder when traversal 119889

1

1198892

0 50 100 150 0 50 100 150050

100150200250300350

Delay d2 Delay d1

J MSE

i

JMSE1

JMSE2

50 100

JMJJ SE1

JMJJ SE2

Figure 13 119869MSE1 and 119869MSE2 surface generated by ODC when 119899 = 5

0 50 100150 0 50 100 150

384

42444648

5

AIC

inde

x

d2 d1

times104

d1 = 30 d2 = 100

AIC = 40159e + 04

d1 = 3330000 dddd222 = 1000

AICAIC 4 0150159e + 0444

Figure 14 AIC surface generated by ODC when 119899 = 5

0 50 100 150 0 50 100 150

0

50

100

150

200

250

d2 d1

J MRS

E

d1 = 30 d2 = 100 JMRSE = 16448d = 30 d = 100 JMJJ RSE = 161 4448

Figure 15 119869MRSE surface generated by ODC when 119899 = 5

0 50100

150

050

100150

050

100150200250300350400

(3 10 239931)

J MRS

E

d2 d150

10050100

Figure 16 119869MRSE surface generated by ODC when 119899 = 5 and 119879119904

=

10 s

0 100 200 300 400 500 600 700 800 900 10001220

1240

1260

1280

1300

1320

1340

1360

1380

Time (s)

Measured valuePredictive value

Kiln

tail

tem

pera

ture

y1

(∘C)

Figure 17 Comparison of calcination temperature measured curveand model predictive curve

10 Mathematical Problems in Engineering

Measured valuePredictive value

0 100 200 300 400 500 600 700 800 900 10001010

1020

1030

1040

1050

1060

1070

1080

1090

Time (s)

Kiln

tail

tem

pera

ture

y2

(∘C)

Figure 18 Comparison of kiln tail temperaturemeasured curve andmodel predictive curve

523 The Third Problem However there is still a problemwhich needs to be resolved In general the model with largetime delay will increase the difficulty of controller designingWe know that when the sampling frequency is higher thanthe actual needed frequency there will be lots of redundantdata And this will raise the model order and the delayTherefore the delay can be reduced by properly decreasingsampling frequency Considering the kiln is a slow time-varying process changing the sampling time from 1 s to 10 swill not affect the model accuracy

From Table 2 it can be seen that when set 119879119904= 10 s the

order and inflexion point is still 119899 = 5 We can also changethe delay to 119889

1= 3010 = 3 119889

2= 10010 = 10 The 119869MRSE

surface can be got as in Figure 16 the results show that 1198891=

3 1198892= 10 This is in agreement with the analysis before

The corresponding rotary kiln calcining zone tempera-ture model is

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896 minus 1198891)

119910 (119896) = 119862119909 (119896) + 119863119906 (119896 minus 1198892)

(17)

and the order 119899 = 5 delay 1198891= 3 119889

2= 10

119860 =

[[[[[

[

09936 00015 minus00062 minus00007 minus00001

00063 09906 00170 minus00017 minus00001

minus00009 00026 09793 minus00147 00209

00000 minus00001 00018 09985 minus00069

minus00000 00001 minus00015 00062 09848

]]]]]

]

119861 = 10minus3

times

[[[[[

[

minus00908 minus00129

minus01650 00957

00680 minus00329

minus00734 00191

minus00093 minus00012

]]]]]

]

119862 = [minus57995 29327 minus05841 minus00157 00260

minus67753 minus23024 minus00247 minus00596 minus00063]

119863 = 10minus15

times [01205 02699

minus01066 00143]

(18)

Compare the measured value and the predicted valuegenerated by the model in Figures 17 and 18

6 Conclusion

In this paper the calcining belt state space model of rotarykiln is built using PO-Moesp subspace method And a noveldouble parameters error performance criterion for the orderchoosing in subspace modelling is introduced Since thepresented method considering the order and delay simulta-neously it can reduce the model order of the delay systemeffectively And also a strategy for stripping the delay factorsfrom the historical data is also proposed The algorithm isverified in identifying an industrial lime kiln In this examplewe solve the practical problem of industrial process withmultidelay and also reduce the order by adjusting samplingtime Further research could shed more light on the issue ofapplying the model online The problem in industrial field ismore complex than simulation environment How to extractproblems from industrial practice and guide the direction ofmodeling research has become the study focus

Conflict of Interests

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

References

[1] M Georgallis P Nowak M Salcudean and I S GartshoreldquoModelling the rotary lime kilnrdquo The Canadian Journal ofChemical Engineering vol 83 no 2 pp 212ndash223 2005

[2] Z Sogut Z Oktay and H Karakoc ldquoMathematical modeling ofheat recovery from a rotary kilnrdquo Applied Thermal Engineeringvol 30 no 8-9 pp 817ndash825 2010

[3] Y H Kim ldquoDevelopment of process model of a rotary kilnfor volatile organic compound recovery from coconut shellrdquoKorean Journal of Chemical Engineering vol 29 no 12 pp 1674ndash1679 2012

[4] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[5] W Weijtjens G de Sitter C Devriendt and P GuillaumeldquoOperational modal parameter estimation of MIMO systemsusing transmissibility functionsrdquo Automatica vol 50 no 2 pp559ndash564 2014

[6] M Imber and V Paschkis ldquoA new theory for a rotary-kiln heatexchangerrdquo International Journal of Heat andMass Transfer vol5 no 7 pp 623ndash638 1962

[7] A Sass ldquoSimulation of heat-transfer phenomena in a rotarykilnrdquo Industrial amp Engineering Chemistry Process Design andDevelopment vol 6 no 4 pp 532ndash535 1967

Mathematical Problems in Engineering 11

[8] S D Shelukar H G K Sundar R Semiat J T Richardson andD Luss ldquoContinuous rotary kiln calcination of yttrium bariumcopper oxide precursor powdersrdquo Industrial and EngineeringChemistry Research vol 33 no 2 pp 421ndash427 1994

[9] Y Yang J Rakhorst M A Reuter and J H L Voncken ldquoAnal-ysis of gas flow and mixing in a rotary kiln waste incineratorrdquoin Proceedings of the 2nd International Conference on CFD inthe Minerals and Process Industries pp 443ndash448 MelbourneAustralia

[10] Y Wang X H Fan and X L Chen ldquoMathematical modelsand expert system for grate-kiln process of iron ore oxide pelletproduction (Part I) mathematical models of grate processrdquoJournal of Central South University of Technology vol 19 no 4pp 1092ndash1097 2012

[11] G Mercere L Bako and S Lecœuche ldquoPropagator-basedmethods for recursive subspace model identificationrdquo SignalProcessing vol 88 no 3 pp 468ndash491 2008

[12] P Misra and M Nikolaou ldquoInput design for model orderdetermination in subspace identificationrdquo AIChE Journal vol49 no 8 pp 2124ndash2132 2003

[13] B Liu B Fang X Liu J Chen Z Huang and X HeldquoLarge margin subspace learning for feature selectionrdquo PatternRecognition vol 46 no 10 pp 2798ndash2806 2013

[14] H Oku and H Kimura ldquoRecursive 4SID algorithms usinggradient type subspace trackingrdquo Automatica vol 38 no 6 pp1035ndash1043 2002

[15] Y Subasi and M Demirekler ldquoQuantitative measure of observ-ability for linear stochastic systemsrdquo Automatica vol 50 no 6pp 1669ndash1674 2014

[16] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[17] X Pan H Zhu F Yang and X Zeng ldquoSubspace trajectorypiecewise-linear model order reduction for nonlinear circuitsrdquoCommunications in Computational Physics vol 14 no 3 pp639ndash663 2013

[18] M Doumlhler and L Mevel ldquoFast multi-order computationof system matrices in subspace-based system identificationrdquoControl Engineering Practice vol 20 no 9 pp 882ndash894 2012

[19] T Breiten and T Damm ldquoKrylov subspace methods for modelorder reduction of bilinear control systemsrdquo Systems and Con-trol Letters vol 59 no 8 pp 443ndash450 2010

[20] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[21] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol 19 no 6 pp 716ndash723 1974

[22] E E Ioannidis ldquoAkaikersquos information criterion correction forthe least-squares autoregressive spectral estimatorrdquo Journal ofTime Series Analysis vol 32 no 6 pp 618ndash630 2011

[23] K Peternell W Scherrer and M Deistler ldquoStatistical analysisof subspace identification methodsrdquo in Proceedings of the 3rdEuropean Control Conference (ECC rsquo95) vol 2 p 1342 1995

[24] D Bauer ldquoOrder estimation in the context of MOESP subspaceidentification methodsrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[25] D Bauer ldquoOrder estimation for subspace methodsrdquo Automat-ica vol 37 no 10 pp 1561ndash1573 2001

[26] J Shalchian A Khaki-Sedigh and A Fatehi ldquoA subspacebased method for time delay estimationrdquo in Proceedings of the

4th International Symposium on Communications Control andSignal Processing (ISCCSP rsquo10) p 4 March 2010

[27] J Lee and T F Edgar ldquoSubspace identification method forsimulation of closed-loop systems with time delaysrdquo AIChEJournal vol 48 no 2 pp 417ndash420 2002

[28] H Zhang T Ma G-B Huang and Z Wang ldquoRobust globalexponential synchronization of uncertain chaotic delayed neu-ral networks via dual-stage impulsive controlrdquo IEEE Transac-tions on Systems Man and Cybernetics B Cybernetics vol 40no 3 pp 831ndash844 2010

[29] P van Overschee and B deMoor ldquoA unifying theorem for threesubspace system identification algorithmsrdquo Automatica vol 31no 12 pp 1853ndash1864 1995

[30] W Favoreel B de Moor and P van Overschee ldquoSubspace statespace system identification for industrial processesrdquo Journal ofProcess Control vol 10 no 2 pp 149ndash155 2000

[31] P van Overschee and B deMoor ldquoN4SID subspace algorithmsfor the identification of combined deterministic-stochasticsystemsrdquo Automatica vol 30 no 1 pp 75ndash93 1994

[32] M Verhaegen ldquoIdentification of the deterministic part ofMIMO state space models given in innovations form frominput-output datardquo Automatica vol 30 no 1 pp 61ndash74 1994

[33] W E Larimore ldquoCanonical variate analysis in identificationfiltering and adaptive controlrdquo in Proceedings of the 29th IEEEConference on Decision and Control pp 596ndash604 December1990

[34] M Viberg ldquoSubspace-based methods for the identification oflinear time-invariant systemsrdquo Automatica vol 31 no 12 pp1835ndash1851 1995

[35] J Wang and S J Qin ldquoA new subspace identification approachbased on principal component analysisrdquo Journal of ProcessControl vol 12 no 8 pp 841ndash855 2002

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

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Preheater

Rotary kiln

Cooler

Figure 1 Technological process chart of rotary kiln

31 Matrices Construct Suppose at time 119896 the past input andoutput data samples are available then construct pastfutureoutput vectors and the Hankel output matrices as follows

119880119901= (

1199060

1199061sdot sdot sdot 119906

119895minus1

1199061

1199062sdot sdot sdot 119906

119895

d

119906119894minus1

119906119894sdot sdot sdot 119906119894+119895minus2

)

(2a)

119884119891= (

119910119894

119910119894+1

sdot sdot sdot 119910119894+119895minus1

119910119894+1

119910119894+2

sdot sdot sdot 119910119894+119895

d

1199102119894minus1

1199102119894

sdot sdot sdot 1199102119894+119895minus2

) (2b)

where 119906119894= [119906

1198941 1199061198942 119906

119894119897] 119910119894= [119910

1198941 1199101198942 119910

119894119898] 119901

represent the past information and 119891 represent the futureinformation 119894 ge 119899 119895 ≫ max(119894119897 119894119898)

The state matrix 119883119901and 119883

119891is defined as follows 119883

119901=

[11990901199091sdot sdot sdot 119909119895minus1]119883119891= [119909119894119909119894+1

sdot sdot sdot 119909119895+119894minus1

]The augmented observation matrix Γ

119894

Γ119894= [119862 119862119860 sdot sdot sdot 119862119860119894]

119879

(3)

Block-triangular matrix Toeplitz matrix119867119894and119867119904

119894

119867119894=

[[[[[[[

[

119863 0 sdot sdot sdot 0

119862119861 119863 d

d 0

119862119894minus2119861 119862119894minus1119861 sdot sdot sdot 119863

]]]]]]]

]

119867119904

119894

=

[[[[[[[

[

0 0 sdot sdot sdot 0

119862 0 d

d 0

119862119894minus2 119862119894minus1 sdot sdot sdot 0

]]]]]]]

]

(4)

The augmented observation inverse matrix of 119860 119861 isΔ119894= [119860119894minus1

119861 119860119894minus2

119861 sdot sdot sdot 119861] And the augmented observationinverse matrix of 119860119870 is Δ119904

119894

= [119860119894minus1

119870 119860119894minus2

119870 sdot sdot sdot 119870]Suppose the input and state matrices are all row full

rank matrices and they are row space orthogonal Then theaugmented inputoutput matrices can be written as follows

119883119891= 119860119894

119883119901+ Δ119894119880119901+ Δ119904

119894

119864119891 (5a)

119884119901= Γ119894119883119901+ 119867119894119880119901+ 119867119904

119894

119864119901 (5b)

119884119891= Γ119894119883119891+ 119867119894119880119891+ 119867119904

119894

119864119891 (5c)

The key step of Moesp method is to estimate the aug-mented observability matrix through the projection future119868119874 data onto past 119868119874 data The augmented observabilitymatrix and state space vector estimation can be got throughSVD decomposition

32 General Algorithm In this section we introduce thegeneral subspace method framework which is profferedby Favoreel et al [30] Subspace identification algorithmconsists of two main steps The first step always performsa weighted projection of the row space of the previouslydefined data Hankel matrices From this projection theaugmented observabilitymatrix Γ

119894and estimate119883

119894of the state

sequence119883119894can be retrievedThen the order is got from SVD

decomposition In the second step the system matrices 119860 119861119862119863 and119876 119878119877 are determined through least squaremethodThe concrete Moesp algorithm is as follows

4 Mathematical Problems in Engineering

(1) Project the 119884119891row space into the orthogonal comple-

ment of the 119880119891row space

119884119891119880perp

119891

= Γ119894119883119891119880perp

119891

+ 119867119889

119894

119880119891119880perp

119891

+ 119867119904

119894

119864119891119880perp

119891

(6)

Since it is assumed that the noise is uncorrelated withthe inputs so 119864

119891119880perp119891

= 119864119891and 119880

119891119880perp119891

= 0 therefore119884119891119880perp119891

= Γ119894119883119891119880perp119891

+ 119867119904119894

119864119891

(2) Select the weighting matrices1198821and119882

2

1198821119884119891

119880perp119891

1198822

=1198821Γ119894119883119891

119880perp119891

1198822

+1198821119867119904

119894

1198641198911198822 (7)

The weighting matrices can be chosen appropriatelyaccording to different subspace methods includingN4SID MOESP CVA basic-4SID and IV-4SID [31ndash34]

Then we can get

119900119894=1198821119884119891

119880perp119891

1198822

=1198821Γ119894119883119891

119880perp119891

1198822

(8)

(3) Carry SVD decomposition

119900119894= (11988011198802) (11987810

0 0)(

119881119879

1

1198811198792

) (9)

And then take the number of nonzero eigenvalue asthe system order rank(119900

119894) = 119899

(4) The augmented observability matrix Γ119894= 119882minus11

119880111987812

1

or119883119894= 119883119894119880perp119891

1198822is derived from the third step

(5) Extract estimate 119860 119861 119862119863 from Γ119894or119883119894

Remark By reference to [30] the weighting matrices1198821and

1198822should satisfy the following three conditions

(1) rank(1198821sdot Γ119894) = rank Γ

119894

(2) rank(119883119894119880perp119891

sdot 1198822) = rank119883

119894

(3) 1198821sdot (119867119904119894

119872119891+ 119873119891) sdot 1198822= 0

The first two conditions guarantee that the rank-119899 prop-erty of Γ

119894119883119894is preserved after projection onto 119880perp

119891

andweighting by119882

1and119882

2 The third condition expresses that

1198822should be uncorrelated with the noise sequences 119908(119896)

and V(119896) By choosing the appropriate weighting matrices1198821and 119882

2 all subspace algorithms for LTI systems can

be interpreted in the above framework including N4SIDMOESP CVA Basic-4SID and IV-4SID

4 The Proposed Method

In the classical system identification theory the actual modelstructure is usually assumed to be known However inpractical it is always not clear Subspace system identificationmethod determines the order of the system by the nonzeroeigenvalue of the augmented observability matrix Howeverthe system nonzero singular values may be very small Thismay lead to the wrong system order and large identificationerror

41 The Order-Delay Double Parameters Error Criterion Themost directly order-selection method is based on the errorperformance criterion This idea is to choose the smallestpossible order that keeps the error below a certain levelThenthe MRSE (mean relative squared error) index is introducedby model error as follows

119869MRSE (119899) =1

119871

119871

sum119896=1

radicsum119899119910

119895=1

(119910119896(119895) minus 119910

119896(119895))2

sum119899119910

119895=1

119910119896(119895)2

(10)

where 119910119896(119895) minus 119910

119896(119895) is the model prediction error and 119871 is

the sample number In [35] use the AIC which was originallydeveloped by Akaike and then adapted by Larimore for SMIGiven a set of samples for a sequence of system order 119899 forexample 119899 isin [0 sdot sdot sdot 20] the order of the model will be the onewhich makes the following AIC index minimum

AIC119899(119899) = 119873 (119898 (1 + ln 2120587) + ln 1003816100381610038161003816Σ119899

1003816100381610038161003816) + 2120575119899119872119899 (11)

where

Σ119899=

1

119873

119873

sum119894=1

119890 (119896) 119890(119896)119879

119890 (119896) = 119910119899(119896) minus 119910

119899(119896)

119872119899= 2119899119898 +

119898 (119898 + 1)

2+ 119899119897 + 119898119897

120575119899=

119873

119873 minus ((119872119899119898) + ((119898 + 1) 2))

(12)

For calculating the AIC(119899) criterion we first suppose theupper bound 119899max of the system order and then calculate theAIC119873(1)AIC

119873(2) AIC

119873(119899max) sequence the appropri-

ate system order 119899 is the one which decrees the AIC indexobviously and the order should be as small as possible TheMRSE and AIC index 119869MRSE(119899) can be analyzed in the samemanner

However the performance index based on a single orderparameter cannot provide an effective solution to the delaysystem which is shown in (1) This led to the problem thatthe original identificationmethod had to increase the order ascost to improvemodel accuracy Here we introduce an order-delay double parameters error criterion which identifies thetwo key structural parameters at the same time That meansthat the index 119869(119899) is changed into the 119869(119899 119889) form

For each given individual 119889 a state space model canbe identified using the Moesp algorithm described in

Mathematical Problems in Engineering 5

Initialization 119899 = 2 119889 = 1 the modelling data after pretreatment 119906(119896 minus 119889max) 119906(119896) 119906(119871) and 119910(119896) 119910(119871)(1) for 119899 = 2 to 119899 = 119899max(2) for 119889 = 1 to 119889 = 119889max(3) Rolling the modelling input data 119906 based on the hypothesis delay 119889 get the data set 119906(119896 minus 119889) 119906(119896) 119906(119871 minus 119889)

119910(119896) 119910(119871)(4) Construct input and output Hankel matrices 119880

119901

119884119901

119880119891

119884119891

(5) Calculate 119860 119861 119862119863 by Moesp method in Section 3 based on Hankel matrices(6) Substitute the 119860 119861 119862119863 and 119889 into the formula (1)(7) Calculate and store the model error and performance index 119869(119899 119889)(8) end for(9) end for(10) Search the inflection point of the 119869(119899 119889) surface

Algorithm 1

y

y

L

L

xk k + Lo

xo k + 1 k + L + 1

Figure 2 Sliding time window sketch map

Section 2 Then its model error can be deserved as Σ119899119889

=

(1119873)sum119873

119894=1

119890(119896)119890(119896)119879 then the AIC(119899 119889) with respect to

individual (119899 119889) as

AIC119873(119899 119889) = 119873 (119898 (1 + ln 2120587) + ln 1003816100381610038161003816Σ119899119889

1003816100381610038161003816) + 2120575119899119872119899 (13)

Also the MRSE criterion 119869MSE2 has the similar form

119869MRSE (119899 119889) =1

119871

119871

sum119896=1

radicsum119899119910

119895=1

(119910119899119889119896

(119895) minus 119910119899119889119896

(119895))2

sum119899119910

119895=1

119910119899119889119896

(119895)2

(14)

Other performance index 119869(sdot) such as SVC IVC NICcriteria mentioned in [25] can also be modified as thismethod

The original performance index just identifies the order119899 Suppose 119899 isin [1 119899max] then 119869(1) 119869(2) 119869(3) 119869(119899max) arecalculated respectively then the inflection point 119899lowast is the best

order and the corresponding systemmatrices119860lowast 119861lowast 119862lowast119863lowastare the best model After improvement we add the delay asother optimal parameters So the system order 119899 isin [1 119899max]and input delay 119889 isin [1 119889max] are all embedded in 119869(119899 119889)Then calculate 119869(1 1) 119869(119899max 1) 119869(1 2) 119869(119899max 2)119869(1 119889max) 119869(119899max 119889max) respectively By searching theminimum point 119899lowast 119889lowast of surface 119869(119899 119889) the best orderdelay and system matrices can all be got Thus taking thedelay as another parameter in the modelling methods it caneffectively avoid high order results in the delay system

42 The Delay Factor Stripping from Historical Data Theintroduction of delay parameters in performance criterionhas resulted to a notable problem The modelling historicaldata matrices have already included delay information It isdifficult to change 119889 in the performance criterion 119869(119899 119889) arti-ficially To solve this problem the sliding-window method isadopted here Sliding-window principle is shown in Figure 2When new samples are added to the window the oldest datainside the window will be discarded

We use sliding window to change delay which is shownin Figure 3 Suppose the output data length is 119871 select theinput data region 119906(119896 minus 119889max) 119906(119896) 119906(119871) Then theinput data can be moved according to different delay from119889 to 119889max

43 The Algorithm Description The detailed procedure ofsubspace identification based on ODC algorithm can beexpressed as Algorithm 1

Then the best combination 119899lowast 119889lowast and the optimum

matching model parameters are all obtained

5 Simulation Results

To demonstrate the superiority of the proposed order selec-tion method in this paper over the conventional methodtheir performance is evaluated through a numerical exampleand an industrial illustrate

6 Mathematical Problems in Engineering

u(k minus 1)

u(k)

u(L)

(d = 0)

y(k)

y(k + 1)

y(L)

u(k minus 1)

u(L minus 1)

u(L)

(d = 1)

y(k)

y(k + 1)

y(L)

u(L minus 1)

u(L)

y(k)

y(k + 1)

y(L)

u(k minus dmax) u(k minus dmax) u(k minus dmax)

u(L minus dmax)

(d = dmax)

middot middot middot

Figure 3 Modeling data sliding sketch map

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

minus1

minus2

minus3

minus4

u1

(a) Input signal 1199061

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

minus1

minus2

minus3

minus4

u2

(b) Input signal 1199062

0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

y1

minus5

minus10

minus15

minus20

(c) Output signal 1199101

0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

y2

minus5

minus10

minus15

(d) Output signal 1199102

Figure 4 PO-Moesp subspace system identification inputoutput signal

51 Example 1 MIMO Process with Input Delay The firstmodel (15) is a MIMO system

119909 (119896 + 1) =[[[

[

0603 0603 0 0

minus0603 0603 0 0

0 0 minus0603 minus0603

0 0 0603 minus0603

]]]

]

119909 (119896)

+[[[

[

11650 minus06965

06268 16961

00751 00591

03516 17971

]]]

]

119906 (119896 minus 5)

119910 (119896) = [02641 minus14462 12460 05774

08717 minus07012 minus06390 minus03600] 119909 (119896)

+ [minus01356 minus12704

minus13493 09846] 119906 (119896 minus 5)

(15)

The time delay is 119889 = 5 and the sample is 119871 = 1000 Weuse the zeromeanwhite noise as the input 119906

1and 1199062sequence

to excite the system The input and output curves are shownin Figure 4

Firstly we use conventional subspace model selectionmethod which is dependant on the number of eigenvaluesFigure 5(a) is the situation of delay = 0 and Figure 5(b) is the

Mathematical Problems in Engineering 7

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Order

Eige

nval

ue

(a) 119889 = 0 situation

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

Order

Eige

nval

ue

(b) 119889 = 5 situation

Figure 5The number of eigenvalues generated when 119889 = 0 and 119889 =5 respectively (a) 119889 = 0 and (b) 119889 = 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

Perfo

rman

ce in

dexJ

Order n

JMRSEJMSE2

JMSE1

Figure 6 Obtain the input delay system order by original errorcriterion methods

situation of delay = 5 According to this strategy the order ofmodel increases from 4 to 13 when the system has delay

Also theMSE (mean squared error)MRSE (mean relativesquared error) and AIC criteria are all tested as shown inFigures 6 and 7Thesemethods all have the distinct inflectionpoint at 13 which is for the 4-order system with input delay

2 4 6 8 10 12 14 16 18

0

1

2

AIC

inde

x

minus6

minus5

minus4

minus3

minus2

minus1

Order n

times104

Figure 7 Obtain the input delay system order by AIC index

0246810 123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

1

Figure 8 The corresponding mean square error surface 119869MSE1generated by ODC

5 Obviously for obtaining the ideal model of a delay systemthese conventional methods have to increase the order

Next the performances of the proposed ODCmethod inthis paper are presented As the two search parameters areavailable so the index performance is shown as a surfaceThe 119869MSE1 of output 1199101 is shown in Figure 8 and the 119869MSE2 ofoutput 119910

2is shown in Figure 9 Three axes are the order 119899

delay 119889 and 119869(sdot) respectivelyTheAIC index surface is shownin Figure 10 The minimum value of these indexes can be goteasily they are also the inflection pointsThe order and delayresults are 119899 = 4 119889 = 5 This is the same with the actualsystem model

The corresponding identified model matrices

119860 =[[[

[

05017 06047 02024 01514

minus06256 05164 03929 minus00828

02228 03763 minus04884 minus06090

01883 minus00936 06069 minus05297

]]]

]

119861 =[[[

[

minus00386 18602

minus10028 00809

minus05982 minus10787

minus00583 minus04779

]]]

]

8 Mathematical Problems in Engineering

02

46

810

123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

2

Figure 9 The corresponding mean square error surface 119869MSE2generated by ODC

02468

10 12

34

56

0

2

AIC

inde

x

minus6

minus4

minus2

Order nDelay d

n = 4 d = 5

times104

Figure 10 The corresponding AIC criterion surface generated byODC

119862 = [minus14276 10033 minus11161 03224

minus11801 minus05779 04116 minus04398]

119863 = [minus01356 minus12704

minus13493 09846]

(16)

For verifying the model the model output error is drawnas 1198901and 1198902in Figure 11

52 Example 2 The Kiln Industrial Illustration To demon-strate the superiority of the proposed order selection methodin this paper over the conventional method their perfor-mance is evaluated through an industrial illustration Thedata come from actual kiln production data of an enterpriseHere the gas flow and the second air flow are selected asthe control input and the calcination temperature and kilntail temperature are taken as output variables The samplingtime is 119879

119904= 1 119904 then use Moesp method modelling the

kiln based on the inputoutput data after preprocessing Herethree main practical problems are solved

521 The First Problem The problem is that these two inputvariables have different delay They need to be identified

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus12

minus10

minus8

minus6

minus4

minus2

times10minus15

e 1

(a) 1198901

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus6

minus4

minus2

times10minus15

e 2

(b) 1198902

Figure 11 Modeling error curves 1198901

(a) and 1198902

(b)

respectively that is to say a triple loop about 1198891 1198892 and 119899

should be carried for solving 119869(119899 1198891 1198892)

In order to obtain the inflection point information moredirectly we identify the order 119899 firstly According to theindustrial field situation choose the possibility maximumvalues which are 119889

1max = 150 1198892max = 150 At first

travel all the possible 1198891 1198892and compute the smallest 119869MSE1

119869MSE2 and 119869MRSE corresponding to the different order 119899Thesecurves are shown in Figure 12 As can be seen from thecurves all of these three error criteria achieve the inflectionpoint at 119899 = 5 so the order is got Then specify 119899 =

5 the triple loop is reduced to double loop which is tocompute 119869MSE1 and 119869MSE2 corresponding to 1198891 and 1198892 between[0 150]

522The Second Problem From Figure 13 we notice anotherproblem that the outputs 119910

1and 119910

2generate different inflec-

tion pointIn order to solve this problem the criterion should choose

MRSE and AIC which take into account 1199101and 119910

2both

together Then get Figures 14 and 15It can be conducted that 119899 = 5 119889

1= 30 119889

2= 100

Mathematical Problems in Engineering 9

Table 2 When 119879119904

= 10 s the order the minmum 119869MRSE and the delay

119899 1 2 3 4 5 6 7 81198891

4 3 3 3 3 3 3 31198892

5 10 10 10 10 10 10 10119869MRSE 6981251 595572 499145 384619 239931 239931 239931 239931

1 2 3 4 5 6 7 8 9 10 11 120

50

100

1 2 3 4 5 6 7 8 9 10 11 120

200

400

1 2 3 4 5 6 7 8 9 10 11 120

02

04

Order n

Order n

Order n

J MRS

EJ M

SE2

J MSE

1

Figure 12 The corresponding minimum 119869(119899 1198891

1198892

) curves of eachorder when traversal 119889

1

1198892

0 50 100 150 0 50 100 150050

100150200250300350

Delay d2 Delay d1

J MSE

i

JMSE1

JMSE2

50 100

JMJJ SE1

JMJJ SE2

Figure 13 119869MSE1 and 119869MSE2 surface generated by ODC when 119899 = 5

0 50 100150 0 50 100 150

384

42444648

5

AIC

inde

x

d2 d1

times104

d1 = 30 d2 = 100

AIC = 40159e + 04

d1 = 3330000 dddd222 = 1000

AICAIC 4 0150159e + 0444

Figure 14 AIC surface generated by ODC when 119899 = 5

0 50 100 150 0 50 100 150

0

50

100

150

200

250

d2 d1

J MRS

E

d1 = 30 d2 = 100 JMRSE = 16448d = 30 d = 100 JMJJ RSE = 161 4448

Figure 15 119869MRSE surface generated by ODC when 119899 = 5

0 50100

150

050

100150

050

100150200250300350400

(3 10 239931)

J MRS

E

d2 d150

10050100

Figure 16 119869MRSE surface generated by ODC when 119899 = 5 and 119879119904

=

10 s

0 100 200 300 400 500 600 700 800 900 10001220

1240

1260

1280

1300

1320

1340

1360

1380

Time (s)

Measured valuePredictive value

Kiln

tail

tem

pera

ture

y1

(∘C)

Figure 17 Comparison of calcination temperature measured curveand model predictive curve

10 Mathematical Problems in Engineering

Measured valuePredictive value

0 100 200 300 400 500 600 700 800 900 10001010

1020

1030

1040

1050

1060

1070

1080

1090

Time (s)

Kiln

tail

tem

pera

ture

y2

(∘C)

Figure 18 Comparison of kiln tail temperaturemeasured curve andmodel predictive curve

523 The Third Problem However there is still a problemwhich needs to be resolved In general the model with largetime delay will increase the difficulty of controller designingWe know that when the sampling frequency is higher thanthe actual needed frequency there will be lots of redundantdata And this will raise the model order and the delayTherefore the delay can be reduced by properly decreasingsampling frequency Considering the kiln is a slow time-varying process changing the sampling time from 1 s to 10 swill not affect the model accuracy

From Table 2 it can be seen that when set 119879119904= 10 s the

order and inflexion point is still 119899 = 5 We can also changethe delay to 119889

1= 3010 = 3 119889

2= 10010 = 10 The 119869MRSE

surface can be got as in Figure 16 the results show that 1198891=

3 1198892= 10 This is in agreement with the analysis before

The corresponding rotary kiln calcining zone tempera-ture model is

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896 minus 1198891)

119910 (119896) = 119862119909 (119896) + 119863119906 (119896 minus 1198892)

(17)

and the order 119899 = 5 delay 1198891= 3 119889

2= 10

119860 =

[[[[[

[

09936 00015 minus00062 minus00007 minus00001

00063 09906 00170 minus00017 minus00001

minus00009 00026 09793 minus00147 00209

00000 minus00001 00018 09985 minus00069

minus00000 00001 minus00015 00062 09848

]]]]]

]

119861 = 10minus3

times

[[[[[

[

minus00908 minus00129

minus01650 00957

00680 minus00329

minus00734 00191

minus00093 minus00012

]]]]]

]

119862 = [minus57995 29327 minus05841 minus00157 00260

minus67753 minus23024 minus00247 minus00596 minus00063]

119863 = 10minus15

times [01205 02699

minus01066 00143]

(18)

Compare the measured value and the predicted valuegenerated by the model in Figures 17 and 18

6 Conclusion

In this paper the calcining belt state space model of rotarykiln is built using PO-Moesp subspace method And a noveldouble parameters error performance criterion for the orderchoosing in subspace modelling is introduced Since thepresented method considering the order and delay simulta-neously it can reduce the model order of the delay systemeffectively And also a strategy for stripping the delay factorsfrom the historical data is also proposed The algorithm isverified in identifying an industrial lime kiln In this examplewe solve the practical problem of industrial process withmultidelay and also reduce the order by adjusting samplingtime Further research could shed more light on the issue ofapplying the model online The problem in industrial field ismore complex than simulation environment How to extractproblems from industrial practice and guide the direction ofmodeling research has become the study focus

Conflict of Interests

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

References

[1] M Georgallis P Nowak M Salcudean and I S GartshoreldquoModelling the rotary lime kilnrdquo The Canadian Journal ofChemical Engineering vol 83 no 2 pp 212ndash223 2005

[2] Z Sogut Z Oktay and H Karakoc ldquoMathematical modeling ofheat recovery from a rotary kilnrdquo Applied Thermal Engineeringvol 30 no 8-9 pp 817ndash825 2010

[3] Y H Kim ldquoDevelopment of process model of a rotary kilnfor volatile organic compound recovery from coconut shellrdquoKorean Journal of Chemical Engineering vol 29 no 12 pp 1674ndash1679 2012

[4] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[5] W Weijtjens G de Sitter C Devriendt and P GuillaumeldquoOperational modal parameter estimation of MIMO systemsusing transmissibility functionsrdquo Automatica vol 50 no 2 pp559ndash564 2014

[6] M Imber and V Paschkis ldquoA new theory for a rotary-kiln heatexchangerrdquo International Journal of Heat andMass Transfer vol5 no 7 pp 623ndash638 1962

[7] A Sass ldquoSimulation of heat-transfer phenomena in a rotarykilnrdquo Industrial amp Engineering Chemistry Process Design andDevelopment vol 6 no 4 pp 532ndash535 1967

Mathematical Problems in Engineering 11

[8] S D Shelukar H G K Sundar R Semiat J T Richardson andD Luss ldquoContinuous rotary kiln calcination of yttrium bariumcopper oxide precursor powdersrdquo Industrial and EngineeringChemistry Research vol 33 no 2 pp 421ndash427 1994

[9] Y Yang J Rakhorst M A Reuter and J H L Voncken ldquoAnal-ysis of gas flow and mixing in a rotary kiln waste incineratorrdquoin Proceedings of the 2nd International Conference on CFD inthe Minerals and Process Industries pp 443ndash448 MelbourneAustralia

[10] Y Wang X H Fan and X L Chen ldquoMathematical modelsand expert system for grate-kiln process of iron ore oxide pelletproduction (Part I) mathematical models of grate processrdquoJournal of Central South University of Technology vol 19 no 4pp 1092ndash1097 2012

[11] G Mercere L Bako and S Lecœuche ldquoPropagator-basedmethods for recursive subspace model identificationrdquo SignalProcessing vol 88 no 3 pp 468ndash491 2008

[12] P Misra and M Nikolaou ldquoInput design for model orderdetermination in subspace identificationrdquo AIChE Journal vol49 no 8 pp 2124ndash2132 2003

[13] B Liu B Fang X Liu J Chen Z Huang and X HeldquoLarge margin subspace learning for feature selectionrdquo PatternRecognition vol 46 no 10 pp 2798ndash2806 2013

[14] H Oku and H Kimura ldquoRecursive 4SID algorithms usinggradient type subspace trackingrdquo Automatica vol 38 no 6 pp1035ndash1043 2002

[15] Y Subasi and M Demirekler ldquoQuantitative measure of observ-ability for linear stochastic systemsrdquo Automatica vol 50 no 6pp 1669ndash1674 2014

[16] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[17] X Pan H Zhu F Yang and X Zeng ldquoSubspace trajectorypiecewise-linear model order reduction for nonlinear circuitsrdquoCommunications in Computational Physics vol 14 no 3 pp639ndash663 2013

[18] M Doumlhler and L Mevel ldquoFast multi-order computationof system matrices in subspace-based system identificationrdquoControl Engineering Practice vol 20 no 9 pp 882ndash894 2012

[19] T Breiten and T Damm ldquoKrylov subspace methods for modelorder reduction of bilinear control systemsrdquo Systems and Con-trol Letters vol 59 no 8 pp 443ndash450 2010

[20] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[21] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol 19 no 6 pp 716ndash723 1974

[22] E E Ioannidis ldquoAkaikersquos information criterion correction forthe least-squares autoregressive spectral estimatorrdquo Journal ofTime Series Analysis vol 32 no 6 pp 618ndash630 2011

[23] K Peternell W Scherrer and M Deistler ldquoStatistical analysisof subspace identification methodsrdquo in Proceedings of the 3rdEuropean Control Conference (ECC rsquo95) vol 2 p 1342 1995

[24] D Bauer ldquoOrder estimation in the context of MOESP subspaceidentification methodsrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[25] D Bauer ldquoOrder estimation for subspace methodsrdquo Automat-ica vol 37 no 10 pp 1561ndash1573 2001

[26] J Shalchian A Khaki-Sedigh and A Fatehi ldquoA subspacebased method for time delay estimationrdquo in Proceedings of the

4th International Symposium on Communications Control andSignal Processing (ISCCSP rsquo10) p 4 March 2010

[27] J Lee and T F Edgar ldquoSubspace identification method forsimulation of closed-loop systems with time delaysrdquo AIChEJournal vol 48 no 2 pp 417ndash420 2002

[28] H Zhang T Ma G-B Huang and Z Wang ldquoRobust globalexponential synchronization of uncertain chaotic delayed neu-ral networks via dual-stage impulsive controlrdquo IEEE Transac-tions on Systems Man and Cybernetics B Cybernetics vol 40no 3 pp 831ndash844 2010

[29] P van Overschee and B deMoor ldquoA unifying theorem for threesubspace system identification algorithmsrdquo Automatica vol 31no 12 pp 1853ndash1864 1995

[30] W Favoreel B de Moor and P van Overschee ldquoSubspace statespace system identification for industrial processesrdquo Journal ofProcess Control vol 10 no 2 pp 149ndash155 2000

[31] P van Overschee and B deMoor ldquoN4SID subspace algorithmsfor the identification of combined deterministic-stochasticsystemsrdquo Automatica vol 30 no 1 pp 75ndash93 1994

[32] M Verhaegen ldquoIdentification of the deterministic part ofMIMO state space models given in innovations form frominput-output datardquo Automatica vol 30 no 1 pp 61ndash74 1994

[33] W E Larimore ldquoCanonical variate analysis in identificationfiltering and adaptive controlrdquo in Proceedings of the 29th IEEEConference on Decision and Control pp 596ndash604 December1990

[34] M Viberg ldquoSubspace-based methods for the identification oflinear time-invariant systemsrdquo Automatica vol 31 no 12 pp1835ndash1851 1995

[35] J Wang and S J Qin ldquoA new subspace identification approachbased on principal component analysisrdquo Journal of ProcessControl vol 12 no 8 pp 841ndash855 2002

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

(1) Project the 119884119891row space into the orthogonal comple-

ment of the 119880119891row space

119884119891119880perp

119891

= Γ119894119883119891119880perp

119891

+ 119867119889

119894

119880119891119880perp

119891

+ 119867119904

119894

119864119891119880perp

119891

(6)

Since it is assumed that the noise is uncorrelated withthe inputs so 119864

119891119880perp119891

= 119864119891and 119880

119891119880perp119891

= 0 therefore119884119891119880perp119891

= Γ119894119883119891119880perp119891

+ 119867119904119894

119864119891

(2) Select the weighting matrices1198821and119882

2

1198821119884119891

119880perp119891

1198822

=1198821Γ119894119883119891

119880perp119891

1198822

+1198821119867119904

119894

1198641198911198822 (7)

The weighting matrices can be chosen appropriatelyaccording to different subspace methods includingN4SID MOESP CVA basic-4SID and IV-4SID [31ndash34]

Then we can get

119900119894=1198821119884119891

119880perp119891

1198822

=1198821Γ119894119883119891

119880perp119891

1198822

(8)

(3) Carry SVD decomposition

119900119894= (11988011198802) (11987810

0 0)(

119881119879

1

1198811198792

) (9)

And then take the number of nonzero eigenvalue asthe system order rank(119900

119894) = 119899

(4) The augmented observability matrix Γ119894= 119882minus11

119880111987812

1

or119883119894= 119883119894119880perp119891

1198822is derived from the third step

(5) Extract estimate 119860 119861 119862119863 from Γ119894or119883119894

Remark By reference to [30] the weighting matrices1198821and

1198822should satisfy the following three conditions

(1) rank(1198821sdot Γ119894) = rank Γ

119894

(2) rank(119883119894119880perp119891

sdot 1198822) = rank119883

119894

(3) 1198821sdot (119867119904119894

119872119891+ 119873119891) sdot 1198822= 0

The first two conditions guarantee that the rank-119899 prop-erty of Γ

119894119883119894is preserved after projection onto 119880perp

119891

andweighting by119882

1and119882

2 The third condition expresses that

1198822should be uncorrelated with the noise sequences 119908(119896)

and V(119896) By choosing the appropriate weighting matrices1198821and 119882

2 all subspace algorithms for LTI systems can

be interpreted in the above framework including N4SIDMOESP CVA Basic-4SID and IV-4SID

4 The Proposed Method

In the classical system identification theory the actual modelstructure is usually assumed to be known However inpractical it is always not clear Subspace system identificationmethod determines the order of the system by the nonzeroeigenvalue of the augmented observability matrix Howeverthe system nonzero singular values may be very small Thismay lead to the wrong system order and large identificationerror

41 The Order-Delay Double Parameters Error Criterion Themost directly order-selection method is based on the errorperformance criterion This idea is to choose the smallestpossible order that keeps the error below a certain levelThenthe MRSE (mean relative squared error) index is introducedby model error as follows

119869MRSE (119899) =1

119871

119871

sum119896=1

radicsum119899119910

119895=1

(119910119896(119895) minus 119910

119896(119895))2

sum119899119910

119895=1

119910119896(119895)2

(10)

where 119910119896(119895) minus 119910

119896(119895) is the model prediction error and 119871 is

the sample number In [35] use the AIC which was originallydeveloped by Akaike and then adapted by Larimore for SMIGiven a set of samples for a sequence of system order 119899 forexample 119899 isin [0 sdot sdot sdot 20] the order of the model will be the onewhich makes the following AIC index minimum

AIC119899(119899) = 119873 (119898 (1 + ln 2120587) + ln 1003816100381610038161003816Σ119899

1003816100381610038161003816) + 2120575119899119872119899 (11)

where

Σ119899=

1

119873

119873

sum119894=1

119890 (119896) 119890(119896)119879

119890 (119896) = 119910119899(119896) minus 119910

119899(119896)

119872119899= 2119899119898 +

119898 (119898 + 1)

2+ 119899119897 + 119898119897

120575119899=

119873

119873 minus ((119872119899119898) + ((119898 + 1) 2))

(12)

For calculating the AIC(119899) criterion we first suppose theupper bound 119899max of the system order and then calculate theAIC119873(1)AIC

119873(2) AIC

119873(119899max) sequence the appropri-

ate system order 119899 is the one which decrees the AIC indexobviously and the order should be as small as possible TheMRSE and AIC index 119869MRSE(119899) can be analyzed in the samemanner

However the performance index based on a single orderparameter cannot provide an effective solution to the delaysystem which is shown in (1) This led to the problem thatthe original identificationmethod had to increase the order ascost to improvemodel accuracy Here we introduce an order-delay double parameters error criterion which identifies thetwo key structural parameters at the same time That meansthat the index 119869(119899) is changed into the 119869(119899 119889) form

For each given individual 119889 a state space model canbe identified using the Moesp algorithm described in

Mathematical Problems in Engineering 5

Initialization 119899 = 2 119889 = 1 the modelling data after pretreatment 119906(119896 minus 119889max) 119906(119896) 119906(119871) and 119910(119896) 119910(119871)(1) for 119899 = 2 to 119899 = 119899max(2) for 119889 = 1 to 119889 = 119889max(3) Rolling the modelling input data 119906 based on the hypothesis delay 119889 get the data set 119906(119896 minus 119889) 119906(119896) 119906(119871 minus 119889)

119910(119896) 119910(119871)(4) Construct input and output Hankel matrices 119880

119901

119884119901

119880119891

119884119891

(5) Calculate 119860 119861 119862119863 by Moesp method in Section 3 based on Hankel matrices(6) Substitute the 119860 119861 119862119863 and 119889 into the formula (1)(7) Calculate and store the model error and performance index 119869(119899 119889)(8) end for(9) end for(10) Search the inflection point of the 119869(119899 119889) surface

Algorithm 1

y

y

L

L

xk k + Lo

xo k + 1 k + L + 1

Figure 2 Sliding time window sketch map

Section 2 Then its model error can be deserved as Σ119899119889

=

(1119873)sum119873

119894=1

119890(119896)119890(119896)119879 then the AIC(119899 119889) with respect to

individual (119899 119889) as

AIC119873(119899 119889) = 119873 (119898 (1 + ln 2120587) + ln 1003816100381610038161003816Σ119899119889

1003816100381610038161003816) + 2120575119899119872119899 (13)

Also the MRSE criterion 119869MSE2 has the similar form

119869MRSE (119899 119889) =1

119871

119871

sum119896=1

radicsum119899119910

119895=1

(119910119899119889119896

(119895) minus 119910119899119889119896

(119895))2

sum119899119910

119895=1

119910119899119889119896

(119895)2

(14)

Other performance index 119869(sdot) such as SVC IVC NICcriteria mentioned in [25] can also be modified as thismethod

The original performance index just identifies the order119899 Suppose 119899 isin [1 119899max] then 119869(1) 119869(2) 119869(3) 119869(119899max) arecalculated respectively then the inflection point 119899lowast is the best

order and the corresponding systemmatrices119860lowast 119861lowast 119862lowast119863lowastare the best model After improvement we add the delay asother optimal parameters So the system order 119899 isin [1 119899max]and input delay 119889 isin [1 119889max] are all embedded in 119869(119899 119889)Then calculate 119869(1 1) 119869(119899max 1) 119869(1 2) 119869(119899max 2)119869(1 119889max) 119869(119899max 119889max) respectively By searching theminimum point 119899lowast 119889lowast of surface 119869(119899 119889) the best orderdelay and system matrices can all be got Thus taking thedelay as another parameter in the modelling methods it caneffectively avoid high order results in the delay system

42 The Delay Factor Stripping from Historical Data Theintroduction of delay parameters in performance criterionhas resulted to a notable problem The modelling historicaldata matrices have already included delay information It isdifficult to change 119889 in the performance criterion 119869(119899 119889) arti-ficially To solve this problem the sliding-window method isadopted here Sliding-window principle is shown in Figure 2When new samples are added to the window the oldest datainside the window will be discarded

We use sliding window to change delay which is shownin Figure 3 Suppose the output data length is 119871 select theinput data region 119906(119896 minus 119889max) 119906(119896) 119906(119871) Then theinput data can be moved according to different delay from119889 to 119889max

43 The Algorithm Description The detailed procedure ofsubspace identification based on ODC algorithm can beexpressed as Algorithm 1

Then the best combination 119899lowast 119889lowast and the optimum

matching model parameters are all obtained

5 Simulation Results

To demonstrate the superiority of the proposed order selec-tion method in this paper over the conventional methodtheir performance is evaluated through a numerical exampleand an industrial illustrate

6 Mathematical Problems in Engineering

u(k minus 1)

u(k)

u(L)

(d = 0)

y(k)

y(k + 1)

y(L)

u(k minus 1)

u(L minus 1)

u(L)

(d = 1)

y(k)

y(k + 1)

y(L)

u(L minus 1)

u(L)

y(k)

y(k + 1)

y(L)

u(k minus dmax) u(k minus dmax) u(k minus dmax)

u(L minus dmax)

(d = dmax)

middot middot middot

Figure 3 Modeling data sliding sketch map

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

minus1

minus2

minus3

minus4

u1

(a) Input signal 1199061

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

minus1

minus2

minus3

minus4

u2

(b) Input signal 1199062

0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

y1

minus5

minus10

minus15

minus20

(c) Output signal 1199101

0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

y2

minus5

minus10

minus15

(d) Output signal 1199102

Figure 4 PO-Moesp subspace system identification inputoutput signal

51 Example 1 MIMO Process with Input Delay The firstmodel (15) is a MIMO system

119909 (119896 + 1) =[[[

[

0603 0603 0 0

minus0603 0603 0 0

0 0 minus0603 minus0603

0 0 0603 minus0603

]]]

]

119909 (119896)

+[[[

[

11650 minus06965

06268 16961

00751 00591

03516 17971

]]]

]

119906 (119896 minus 5)

119910 (119896) = [02641 minus14462 12460 05774

08717 minus07012 minus06390 minus03600] 119909 (119896)

+ [minus01356 minus12704

minus13493 09846] 119906 (119896 minus 5)

(15)

The time delay is 119889 = 5 and the sample is 119871 = 1000 Weuse the zeromeanwhite noise as the input 119906

1and 1199062sequence

to excite the system The input and output curves are shownin Figure 4

Firstly we use conventional subspace model selectionmethod which is dependant on the number of eigenvaluesFigure 5(a) is the situation of delay = 0 and Figure 5(b) is the

Mathematical Problems in Engineering 7

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Order

Eige

nval

ue

(a) 119889 = 0 situation

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

Order

Eige

nval

ue

(b) 119889 = 5 situation

Figure 5The number of eigenvalues generated when 119889 = 0 and 119889 =5 respectively (a) 119889 = 0 and (b) 119889 = 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

Perfo

rman

ce in

dexJ

Order n

JMRSEJMSE2

JMSE1

Figure 6 Obtain the input delay system order by original errorcriterion methods

situation of delay = 5 According to this strategy the order ofmodel increases from 4 to 13 when the system has delay

Also theMSE (mean squared error)MRSE (mean relativesquared error) and AIC criteria are all tested as shown inFigures 6 and 7Thesemethods all have the distinct inflectionpoint at 13 which is for the 4-order system with input delay

2 4 6 8 10 12 14 16 18

0

1

2

AIC

inde

x

minus6

minus5

minus4

minus3

minus2

minus1

Order n

times104

Figure 7 Obtain the input delay system order by AIC index

0246810 123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

1

Figure 8 The corresponding mean square error surface 119869MSE1generated by ODC

5 Obviously for obtaining the ideal model of a delay systemthese conventional methods have to increase the order

Next the performances of the proposed ODCmethod inthis paper are presented As the two search parameters areavailable so the index performance is shown as a surfaceThe 119869MSE1 of output 1199101 is shown in Figure 8 and the 119869MSE2 ofoutput 119910

2is shown in Figure 9 Three axes are the order 119899

delay 119889 and 119869(sdot) respectivelyTheAIC index surface is shownin Figure 10 The minimum value of these indexes can be goteasily they are also the inflection pointsThe order and delayresults are 119899 = 4 119889 = 5 This is the same with the actualsystem model

The corresponding identified model matrices

119860 =[[[

[

05017 06047 02024 01514

minus06256 05164 03929 minus00828

02228 03763 minus04884 minus06090

01883 minus00936 06069 minus05297

]]]

]

119861 =[[[

[

minus00386 18602

minus10028 00809

minus05982 minus10787

minus00583 minus04779

]]]

]

8 Mathematical Problems in Engineering

02

46

810

123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

2

Figure 9 The corresponding mean square error surface 119869MSE2generated by ODC

02468

10 12

34

56

0

2

AIC

inde

x

minus6

minus4

minus2

Order nDelay d

n = 4 d = 5

times104

Figure 10 The corresponding AIC criterion surface generated byODC

119862 = [minus14276 10033 minus11161 03224

minus11801 minus05779 04116 minus04398]

119863 = [minus01356 minus12704

minus13493 09846]

(16)

For verifying the model the model output error is drawnas 1198901and 1198902in Figure 11

52 Example 2 The Kiln Industrial Illustration To demon-strate the superiority of the proposed order selection methodin this paper over the conventional method their perfor-mance is evaluated through an industrial illustration Thedata come from actual kiln production data of an enterpriseHere the gas flow and the second air flow are selected asthe control input and the calcination temperature and kilntail temperature are taken as output variables The samplingtime is 119879

119904= 1 119904 then use Moesp method modelling the

kiln based on the inputoutput data after preprocessing Herethree main practical problems are solved

521 The First Problem The problem is that these two inputvariables have different delay They need to be identified

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus12

minus10

minus8

minus6

minus4

minus2

times10minus15

e 1

(a) 1198901

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus6

minus4

minus2

times10minus15

e 2

(b) 1198902

Figure 11 Modeling error curves 1198901

(a) and 1198902

(b)

respectively that is to say a triple loop about 1198891 1198892 and 119899

should be carried for solving 119869(119899 1198891 1198892)

In order to obtain the inflection point information moredirectly we identify the order 119899 firstly According to theindustrial field situation choose the possibility maximumvalues which are 119889

1max = 150 1198892max = 150 At first

travel all the possible 1198891 1198892and compute the smallest 119869MSE1

119869MSE2 and 119869MRSE corresponding to the different order 119899Thesecurves are shown in Figure 12 As can be seen from thecurves all of these three error criteria achieve the inflectionpoint at 119899 = 5 so the order is got Then specify 119899 =

5 the triple loop is reduced to double loop which is tocompute 119869MSE1 and 119869MSE2 corresponding to 1198891 and 1198892 between[0 150]

522The Second Problem From Figure 13 we notice anotherproblem that the outputs 119910

1and 119910

2generate different inflec-

tion pointIn order to solve this problem the criterion should choose

MRSE and AIC which take into account 1199101and 119910

2both

together Then get Figures 14 and 15It can be conducted that 119899 = 5 119889

1= 30 119889

2= 100

Mathematical Problems in Engineering 9

Table 2 When 119879119904

= 10 s the order the minmum 119869MRSE and the delay

119899 1 2 3 4 5 6 7 81198891

4 3 3 3 3 3 3 31198892

5 10 10 10 10 10 10 10119869MRSE 6981251 595572 499145 384619 239931 239931 239931 239931

1 2 3 4 5 6 7 8 9 10 11 120

50

100

1 2 3 4 5 6 7 8 9 10 11 120

200

400

1 2 3 4 5 6 7 8 9 10 11 120

02

04

Order n

Order n

Order n

J MRS

EJ M

SE2

J MSE

1

Figure 12 The corresponding minimum 119869(119899 1198891

1198892

) curves of eachorder when traversal 119889

1

1198892

0 50 100 150 0 50 100 150050

100150200250300350

Delay d2 Delay d1

J MSE

i

JMSE1

JMSE2

50 100

JMJJ SE1

JMJJ SE2

Figure 13 119869MSE1 and 119869MSE2 surface generated by ODC when 119899 = 5

0 50 100150 0 50 100 150

384

42444648

5

AIC

inde

x

d2 d1

times104

d1 = 30 d2 = 100

AIC = 40159e + 04

d1 = 3330000 dddd222 = 1000

AICAIC 4 0150159e + 0444

Figure 14 AIC surface generated by ODC when 119899 = 5

0 50 100 150 0 50 100 150

0

50

100

150

200

250

d2 d1

J MRS

E

d1 = 30 d2 = 100 JMRSE = 16448d = 30 d = 100 JMJJ RSE = 161 4448

Figure 15 119869MRSE surface generated by ODC when 119899 = 5

0 50100

150

050

100150

050

100150200250300350400

(3 10 239931)

J MRS

E

d2 d150

10050100

Figure 16 119869MRSE surface generated by ODC when 119899 = 5 and 119879119904

=

10 s

0 100 200 300 400 500 600 700 800 900 10001220

1240

1260

1280

1300

1320

1340

1360

1380

Time (s)

Measured valuePredictive value

Kiln

tail

tem

pera

ture

y1

(∘C)

Figure 17 Comparison of calcination temperature measured curveand model predictive curve

10 Mathematical Problems in Engineering

Measured valuePredictive value

0 100 200 300 400 500 600 700 800 900 10001010

1020

1030

1040

1050

1060

1070

1080

1090

Time (s)

Kiln

tail

tem

pera

ture

y2

(∘C)

Figure 18 Comparison of kiln tail temperaturemeasured curve andmodel predictive curve

523 The Third Problem However there is still a problemwhich needs to be resolved In general the model with largetime delay will increase the difficulty of controller designingWe know that when the sampling frequency is higher thanthe actual needed frequency there will be lots of redundantdata And this will raise the model order and the delayTherefore the delay can be reduced by properly decreasingsampling frequency Considering the kiln is a slow time-varying process changing the sampling time from 1 s to 10 swill not affect the model accuracy

From Table 2 it can be seen that when set 119879119904= 10 s the

order and inflexion point is still 119899 = 5 We can also changethe delay to 119889

1= 3010 = 3 119889

2= 10010 = 10 The 119869MRSE

surface can be got as in Figure 16 the results show that 1198891=

3 1198892= 10 This is in agreement with the analysis before

The corresponding rotary kiln calcining zone tempera-ture model is

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896 minus 1198891)

119910 (119896) = 119862119909 (119896) + 119863119906 (119896 minus 1198892)

(17)

and the order 119899 = 5 delay 1198891= 3 119889

2= 10

119860 =

[[[[[

[

09936 00015 minus00062 minus00007 minus00001

00063 09906 00170 minus00017 minus00001

minus00009 00026 09793 minus00147 00209

00000 minus00001 00018 09985 minus00069

minus00000 00001 minus00015 00062 09848

]]]]]

]

119861 = 10minus3

times

[[[[[

[

minus00908 minus00129

minus01650 00957

00680 minus00329

minus00734 00191

minus00093 minus00012

]]]]]

]

119862 = [minus57995 29327 minus05841 minus00157 00260

minus67753 minus23024 minus00247 minus00596 minus00063]

119863 = 10minus15

times [01205 02699

minus01066 00143]

(18)

Compare the measured value and the predicted valuegenerated by the model in Figures 17 and 18

6 Conclusion

In this paper the calcining belt state space model of rotarykiln is built using PO-Moesp subspace method And a noveldouble parameters error performance criterion for the orderchoosing in subspace modelling is introduced Since thepresented method considering the order and delay simulta-neously it can reduce the model order of the delay systemeffectively And also a strategy for stripping the delay factorsfrom the historical data is also proposed The algorithm isverified in identifying an industrial lime kiln In this examplewe solve the practical problem of industrial process withmultidelay and also reduce the order by adjusting samplingtime Further research could shed more light on the issue ofapplying the model online The problem in industrial field ismore complex than simulation environment How to extractproblems from industrial practice and guide the direction ofmodeling research has become the study focus

Conflict of Interests

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

References

[1] M Georgallis P Nowak M Salcudean and I S GartshoreldquoModelling the rotary lime kilnrdquo The Canadian Journal ofChemical Engineering vol 83 no 2 pp 212ndash223 2005

[2] Z Sogut Z Oktay and H Karakoc ldquoMathematical modeling ofheat recovery from a rotary kilnrdquo Applied Thermal Engineeringvol 30 no 8-9 pp 817ndash825 2010

[3] Y H Kim ldquoDevelopment of process model of a rotary kilnfor volatile organic compound recovery from coconut shellrdquoKorean Journal of Chemical Engineering vol 29 no 12 pp 1674ndash1679 2012

[4] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[5] W Weijtjens G de Sitter C Devriendt and P GuillaumeldquoOperational modal parameter estimation of MIMO systemsusing transmissibility functionsrdquo Automatica vol 50 no 2 pp559ndash564 2014

[6] M Imber and V Paschkis ldquoA new theory for a rotary-kiln heatexchangerrdquo International Journal of Heat andMass Transfer vol5 no 7 pp 623ndash638 1962

[7] A Sass ldquoSimulation of heat-transfer phenomena in a rotarykilnrdquo Industrial amp Engineering Chemistry Process Design andDevelopment vol 6 no 4 pp 532ndash535 1967

Mathematical Problems in Engineering 11

[8] S D Shelukar H G K Sundar R Semiat J T Richardson andD Luss ldquoContinuous rotary kiln calcination of yttrium bariumcopper oxide precursor powdersrdquo Industrial and EngineeringChemistry Research vol 33 no 2 pp 421ndash427 1994

[9] Y Yang J Rakhorst M A Reuter and J H L Voncken ldquoAnal-ysis of gas flow and mixing in a rotary kiln waste incineratorrdquoin Proceedings of the 2nd International Conference on CFD inthe Minerals and Process Industries pp 443ndash448 MelbourneAustralia

[10] Y Wang X H Fan and X L Chen ldquoMathematical modelsand expert system for grate-kiln process of iron ore oxide pelletproduction (Part I) mathematical models of grate processrdquoJournal of Central South University of Technology vol 19 no 4pp 1092ndash1097 2012

[11] G Mercere L Bako and S Lecœuche ldquoPropagator-basedmethods for recursive subspace model identificationrdquo SignalProcessing vol 88 no 3 pp 468ndash491 2008

[12] P Misra and M Nikolaou ldquoInput design for model orderdetermination in subspace identificationrdquo AIChE Journal vol49 no 8 pp 2124ndash2132 2003

[13] B Liu B Fang X Liu J Chen Z Huang and X HeldquoLarge margin subspace learning for feature selectionrdquo PatternRecognition vol 46 no 10 pp 2798ndash2806 2013

[14] H Oku and H Kimura ldquoRecursive 4SID algorithms usinggradient type subspace trackingrdquo Automatica vol 38 no 6 pp1035ndash1043 2002

[15] Y Subasi and M Demirekler ldquoQuantitative measure of observ-ability for linear stochastic systemsrdquo Automatica vol 50 no 6pp 1669ndash1674 2014

[16] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[17] X Pan H Zhu F Yang and X Zeng ldquoSubspace trajectorypiecewise-linear model order reduction for nonlinear circuitsrdquoCommunications in Computational Physics vol 14 no 3 pp639ndash663 2013

[18] M Doumlhler and L Mevel ldquoFast multi-order computationof system matrices in subspace-based system identificationrdquoControl Engineering Practice vol 20 no 9 pp 882ndash894 2012

[19] T Breiten and T Damm ldquoKrylov subspace methods for modelorder reduction of bilinear control systemsrdquo Systems and Con-trol Letters vol 59 no 8 pp 443ndash450 2010

[20] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[21] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol 19 no 6 pp 716ndash723 1974

[22] E E Ioannidis ldquoAkaikersquos information criterion correction forthe least-squares autoregressive spectral estimatorrdquo Journal ofTime Series Analysis vol 32 no 6 pp 618ndash630 2011

[23] K Peternell W Scherrer and M Deistler ldquoStatistical analysisof subspace identification methodsrdquo in Proceedings of the 3rdEuropean Control Conference (ECC rsquo95) vol 2 p 1342 1995

[24] D Bauer ldquoOrder estimation in the context of MOESP subspaceidentification methodsrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[25] D Bauer ldquoOrder estimation for subspace methodsrdquo Automat-ica vol 37 no 10 pp 1561ndash1573 2001

[26] J Shalchian A Khaki-Sedigh and A Fatehi ldquoA subspacebased method for time delay estimationrdquo in Proceedings of the

4th International Symposium on Communications Control andSignal Processing (ISCCSP rsquo10) p 4 March 2010

[27] J Lee and T F Edgar ldquoSubspace identification method forsimulation of closed-loop systems with time delaysrdquo AIChEJournal vol 48 no 2 pp 417ndash420 2002

[28] H Zhang T Ma G-B Huang and Z Wang ldquoRobust globalexponential synchronization of uncertain chaotic delayed neu-ral networks via dual-stage impulsive controlrdquo IEEE Transac-tions on Systems Man and Cybernetics B Cybernetics vol 40no 3 pp 831ndash844 2010

[29] P van Overschee and B deMoor ldquoA unifying theorem for threesubspace system identification algorithmsrdquo Automatica vol 31no 12 pp 1853ndash1864 1995

[30] W Favoreel B de Moor and P van Overschee ldquoSubspace statespace system identification for industrial processesrdquo Journal ofProcess Control vol 10 no 2 pp 149ndash155 2000

[31] P van Overschee and B deMoor ldquoN4SID subspace algorithmsfor the identification of combined deterministic-stochasticsystemsrdquo Automatica vol 30 no 1 pp 75ndash93 1994

[32] M Verhaegen ldquoIdentification of the deterministic part ofMIMO state space models given in innovations form frominput-output datardquo Automatica vol 30 no 1 pp 61ndash74 1994

[33] W E Larimore ldquoCanonical variate analysis in identificationfiltering and adaptive controlrdquo in Proceedings of the 29th IEEEConference on Decision and Control pp 596ndash604 December1990

[34] M Viberg ldquoSubspace-based methods for the identification oflinear time-invariant systemsrdquo Automatica vol 31 no 12 pp1835ndash1851 1995

[35] J Wang and S J Qin ldquoA new subspace identification approachbased on principal component analysisrdquo Journal of ProcessControl vol 12 no 8 pp 841ndash855 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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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

Initialization 119899 = 2 119889 = 1 the modelling data after pretreatment 119906(119896 minus 119889max) 119906(119896) 119906(119871) and 119910(119896) 119910(119871)(1) for 119899 = 2 to 119899 = 119899max(2) for 119889 = 1 to 119889 = 119889max(3) Rolling the modelling input data 119906 based on the hypothesis delay 119889 get the data set 119906(119896 minus 119889) 119906(119896) 119906(119871 minus 119889)

119910(119896) 119910(119871)(4) Construct input and output Hankel matrices 119880

119901

119884119901

119880119891

119884119891

(5) Calculate 119860 119861 119862119863 by Moesp method in Section 3 based on Hankel matrices(6) Substitute the 119860 119861 119862119863 and 119889 into the formula (1)(7) Calculate and store the model error and performance index 119869(119899 119889)(8) end for(9) end for(10) Search the inflection point of the 119869(119899 119889) surface

Algorithm 1

y

y

L

L

xk k + Lo

xo k + 1 k + L + 1

Figure 2 Sliding time window sketch map

Section 2 Then its model error can be deserved as Σ119899119889

=

(1119873)sum119873

119894=1

119890(119896)119890(119896)119879 then the AIC(119899 119889) with respect to

individual (119899 119889) as

AIC119873(119899 119889) = 119873 (119898 (1 + ln 2120587) + ln 1003816100381610038161003816Σ119899119889

1003816100381610038161003816) + 2120575119899119872119899 (13)

Also the MRSE criterion 119869MSE2 has the similar form

119869MRSE (119899 119889) =1

119871

119871

sum119896=1

radicsum119899119910

119895=1

(119910119899119889119896

(119895) minus 119910119899119889119896

(119895))2

sum119899119910

119895=1

119910119899119889119896

(119895)2

(14)

Other performance index 119869(sdot) such as SVC IVC NICcriteria mentioned in [25] can also be modified as thismethod

The original performance index just identifies the order119899 Suppose 119899 isin [1 119899max] then 119869(1) 119869(2) 119869(3) 119869(119899max) arecalculated respectively then the inflection point 119899lowast is the best

order and the corresponding systemmatrices119860lowast 119861lowast 119862lowast119863lowastare the best model After improvement we add the delay asother optimal parameters So the system order 119899 isin [1 119899max]and input delay 119889 isin [1 119889max] are all embedded in 119869(119899 119889)Then calculate 119869(1 1) 119869(119899max 1) 119869(1 2) 119869(119899max 2)119869(1 119889max) 119869(119899max 119889max) respectively By searching theminimum point 119899lowast 119889lowast of surface 119869(119899 119889) the best orderdelay and system matrices can all be got Thus taking thedelay as another parameter in the modelling methods it caneffectively avoid high order results in the delay system

42 The Delay Factor Stripping from Historical Data Theintroduction of delay parameters in performance criterionhas resulted to a notable problem The modelling historicaldata matrices have already included delay information It isdifficult to change 119889 in the performance criterion 119869(119899 119889) arti-ficially To solve this problem the sliding-window method isadopted here Sliding-window principle is shown in Figure 2When new samples are added to the window the oldest datainside the window will be discarded

We use sliding window to change delay which is shownin Figure 3 Suppose the output data length is 119871 select theinput data region 119906(119896 minus 119889max) 119906(119896) 119906(119871) Then theinput data can be moved according to different delay from119889 to 119889max

43 The Algorithm Description The detailed procedure ofsubspace identification based on ODC algorithm can beexpressed as Algorithm 1

Then the best combination 119899lowast 119889lowast and the optimum

matching model parameters are all obtained

5 Simulation Results

To demonstrate the superiority of the proposed order selec-tion method in this paper over the conventional methodtheir performance is evaluated through a numerical exampleand an industrial illustrate

6 Mathematical Problems in Engineering

u(k minus 1)

u(k)

u(L)

(d = 0)

y(k)

y(k + 1)

y(L)

u(k minus 1)

u(L minus 1)

u(L)

(d = 1)

y(k)

y(k + 1)

y(L)

u(L minus 1)

u(L)

y(k)

y(k + 1)

y(L)

u(k minus dmax) u(k minus dmax) u(k minus dmax)

u(L minus dmax)

(d = dmax)

middot middot middot

Figure 3 Modeling data sliding sketch map

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

minus1

minus2

minus3

minus4

u1

(a) Input signal 1199061

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

minus1

minus2

minus3

minus4

u2

(b) Input signal 1199062

0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

y1

minus5

minus10

minus15

minus20

(c) Output signal 1199101

0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

y2

minus5

minus10

minus15

(d) Output signal 1199102

Figure 4 PO-Moesp subspace system identification inputoutput signal

51 Example 1 MIMO Process with Input Delay The firstmodel (15) is a MIMO system

119909 (119896 + 1) =[[[

[

0603 0603 0 0

minus0603 0603 0 0

0 0 minus0603 minus0603

0 0 0603 minus0603

]]]

]

119909 (119896)

+[[[

[

11650 minus06965

06268 16961

00751 00591

03516 17971

]]]

]

119906 (119896 minus 5)

119910 (119896) = [02641 minus14462 12460 05774

08717 minus07012 minus06390 minus03600] 119909 (119896)

+ [minus01356 minus12704

minus13493 09846] 119906 (119896 minus 5)

(15)

The time delay is 119889 = 5 and the sample is 119871 = 1000 Weuse the zeromeanwhite noise as the input 119906

1and 1199062sequence

to excite the system The input and output curves are shownin Figure 4

Firstly we use conventional subspace model selectionmethod which is dependant on the number of eigenvaluesFigure 5(a) is the situation of delay = 0 and Figure 5(b) is the

Mathematical Problems in Engineering 7

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Order

Eige

nval

ue

(a) 119889 = 0 situation

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

Order

Eige

nval

ue

(b) 119889 = 5 situation

Figure 5The number of eigenvalues generated when 119889 = 0 and 119889 =5 respectively (a) 119889 = 0 and (b) 119889 = 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

Perfo

rman

ce in

dexJ

Order n

JMRSEJMSE2

JMSE1

Figure 6 Obtain the input delay system order by original errorcriterion methods

situation of delay = 5 According to this strategy the order ofmodel increases from 4 to 13 when the system has delay

Also theMSE (mean squared error)MRSE (mean relativesquared error) and AIC criteria are all tested as shown inFigures 6 and 7Thesemethods all have the distinct inflectionpoint at 13 which is for the 4-order system with input delay

2 4 6 8 10 12 14 16 18

0

1

2

AIC

inde

x

minus6

minus5

minus4

minus3

minus2

minus1

Order n

times104

Figure 7 Obtain the input delay system order by AIC index

0246810 123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

1

Figure 8 The corresponding mean square error surface 119869MSE1generated by ODC

5 Obviously for obtaining the ideal model of a delay systemthese conventional methods have to increase the order

Next the performances of the proposed ODCmethod inthis paper are presented As the two search parameters areavailable so the index performance is shown as a surfaceThe 119869MSE1 of output 1199101 is shown in Figure 8 and the 119869MSE2 ofoutput 119910

2is shown in Figure 9 Three axes are the order 119899

delay 119889 and 119869(sdot) respectivelyTheAIC index surface is shownin Figure 10 The minimum value of these indexes can be goteasily they are also the inflection pointsThe order and delayresults are 119899 = 4 119889 = 5 This is the same with the actualsystem model

The corresponding identified model matrices

119860 =[[[

[

05017 06047 02024 01514

minus06256 05164 03929 minus00828

02228 03763 minus04884 minus06090

01883 minus00936 06069 minus05297

]]]

]

119861 =[[[

[

minus00386 18602

minus10028 00809

minus05982 minus10787

minus00583 minus04779

]]]

]

8 Mathematical Problems in Engineering

02

46

810

123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

2

Figure 9 The corresponding mean square error surface 119869MSE2generated by ODC

02468

10 12

34

56

0

2

AIC

inde

x

minus6

minus4

minus2

Order nDelay d

n = 4 d = 5

times104

Figure 10 The corresponding AIC criterion surface generated byODC

119862 = [minus14276 10033 minus11161 03224

minus11801 minus05779 04116 minus04398]

119863 = [minus01356 minus12704

minus13493 09846]

(16)

For verifying the model the model output error is drawnas 1198901and 1198902in Figure 11

52 Example 2 The Kiln Industrial Illustration To demon-strate the superiority of the proposed order selection methodin this paper over the conventional method their perfor-mance is evaluated through an industrial illustration Thedata come from actual kiln production data of an enterpriseHere the gas flow and the second air flow are selected asthe control input and the calcination temperature and kilntail temperature are taken as output variables The samplingtime is 119879

119904= 1 119904 then use Moesp method modelling the

kiln based on the inputoutput data after preprocessing Herethree main practical problems are solved

521 The First Problem The problem is that these two inputvariables have different delay They need to be identified

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus12

minus10

minus8

minus6

minus4

minus2

times10minus15

e 1

(a) 1198901

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus6

minus4

minus2

times10minus15

e 2

(b) 1198902

Figure 11 Modeling error curves 1198901

(a) and 1198902

(b)

respectively that is to say a triple loop about 1198891 1198892 and 119899

should be carried for solving 119869(119899 1198891 1198892)

In order to obtain the inflection point information moredirectly we identify the order 119899 firstly According to theindustrial field situation choose the possibility maximumvalues which are 119889

1max = 150 1198892max = 150 At first

travel all the possible 1198891 1198892and compute the smallest 119869MSE1

119869MSE2 and 119869MRSE corresponding to the different order 119899Thesecurves are shown in Figure 12 As can be seen from thecurves all of these three error criteria achieve the inflectionpoint at 119899 = 5 so the order is got Then specify 119899 =

5 the triple loop is reduced to double loop which is tocompute 119869MSE1 and 119869MSE2 corresponding to 1198891 and 1198892 between[0 150]

522The Second Problem From Figure 13 we notice anotherproblem that the outputs 119910

1and 119910

2generate different inflec-

tion pointIn order to solve this problem the criterion should choose

MRSE and AIC which take into account 1199101and 119910

2both

together Then get Figures 14 and 15It can be conducted that 119899 = 5 119889

1= 30 119889

2= 100

Mathematical Problems in Engineering 9

Table 2 When 119879119904

= 10 s the order the minmum 119869MRSE and the delay

119899 1 2 3 4 5 6 7 81198891

4 3 3 3 3 3 3 31198892

5 10 10 10 10 10 10 10119869MRSE 6981251 595572 499145 384619 239931 239931 239931 239931

1 2 3 4 5 6 7 8 9 10 11 120

50

100

1 2 3 4 5 6 7 8 9 10 11 120

200

400

1 2 3 4 5 6 7 8 9 10 11 120

02

04

Order n

Order n

Order n

J MRS

EJ M

SE2

J MSE

1

Figure 12 The corresponding minimum 119869(119899 1198891

1198892

) curves of eachorder when traversal 119889

1

1198892

0 50 100 150 0 50 100 150050

100150200250300350

Delay d2 Delay d1

J MSE

i

JMSE1

JMSE2

50 100

JMJJ SE1

JMJJ SE2

Figure 13 119869MSE1 and 119869MSE2 surface generated by ODC when 119899 = 5

0 50 100150 0 50 100 150

384

42444648

5

AIC

inde

x

d2 d1

times104

d1 = 30 d2 = 100

AIC = 40159e + 04

d1 = 3330000 dddd222 = 1000

AICAIC 4 0150159e + 0444

Figure 14 AIC surface generated by ODC when 119899 = 5

0 50 100 150 0 50 100 150

0

50

100

150

200

250

d2 d1

J MRS

E

d1 = 30 d2 = 100 JMRSE = 16448d = 30 d = 100 JMJJ RSE = 161 4448

Figure 15 119869MRSE surface generated by ODC when 119899 = 5

0 50100

150

050

100150

050

100150200250300350400

(3 10 239931)

J MRS

E

d2 d150

10050100

Figure 16 119869MRSE surface generated by ODC when 119899 = 5 and 119879119904

=

10 s

0 100 200 300 400 500 600 700 800 900 10001220

1240

1260

1280

1300

1320

1340

1360

1380

Time (s)

Measured valuePredictive value

Kiln

tail

tem

pera

ture

y1

(∘C)

Figure 17 Comparison of calcination temperature measured curveand model predictive curve

10 Mathematical Problems in Engineering

Measured valuePredictive value

0 100 200 300 400 500 600 700 800 900 10001010

1020

1030

1040

1050

1060

1070

1080

1090

Time (s)

Kiln

tail

tem

pera

ture

y2

(∘C)

Figure 18 Comparison of kiln tail temperaturemeasured curve andmodel predictive curve

523 The Third Problem However there is still a problemwhich needs to be resolved In general the model with largetime delay will increase the difficulty of controller designingWe know that when the sampling frequency is higher thanthe actual needed frequency there will be lots of redundantdata And this will raise the model order and the delayTherefore the delay can be reduced by properly decreasingsampling frequency Considering the kiln is a slow time-varying process changing the sampling time from 1 s to 10 swill not affect the model accuracy

From Table 2 it can be seen that when set 119879119904= 10 s the

order and inflexion point is still 119899 = 5 We can also changethe delay to 119889

1= 3010 = 3 119889

2= 10010 = 10 The 119869MRSE

surface can be got as in Figure 16 the results show that 1198891=

3 1198892= 10 This is in agreement with the analysis before

The corresponding rotary kiln calcining zone tempera-ture model is

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896 minus 1198891)

119910 (119896) = 119862119909 (119896) + 119863119906 (119896 minus 1198892)

(17)

and the order 119899 = 5 delay 1198891= 3 119889

2= 10

119860 =

[[[[[

[

09936 00015 minus00062 minus00007 minus00001

00063 09906 00170 minus00017 minus00001

minus00009 00026 09793 minus00147 00209

00000 minus00001 00018 09985 minus00069

minus00000 00001 minus00015 00062 09848

]]]]]

]

119861 = 10minus3

times

[[[[[

[

minus00908 minus00129

minus01650 00957

00680 minus00329

minus00734 00191

minus00093 minus00012

]]]]]

]

119862 = [minus57995 29327 minus05841 minus00157 00260

minus67753 minus23024 minus00247 minus00596 minus00063]

119863 = 10minus15

times [01205 02699

minus01066 00143]

(18)

Compare the measured value and the predicted valuegenerated by the model in Figures 17 and 18

6 Conclusion

In this paper the calcining belt state space model of rotarykiln is built using PO-Moesp subspace method And a noveldouble parameters error performance criterion for the orderchoosing in subspace modelling is introduced Since thepresented method considering the order and delay simulta-neously it can reduce the model order of the delay systemeffectively And also a strategy for stripping the delay factorsfrom the historical data is also proposed The algorithm isverified in identifying an industrial lime kiln In this examplewe solve the practical problem of industrial process withmultidelay and also reduce the order by adjusting samplingtime Further research could shed more light on the issue ofapplying the model online The problem in industrial field ismore complex than simulation environment How to extractproblems from industrial practice and guide the direction ofmodeling research has become the study focus

Conflict of Interests

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

References

[1] M Georgallis P Nowak M Salcudean and I S GartshoreldquoModelling the rotary lime kilnrdquo The Canadian Journal ofChemical Engineering vol 83 no 2 pp 212ndash223 2005

[2] Z Sogut Z Oktay and H Karakoc ldquoMathematical modeling ofheat recovery from a rotary kilnrdquo Applied Thermal Engineeringvol 30 no 8-9 pp 817ndash825 2010

[3] Y H Kim ldquoDevelopment of process model of a rotary kilnfor volatile organic compound recovery from coconut shellrdquoKorean Journal of Chemical Engineering vol 29 no 12 pp 1674ndash1679 2012

[4] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[5] W Weijtjens G de Sitter C Devriendt and P GuillaumeldquoOperational modal parameter estimation of MIMO systemsusing transmissibility functionsrdquo Automatica vol 50 no 2 pp559ndash564 2014

[6] M Imber and V Paschkis ldquoA new theory for a rotary-kiln heatexchangerrdquo International Journal of Heat andMass Transfer vol5 no 7 pp 623ndash638 1962

[7] A Sass ldquoSimulation of heat-transfer phenomena in a rotarykilnrdquo Industrial amp Engineering Chemistry Process Design andDevelopment vol 6 no 4 pp 532ndash535 1967

Mathematical Problems in Engineering 11

[8] S D Shelukar H G K Sundar R Semiat J T Richardson andD Luss ldquoContinuous rotary kiln calcination of yttrium bariumcopper oxide precursor powdersrdquo Industrial and EngineeringChemistry Research vol 33 no 2 pp 421ndash427 1994

[9] Y Yang J Rakhorst M A Reuter and J H L Voncken ldquoAnal-ysis of gas flow and mixing in a rotary kiln waste incineratorrdquoin Proceedings of the 2nd International Conference on CFD inthe Minerals and Process Industries pp 443ndash448 MelbourneAustralia

[10] Y Wang X H Fan and X L Chen ldquoMathematical modelsand expert system for grate-kiln process of iron ore oxide pelletproduction (Part I) mathematical models of grate processrdquoJournal of Central South University of Technology vol 19 no 4pp 1092ndash1097 2012

[11] G Mercere L Bako and S Lecœuche ldquoPropagator-basedmethods for recursive subspace model identificationrdquo SignalProcessing vol 88 no 3 pp 468ndash491 2008

[12] P Misra and M Nikolaou ldquoInput design for model orderdetermination in subspace identificationrdquo AIChE Journal vol49 no 8 pp 2124ndash2132 2003

[13] B Liu B Fang X Liu J Chen Z Huang and X HeldquoLarge margin subspace learning for feature selectionrdquo PatternRecognition vol 46 no 10 pp 2798ndash2806 2013

[14] H Oku and H Kimura ldquoRecursive 4SID algorithms usinggradient type subspace trackingrdquo Automatica vol 38 no 6 pp1035ndash1043 2002

[15] Y Subasi and M Demirekler ldquoQuantitative measure of observ-ability for linear stochastic systemsrdquo Automatica vol 50 no 6pp 1669ndash1674 2014

[16] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[17] X Pan H Zhu F Yang and X Zeng ldquoSubspace trajectorypiecewise-linear model order reduction for nonlinear circuitsrdquoCommunications in Computational Physics vol 14 no 3 pp639ndash663 2013

[18] M Doumlhler and L Mevel ldquoFast multi-order computationof system matrices in subspace-based system identificationrdquoControl Engineering Practice vol 20 no 9 pp 882ndash894 2012

[19] T Breiten and T Damm ldquoKrylov subspace methods for modelorder reduction of bilinear control systemsrdquo Systems and Con-trol Letters vol 59 no 8 pp 443ndash450 2010

[20] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[21] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol 19 no 6 pp 716ndash723 1974

[22] E E Ioannidis ldquoAkaikersquos information criterion correction forthe least-squares autoregressive spectral estimatorrdquo Journal ofTime Series Analysis vol 32 no 6 pp 618ndash630 2011

[23] K Peternell W Scherrer and M Deistler ldquoStatistical analysisof subspace identification methodsrdquo in Proceedings of the 3rdEuropean Control Conference (ECC rsquo95) vol 2 p 1342 1995

[24] D Bauer ldquoOrder estimation in the context of MOESP subspaceidentification methodsrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[25] D Bauer ldquoOrder estimation for subspace methodsrdquo Automat-ica vol 37 no 10 pp 1561ndash1573 2001

[26] J Shalchian A Khaki-Sedigh and A Fatehi ldquoA subspacebased method for time delay estimationrdquo in Proceedings of the

4th International Symposium on Communications Control andSignal Processing (ISCCSP rsquo10) p 4 March 2010

[27] J Lee and T F Edgar ldquoSubspace identification method forsimulation of closed-loop systems with time delaysrdquo AIChEJournal vol 48 no 2 pp 417ndash420 2002

[28] H Zhang T Ma G-B Huang and Z Wang ldquoRobust globalexponential synchronization of uncertain chaotic delayed neu-ral networks via dual-stage impulsive controlrdquo IEEE Transac-tions on Systems Man and Cybernetics B Cybernetics vol 40no 3 pp 831ndash844 2010

[29] P van Overschee and B deMoor ldquoA unifying theorem for threesubspace system identification algorithmsrdquo Automatica vol 31no 12 pp 1853ndash1864 1995

[30] W Favoreel B de Moor and P van Overschee ldquoSubspace statespace system identification for industrial processesrdquo Journal ofProcess Control vol 10 no 2 pp 149ndash155 2000

[31] P van Overschee and B deMoor ldquoN4SID subspace algorithmsfor the identification of combined deterministic-stochasticsystemsrdquo Automatica vol 30 no 1 pp 75ndash93 1994

[32] M Verhaegen ldquoIdentification of the deterministic part ofMIMO state space models given in innovations form frominput-output datardquo Automatica vol 30 no 1 pp 61ndash74 1994

[33] W E Larimore ldquoCanonical variate analysis in identificationfiltering and adaptive controlrdquo in Proceedings of the 29th IEEEConference on Decision and Control pp 596ndash604 December1990

[34] M Viberg ldquoSubspace-based methods for the identification oflinear time-invariant systemsrdquo Automatica vol 31 no 12 pp1835ndash1851 1995

[35] J Wang and S J Qin ldquoA new subspace identification approachbased on principal component analysisrdquo Journal of ProcessControl vol 12 no 8 pp 841ndash855 2002

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

u(k minus 1)

u(k)

u(L)

(d = 0)

y(k)

y(k + 1)

y(L)

u(k minus 1)

u(L minus 1)

u(L)

(d = 1)

y(k)

y(k + 1)

y(L)

u(L minus 1)

u(L)

y(k)

y(k + 1)

y(L)

u(k minus dmax) u(k minus dmax) u(k minus dmax)

u(L minus dmax)

(d = dmax)

middot middot middot

Figure 3 Modeling data sliding sketch map

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

minus1

minus2

minus3

minus4

u1

(a) Input signal 1199061

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

minus1

minus2

minus3

minus4

u2

(b) Input signal 1199062

0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

y1

minus5

minus10

minus15

minus20

(c) Output signal 1199101

0 100 200 300 400 500 600 700 800 900 1000

0

5

10

15

20

y2

minus5

minus10

minus15

(d) Output signal 1199102

Figure 4 PO-Moesp subspace system identification inputoutput signal

51 Example 1 MIMO Process with Input Delay The firstmodel (15) is a MIMO system

119909 (119896 + 1) =[[[

[

0603 0603 0 0

minus0603 0603 0 0

0 0 minus0603 minus0603

0 0 0603 minus0603

]]]

]

119909 (119896)

+[[[

[

11650 minus06965

06268 16961

00751 00591

03516 17971

]]]

]

119906 (119896 minus 5)

119910 (119896) = [02641 minus14462 12460 05774

08717 minus07012 minus06390 minus03600] 119909 (119896)

+ [minus01356 minus12704

minus13493 09846] 119906 (119896 minus 5)

(15)

The time delay is 119889 = 5 and the sample is 119871 = 1000 Weuse the zeromeanwhite noise as the input 119906

1and 1199062sequence

to excite the system The input and output curves are shownin Figure 4

Firstly we use conventional subspace model selectionmethod which is dependant on the number of eigenvaluesFigure 5(a) is the situation of delay = 0 and Figure 5(b) is the

Mathematical Problems in Engineering 7

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Order

Eige

nval

ue

(a) 119889 = 0 situation

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

Order

Eige

nval

ue

(b) 119889 = 5 situation

Figure 5The number of eigenvalues generated when 119889 = 0 and 119889 =5 respectively (a) 119889 = 0 and (b) 119889 = 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

Perfo

rman

ce in

dexJ

Order n

JMRSEJMSE2

JMSE1

Figure 6 Obtain the input delay system order by original errorcriterion methods

situation of delay = 5 According to this strategy the order ofmodel increases from 4 to 13 when the system has delay

Also theMSE (mean squared error)MRSE (mean relativesquared error) and AIC criteria are all tested as shown inFigures 6 and 7Thesemethods all have the distinct inflectionpoint at 13 which is for the 4-order system with input delay

2 4 6 8 10 12 14 16 18

0

1

2

AIC

inde

x

minus6

minus5

minus4

minus3

minus2

minus1

Order n

times104

Figure 7 Obtain the input delay system order by AIC index

0246810 123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

1

Figure 8 The corresponding mean square error surface 119869MSE1generated by ODC

5 Obviously for obtaining the ideal model of a delay systemthese conventional methods have to increase the order

Next the performances of the proposed ODCmethod inthis paper are presented As the two search parameters areavailable so the index performance is shown as a surfaceThe 119869MSE1 of output 1199101 is shown in Figure 8 and the 119869MSE2 ofoutput 119910

2is shown in Figure 9 Three axes are the order 119899

delay 119889 and 119869(sdot) respectivelyTheAIC index surface is shownin Figure 10 The minimum value of these indexes can be goteasily they are also the inflection pointsThe order and delayresults are 119899 = 4 119889 = 5 This is the same with the actualsystem model

The corresponding identified model matrices

119860 =[[[

[

05017 06047 02024 01514

minus06256 05164 03929 minus00828

02228 03763 minus04884 minus06090

01883 minus00936 06069 minus05297

]]]

]

119861 =[[[

[

minus00386 18602

minus10028 00809

minus05982 minus10787

minus00583 minus04779

]]]

]

8 Mathematical Problems in Engineering

02

46

810

123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

2

Figure 9 The corresponding mean square error surface 119869MSE2generated by ODC

02468

10 12

34

56

0

2

AIC

inde

x

minus6

minus4

minus2

Order nDelay d

n = 4 d = 5

times104

Figure 10 The corresponding AIC criterion surface generated byODC

119862 = [minus14276 10033 minus11161 03224

minus11801 minus05779 04116 minus04398]

119863 = [minus01356 minus12704

minus13493 09846]

(16)

For verifying the model the model output error is drawnas 1198901and 1198902in Figure 11

52 Example 2 The Kiln Industrial Illustration To demon-strate the superiority of the proposed order selection methodin this paper over the conventional method their perfor-mance is evaluated through an industrial illustration Thedata come from actual kiln production data of an enterpriseHere the gas flow and the second air flow are selected asthe control input and the calcination temperature and kilntail temperature are taken as output variables The samplingtime is 119879

119904= 1 119904 then use Moesp method modelling the

kiln based on the inputoutput data after preprocessing Herethree main practical problems are solved

521 The First Problem The problem is that these two inputvariables have different delay They need to be identified

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus12

minus10

minus8

minus6

minus4

minus2

times10minus15

e 1

(a) 1198901

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus6

minus4

minus2

times10minus15

e 2

(b) 1198902

Figure 11 Modeling error curves 1198901

(a) and 1198902

(b)

respectively that is to say a triple loop about 1198891 1198892 and 119899

should be carried for solving 119869(119899 1198891 1198892)

In order to obtain the inflection point information moredirectly we identify the order 119899 firstly According to theindustrial field situation choose the possibility maximumvalues which are 119889

1max = 150 1198892max = 150 At first

travel all the possible 1198891 1198892and compute the smallest 119869MSE1

119869MSE2 and 119869MRSE corresponding to the different order 119899Thesecurves are shown in Figure 12 As can be seen from thecurves all of these three error criteria achieve the inflectionpoint at 119899 = 5 so the order is got Then specify 119899 =

5 the triple loop is reduced to double loop which is tocompute 119869MSE1 and 119869MSE2 corresponding to 1198891 and 1198892 between[0 150]

522The Second Problem From Figure 13 we notice anotherproblem that the outputs 119910

1and 119910

2generate different inflec-

tion pointIn order to solve this problem the criterion should choose

MRSE and AIC which take into account 1199101and 119910

2both

together Then get Figures 14 and 15It can be conducted that 119899 = 5 119889

1= 30 119889

2= 100

Mathematical Problems in Engineering 9

Table 2 When 119879119904

= 10 s the order the minmum 119869MRSE and the delay

119899 1 2 3 4 5 6 7 81198891

4 3 3 3 3 3 3 31198892

5 10 10 10 10 10 10 10119869MRSE 6981251 595572 499145 384619 239931 239931 239931 239931

1 2 3 4 5 6 7 8 9 10 11 120

50

100

1 2 3 4 5 6 7 8 9 10 11 120

200

400

1 2 3 4 5 6 7 8 9 10 11 120

02

04

Order n

Order n

Order n

J MRS

EJ M

SE2

J MSE

1

Figure 12 The corresponding minimum 119869(119899 1198891

1198892

) curves of eachorder when traversal 119889

1

1198892

0 50 100 150 0 50 100 150050

100150200250300350

Delay d2 Delay d1

J MSE

i

JMSE1

JMSE2

50 100

JMJJ SE1

JMJJ SE2

Figure 13 119869MSE1 and 119869MSE2 surface generated by ODC when 119899 = 5

0 50 100150 0 50 100 150

384

42444648

5

AIC

inde

x

d2 d1

times104

d1 = 30 d2 = 100

AIC = 40159e + 04

d1 = 3330000 dddd222 = 1000

AICAIC 4 0150159e + 0444

Figure 14 AIC surface generated by ODC when 119899 = 5

0 50 100 150 0 50 100 150

0

50

100

150

200

250

d2 d1

J MRS

E

d1 = 30 d2 = 100 JMRSE = 16448d = 30 d = 100 JMJJ RSE = 161 4448

Figure 15 119869MRSE surface generated by ODC when 119899 = 5

0 50100

150

050

100150

050

100150200250300350400

(3 10 239931)

J MRS

E

d2 d150

10050100

Figure 16 119869MRSE surface generated by ODC when 119899 = 5 and 119879119904

=

10 s

0 100 200 300 400 500 600 700 800 900 10001220

1240

1260

1280

1300

1320

1340

1360

1380

Time (s)

Measured valuePredictive value

Kiln

tail

tem

pera

ture

y1

(∘C)

Figure 17 Comparison of calcination temperature measured curveand model predictive curve

10 Mathematical Problems in Engineering

Measured valuePredictive value

0 100 200 300 400 500 600 700 800 900 10001010

1020

1030

1040

1050

1060

1070

1080

1090

Time (s)

Kiln

tail

tem

pera

ture

y2

(∘C)

Figure 18 Comparison of kiln tail temperaturemeasured curve andmodel predictive curve

523 The Third Problem However there is still a problemwhich needs to be resolved In general the model with largetime delay will increase the difficulty of controller designingWe know that when the sampling frequency is higher thanthe actual needed frequency there will be lots of redundantdata And this will raise the model order and the delayTherefore the delay can be reduced by properly decreasingsampling frequency Considering the kiln is a slow time-varying process changing the sampling time from 1 s to 10 swill not affect the model accuracy

From Table 2 it can be seen that when set 119879119904= 10 s the

order and inflexion point is still 119899 = 5 We can also changethe delay to 119889

1= 3010 = 3 119889

2= 10010 = 10 The 119869MRSE

surface can be got as in Figure 16 the results show that 1198891=

3 1198892= 10 This is in agreement with the analysis before

The corresponding rotary kiln calcining zone tempera-ture model is

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896 minus 1198891)

119910 (119896) = 119862119909 (119896) + 119863119906 (119896 minus 1198892)

(17)

and the order 119899 = 5 delay 1198891= 3 119889

2= 10

119860 =

[[[[[

[

09936 00015 minus00062 minus00007 minus00001

00063 09906 00170 minus00017 minus00001

minus00009 00026 09793 minus00147 00209

00000 minus00001 00018 09985 minus00069

minus00000 00001 minus00015 00062 09848

]]]]]

]

119861 = 10minus3

times

[[[[[

[

minus00908 minus00129

minus01650 00957

00680 minus00329

minus00734 00191

minus00093 minus00012

]]]]]

]

119862 = [minus57995 29327 minus05841 minus00157 00260

minus67753 minus23024 minus00247 minus00596 minus00063]

119863 = 10minus15

times [01205 02699

minus01066 00143]

(18)

Compare the measured value and the predicted valuegenerated by the model in Figures 17 and 18

6 Conclusion

In this paper the calcining belt state space model of rotarykiln is built using PO-Moesp subspace method And a noveldouble parameters error performance criterion for the orderchoosing in subspace modelling is introduced Since thepresented method considering the order and delay simulta-neously it can reduce the model order of the delay systemeffectively And also a strategy for stripping the delay factorsfrom the historical data is also proposed The algorithm isverified in identifying an industrial lime kiln In this examplewe solve the practical problem of industrial process withmultidelay and also reduce the order by adjusting samplingtime Further research could shed more light on the issue ofapplying the model online The problem in industrial field ismore complex than simulation environment How to extractproblems from industrial practice and guide the direction ofmodeling research has become the study focus

Conflict of Interests

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

References

[1] M Georgallis P Nowak M Salcudean and I S GartshoreldquoModelling the rotary lime kilnrdquo The Canadian Journal ofChemical Engineering vol 83 no 2 pp 212ndash223 2005

[2] Z Sogut Z Oktay and H Karakoc ldquoMathematical modeling ofheat recovery from a rotary kilnrdquo Applied Thermal Engineeringvol 30 no 8-9 pp 817ndash825 2010

[3] Y H Kim ldquoDevelopment of process model of a rotary kilnfor volatile organic compound recovery from coconut shellrdquoKorean Journal of Chemical Engineering vol 29 no 12 pp 1674ndash1679 2012

[4] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[5] W Weijtjens G de Sitter C Devriendt and P GuillaumeldquoOperational modal parameter estimation of MIMO systemsusing transmissibility functionsrdquo Automatica vol 50 no 2 pp559ndash564 2014

[6] M Imber and V Paschkis ldquoA new theory for a rotary-kiln heatexchangerrdquo International Journal of Heat andMass Transfer vol5 no 7 pp 623ndash638 1962

[7] A Sass ldquoSimulation of heat-transfer phenomena in a rotarykilnrdquo Industrial amp Engineering Chemistry Process Design andDevelopment vol 6 no 4 pp 532ndash535 1967

Mathematical Problems in Engineering 11

[8] S D Shelukar H G K Sundar R Semiat J T Richardson andD Luss ldquoContinuous rotary kiln calcination of yttrium bariumcopper oxide precursor powdersrdquo Industrial and EngineeringChemistry Research vol 33 no 2 pp 421ndash427 1994

[9] Y Yang J Rakhorst M A Reuter and J H L Voncken ldquoAnal-ysis of gas flow and mixing in a rotary kiln waste incineratorrdquoin Proceedings of the 2nd International Conference on CFD inthe Minerals and Process Industries pp 443ndash448 MelbourneAustralia

[10] Y Wang X H Fan and X L Chen ldquoMathematical modelsand expert system for grate-kiln process of iron ore oxide pelletproduction (Part I) mathematical models of grate processrdquoJournal of Central South University of Technology vol 19 no 4pp 1092ndash1097 2012

[11] G Mercere L Bako and S Lecœuche ldquoPropagator-basedmethods for recursive subspace model identificationrdquo SignalProcessing vol 88 no 3 pp 468ndash491 2008

[12] P Misra and M Nikolaou ldquoInput design for model orderdetermination in subspace identificationrdquo AIChE Journal vol49 no 8 pp 2124ndash2132 2003

[13] B Liu B Fang X Liu J Chen Z Huang and X HeldquoLarge margin subspace learning for feature selectionrdquo PatternRecognition vol 46 no 10 pp 2798ndash2806 2013

[14] H Oku and H Kimura ldquoRecursive 4SID algorithms usinggradient type subspace trackingrdquo Automatica vol 38 no 6 pp1035ndash1043 2002

[15] Y Subasi and M Demirekler ldquoQuantitative measure of observ-ability for linear stochastic systemsrdquo Automatica vol 50 no 6pp 1669ndash1674 2014

[16] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[17] X Pan H Zhu F Yang and X Zeng ldquoSubspace trajectorypiecewise-linear model order reduction for nonlinear circuitsrdquoCommunications in Computational Physics vol 14 no 3 pp639ndash663 2013

[18] M Doumlhler and L Mevel ldquoFast multi-order computationof system matrices in subspace-based system identificationrdquoControl Engineering Practice vol 20 no 9 pp 882ndash894 2012

[19] T Breiten and T Damm ldquoKrylov subspace methods for modelorder reduction of bilinear control systemsrdquo Systems and Con-trol Letters vol 59 no 8 pp 443ndash450 2010

[20] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[21] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol 19 no 6 pp 716ndash723 1974

[22] E E Ioannidis ldquoAkaikersquos information criterion correction forthe least-squares autoregressive spectral estimatorrdquo Journal ofTime Series Analysis vol 32 no 6 pp 618ndash630 2011

[23] K Peternell W Scherrer and M Deistler ldquoStatistical analysisof subspace identification methodsrdquo in Proceedings of the 3rdEuropean Control Conference (ECC rsquo95) vol 2 p 1342 1995

[24] D Bauer ldquoOrder estimation in the context of MOESP subspaceidentification methodsrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[25] D Bauer ldquoOrder estimation for subspace methodsrdquo Automat-ica vol 37 no 10 pp 1561ndash1573 2001

[26] J Shalchian A Khaki-Sedigh and A Fatehi ldquoA subspacebased method for time delay estimationrdquo in Proceedings of the

4th International Symposium on Communications Control andSignal Processing (ISCCSP rsquo10) p 4 March 2010

[27] J Lee and T F Edgar ldquoSubspace identification method forsimulation of closed-loop systems with time delaysrdquo AIChEJournal vol 48 no 2 pp 417ndash420 2002

[28] H Zhang T Ma G-B Huang and Z Wang ldquoRobust globalexponential synchronization of uncertain chaotic delayed neu-ral networks via dual-stage impulsive controlrdquo IEEE Transac-tions on Systems Man and Cybernetics B Cybernetics vol 40no 3 pp 831ndash844 2010

[29] P van Overschee and B deMoor ldquoA unifying theorem for threesubspace system identification algorithmsrdquo Automatica vol 31no 12 pp 1853ndash1864 1995

[30] W Favoreel B de Moor and P van Overschee ldquoSubspace statespace system identification for industrial processesrdquo Journal ofProcess Control vol 10 no 2 pp 149ndash155 2000

[31] P van Overschee and B deMoor ldquoN4SID subspace algorithmsfor the identification of combined deterministic-stochasticsystemsrdquo Automatica vol 30 no 1 pp 75ndash93 1994

[32] M Verhaegen ldquoIdentification of the deterministic part ofMIMO state space models given in innovations form frominput-output datardquo Automatica vol 30 no 1 pp 61ndash74 1994

[33] W E Larimore ldquoCanonical variate analysis in identificationfiltering and adaptive controlrdquo in Proceedings of the 29th IEEEConference on Decision and Control pp 596ndash604 December1990

[34] M Viberg ldquoSubspace-based methods for the identification oflinear time-invariant systemsrdquo Automatica vol 31 no 12 pp1835ndash1851 1995

[35] J Wang and S J Qin ldquoA new subspace identification approachbased on principal component analysisrdquo Journal of ProcessControl vol 12 no 8 pp 841ndash855 2002

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 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

Order

Eige

nval

ue

(a) 119889 = 0 situation

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

Order

Eige

nval

ue

(b) 119889 = 5 situation

Figure 5The number of eigenvalues generated when 119889 = 0 and 119889 =5 respectively (a) 119889 = 0 and (b) 119889 = 5

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

Perfo

rman

ce in

dexJ

Order n

JMRSEJMSE2

JMSE1

Figure 6 Obtain the input delay system order by original errorcriterion methods

situation of delay = 5 According to this strategy the order ofmodel increases from 4 to 13 when the system has delay

Also theMSE (mean squared error)MRSE (mean relativesquared error) and AIC criteria are all tested as shown inFigures 6 and 7Thesemethods all have the distinct inflectionpoint at 13 which is for the 4-order system with input delay

2 4 6 8 10 12 14 16 18

0

1

2

AIC

inde

x

minus6

minus5

minus4

minus3

minus2

minus1

Order n

times104

Figure 7 Obtain the input delay system order by AIC index

0246810 123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

1

Figure 8 The corresponding mean square error surface 119869MSE1generated by ODC

5 Obviously for obtaining the ideal model of a delay systemthese conventional methods have to increase the order

Next the performances of the proposed ODCmethod inthis paper are presented As the two search parameters areavailable so the index performance is shown as a surfaceThe 119869MSE1 of output 1199101 is shown in Figure 8 and the 119869MSE2 ofoutput 119910

2is shown in Figure 9 Three axes are the order 119899

delay 119889 and 119869(sdot) respectivelyTheAIC index surface is shownin Figure 10 The minimum value of these indexes can be goteasily they are also the inflection pointsThe order and delayresults are 119899 = 4 119889 = 5 This is the same with the actualsystem model

The corresponding identified model matrices

119860 =[[[

[

05017 06047 02024 01514

minus06256 05164 03929 minus00828

02228 03763 minus04884 minus06090

01883 minus00936 06069 minus05297

]]]

]

119861 =[[[

[

minus00386 18602

minus10028 00809

minus05982 minus10787

minus00583 minus04779

]]]

]

8 Mathematical Problems in Engineering

02

46

810

123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

2

Figure 9 The corresponding mean square error surface 119869MSE2generated by ODC

02468

10 12

34

56

0

2

AIC

inde

x

minus6

minus4

minus2

Order nDelay d

n = 4 d = 5

times104

Figure 10 The corresponding AIC criterion surface generated byODC

119862 = [minus14276 10033 minus11161 03224

minus11801 minus05779 04116 minus04398]

119863 = [minus01356 minus12704

minus13493 09846]

(16)

For verifying the model the model output error is drawnas 1198901and 1198902in Figure 11

52 Example 2 The Kiln Industrial Illustration To demon-strate the superiority of the proposed order selection methodin this paper over the conventional method their perfor-mance is evaluated through an industrial illustration Thedata come from actual kiln production data of an enterpriseHere the gas flow and the second air flow are selected asthe control input and the calcination temperature and kilntail temperature are taken as output variables The samplingtime is 119879

119904= 1 119904 then use Moesp method modelling the

kiln based on the inputoutput data after preprocessing Herethree main practical problems are solved

521 The First Problem The problem is that these two inputvariables have different delay They need to be identified

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus12

minus10

minus8

minus6

minus4

minus2

times10minus15

e 1

(a) 1198901

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus6

minus4

minus2

times10minus15

e 2

(b) 1198902

Figure 11 Modeling error curves 1198901

(a) and 1198902

(b)

respectively that is to say a triple loop about 1198891 1198892 and 119899

should be carried for solving 119869(119899 1198891 1198892)

In order to obtain the inflection point information moredirectly we identify the order 119899 firstly According to theindustrial field situation choose the possibility maximumvalues which are 119889

1max = 150 1198892max = 150 At first

travel all the possible 1198891 1198892and compute the smallest 119869MSE1

119869MSE2 and 119869MRSE corresponding to the different order 119899Thesecurves are shown in Figure 12 As can be seen from thecurves all of these three error criteria achieve the inflectionpoint at 119899 = 5 so the order is got Then specify 119899 =

5 the triple loop is reduced to double loop which is tocompute 119869MSE1 and 119869MSE2 corresponding to 1198891 and 1198892 between[0 150]

522The Second Problem From Figure 13 we notice anotherproblem that the outputs 119910

1and 119910

2generate different inflec-

tion pointIn order to solve this problem the criterion should choose

MRSE and AIC which take into account 1199101and 119910

2both

together Then get Figures 14 and 15It can be conducted that 119899 = 5 119889

1= 30 119889

2= 100

Mathematical Problems in Engineering 9

Table 2 When 119879119904

= 10 s the order the minmum 119869MRSE and the delay

119899 1 2 3 4 5 6 7 81198891

4 3 3 3 3 3 3 31198892

5 10 10 10 10 10 10 10119869MRSE 6981251 595572 499145 384619 239931 239931 239931 239931

1 2 3 4 5 6 7 8 9 10 11 120

50

100

1 2 3 4 5 6 7 8 9 10 11 120

200

400

1 2 3 4 5 6 7 8 9 10 11 120

02

04

Order n

Order n

Order n

J MRS

EJ M

SE2

J MSE

1

Figure 12 The corresponding minimum 119869(119899 1198891

1198892

) curves of eachorder when traversal 119889

1

1198892

0 50 100 150 0 50 100 150050

100150200250300350

Delay d2 Delay d1

J MSE

i

JMSE1

JMSE2

50 100

JMJJ SE1

JMJJ SE2

Figure 13 119869MSE1 and 119869MSE2 surface generated by ODC when 119899 = 5

0 50 100150 0 50 100 150

384

42444648

5

AIC

inde

x

d2 d1

times104

d1 = 30 d2 = 100

AIC = 40159e + 04

d1 = 3330000 dddd222 = 1000

AICAIC 4 0150159e + 0444

Figure 14 AIC surface generated by ODC when 119899 = 5

0 50 100 150 0 50 100 150

0

50

100

150

200

250

d2 d1

J MRS

E

d1 = 30 d2 = 100 JMRSE = 16448d = 30 d = 100 JMJJ RSE = 161 4448

Figure 15 119869MRSE surface generated by ODC when 119899 = 5

0 50100

150

050

100150

050

100150200250300350400

(3 10 239931)

J MRS

E

d2 d150

10050100

Figure 16 119869MRSE surface generated by ODC when 119899 = 5 and 119879119904

=

10 s

0 100 200 300 400 500 600 700 800 900 10001220

1240

1260

1280

1300

1320

1340

1360

1380

Time (s)

Measured valuePredictive value

Kiln

tail

tem

pera

ture

y1

(∘C)

Figure 17 Comparison of calcination temperature measured curveand model predictive curve

10 Mathematical Problems in Engineering

Measured valuePredictive value

0 100 200 300 400 500 600 700 800 900 10001010

1020

1030

1040

1050

1060

1070

1080

1090

Time (s)

Kiln

tail

tem

pera

ture

y2

(∘C)

Figure 18 Comparison of kiln tail temperaturemeasured curve andmodel predictive curve

523 The Third Problem However there is still a problemwhich needs to be resolved In general the model with largetime delay will increase the difficulty of controller designingWe know that when the sampling frequency is higher thanthe actual needed frequency there will be lots of redundantdata And this will raise the model order and the delayTherefore the delay can be reduced by properly decreasingsampling frequency Considering the kiln is a slow time-varying process changing the sampling time from 1 s to 10 swill not affect the model accuracy

From Table 2 it can be seen that when set 119879119904= 10 s the

order and inflexion point is still 119899 = 5 We can also changethe delay to 119889

1= 3010 = 3 119889

2= 10010 = 10 The 119869MRSE

surface can be got as in Figure 16 the results show that 1198891=

3 1198892= 10 This is in agreement with the analysis before

The corresponding rotary kiln calcining zone tempera-ture model is

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896 minus 1198891)

119910 (119896) = 119862119909 (119896) + 119863119906 (119896 minus 1198892)

(17)

and the order 119899 = 5 delay 1198891= 3 119889

2= 10

119860 =

[[[[[

[

09936 00015 minus00062 minus00007 minus00001

00063 09906 00170 minus00017 minus00001

minus00009 00026 09793 minus00147 00209

00000 minus00001 00018 09985 minus00069

minus00000 00001 minus00015 00062 09848

]]]]]

]

119861 = 10minus3

times

[[[[[

[

minus00908 minus00129

minus01650 00957

00680 minus00329

minus00734 00191

minus00093 minus00012

]]]]]

]

119862 = [minus57995 29327 minus05841 minus00157 00260

minus67753 minus23024 minus00247 minus00596 minus00063]

119863 = 10minus15

times [01205 02699

minus01066 00143]

(18)

Compare the measured value and the predicted valuegenerated by the model in Figures 17 and 18

6 Conclusion

In this paper the calcining belt state space model of rotarykiln is built using PO-Moesp subspace method And a noveldouble parameters error performance criterion for the orderchoosing in subspace modelling is introduced Since thepresented method considering the order and delay simulta-neously it can reduce the model order of the delay systemeffectively And also a strategy for stripping the delay factorsfrom the historical data is also proposed The algorithm isverified in identifying an industrial lime kiln In this examplewe solve the practical problem of industrial process withmultidelay and also reduce the order by adjusting samplingtime Further research could shed more light on the issue ofapplying the model online The problem in industrial field ismore complex than simulation environment How to extractproblems from industrial practice and guide the direction ofmodeling research has become the study focus

Conflict of Interests

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

References

[1] M Georgallis P Nowak M Salcudean and I S GartshoreldquoModelling the rotary lime kilnrdquo The Canadian Journal ofChemical Engineering vol 83 no 2 pp 212ndash223 2005

[2] Z Sogut Z Oktay and H Karakoc ldquoMathematical modeling ofheat recovery from a rotary kilnrdquo Applied Thermal Engineeringvol 30 no 8-9 pp 817ndash825 2010

[3] Y H Kim ldquoDevelopment of process model of a rotary kilnfor volatile organic compound recovery from coconut shellrdquoKorean Journal of Chemical Engineering vol 29 no 12 pp 1674ndash1679 2012

[4] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[5] W Weijtjens G de Sitter C Devriendt and P GuillaumeldquoOperational modal parameter estimation of MIMO systemsusing transmissibility functionsrdquo Automatica vol 50 no 2 pp559ndash564 2014

[6] M Imber and V Paschkis ldquoA new theory for a rotary-kiln heatexchangerrdquo International Journal of Heat andMass Transfer vol5 no 7 pp 623ndash638 1962

[7] A Sass ldquoSimulation of heat-transfer phenomena in a rotarykilnrdquo Industrial amp Engineering Chemistry Process Design andDevelopment vol 6 no 4 pp 532ndash535 1967

Mathematical Problems in Engineering 11

[8] S D Shelukar H G K Sundar R Semiat J T Richardson andD Luss ldquoContinuous rotary kiln calcination of yttrium bariumcopper oxide precursor powdersrdquo Industrial and EngineeringChemistry Research vol 33 no 2 pp 421ndash427 1994

[9] Y Yang J Rakhorst M A Reuter and J H L Voncken ldquoAnal-ysis of gas flow and mixing in a rotary kiln waste incineratorrdquoin Proceedings of the 2nd International Conference on CFD inthe Minerals and Process Industries pp 443ndash448 MelbourneAustralia

[10] Y Wang X H Fan and X L Chen ldquoMathematical modelsand expert system for grate-kiln process of iron ore oxide pelletproduction (Part I) mathematical models of grate processrdquoJournal of Central South University of Technology vol 19 no 4pp 1092ndash1097 2012

[11] G Mercere L Bako and S Lecœuche ldquoPropagator-basedmethods for recursive subspace model identificationrdquo SignalProcessing vol 88 no 3 pp 468ndash491 2008

[12] P Misra and M Nikolaou ldquoInput design for model orderdetermination in subspace identificationrdquo AIChE Journal vol49 no 8 pp 2124ndash2132 2003

[13] B Liu B Fang X Liu J Chen Z Huang and X HeldquoLarge margin subspace learning for feature selectionrdquo PatternRecognition vol 46 no 10 pp 2798ndash2806 2013

[14] H Oku and H Kimura ldquoRecursive 4SID algorithms usinggradient type subspace trackingrdquo Automatica vol 38 no 6 pp1035ndash1043 2002

[15] Y Subasi and M Demirekler ldquoQuantitative measure of observ-ability for linear stochastic systemsrdquo Automatica vol 50 no 6pp 1669ndash1674 2014

[16] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[17] X Pan H Zhu F Yang and X Zeng ldquoSubspace trajectorypiecewise-linear model order reduction for nonlinear circuitsrdquoCommunications in Computational Physics vol 14 no 3 pp639ndash663 2013

[18] M Doumlhler and L Mevel ldquoFast multi-order computationof system matrices in subspace-based system identificationrdquoControl Engineering Practice vol 20 no 9 pp 882ndash894 2012

[19] T Breiten and T Damm ldquoKrylov subspace methods for modelorder reduction of bilinear control systemsrdquo Systems and Con-trol Letters vol 59 no 8 pp 443ndash450 2010

[20] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[21] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol 19 no 6 pp 716ndash723 1974

[22] E E Ioannidis ldquoAkaikersquos information criterion correction forthe least-squares autoregressive spectral estimatorrdquo Journal ofTime Series Analysis vol 32 no 6 pp 618ndash630 2011

[23] K Peternell W Scherrer and M Deistler ldquoStatistical analysisof subspace identification methodsrdquo in Proceedings of the 3rdEuropean Control Conference (ECC rsquo95) vol 2 p 1342 1995

[24] D Bauer ldquoOrder estimation in the context of MOESP subspaceidentification methodsrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[25] D Bauer ldquoOrder estimation for subspace methodsrdquo Automat-ica vol 37 no 10 pp 1561ndash1573 2001

[26] J Shalchian A Khaki-Sedigh and A Fatehi ldquoA subspacebased method for time delay estimationrdquo in Proceedings of the

4th International Symposium on Communications Control andSignal Processing (ISCCSP rsquo10) p 4 March 2010

[27] J Lee and T F Edgar ldquoSubspace identification method forsimulation of closed-loop systems with time delaysrdquo AIChEJournal vol 48 no 2 pp 417ndash420 2002

[28] H Zhang T Ma G-B Huang and Z Wang ldquoRobust globalexponential synchronization of uncertain chaotic delayed neu-ral networks via dual-stage impulsive controlrdquo IEEE Transac-tions on Systems Man and Cybernetics B Cybernetics vol 40no 3 pp 831ndash844 2010

[29] P van Overschee and B deMoor ldquoA unifying theorem for threesubspace system identification algorithmsrdquo Automatica vol 31no 12 pp 1853ndash1864 1995

[30] W Favoreel B de Moor and P van Overschee ldquoSubspace statespace system identification for industrial processesrdquo Journal ofProcess Control vol 10 no 2 pp 149ndash155 2000

[31] P van Overschee and B deMoor ldquoN4SID subspace algorithmsfor the identification of combined deterministic-stochasticsystemsrdquo Automatica vol 30 no 1 pp 75ndash93 1994

[32] M Verhaegen ldquoIdentification of the deterministic part ofMIMO state space models given in innovations form frominput-output datardquo Automatica vol 30 no 1 pp 61ndash74 1994

[33] W E Larimore ldquoCanonical variate analysis in identificationfiltering and adaptive controlrdquo in Proceedings of the 29th IEEEConference on Decision and Control pp 596ndash604 December1990

[34] M Viberg ldquoSubspace-based methods for the identification oflinear time-invariant systemsrdquo Automatica vol 31 no 12 pp1835ndash1851 1995

[35] J Wang and S J Qin ldquoA new subspace identification approachbased on principal component analysisrdquo Journal of ProcessControl vol 12 no 8 pp 841ndash855 2002

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

02

46

810

123456

005

115

225

335

Order nDelay d

n = 4 d = 5

J MSE

2

Figure 9 The corresponding mean square error surface 119869MSE2generated by ODC

02468

10 12

34

56

0

2

AIC

inde

x

minus6

minus4

minus2

Order nDelay d

n = 4 d = 5

times104

Figure 10 The corresponding AIC criterion surface generated byODC

119862 = [minus14276 10033 minus11161 03224

minus11801 minus05779 04116 minus04398]

119863 = [minus01356 minus12704

minus13493 09846]

(16)

For verifying the model the model output error is drawnas 1198901and 1198902in Figure 11

52 Example 2 The Kiln Industrial Illustration To demon-strate the superiority of the proposed order selection methodin this paper over the conventional method their perfor-mance is evaluated through an industrial illustration Thedata come from actual kiln production data of an enterpriseHere the gas flow and the second air flow are selected asthe control input and the calcination temperature and kilntail temperature are taken as output variables The samplingtime is 119879

119904= 1 119904 then use Moesp method modelling the

kiln based on the inputoutput data after preprocessing Herethree main practical problems are solved

521 The First Problem The problem is that these two inputvariables have different delay They need to be identified

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus12

minus10

minus8

minus6

minus4

minus2

times10minus15

e 1

(a) 1198901

0 100 200 300 400 500 600 700 800 900 1000

0

2

4

6

Samples

minus6

minus4

minus2

times10minus15

e 2

(b) 1198902

Figure 11 Modeling error curves 1198901

(a) and 1198902

(b)

respectively that is to say a triple loop about 1198891 1198892 and 119899

should be carried for solving 119869(119899 1198891 1198892)

In order to obtain the inflection point information moredirectly we identify the order 119899 firstly According to theindustrial field situation choose the possibility maximumvalues which are 119889

1max = 150 1198892max = 150 At first

travel all the possible 1198891 1198892and compute the smallest 119869MSE1

119869MSE2 and 119869MRSE corresponding to the different order 119899Thesecurves are shown in Figure 12 As can be seen from thecurves all of these three error criteria achieve the inflectionpoint at 119899 = 5 so the order is got Then specify 119899 =

5 the triple loop is reduced to double loop which is tocompute 119869MSE1 and 119869MSE2 corresponding to 1198891 and 1198892 between[0 150]

522The Second Problem From Figure 13 we notice anotherproblem that the outputs 119910

1and 119910

2generate different inflec-

tion pointIn order to solve this problem the criterion should choose

MRSE and AIC which take into account 1199101and 119910

2both

together Then get Figures 14 and 15It can be conducted that 119899 = 5 119889

1= 30 119889

2= 100

Mathematical Problems in Engineering 9

Table 2 When 119879119904

= 10 s the order the minmum 119869MRSE and the delay

119899 1 2 3 4 5 6 7 81198891

4 3 3 3 3 3 3 31198892

5 10 10 10 10 10 10 10119869MRSE 6981251 595572 499145 384619 239931 239931 239931 239931

1 2 3 4 5 6 7 8 9 10 11 120

50

100

1 2 3 4 5 6 7 8 9 10 11 120

200

400

1 2 3 4 5 6 7 8 9 10 11 120

02

04

Order n

Order n

Order n

J MRS

EJ M

SE2

J MSE

1

Figure 12 The corresponding minimum 119869(119899 1198891

1198892

) curves of eachorder when traversal 119889

1

1198892

0 50 100 150 0 50 100 150050

100150200250300350

Delay d2 Delay d1

J MSE

i

JMSE1

JMSE2

50 100

JMJJ SE1

JMJJ SE2

Figure 13 119869MSE1 and 119869MSE2 surface generated by ODC when 119899 = 5

0 50 100150 0 50 100 150

384

42444648

5

AIC

inde

x

d2 d1

times104

d1 = 30 d2 = 100

AIC = 40159e + 04

d1 = 3330000 dddd222 = 1000

AICAIC 4 0150159e + 0444

Figure 14 AIC surface generated by ODC when 119899 = 5

0 50 100 150 0 50 100 150

0

50

100

150

200

250

d2 d1

J MRS

E

d1 = 30 d2 = 100 JMRSE = 16448d = 30 d = 100 JMJJ RSE = 161 4448

Figure 15 119869MRSE surface generated by ODC when 119899 = 5

0 50100

150

050

100150

050

100150200250300350400

(3 10 239931)

J MRS

E

d2 d150

10050100

Figure 16 119869MRSE surface generated by ODC when 119899 = 5 and 119879119904

=

10 s

0 100 200 300 400 500 600 700 800 900 10001220

1240

1260

1280

1300

1320

1340

1360

1380

Time (s)

Measured valuePredictive value

Kiln

tail

tem

pera

ture

y1

(∘C)

Figure 17 Comparison of calcination temperature measured curveand model predictive curve

10 Mathematical Problems in Engineering

Measured valuePredictive value

0 100 200 300 400 500 600 700 800 900 10001010

1020

1030

1040

1050

1060

1070

1080

1090

Time (s)

Kiln

tail

tem

pera

ture

y2

(∘C)

Figure 18 Comparison of kiln tail temperaturemeasured curve andmodel predictive curve

523 The Third Problem However there is still a problemwhich needs to be resolved In general the model with largetime delay will increase the difficulty of controller designingWe know that when the sampling frequency is higher thanthe actual needed frequency there will be lots of redundantdata And this will raise the model order and the delayTherefore the delay can be reduced by properly decreasingsampling frequency Considering the kiln is a slow time-varying process changing the sampling time from 1 s to 10 swill not affect the model accuracy

From Table 2 it can be seen that when set 119879119904= 10 s the

order and inflexion point is still 119899 = 5 We can also changethe delay to 119889

1= 3010 = 3 119889

2= 10010 = 10 The 119869MRSE

surface can be got as in Figure 16 the results show that 1198891=

3 1198892= 10 This is in agreement with the analysis before

The corresponding rotary kiln calcining zone tempera-ture model is

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896 minus 1198891)

119910 (119896) = 119862119909 (119896) + 119863119906 (119896 minus 1198892)

(17)

and the order 119899 = 5 delay 1198891= 3 119889

2= 10

119860 =

[[[[[

[

09936 00015 minus00062 minus00007 minus00001

00063 09906 00170 minus00017 minus00001

minus00009 00026 09793 minus00147 00209

00000 minus00001 00018 09985 minus00069

minus00000 00001 minus00015 00062 09848

]]]]]

]

119861 = 10minus3

times

[[[[[

[

minus00908 minus00129

minus01650 00957

00680 minus00329

minus00734 00191

minus00093 minus00012

]]]]]

]

119862 = [minus57995 29327 minus05841 minus00157 00260

minus67753 minus23024 minus00247 minus00596 minus00063]

119863 = 10minus15

times [01205 02699

minus01066 00143]

(18)

Compare the measured value and the predicted valuegenerated by the model in Figures 17 and 18

6 Conclusion

In this paper the calcining belt state space model of rotarykiln is built using PO-Moesp subspace method And a noveldouble parameters error performance criterion for the orderchoosing in subspace modelling is introduced Since thepresented method considering the order and delay simulta-neously it can reduce the model order of the delay systemeffectively And also a strategy for stripping the delay factorsfrom the historical data is also proposed The algorithm isverified in identifying an industrial lime kiln In this examplewe solve the practical problem of industrial process withmultidelay and also reduce the order by adjusting samplingtime Further research could shed more light on the issue ofapplying the model online The problem in industrial field ismore complex than simulation environment How to extractproblems from industrial practice and guide the direction ofmodeling research has become the study focus

Conflict of Interests

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

References

[1] M Georgallis P Nowak M Salcudean and I S GartshoreldquoModelling the rotary lime kilnrdquo The Canadian Journal ofChemical Engineering vol 83 no 2 pp 212ndash223 2005

[2] Z Sogut Z Oktay and H Karakoc ldquoMathematical modeling ofheat recovery from a rotary kilnrdquo Applied Thermal Engineeringvol 30 no 8-9 pp 817ndash825 2010

[3] Y H Kim ldquoDevelopment of process model of a rotary kilnfor volatile organic compound recovery from coconut shellrdquoKorean Journal of Chemical Engineering vol 29 no 12 pp 1674ndash1679 2012

[4] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[5] W Weijtjens G de Sitter C Devriendt and P GuillaumeldquoOperational modal parameter estimation of MIMO systemsusing transmissibility functionsrdquo Automatica vol 50 no 2 pp559ndash564 2014

[6] M Imber and V Paschkis ldquoA new theory for a rotary-kiln heatexchangerrdquo International Journal of Heat andMass Transfer vol5 no 7 pp 623ndash638 1962

[7] A Sass ldquoSimulation of heat-transfer phenomena in a rotarykilnrdquo Industrial amp Engineering Chemistry Process Design andDevelopment vol 6 no 4 pp 532ndash535 1967

Mathematical Problems in Engineering 11

[8] S D Shelukar H G K Sundar R Semiat J T Richardson andD Luss ldquoContinuous rotary kiln calcination of yttrium bariumcopper oxide precursor powdersrdquo Industrial and EngineeringChemistry Research vol 33 no 2 pp 421ndash427 1994

[9] Y Yang J Rakhorst M A Reuter and J H L Voncken ldquoAnal-ysis of gas flow and mixing in a rotary kiln waste incineratorrdquoin Proceedings of the 2nd International Conference on CFD inthe Minerals and Process Industries pp 443ndash448 MelbourneAustralia

[10] Y Wang X H Fan and X L Chen ldquoMathematical modelsand expert system for grate-kiln process of iron ore oxide pelletproduction (Part I) mathematical models of grate processrdquoJournal of Central South University of Technology vol 19 no 4pp 1092ndash1097 2012

[11] G Mercere L Bako and S Lecœuche ldquoPropagator-basedmethods for recursive subspace model identificationrdquo SignalProcessing vol 88 no 3 pp 468ndash491 2008

[12] P Misra and M Nikolaou ldquoInput design for model orderdetermination in subspace identificationrdquo AIChE Journal vol49 no 8 pp 2124ndash2132 2003

[13] B Liu B Fang X Liu J Chen Z Huang and X HeldquoLarge margin subspace learning for feature selectionrdquo PatternRecognition vol 46 no 10 pp 2798ndash2806 2013

[14] H Oku and H Kimura ldquoRecursive 4SID algorithms usinggradient type subspace trackingrdquo Automatica vol 38 no 6 pp1035ndash1043 2002

[15] Y Subasi and M Demirekler ldquoQuantitative measure of observ-ability for linear stochastic systemsrdquo Automatica vol 50 no 6pp 1669ndash1674 2014

[16] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[17] X Pan H Zhu F Yang and X Zeng ldquoSubspace trajectorypiecewise-linear model order reduction for nonlinear circuitsrdquoCommunications in Computational Physics vol 14 no 3 pp639ndash663 2013

[18] M Doumlhler and L Mevel ldquoFast multi-order computationof system matrices in subspace-based system identificationrdquoControl Engineering Practice vol 20 no 9 pp 882ndash894 2012

[19] T Breiten and T Damm ldquoKrylov subspace methods for modelorder reduction of bilinear control systemsrdquo Systems and Con-trol Letters vol 59 no 8 pp 443ndash450 2010

[20] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[21] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol 19 no 6 pp 716ndash723 1974

[22] E E Ioannidis ldquoAkaikersquos information criterion correction forthe least-squares autoregressive spectral estimatorrdquo Journal ofTime Series Analysis vol 32 no 6 pp 618ndash630 2011

[23] K Peternell W Scherrer and M Deistler ldquoStatistical analysisof subspace identification methodsrdquo in Proceedings of the 3rdEuropean Control Conference (ECC rsquo95) vol 2 p 1342 1995

[24] D Bauer ldquoOrder estimation in the context of MOESP subspaceidentification methodsrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[25] D Bauer ldquoOrder estimation for subspace methodsrdquo Automat-ica vol 37 no 10 pp 1561ndash1573 2001

[26] J Shalchian A Khaki-Sedigh and A Fatehi ldquoA subspacebased method for time delay estimationrdquo in Proceedings of the

4th International Symposium on Communications Control andSignal Processing (ISCCSP rsquo10) p 4 March 2010

[27] J Lee and T F Edgar ldquoSubspace identification method forsimulation of closed-loop systems with time delaysrdquo AIChEJournal vol 48 no 2 pp 417ndash420 2002

[28] H Zhang T Ma G-B Huang and Z Wang ldquoRobust globalexponential synchronization of uncertain chaotic delayed neu-ral networks via dual-stage impulsive controlrdquo IEEE Transac-tions on Systems Man and Cybernetics B Cybernetics vol 40no 3 pp 831ndash844 2010

[29] P van Overschee and B deMoor ldquoA unifying theorem for threesubspace system identification algorithmsrdquo Automatica vol 31no 12 pp 1853ndash1864 1995

[30] W Favoreel B de Moor and P van Overschee ldquoSubspace statespace system identification for industrial processesrdquo Journal ofProcess Control vol 10 no 2 pp 149ndash155 2000

[31] P van Overschee and B deMoor ldquoN4SID subspace algorithmsfor the identification of combined deterministic-stochasticsystemsrdquo Automatica vol 30 no 1 pp 75ndash93 1994

[32] M Verhaegen ldquoIdentification of the deterministic part ofMIMO state space models given in innovations form frominput-output datardquo Automatica vol 30 no 1 pp 61ndash74 1994

[33] W E Larimore ldquoCanonical variate analysis in identificationfiltering and adaptive controlrdquo in Proceedings of the 29th IEEEConference on Decision and Control pp 596ndash604 December1990

[34] M Viberg ldquoSubspace-based methods for the identification oflinear time-invariant systemsrdquo Automatica vol 31 no 12 pp1835ndash1851 1995

[35] J Wang and S J Qin ldquoA new subspace identification approachbased on principal component analysisrdquo Journal of ProcessControl vol 12 no 8 pp 841ndash855 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 2 When 119879119904

= 10 s the order the minmum 119869MRSE and the delay

119899 1 2 3 4 5 6 7 81198891

4 3 3 3 3 3 3 31198892

5 10 10 10 10 10 10 10119869MRSE 6981251 595572 499145 384619 239931 239931 239931 239931

1 2 3 4 5 6 7 8 9 10 11 120

50

100

1 2 3 4 5 6 7 8 9 10 11 120

200

400

1 2 3 4 5 6 7 8 9 10 11 120

02

04

Order n

Order n

Order n

J MRS

EJ M

SE2

J MSE

1

Figure 12 The corresponding minimum 119869(119899 1198891

1198892

) curves of eachorder when traversal 119889

1

1198892

0 50 100 150 0 50 100 150050

100150200250300350

Delay d2 Delay d1

J MSE

i

JMSE1

JMSE2

50 100

JMJJ SE1

JMJJ SE2

Figure 13 119869MSE1 and 119869MSE2 surface generated by ODC when 119899 = 5

0 50 100150 0 50 100 150

384

42444648

5

AIC

inde

x

d2 d1

times104

d1 = 30 d2 = 100

AIC = 40159e + 04

d1 = 3330000 dddd222 = 1000

AICAIC 4 0150159e + 0444

Figure 14 AIC surface generated by ODC when 119899 = 5

0 50 100 150 0 50 100 150

0

50

100

150

200

250

d2 d1

J MRS

E

d1 = 30 d2 = 100 JMRSE = 16448d = 30 d = 100 JMJJ RSE = 161 4448

Figure 15 119869MRSE surface generated by ODC when 119899 = 5

0 50100

150

050

100150

050

100150200250300350400

(3 10 239931)

J MRS

E

d2 d150

10050100

Figure 16 119869MRSE surface generated by ODC when 119899 = 5 and 119879119904

=

10 s

0 100 200 300 400 500 600 700 800 900 10001220

1240

1260

1280

1300

1320

1340

1360

1380

Time (s)

Measured valuePredictive value

Kiln

tail

tem

pera

ture

y1

(∘C)

Figure 17 Comparison of calcination temperature measured curveand model predictive curve

10 Mathematical Problems in Engineering

Measured valuePredictive value

0 100 200 300 400 500 600 700 800 900 10001010

1020

1030

1040

1050

1060

1070

1080

1090

Time (s)

Kiln

tail

tem

pera

ture

y2

(∘C)

Figure 18 Comparison of kiln tail temperaturemeasured curve andmodel predictive curve

523 The Third Problem However there is still a problemwhich needs to be resolved In general the model with largetime delay will increase the difficulty of controller designingWe know that when the sampling frequency is higher thanthe actual needed frequency there will be lots of redundantdata And this will raise the model order and the delayTherefore the delay can be reduced by properly decreasingsampling frequency Considering the kiln is a slow time-varying process changing the sampling time from 1 s to 10 swill not affect the model accuracy

From Table 2 it can be seen that when set 119879119904= 10 s the

order and inflexion point is still 119899 = 5 We can also changethe delay to 119889

1= 3010 = 3 119889

2= 10010 = 10 The 119869MRSE

surface can be got as in Figure 16 the results show that 1198891=

3 1198892= 10 This is in agreement with the analysis before

The corresponding rotary kiln calcining zone tempera-ture model is

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896 minus 1198891)

119910 (119896) = 119862119909 (119896) + 119863119906 (119896 minus 1198892)

(17)

and the order 119899 = 5 delay 1198891= 3 119889

2= 10

119860 =

[[[[[

[

09936 00015 minus00062 minus00007 minus00001

00063 09906 00170 minus00017 minus00001

minus00009 00026 09793 minus00147 00209

00000 minus00001 00018 09985 minus00069

minus00000 00001 minus00015 00062 09848

]]]]]

]

119861 = 10minus3

times

[[[[[

[

minus00908 minus00129

minus01650 00957

00680 minus00329

minus00734 00191

minus00093 minus00012

]]]]]

]

119862 = [minus57995 29327 minus05841 minus00157 00260

minus67753 minus23024 minus00247 minus00596 minus00063]

119863 = 10minus15

times [01205 02699

minus01066 00143]

(18)

Compare the measured value and the predicted valuegenerated by the model in Figures 17 and 18

6 Conclusion

In this paper the calcining belt state space model of rotarykiln is built using PO-Moesp subspace method And a noveldouble parameters error performance criterion for the orderchoosing in subspace modelling is introduced Since thepresented method considering the order and delay simulta-neously it can reduce the model order of the delay systemeffectively And also a strategy for stripping the delay factorsfrom the historical data is also proposed The algorithm isverified in identifying an industrial lime kiln In this examplewe solve the practical problem of industrial process withmultidelay and also reduce the order by adjusting samplingtime Further research could shed more light on the issue ofapplying the model online The problem in industrial field ismore complex than simulation environment How to extractproblems from industrial practice and guide the direction ofmodeling research has become the study focus

Conflict of Interests

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

References

[1] M Georgallis P Nowak M Salcudean and I S GartshoreldquoModelling the rotary lime kilnrdquo The Canadian Journal ofChemical Engineering vol 83 no 2 pp 212ndash223 2005

[2] Z Sogut Z Oktay and H Karakoc ldquoMathematical modeling ofheat recovery from a rotary kilnrdquo Applied Thermal Engineeringvol 30 no 8-9 pp 817ndash825 2010

[3] Y H Kim ldquoDevelopment of process model of a rotary kilnfor volatile organic compound recovery from coconut shellrdquoKorean Journal of Chemical Engineering vol 29 no 12 pp 1674ndash1679 2012

[4] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[5] W Weijtjens G de Sitter C Devriendt and P GuillaumeldquoOperational modal parameter estimation of MIMO systemsusing transmissibility functionsrdquo Automatica vol 50 no 2 pp559ndash564 2014

[6] M Imber and V Paschkis ldquoA new theory for a rotary-kiln heatexchangerrdquo International Journal of Heat andMass Transfer vol5 no 7 pp 623ndash638 1962

[7] A Sass ldquoSimulation of heat-transfer phenomena in a rotarykilnrdquo Industrial amp Engineering Chemistry Process Design andDevelopment vol 6 no 4 pp 532ndash535 1967

Mathematical Problems in Engineering 11

[8] S D Shelukar H G K Sundar R Semiat J T Richardson andD Luss ldquoContinuous rotary kiln calcination of yttrium bariumcopper oxide precursor powdersrdquo Industrial and EngineeringChemistry Research vol 33 no 2 pp 421ndash427 1994

[9] Y Yang J Rakhorst M A Reuter and J H L Voncken ldquoAnal-ysis of gas flow and mixing in a rotary kiln waste incineratorrdquoin Proceedings of the 2nd International Conference on CFD inthe Minerals and Process Industries pp 443ndash448 MelbourneAustralia

[10] Y Wang X H Fan and X L Chen ldquoMathematical modelsand expert system for grate-kiln process of iron ore oxide pelletproduction (Part I) mathematical models of grate processrdquoJournal of Central South University of Technology vol 19 no 4pp 1092ndash1097 2012

[11] G Mercere L Bako and S Lecœuche ldquoPropagator-basedmethods for recursive subspace model identificationrdquo SignalProcessing vol 88 no 3 pp 468ndash491 2008

[12] P Misra and M Nikolaou ldquoInput design for model orderdetermination in subspace identificationrdquo AIChE Journal vol49 no 8 pp 2124ndash2132 2003

[13] B Liu B Fang X Liu J Chen Z Huang and X HeldquoLarge margin subspace learning for feature selectionrdquo PatternRecognition vol 46 no 10 pp 2798ndash2806 2013

[14] H Oku and H Kimura ldquoRecursive 4SID algorithms usinggradient type subspace trackingrdquo Automatica vol 38 no 6 pp1035ndash1043 2002

[15] Y Subasi and M Demirekler ldquoQuantitative measure of observ-ability for linear stochastic systemsrdquo Automatica vol 50 no 6pp 1669ndash1674 2014

[16] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[17] X Pan H Zhu F Yang and X Zeng ldquoSubspace trajectorypiecewise-linear model order reduction for nonlinear circuitsrdquoCommunications in Computational Physics vol 14 no 3 pp639ndash663 2013

[18] M Doumlhler and L Mevel ldquoFast multi-order computationof system matrices in subspace-based system identificationrdquoControl Engineering Practice vol 20 no 9 pp 882ndash894 2012

[19] T Breiten and T Damm ldquoKrylov subspace methods for modelorder reduction of bilinear control systemsrdquo Systems and Con-trol Letters vol 59 no 8 pp 443ndash450 2010

[20] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[21] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol 19 no 6 pp 716ndash723 1974

[22] E E Ioannidis ldquoAkaikersquos information criterion correction forthe least-squares autoregressive spectral estimatorrdquo Journal ofTime Series Analysis vol 32 no 6 pp 618ndash630 2011

[23] K Peternell W Scherrer and M Deistler ldquoStatistical analysisof subspace identification methodsrdquo in Proceedings of the 3rdEuropean Control Conference (ECC rsquo95) vol 2 p 1342 1995

[24] D Bauer ldquoOrder estimation in the context of MOESP subspaceidentification methodsrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[25] D Bauer ldquoOrder estimation for subspace methodsrdquo Automat-ica vol 37 no 10 pp 1561ndash1573 2001

[26] J Shalchian A Khaki-Sedigh and A Fatehi ldquoA subspacebased method for time delay estimationrdquo in Proceedings of the

4th International Symposium on Communications Control andSignal Processing (ISCCSP rsquo10) p 4 March 2010

[27] J Lee and T F Edgar ldquoSubspace identification method forsimulation of closed-loop systems with time delaysrdquo AIChEJournal vol 48 no 2 pp 417ndash420 2002

[28] H Zhang T Ma G-B Huang and Z Wang ldquoRobust globalexponential synchronization of uncertain chaotic delayed neu-ral networks via dual-stage impulsive controlrdquo IEEE Transac-tions on Systems Man and Cybernetics B Cybernetics vol 40no 3 pp 831ndash844 2010

[29] P van Overschee and B deMoor ldquoA unifying theorem for threesubspace system identification algorithmsrdquo Automatica vol 31no 12 pp 1853ndash1864 1995

[30] W Favoreel B de Moor and P van Overschee ldquoSubspace statespace system identification for industrial processesrdquo Journal ofProcess Control vol 10 no 2 pp 149ndash155 2000

[31] P van Overschee and B deMoor ldquoN4SID subspace algorithmsfor the identification of combined deterministic-stochasticsystemsrdquo Automatica vol 30 no 1 pp 75ndash93 1994

[32] M Verhaegen ldquoIdentification of the deterministic part ofMIMO state space models given in innovations form frominput-output datardquo Automatica vol 30 no 1 pp 61ndash74 1994

[33] W E Larimore ldquoCanonical variate analysis in identificationfiltering and adaptive controlrdquo in Proceedings of the 29th IEEEConference on Decision and Control pp 596ndash604 December1990

[34] M Viberg ldquoSubspace-based methods for the identification oflinear time-invariant systemsrdquo Automatica vol 31 no 12 pp1835ndash1851 1995

[35] J Wang and S J Qin ldquoA new subspace identification approachbased on principal component analysisrdquo Journal of ProcessControl vol 12 no 8 pp 841ndash855 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Measured valuePredictive value

0 100 200 300 400 500 600 700 800 900 10001010

1020

1030

1040

1050

1060

1070

1080

1090

Time (s)

Kiln

tail

tem

pera

ture

y2

(∘C)

Figure 18 Comparison of kiln tail temperaturemeasured curve andmodel predictive curve

523 The Third Problem However there is still a problemwhich needs to be resolved In general the model with largetime delay will increase the difficulty of controller designingWe know that when the sampling frequency is higher thanthe actual needed frequency there will be lots of redundantdata And this will raise the model order and the delayTherefore the delay can be reduced by properly decreasingsampling frequency Considering the kiln is a slow time-varying process changing the sampling time from 1 s to 10 swill not affect the model accuracy

From Table 2 it can be seen that when set 119879119904= 10 s the

order and inflexion point is still 119899 = 5 We can also changethe delay to 119889

1= 3010 = 3 119889

2= 10010 = 10 The 119869MRSE

surface can be got as in Figure 16 the results show that 1198891=

3 1198892= 10 This is in agreement with the analysis before

The corresponding rotary kiln calcining zone tempera-ture model is

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896 minus 1198891)

119910 (119896) = 119862119909 (119896) + 119863119906 (119896 minus 1198892)

(17)

and the order 119899 = 5 delay 1198891= 3 119889

2= 10

119860 =

[[[[[

[

09936 00015 minus00062 minus00007 minus00001

00063 09906 00170 minus00017 minus00001

minus00009 00026 09793 minus00147 00209

00000 minus00001 00018 09985 minus00069

minus00000 00001 minus00015 00062 09848

]]]]]

]

119861 = 10minus3

times

[[[[[

[

minus00908 minus00129

minus01650 00957

00680 minus00329

minus00734 00191

minus00093 minus00012

]]]]]

]

119862 = [minus57995 29327 minus05841 minus00157 00260

minus67753 minus23024 minus00247 minus00596 minus00063]

119863 = 10minus15

times [01205 02699

minus01066 00143]

(18)

Compare the measured value and the predicted valuegenerated by the model in Figures 17 and 18

6 Conclusion

In this paper the calcining belt state space model of rotarykiln is built using PO-Moesp subspace method And a noveldouble parameters error performance criterion for the orderchoosing in subspace modelling is introduced Since thepresented method considering the order and delay simulta-neously it can reduce the model order of the delay systemeffectively And also a strategy for stripping the delay factorsfrom the historical data is also proposed The algorithm isverified in identifying an industrial lime kiln In this examplewe solve the practical problem of industrial process withmultidelay and also reduce the order by adjusting samplingtime Further research could shed more light on the issue ofapplying the model online The problem in industrial field ismore complex than simulation environment How to extractproblems from industrial practice and guide the direction ofmodeling research has become the study focus

Conflict of Interests

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

References

[1] M Georgallis P Nowak M Salcudean and I S GartshoreldquoModelling the rotary lime kilnrdquo The Canadian Journal ofChemical Engineering vol 83 no 2 pp 212ndash223 2005

[2] Z Sogut Z Oktay and H Karakoc ldquoMathematical modeling ofheat recovery from a rotary kilnrdquo Applied Thermal Engineeringvol 30 no 8-9 pp 817ndash825 2010

[3] Y H Kim ldquoDevelopment of process model of a rotary kilnfor volatile organic compound recovery from coconut shellrdquoKorean Journal of Chemical Engineering vol 29 no 12 pp 1674ndash1679 2012

[4] H Zhang and Y Quan ldquoModeling identification and controlof a class of nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 9 no 2 pp 349ndash354 2001

[5] W Weijtjens G de Sitter C Devriendt and P GuillaumeldquoOperational modal parameter estimation of MIMO systemsusing transmissibility functionsrdquo Automatica vol 50 no 2 pp559ndash564 2014

[6] M Imber and V Paschkis ldquoA new theory for a rotary-kiln heatexchangerrdquo International Journal of Heat andMass Transfer vol5 no 7 pp 623ndash638 1962

[7] A Sass ldquoSimulation of heat-transfer phenomena in a rotarykilnrdquo Industrial amp Engineering Chemistry Process Design andDevelopment vol 6 no 4 pp 532ndash535 1967

Mathematical Problems in Engineering 11

[8] S D Shelukar H G K Sundar R Semiat J T Richardson andD Luss ldquoContinuous rotary kiln calcination of yttrium bariumcopper oxide precursor powdersrdquo Industrial and EngineeringChemistry Research vol 33 no 2 pp 421ndash427 1994

[9] Y Yang J Rakhorst M A Reuter and J H L Voncken ldquoAnal-ysis of gas flow and mixing in a rotary kiln waste incineratorrdquoin Proceedings of the 2nd International Conference on CFD inthe Minerals and Process Industries pp 443ndash448 MelbourneAustralia

[10] Y Wang X H Fan and X L Chen ldquoMathematical modelsand expert system for grate-kiln process of iron ore oxide pelletproduction (Part I) mathematical models of grate processrdquoJournal of Central South University of Technology vol 19 no 4pp 1092ndash1097 2012

[11] G Mercere L Bako and S Lecœuche ldquoPropagator-basedmethods for recursive subspace model identificationrdquo SignalProcessing vol 88 no 3 pp 468ndash491 2008

[12] P Misra and M Nikolaou ldquoInput design for model orderdetermination in subspace identificationrdquo AIChE Journal vol49 no 8 pp 2124ndash2132 2003

[13] B Liu B Fang X Liu J Chen Z Huang and X HeldquoLarge margin subspace learning for feature selectionrdquo PatternRecognition vol 46 no 10 pp 2798ndash2806 2013

[14] H Oku and H Kimura ldquoRecursive 4SID algorithms usinggradient type subspace trackingrdquo Automatica vol 38 no 6 pp1035ndash1043 2002

[15] Y Subasi and M Demirekler ldquoQuantitative measure of observ-ability for linear stochastic systemsrdquo Automatica vol 50 no 6pp 1669ndash1674 2014

[16] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[17] X Pan H Zhu F Yang and X Zeng ldquoSubspace trajectorypiecewise-linear model order reduction for nonlinear circuitsrdquoCommunications in Computational Physics vol 14 no 3 pp639ndash663 2013

[18] M Doumlhler and L Mevel ldquoFast multi-order computationof system matrices in subspace-based system identificationrdquoControl Engineering Practice vol 20 no 9 pp 882ndash894 2012

[19] T Breiten and T Damm ldquoKrylov subspace methods for modelorder reduction of bilinear control systemsrdquo Systems and Con-trol Letters vol 59 no 8 pp 443ndash450 2010

[20] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[21] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol 19 no 6 pp 716ndash723 1974

[22] E E Ioannidis ldquoAkaikersquos information criterion correction forthe least-squares autoregressive spectral estimatorrdquo Journal ofTime Series Analysis vol 32 no 6 pp 618ndash630 2011

[23] K Peternell W Scherrer and M Deistler ldquoStatistical analysisof subspace identification methodsrdquo in Proceedings of the 3rdEuropean Control Conference (ECC rsquo95) vol 2 p 1342 1995

[24] D Bauer ldquoOrder estimation in the context of MOESP subspaceidentification methodsrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[25] D Bauer ldquoOrder estimation for subspace methodsrdquo Automat-ica vol 37 no 10 pp 1561ndash1573 2001

[26] J Shalchian A Khaki-Sedigh and A Fatehi ldquoA subspacebased method for time delay estimationrdquo in Proceedings of the

4th International Symposium on Communications Control andSignal Processing (ISCCSP rsquo10) p 4 March 2010

[27] J Lee and T F Edgar ldquoSubspace identification method forsimulation of closed-loop systems with time delaysrdquo AIChEJournal vol 48 no 2 pp 417ndash420 2002

[28] H Zhang T Ma G-B Huang and Z Wang ldquoRobust globalexponential synchronization of uncertain chaotic delayed neu-ral networks via dual-stage impulsive controlrdquo IEEE Transac-tions on Systems Man and Cybernetics B Cybernetics vol 40no 3 pp 831ndash844 2010

[29] P van Overschee and B deMoor ldquoA unifying theorem for threesubspace system identification algorithmsrdquo Automatica vol 31no 12 pp 1853ndash1864 1995

[30] W Favoreel B de Moor and P van Overschee ldquoSubspace statespace system identification for industrial processesrdquo Journal ofProcess Control vol 10 no 2 pp 149ndash155 2000

[31] P van Overschee and B deMoor ldquoN4SID subspace algorithmsfor the identification of combined deterministic-stochasticsystemsrdquo Automatica vol 30 no 1 pp 75ndash93 1994

[32] M Verhaegen ldquoIdentification of the deterministic part ofMIMO state space models given in innovations form frominput-output datardquo Automatica vol 30 no 1 pp 61ndash74 1994

[33] W E Larimore ldquoCanonical variate analysis in identificationfiltering and adaptive controlrdquo in Proceedings of the 29th IEEEConference on Decision and Control pp 596ndash604 December1990

[34] M Viberg ldquoSubspace-based methods for the identification oflinear time-invariant systemsrdquo Automatica vol 31 no 12 pp1835ndash1851 1995

[35] J Wang and S J Qin ldquoA new subspace identification approachbased on principal component analysisrdquo Journal of ProcessControl vol 12 no 8 pp 841ndash855 2002

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 11

[8] S D Shelukar H G K Sundar R Semiat J T Richardson andD Luss ldquoContinuous rotary kiln calcination of yttrium bariumcopper oxide precursor powdersrdquo Industrial and EngineeringChemistry Research vol 33 no 2 pp 421ndash427 1994

[9] Y Yang J Rakhorst M A Reuter and J H L Voncken ldquoAnal-ysis of gas flow and mixing in a rotary kiln waste incineratorrdquoin Proceedings of the 2nd International Conference on CFD inthe Minerals and Process Industries pp 443ndash448 MelbourneAustralia

[10] Y Wang X H Fan and X L Chen ldquoMathematical modelsand expert system for grate-kiln process of iron ore oxide pelletproduction (Part I) mathematical models of grate processrdquoJournal of Central South University of Technology vol 19 no 4pp 1092ndash1097 2012

[11] G Mercere L Bako and S Lecœuche ldquoPropagator-basedmethods for recursive subspace model identificationrdquo SignalProcessing vol 88 no 3 pp 468ndash491 2008

[12] P Misra and M Nikolaou ldquoInput design for model orderdetermination in subspace identificationrdquo AIChE Journal vol49 no 8 pp 2124ndash2132 2003

[13] B Liu B Fang X Liu J Chen Z Huang and X HeldquoLarge margin subspace learning for feature selectionrdquo PatternRecognition vol 46 no 10 pp 2798ndash2806 2013

[14] H Oku and H Kimura ldquoRecursive 4SID algorithms usinggradient type subspace trackingrdquo Automatica vol 38 no 6 pp1035ndash1043 2002

[15] Y Subasi and M Demirekler ldquoQuantitative measure of observ-ability for linear stochastic systemsrdquo Automatica vol 50 no 6pp 1669ndash1674 2014

[16] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[17] X Pan H Zhu F Yang and X Zeng ldquoSubspace trajectorypiecewise-linear model order reduction for nonlinear circuitsrdquoCommunications in Computational Physics vol 14 no 3 pp639ndash663 2013

[18] M Doumlhler and L Mevel ldquoFast multi-order computationof system matrices in subspace-based system identificationrdquoControl Engineering Practice vol 20 no 9 pp 882ndash894 2012

[19] T Breiten and T Damm ldquoKrylov subspace methods for modelorder reduction of bilinear control systemsrdquo Systems and Con-trol Letters vol 59 no 8 pp 443ndash450 2010

[20] A Garcıa-Hiernaux J Casals and M Jerez ldquoEstimating thesystem order by subspace methodsrdquo Computational Statisticsvol 27 no 3 pp 411ndash425 2012

[21] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol 19 no 6 pp 716ndash723 1974

[22] E E Ioannidis ldquoAkaikersquos information criterion correction forthe least-squares autoregressive spectral estimatorrdquo Journal ofTime Series Analysis vol 32 no 6 pp 618ndash630 2011

[23] K Peternell W Scherrer and M Deistler ldquoStatistical analysisof subspace identification methodsrdquo in Proceedings of the 3rdEuropean Control Conference (ECC rsquo95) vol 2 p 1342 1995

[24] D Bauer ldquoOrder estimation in the context of MOESP subspaceidentification methodsrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[25] D Bauer ldquoOrder estimation for subspace methodsrdquo Automat-ica vol 37 no 10 pp 1561ndash1573 2001

[26] J Shalchian A Khaki-Sedigh and A Fatehi ldquoA subspacebased method for time delay estimationrdquo in Proceedings of the

4th International Symposium on Communications Control andSignal Processing (ISCCSP rsquo10) p 4 March 2010

[27] J Lee and T F Edgar ldquoSubspace identification method forsimulation of closed-loop systems with time delaysrdquo AIChEJournal vol 48 no 2 pp 417ndash420 2002

[28] H Zhang T Ma G-B Huang and Z Wang ldquoRobust globalexponential synchronization of uncertain chaotic delayed neu-ral networks via dual-stage impulsive controlrdquo IEEE Transac-tions on Systems Man and Cybernetics B Cybernetics vol 40no 3 pp 831ndash844 2010

[29] P van Overschee and B deMoor ldquoA unifying theorem for threesubspace system identification algorithmsrdquo Automatica vol 31no 12 pp 1853ndash1864 1995

[30] W Favoreel B de Moor and P van Overschee ldquoSubspace statespace system identification for industrial processesrdquo Journal ofProcess Control vol 10 no 2 pp 149ndash155 2000

[31] P van Overschee and B deMoor ldquoN4SID subspace algorithmsfor the identification of combined deterministic-stochasticsystemsrdquo Automatica vol 30 no 1 pp 75ndash93 1994

[32] M Verhaegen ldquoIdentification of the deterministic part ofMIMO state space models given in innovations form frominput-output datardquo Automatica vol 30 no 1 pp 61ndash74 1994

[33] W E Larimore ldquoCanonical variate analysis in identificationfiltering and adaptive controlrdquo in Proceedings of the 29th IEEEConference on Decision and Control pp 596ndash604 December1990

[34] M Viberg ldquoSubspace-based methods for the identification oflinear time-invariant systemsrdquo Automatica vol 31 no 12 pp1835ndash1851 1995

[35] J Wang and S J Qin ldquoA new subspace identification approachbased on principal component analysisrdquo Journal of ProcessControl vol 12 no 8 pp 841ndash855 2002

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

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