research article modelling of lime kiln using subspace
TRANSCRIPT
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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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
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
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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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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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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 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