Proc IMechE Part I: J Systems and Control Engineering...

Original Article

Proc IMechE Part I:J Systems and Control Engineering227(5) 482–494� IMechE 2013Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0959651813479863pii.sagepub.com

Automatic model bank selection inmultiple model identification of gasturbine dynamics

SeyedM Hosseini1, Alireza Fatehi1, Ali K Sedigh1 and Tor A Johansen2

AbstractA multiple model structure of a prototype industrial gas turbine system is constructed under normal operation using asystematic method that incorporates non-linearity measure and H-gap metric tools with the multiple models technique.First, two new non-linearity indices for multiple input–multiple output systems are introduced and employed for decom-posing the operating space of a gas turbine into some linear and non-linear modes. The non-linear modes may be furtherpartitioned into some linear modes. The input and output data in each of the linear modes are used to construct an ini-tial multiple model structure. In order to avoid the increase of the number of linear local models, the H-gap metric isextended to multiple input–multiple output systems and used to measure the similarity between linear local models andto merge the similar models. As a result, an algorithm is proposed for construction of multiple linear local models. Thealgorithm is employed for the identification of a single-shaft prototype industrial gas turbine.

KeywordsGas turbine, identification, multiple model, non-linearity measure, H-gap metric

Date received: 6 August 2012; accepted: 8 December 2012

Introduction

The modelling and identification of gas turbine systemsis an active research area that is widely used in control,performance monitoring, fault detection and diagnosisof gas turbines. Generally, there are two approaches inmodelling a plant: first, principle modelling that isbased on physical dynamic equations of the plant andsystem identification, which is a data-driven modelling.System identification has been extensively used in appli-cation of industrial plants, and there are many publica-tions in this regard.1–5

Initially, Evans et al.6 focused on frequency-domainidentification techniques for linear modelling of a gasturbine. Then Arkov et al.7 employed extended leastsquares with optimal-smoothing for time-varying linearmodelling. However, the non-linear variations of theengine dynamics make the linear models unable to cap-ture the complicated engine behaviour. Hence, demandsfor a reliable accurate non-linear model have increasedin recent years. In this context, two practical and usefulapproaches are considered. In the first approach, a glo-bal non-linear auto-regressive with exogenous input(NARX) or non-linear auto-regressive moving averagewith exogenous input (NARMAX) model is derived

that is able to identify and handle the entire engine’sworking space.8–10

In 2004, a gas turbine system is identified by Takagi-Sugeno fuzzy model.11 It has been shown that this kindof modelling provides acceptable accuracy. Afterward,in 2008, a linear model is constructed for a given gasturbine system, and a robust observer is designed basedon that model.12 Moreover, in 2012, the non-linearidentification based on NARX method is used for agas turbine system.

In the second approach, the multiple model (MM)technique is used. The MM approach has a rich historyof application in modelling and identification.13,14 Also,it has received a great deal of attention in gas turbine

1Department of Electrical & Computer Engineering, K. N. Toosi

University of Technology, Tehran, Iran2Department of Engineering Cybernetics, Norwegian University of

Science and Technology, Trondheim, Norway

Corresponding author:

SeyedM Hosseini, Advance Process Automation and Control (APAC)

Research Group, Industrial Control Center of Excellence, Department of

Electrical & Computer Engineering, K. N. Toosi University of Technology,

P.O. Box 16315-1355, Tehran, Iran.

Email: [email protected], [email protected]

modelling and fault detection12,15 due to its great poten-tial in decomposing a complex problem into simplersub-problems.

A key issue in non-linear modelling with the MMtechnique is the model set design, which includes deter-mination of the number of local models and their con-struction.16 For this purpose, it is necessary todetermine the distribution of local models in the work-ing space of the plant. A useful metric for managing thenumber of local models is the gap metric. Galan et al.17

used the gap metric within the MM-based control struc-ture using an offline algorithm, and then Jingjinget al.14 used the gap metric to minimize the number oflocal models in a multiple linear model approach.

Hosseini et al.18 proposed a systematic method formultiple linear modelling of a non-linear single input–single output (SISO) system by using a non-linearitymeasure and the H-gap metric. In the proposedmethod, at the first step, the total operating space ispartitioned by non-linearity indices into some linearmodes that describe the overall behaviour of the sys-tem. There are many indices to measure the non-linearity degree of a system.19–21 However, two newnon-linearity indices were presented in order to decom-pose the operating regime. The initial model bank isthen constructed based on the obtained linear modes.Afterward, it is modified by measuring the similaritybetween each two linear models by the H-gap metricmethod.

Here, an organized method for the non-linear identi-fication of gas turbine based on the MM approach ispresented by extending the non-linearity measure andthe H-gap metric tools for multiple input–multiple out-put (MIMO) systems. In order to show the functional-ity of the presented method, it has been compared withNARX model, which is one of the well-known methodspresented for gas turbine identification.8,22–24 It isshown, in simulation study, that for highly non-linearoutputs, a NARX model has less accuracy than thesuggested MM.

The main contributions of this article are summar-ized as follows.

1. Extension of coherence- and bicoherence-basednon-linearity indices and H-gap metric to theMIMO systems;

2. Generating an automatic model bank selectionalgorithm in order to identify MIMO non-linearsystems;

3. Applying the developed method to identify anindustrial gas turbine system.

The rest of the article is organized as follows. In sec-tion ‘Gas turbine modelling preliminaries’, some preli-minaries of gas turbine systems are presented. Section‘Non-linearity measure’ involves the basic mathemati-cal properties of the linear cross-correlation and higherorder auto-spectrum methods for MIMO systems, and

two new indices are then defined and investigated.Then the H-gap metric is extended to MIMO systemsin section ‘Gap-based metrics’. In section ‘Model bankselection algorithm’, the main properties of the MMapproach is studied, and an algorithm for model bankselection is presented. Section ‘Simulation results’involves several simulation results in order to test theeffectiveness of the proposed method, followed by con-clusions in section ‘Conclusions’. The notations usedthroughout this article are listed in Appendix 1.

Gas turbine modelling preliminaries

A typical gas turbine is made up of three main compo-nents: a compressor, a combustion chamber and a tur-bine. Engine models are required in the developmentand proper operation of a gas turbine.9 These modelsare used to

� predict the performance of the engine;� design and test the performance of the engine con-

trol systems;� employ fault detection techniques.

In physical modelling approach, the mass-balanceequivalence and thermodynamic principles are derivedand provide important insights into engine behaviour.But mathematical models using computational tech-niques and thermodynamics relations cannot be directlyused for the control system design or stability studiesdue to their complexity and non-linear characteristic.25

Moreover, there are some factors such as generating theproper compressor and turbine maps, the effect ofBleed valve and inlet guide vane (IGV) valve on thecompressor map that make the physical modellingproblem become more complex.26,27 One solution is toprovide a simplified mathematical model that consistsof a set of algebraic equations and simple time delaycontrollers.28 However, the applications of gas turbinesystems depend greatly on the accuracy of the systemmodel and governor parameters.

Alternatively, system identification techniques canbe employed to derive up-to-date practical models ofthe gas turbines. There are several articles dedicated tothe gas turbine system modelling based on the identifi-cation techniques.6,29

Non-linearity measure

Before proceeding with any kind of identification, it isuseful to determine whether the system behaviour islinear or non-linear. So measuring the degree of non-linearity of a process under various operating condi-tions is important. In control theory, it is said that asystem is non-linear if it cannot be modelled by a lineartransfer function.29 There are plenty of methods tomeasure the non-linearity. Non-linearity measure

Hosseini et al. 483

methods are divided into signal-based and model-basedmethods.29

The signal-based methods are considered in this arti-cle and are based on the analysis of the output and orinput data. Correlation-based analysis and higher orderspectral analysis are the two useful methods to measurethe non-linearity of dynamical systems.21 In this sec-tion, we extend these methods to measure the non-linearity of MIMO systems.

Assume a linear plant with p inputs and r outputs,which is described by

z1(k)z2(k)

..

.

zr(k)

26664

37775

|fflfflfflfflffl{zfflfflfflfflffl}z(k) r31

=

H11(q) � � � H1p(q)

..

. . .. ..

.

Hr1(q) � � � Hrp(q)

264

375

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}H(q) r3p

:

u1(k)u2(k)

..

.

up(k)

26664

37775

|fflfflfflfflffl{zfflfflfflfflffl}u(k) p31|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

y(k)

+

w1(k)w2(k)

..

.

wr(k)

26664

37775

|fflfflfflfflfflffl{zfflfflfflfflfflffl}w(k) r31

ð1Þ

where w, u, z andH are noise vector, input vector, out-put vector and transfer function matrix, respectively.The sequence w(k)f g is random with zero mean andcovariance matrix E wwT½ �=L.

Coherence non-linearity measure

It has been shown that for a SISO linear time invariant(LTI) system without noise, the squared coherencefunction (g2

yu) between the input and the output is equalto 121

g2yu(jv)=

fyu(jv)�� 2

fuu jvð Þ:fyy(jv)=1 ð2Þ

Now assume that u and y are input and output ran-dom vectors of a MIMO system, respectively. The auto-correlation and cross-correlation functions betweenthese vectors are defined in the study by Priestley30 as

Ruu tð Þ=E u tð ÞuT t� tð Þ� �

ð3ÞRyu tð Þ=E y tð ÞuT t� tð Þ

� �ð4Þ

Considering the frequency domain of above correlationfunctions, the following lemma can be presented.

Lemma 1. For a MIMO linear system, which isdescribed by equation (1), the following matrix coher-ence function G vð Þ is the identity matrix

G vð Þ=E½fyy vð Þ��1E½fyu vð Þ�E½fuu vð Þ�y�1E½fyu vð Þ�y

Proof. See Appendix 2.

Hence, a new non-linearity index for a MIMO sys-tem can be defined as below

NLI1= maxv

G vð Þk k2F�1� �

ð5Þ

If a system is LTI, then NLI1=0.

Bicoherence non-linearity measure

Higher order spectral analysis is another technique thatcan be used to detect the system non-linearity. In thestudy by Priestley,30 it has been shown that if the inputof a linear stochastic process is Gaussian, then all spec-tra of odd order are identically zero. Consequently, if aprocess presents a non-zero higher order spectrum, thiscould be due to two reasons: the process is linear, butthe input is non-Gaussian, or the process has somenon-linearity components.

Assume equation (1) for a system without measure-ment noise. Then

Yr31 vð Þ=Hr3p vð ÞUp31 vð Þ ð6Þ

For a zero mean vector time series y tð Þ=y1y2 . . . yr½ �T, the third-order cumulant is defined in thestudy by Mendel.31

Cy

3 t1, t2ð Þ=cum y tð Þ, y t+ t1ð Þ, y t+ t2ð Þh i

=E y tð Þ � y t+ t1ð Þ � y t+ t2ð Þh i

ð7Þ

Now, we define the multivariate bispectrum as

Bisy v1;v2ð Þ=DDFT cy

3(t1, t2)� �

=E Y v1ð Þ � Y v2ð Þ � Y� v1 +v2ð Þ½ � ð8Þ

Lemma 2. For a MIMO LTI system with independentGaussian and zero mean inputs without additive noise

Bisy v1,v2ð Þ=0


Lemma 3. If A 2 Rm3n, B 2 Rr3s, C 2 Rn3p, then aCauchy–Schwarz-like inequality based on theKronecker and Hadamard products can be defined as

E A� B� C½ �j j2��

E A� A�ð Þ � B� B�ð Þ½ � � E C� C�ð Þ½ �k k41


Assuming A=Y(v1),B=Y v2ð Þ,C=Y� v1 +v2ð Þ,considering the result of Lemma 3, a normalized ver-sion of multivariate bispectrum (bicoherence) isobtained as follows

484 Proc IMechE Part I: J Systems and Control Engineering 227(5)

It is easily shown that for a multiple input–singleoutput (MISO) system, which is used in our simulationstudies, the multivariate bicoherence function changesto its simpler well-known form as

bic2y v1,v2ð Þ= E Y v1ð ÞY v2ð ÞY� v1 +v2ð Þ½ �j j2

E Y v1ð Þj j2 Y v2ð Þj j2h i

:E Y v1 +v2ð Þj j2h i

ð10Þ

It is proven in the study by Priestley30 that for anyLTI system, with Gaussian input, the bicoherence func-tion is equal to zero and independent of frequency. Thisproperty allows the following indices to be defined forthe non-linearity measure

NLI2= maxv1,v2

bic2y v1,v2ð Þ ð11Þ

It is worth mentioning that there are some differ-ences between the above indices as listed in thefollowing:

1. The NLI1 uses the input/output signal while theNLI2 employs just the output signal (time series).

2. The NLI1 uses less data than the NLI2 index.3. Since NLI2 involves higher order moment of a sig-

nal, it is more accurate and efficient than the NLI1index.

In relation to the above distinctions, both NLI1 andNLI2 indices are employed here to measure the non-linearity degree of engine dynamics.

Gap-based metrics

The gap metric is an appropriate tool to measure thedistance between two linear systems and is superior tothe norm-based methods.32 Consider a finite dimen-sional linear operator Pi defined in the H2 space. Thetransfer function Pi can have the normalized rightcoprime factorization (GT

i = Ni Mi½ �) and leftcoprime factorization ( ~Gi = � ~Mi

~Ni

� �) given by

Pi =NiMi�1 = ~Mi

�1 ~Ni ð12Þ

Georgiou33 showed that the gap metric can be com-puted as

dg(P1,P2)= max ~dg(P1,P2),~dg(P2,P1)� �

ð13Þ

where

~dg(P1,P2)= infQ2H‘

N1

M1

� N2

M2

Q

��

‘

ð14Þ

It is always true that 04dg(P1,P2)41. If the gapmetric of two systems is close to zero, this indicates thatthe two linear systems are similar.

v-gap metric

Based on the gap metric structure, a related metric onthe linear operators known as the v-gap metric has beenintroduced in the study by Vinnicombe.34 This metricinduces the same topology that the gap metric wasdeveloped on, though the v-gap metric is less conserva-tive than the gap metric. Vinnicombe34 presents a defi-nition of the v-gap metric as shown in the following

dv P1,P2ð Þ

=~G2G1

�� ‘

if det G�2G1

� �6¼ 0&wnodet(G�2G1)=0

1 otherwise

(

ð15Þ

~G2G1

�� ‘= I+P2P

�2

� �� 12(P2 � P1) I+P�1P1

� ��12

�� ‘

ð16Þ

where wno is the number of counter-clockwise encircle-ment around the origin by a given transfer functionG sð Þ evaluated on the Nyquist contour G.

H-gap metric

Although the gap and the v-gap metrics are useful solu-tions for measuring the distance between two linear sys-tems, they may cause inaccuracies as these metricsmerely consider the worst case frequency. Therefore,values at other frequencies are not considered.34 Hence,a new metric is defined in the study by Hosseiniet al.18,35 This metric involves new construction follow-ing the same topology as the v-gap metric; however, ituses the H2 norm instead of H‘ norm.

Now, we define the H-gap metric for two MIMO lin-ear systems P1,P2 as

dhv(P1,P2)= infQ2H‘

wnodet Qð Þ=0

G1 � G2Qk k2max �s

vG1 � G2

�Qð Þ��

2

ð17Þ

where �Q is the value that minimizes the distancebetween coprime factorizations of P1;P2.

Lemma 4. If G1,G2 are the normalized right coprimefactorization of P1,P2, respectively, and Q 2 H‘, then

infQ2H‘

wnodet Qð Þ=0


vG1 � G2

�Qð Þ��

2

=~G2G1

�� 2

maxv

�s ~G2G1

� �� 2

bic2y v1,v2ð Þ=E Y v1ð Þ � Y v2ð Þ � Y� v1 +v2ð Þ½ �j j2

�� E Y v1ð Þ � Y v1ð Þ�ð Þ � Y v2ð Þ � Y v2ð Þ�ð Þ½ � � E Y� v1 +v2ð Þ � Y v1 +v2ð Þ½ �k k41 ð9Þ

Hosseini et al. 485


Considering equation (17) and Lemma 4, the newmetric is reformulated as

dhv(P1,P2)=~G2G1

�� 2

maxv

�s ~G2G1

� �� 2

ð18Þ

Since there is no consideration to the upper value of�s ~G2G1

� �, it is difficult to use it as a distance measure.

So, without loss of generality, the denominator of equa-tion (18) is replaced with

dhv(P1,P2)=~G2G1

�� 2

1+ maxv

�s ~G2G1ð Þ2

��2

ð19Þ

The H-gap metric involves the same winding numberconditions as the v-gap metric.36 Using the Parseval’stheorem, a straightforward form of this metric is

dhvðP1;P2Þ ¼12p

R vH

vLtr Disð�jvÞTDisðjvÞh i

dv� �1

2

12p

R vH

vL

1þmaxv

�s tr Disð�jvÞTDisðjvÞ½ �ð Þ� �

2dv

0@

1A

12

DisðvÞ ¼ Iþ P2P�2

� ��12ðP2 � P1Þ Iþ P�1P1

� ��12

8>>>>>>>><>>>>>>>>:

ð20Þ

where vL and vH are chosen based on the P1 and P2

bandwidth. It is always true that 04dhv41. In thesimulation study, we will demonstrate how the H-gapmetric can produce better results in comparison withthe other two metrics.

Model bank selection algorithm

The MM approach

The MM method relies on a divide and conquer tech-nique. In this technique, the total operating space of asystem is divided into some smaller local sub-spaces.Next, a local model is constructed for each sub-space.Finally, all of the local models are combined to generatethe total system response (Figure 1). The main advan-tage of the MM approach is that the complex modellingprocedure is performed easily in the form of simplerlocal models.

Model set design is one of the main issues in MMtechnique. The aim of model set design is to determinethe suitable local model set in order to estimate the out-put response. In this context, two basic questionsshould be addressed. The first one is how to design orselect local linear models and the second one is how thenumber of effective local models is determined. It isshown in the study by Li16 that the use of over- orunder-numbered models is problematic. A criterion istherefore required to decide whether a new modelshould be added to the model set or not. We proposeoperating regions decomposition by non-linearly

measure and then comparing the models in theseregions by H-gap metric. In the following section, asystematic method for model bank selection isintroduced.

Model bank selection algorithm

In this section, an algorithm is presented that usesNLI1, NLI2 and the H-gap metric tools for designing asuitable model bank for a MIMO non-linear system.

Main algorithm. Initially, to construct the models, anexcitation signal is needed that should be effective forboth non-linearity detection (Gaussian signal) and sys-tem identification (persistently exciting).

In the next stage, the data need to be segmented inorder to measure the non-linearity. First, a large amountof data are gathered then using the (NLI2), it is decidedwhether the segmented data are linear or non-linear.Although the NLI2 index is zero for any LTI systembecause of some numerical errors, it will not be equal tozero in practice. So, a threshold should be determined.The threshold (TNLI2) is selected based on the segmenteddata length, type of input and signal-to-noise ratio.29

After decomposing the operating space into somelinear and non-linear modes, the non-linear modes maybe further partitioned into some linear modes throughsub-algorithm 1.

Sub-algorithm 1. In this sub-algorithm, each segmentwith non-linear behaviour is further partitioned. Sincethe bicoherence test needs a large amount of data, thelinear cross-correlation method (NLI1) is applied as ituses less data and reduces the complexity, although theaccuracy may slightly deteriorate. Similarly, anotherthreshold (TNLI1) is used for this non-linearity index.

Now a supervisor is needed to find the effective localmodels.

Sub-algorithm 2. In this sub-algorithm, some of theunnecessary linear local models are omitted, and thefinal model bank with a reduced number of models is

Figure 1. The multiple model approach.


selected. Moreover, the sum of squared error (SSE)index is used as a performance index.

Finally, all the linear local models are assumed in agiven model bank, and ineffective models are removedthrough sub-algorithm 2 once more, and then the finalmodel bank is constructed (Figures 2–4).

Simulation results

This section first describes the dynamic non-linearmodel of a typical industrial gas turbine. The model is

modular in structure and avoids unnecessary detailswhile maintaining an acceptable accuracy.37 Thedynamic behaviour of each module of turbine is formu-lated by using the thermodynamic relations and mass-balance equations. The simulation of the gas turbinewas carried on by integrating the differential equationsand solving the static equations with the variable valuescalculated at each time instant.12

According to Figure 5, Mf (fuel flow), Ta and Pa

(ambient temperature and pressure) and lower heatingvalue (LHV) are boundary condition inputs, Pe (electricpower), T5 and m5 (turbine exhaust temperature (TET)and mass flow) are the outputs. The engine load regula-tion is performed by means of fuel flow rate and IGVangle inputs to keep the turbine outlet temperature con-stant. The non-linear model has been calibrated bymeans of a reference steady-state condition data of areal industrial gas turbine and has been used to simu-late various machine transients.37

This simulator is considered as a MIMO non-linearsystem with a linear proportional–integral–derivative(PID) controller (governor) for load regulation. The nom-inal values of inputs are listed in Table 1. Although itincludes several outputs, just three outputs are consideredin our simulation studies. These outputs are the shaftspeed, the compressor torque and TET (Figure 6).

To construct the MM by the proposed algorithm,the segment data length is considered 4096 samples,

Figure 2. Main algorithm.

Figure 3. Sub-algorithm 1.

Hosseini et al. 487

two independent pseudo random binary signals(PRBS) with proper amplitude and frequency areadded to the inlet air and fuel flow and the SSE is con-sidered as a suitable performance index of the modelbank to validate the obtained results. Also, a thresholdof TSSE =110 is selected to check the performance ofthe model bank. Each model is a MISO linear model as

yðtÞ ¼Xnyi¼1

aiyðt� iÞ þXnkk¼1

Xnui¼1

bi;kukðt� kÞ ð21Þ

The parameters are estimated using the least squaresalgorithm. Moreover, in order to select the order ofauto-regressive with exogenous input (ARX) models,the Bayesian information criterion (BIC) is used asfollows5

BIC=N lnSSE

N

�+ k ln Nð Þ ð22Þ

where N, SSE and k are the sample size, the loss func-tion and the number of parameters, respectively. TheBIC index makes a trade-off between complexity andperformance of a model. The model order that mini-mizes the BIC criteria is selected as the best order. Byconsidering the total MIMO gas turbine system as threeMISO sub-systems, the simulation study is done basedon each individual output.

Considering the available literatures,8,22–24 theNARX or NARMAX method is one of the most usefulmethods in the context of gas turbine identification.Hence, we employ this method in order to comparewith the presented method in our simulation study.

Figure 4. Sub-algorithm 2.


At the first stage, the linearity or non-linearity of theengine for each of the selected outputs is investigated(Table 2). Based on Table 2, vt and T5 are fairly linearwhile the dynamic of Qc is highly non-linear.

Identification of the TET

The TET (T5) is one of the main outputs of the gas tur-bine system that effects overall efficiency; moreover, itis used in control temperature and performance moni-toring. Based on Table 2, the T5 output behaviour islinear, so the required performance can be attainedwith a single ARX model. Using BIC criteria for orders

up to 3 gives the following results: na =3, nb = 3 3½ �,nk = 0 0½ � and SSE=29:47\TSSE (Figure 7). Byincreasing the model order, no meaningful change isobtained as na =5, nb = 5 5½ �, nk = 0 0½ � !SSE=24:348.

Identification of the shaft speed (vt)

The control construction of the engine is fundamentallybased on the shaft speed output. As shown in Table 2,the vt output behaves linearly so it is sufficient to finda linear model in order to accurately estimate the out-put. Using the BIC index, up to order three, providesus na =3, nb = 2 2½ �, nk = 0 0½ � and SSE=31:76(Figure 8). Similarly, increasing the model order pro-vides no meaningful change as na =6, nb = 3 3½ �,nk = 0 0½ �, with SSE=31:49.

Identification of compressor torque (Qc)

Compressor torque shows the characteristic of thecompressor. It also attracts many researchers’ attention

Figure 5. A schematic lay out of the gas turbine prototype.12

IGV: inlet guide vane; PID: proportional–integral–derivative; LHV: lower heating value; CM: compressor map; TM: turbine map; ID: intake duct; C:

compressor; CC: combustion chamber; T: turbine; ED: exhaust duct; EG: electric generator.

Table 1. Input variables with their nominal values.

Input variables Nominal value

Ambient air temperature 273:6 KAmbient air pressure 1:009e5 N=m2

Fuel flow 0:216 kg=sValve angle 63:158

Figure 6. Input/output block diagram of gas turbine.

Table 2. Non-linearity degree of each individual output.

Output Non-linearitydegree (NLI2)

T5 0.16vt 0.19Qc 0.6

Hosseini et al. 489

since it is considered as one of the sensitive outputs in agas turbine fault diagnosis system. As depicted inTable 2, the behaviour of this output is highly non-linear; it needs a non-linear model to achieve a desiredperformance. We will show that non-linear modelsbased on MMs have better performance than a singlenon-linear ARX model.

Similar to the previous examples, using the BIC cri-teria for orders up to 6 gives the best orders as: na =3,nb = 2 2½ �, nk = 5 5½ � and SSE=207:1. Collectingenough input and output data and implementing theproposed algorithm of section ‘Model bank selectionalgorithm’, leads to the results of Figure 9. It shows thatthe H-gap metric has slightly better functionality incomparison with the gap metric and the v-gap metrics.Moreover, considering sub-algorithm 2, the selectedmodel bank with 26 linear local models satisfies theminimum requirement (SSE=64:18\TSSE). Then,the NARX method is applied to the same problem.

This model has been used in identification of gas tur-bine systems by many researchers. To do so, a dynamicNN with parallel (simulator) structure is used. For eachNN structure, the model is trained five times from dif-ferent initial parameters; the best one is selected as thefinal model. Using train and test data, a NARX modelis obtained as the best NN model with 11 delays forinput vector, 13 delays for output vector and 11 neu-rons in its single hidden layer. The obtained results,which are depicted in Table 3 and Figure 10, show thatthe performance of the MM structure is better than thatof the NARX one; moreover, the number of free para-meters of MM is much less than the NARX.

Global modelling of gas turbine

The purpose of this section is to test the proposedmethod for global modelling in multiple operatingpoints. The shaft speed output is selected with two dif-ferent operating points. The operating points are alter-natively changed between 1496 rad/s and 1765 rad/sfrom t=0 s to t = 160 s, and then it changed between1765 rad/s and 1975 rad/s from t=160 s to t=281 s.Using the proposed algorithm generates a model bankwith 16 linear local models with orders na =3,nb = 2 2½ �, nk = 0 0½ � (Table 4).

Figure 7. Real output and identified output of TET.SSE: sum of squared error.

Figure 8. Real output and identified output of vt.SSE: sum of squared error.

Table 3. Comparing between MM and NARX methods.

Output(Qc)

No. oflocalmodels

No. of free parameters SSE

MM 26 26 3 7 = 182 64.18NARX 1 11 3 (2 3 11 + 1

3 13 + 2) + 1 = 408108.73

SSE: sum of squared error; MM: multiple model; NARX: non-linear

auto-regressive with exogenous input.

Figure 9. SSE variations with the number of linear localmodels.SSE: sum of squared error.


Similarly, a single NARX model, based on the NN,has also been designed. As a result, a single multilayerperceptron (MLP) network is obtained as the bestNARX model with three delays of input vector, twodelays of output and 7 neurons (Table 5). The result ofTable 5 is based on five times training of each NN fromdifferent initial parameters.

Considering the results of Tables 4 and 5, Figure 11reveals that, with similar number of parameters, theperformance of the proposed method is much betterthan the NARX method. However, as depicted inTable 6, the accuracy of the MM structure with fourlocal models is approximately the same as that of theNN model with 11 neurons. It means the same

accuracy can be obtained with less complexity by theproposed MM structure.

Conclusion

The MM technique is an appropriate tool for dealingwith extensive non-linear systems with wide operatingpoints. The main point of this article consists of pro-viding a systematic method which incorporatesnon-linearity measure and H-gap metric tools withMM technique in order to construct a proper modelbank with a reduced number of local models, while nomathematical model is available.

Figure 10. Multiple model error and NARX error.SSE: sum of squared error; MM: multiple model; NARX: non-linear auto-

regressive with exogenous input.

Table 5. SSE variations with different number of neurons.

No. of Neurons 1 2 3 4 5 6 7 8 9 10 11 12

SSE (NARX) 226.1 159.6 113.1 126.1 158.0 120.2 107.9 205.5 136.7 105.5 86.76 188.9

SSE: sum of squared error; NARX: non-linear auto-regressive with exogenous input.

Table 4. SSE variations with different number of local models.

No. of local models 1 2 3 4 5 6 9 16 17 35 42

SSE (MM) 264.08 111.28 111.21 84.912 77.599 72.199 62.733 44.398 49.398 38.149 36.266

SSE: sum of squared error; MM: multiple model.

Figure 11. Real output versus identified output with NARXmodel and suggested multiple model.SSE: sum of squared error; MM: multiple model; NARX: non-linear auto-

regressive with exogenous input.

Table 6. Comparing between MM and NARX methods.

Output (Wt) No. of local models No. of free parameters SSE

MM 4 4 3 7=28 84.91NARX 1 11 3 (2 3 3 + 1 3 2 + 2) + 1 = 111 86.76

SSE: sum of squared error; MM: multiple model; NARX: non-linear auto-regressive with exogenous input.

Hosseini et al. 491

The linear cross-correlation and higher orderspectrum-based methods are extended for MIMO sys-tems and employed for non-linearity detection and par-titioning the total operating space into some linear sub-spaces. In order to avoid the increase of the number oflinear local models, the H-gap metric for multivariablesystems is defined and used to compare the gap betweenthe models. To illustrate the usefulness of the suggestedmethod, the simulation data of a single-shaft industrialgas turbine plant is used. Moreover, it has been com-pared with a global non-linear modelling (NARX),which shows its superiority to this model.

As a result, the proposed method provides a sys-tematic solution for modelling of a large class of non-linear systems with high performance and low complex-ity. It is fairly easy to use and involves some practicalviewpoints for engineers to implement it in industrialapplications.

Funding

This research received no specific grant from any fund-ing agency in the public, commercial, or not-for-profitsectors.

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Appendix 1

Notation

DDFT double discrete Fourier transformE expected valueFT Fourier transformY jvð Þ Fourier transform of y tð Þ:j j magnitude� complex conjugates standard deviation:k kF Frobenius norm

y transpose conjugate�s upper singular value� Kronecker product� Hadamard productcum(t1, ::, tn�1) nth order cumulantfyu jvð Þ cross-correlation functionfyy jvð Þ autocorrelation functiongyu jvð Þ coherence functionBisy v1,v2ð Þ bispectrum of y(t)bicy v1,v2ð Þ bicoherence of y(t)Py vð Þ power spectrum of y(t)wno winding numberGT

i = Ni Mi½ � normalized right coprimefactorization

~Gi = � ~Mi~Ni

� �normalized left coprime

Appendix 2

Proof of Lemma 1. Based on equations, (1) and (3) to (4)

Y vð Þ=H vð ÞU vð Þ ð23Þ

fuu vð Þ=DFT Ruu tð Þð Þ=U vð ÞUy vð Þ ð24Þ

fyu vð Þ=DFT Ryu tð Þ� �

=Y vð ÞUy vð Þ

= H vð ÞU vð Þð ÞUy vð Þ=H vð Þfuu vð Þ ð25Þ

fyy vð Þ=Y vð ÞYy vð Þ= H vð ÞU vð Þð Þ Uy vð ÞHy vð Þ� �

=H vð Þfuu vð ÞHy vð Þ ð26Þ

Based on equations (25) and (26)

f�1yy vð Þfyu vð Þf�yuu vð Þfyyu vð Þ= I ð27Þ

In order to estimate the coherence function withfinite data, the expected value operator is used. So

E fyu vð Þ� �

=H vð ÞE fuu vð Þ½ � ð28Þ

E fyy vð Þ� �

=H vð ÞE fuu vð Þ½ �Hy vð Þ

=E fyu vð Þ� �

E fuu vð Þ½ �y�1E fyu vð Þ� �yð29Þ

! G vð Þ=E fyy vð Þ� ��1

E fyu vð Þ� �

E fuu vð Þ½ �y�1

E fyu vð Þ� �y

= Ir3r 8v ð30Þ

Therefore, if the system is linear, then G vð Þ= I.

Proof of Lemma 2. Substituting equation (6) into equa-tion (8) gives

Bisy v1;v2ð Þ=E H v1ð ÞU v1ð Þ �H v2ð ÞU v2ð Þ�½

H� v1 +v2ð ÞU� v1 +v2ð Þ� ð31Þ

The Kronecker product when A 2 Rm3n,B 2 Rr3s,C 2 Rn3p,D 2 Rs3t has the followingproperty38

ACð Þ � BDð Þ= A� Bð Þ C�Dð Þ 2 Rmr3pt ð32Þ

Therefore, from equation (31), it is deduced that

Bisy v1,v2ð Þ=H v1ð Þ �H v2ð Þ �H� v1 +v2ð Þ

E U v1ð Þ �U v2ð Þ �U� v1 +v2ð Þ½ � ð33Þ

It has been shown that for a Gaussian input, theexpected value of an odd order moment of a signal iszero.29 Therefore

E Ui(vi)Ui(vj)Ui(vk)� �

=0 i, j, k=1, 2, . . . p ð34Þ

Using the independently zero mean property ofinputs gives

E Ui(vi)Uj(vj)Uk(vk)� �

=E Ui(vi)½ �E Uj(vj)� �

E Uk(vk)½ �=0 i, j, k=1, 2, . . . p ð35Þ

By incorporating equations (34) and (35)

E U v1ð Þ �U v2ð Þ �U� v1 +v2ð Þ½ �=0! Bisy v1,v2ð Þ=0 ð36Þ

Proof of Lemma 3. Considering the Kronecker productdefinition,38 it is easy to show that

A� B� Cj j2 = Aj j2 � Bj j2 � Cj j2 ð37Þ

Hosseini et al. 493

E A� B½ �=E A½ � � E B½ � ð38Þ

Assume D=A� B� C, using the Cauchy–Schwarzinequality

E D½ �j j24E Dj j2h i

ð39Þ

Therefore, based on equations (37) to (39)

E A� B� C½ �j j24E Aj j2 � Bj j2h i

� E Cj j2h i

ð40Þ

Using the Hadamard product definition,38

( Aj j2 =A� A�), the following Cauchy–Schwarz-likeinequality is defined as

E A�B�C½ �j j24E A�A�ð Þ� B�B�ð Þ½ ��E C�C�ð Þ½ �ð41Þ

Employing the norm operator at both sides of theabove inequality now provides us the followinginequality

E A� B� C½ �j j2��

E A� A�ð Þ � B� B�ð Þ½ � � E C� C�ð Þ½ �k k41 ð42Þ

Proof of Lemma 4. It is true that if R and S are tworational transfer function matrices and S�S= I, thenSRk k2 = Rk k2.

35

Moreover, it is easily shown that

maxvL4v4vH

�s SRð Þð Þ��

��2

=

ðvH

vL

max �s SRð Þð Þj j2dv

=

ðvH

vL

max �s Rð Þð Þj j2dv= maxvL4v4vH

�s Rð Þð Þ��

��2

ð43Þ

Now assuming S=G�2~G2

, it can be shown that

S�S= I. Then

G1 � G2Qk k2max

v�s G1 � G2

�Qð Þ��

2

=

G�2~G2

G1 � G2Q½ �

��2

maxv

�sG�2~G2

G1 � G2

�Q½ � ��

��2

=

G�2G1 �Q

~G2G1

��2

max �sv

G�2G1 � �Q

~G2G1

��2

ð44Þ

If Q=G�2G1, then �Q=Q and the infimum of the left-hand side of the above inequality is attained. Therefore

infQ2H‘


vG1 � G2

�Qð Þ��

2

=~G2G1

�� 2

maxv

�s ~G2G1

� �� 2

ð45Þ


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