J. Automation & Systems Engineering 10 Regular ...

J. Automation & Systems Engineering 10-4 10 (2016): 221-230 Regular paper Parametric Identification of Multivariable Industrial System Using Left Matrix Fraction Description Bachir Nail1, Abdellah Kouzou1,Ahmed Hafaifa1, Choayb Djeddi1 1

The Applied Automation and Industrial Diagnostics Laboratory LAADI, University of DJELFA, Algeria.

Abstract-In this paper we are applied the theory of Linear Multivariable system identification based on the Left Matrix Fraction Description (LMFD) to find a approximate mathematical model using a recent measured inputs outputs real data of turbo compressor gas turbine. tur The approximate model is obtained via the MIMO Least Square(MLS) estimator based on the Left MFD, to select the best model order we check each one with two validation criteria AIC and FPE , finally we acquired stable valid and reliable model in Multivariable Block Observable canonical Form. Keywords:System identification, Linear system, MIMO Least Square (MLS), Left matrix

fraction description (LMFD), Turbo Compressor. 1. INTRODUCTION Before applying any control law to an industrial process we must have a mathematical model of it. The use of physical or mechanical laws is not always obvious, and is not the objective in our work, However the System identification deals with the problem of obtaining "approximate" models of dynamic systems from real r measured input-output data, which is often sufficient to achieve control goals in advanced engineering applications. applications The majority of the researches they have been done about the Modelling Model of turbocompressor gas turbine Using largely the artificial iintelligence theory (Fuzzy Logic , ANFIS and Neural Network ...), for the following grounds easy to apply and unconditioned constraints in the quality and the characterisation of the data (smoothing, convergence and number of samples is not required) , the disadvantages on these models, the difficulty of controlling, and the problem of instability, also the limitations of applying the theory of advanced control for the reason of the lack of the mathematical model, there are many results have been published in n recent years on modelling model of gas turbine and turbo compressor : Gas turbine modelling ing based on fuzzy clustering algorithm using experimental data[1], Fuzzy Modelling ing and Control of Centrifugal Compressor Used in Gas Pipelines Systems [2], Gas turbine parameters rameters modelling based on fuzzy logic and artificial neural networks: Solar TITAN 130N investigation[3], investigation[3] and others [4,5]. A recent papers have been published in MIMO System Identification methods using Left Matrix Fraction Description such as : An optimal opt instrumental variable identification method for LMFD models[6], Extending the SRIV algorithm to LMFD models, A matrix fraction description-based based identification algorithm for MIMO ARMAX models[7], A matrix fraction description-based based identification algorithm algo for MIMO ARMAX models[8], Identification of discrete-time time MIMO systems using a class of observable canonicalcanonical form[9], Multivariable least squares frequency domain identification using polynomial matrix fraction descriptions[10], riptions[10], and others [11,12,13,14,15,16]. [11,12,13

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B. Nail et al.: Parametric Identification of Multivariable Industrial System Using Left Matrix Fraction Description

This work aims to identify and modelling the behaviour of turbo-compressor gas turbine, installed in the natural gas field HassiR'Mel located 550 kilometres south of Algiers, the parametric multivariable model is obtained via the MIMO least square estimator based on Left matrix fraction description theory (LMFD) using experimental data, this multivariable model consisting of two inputs T1, P1 (Aspiration of Temperature and Pressure) and two outputs P2, T2 (Refoulement of Pressure and Temperature) The real data acquisition in a period of 1208 hours, to validate the model order two criteria have been used AIC and FPE, and we take into account the obtained validated model is left coprime. The paper is organized as follow: firstly as an introductory we have discussed historically the recent development about the modelling and the parametric identification of Turbocompressor gas turbine, then it is followed by some preliminaries about Matrix Faction Description (MFD) and as a third section the theory of MIMO Least Square based on MFD, after that theapplication and a discussion of the obtained results. Finally comments, perspectives and a conclusion will finish the paper.

2.

Matrix Fraction Description based method

Matrix Faction Description (MFD) is a representation of a matrix transfer function of a multivariable system as a ratio of two polynomial matrices. An introduction to matrix polynomials and MFD properties are given in [17, 18, 19, 20, and 21]. The MFD approach is based on the fact that the Transfer Function Matrices G (q −1 ) and H (q −1 ) of a MIMO system described by the vector difference equation. y[ k ] = G ( q − 1 ) u [ k ] + H ( q − 1 ) e[ k ] (1) Can be represented as ratio of two polynomial matrices.However, because matrices do not commute in general, we note that there are two representations for the transfer function matrix G ( q −1 ) or H (q −1 ) as a ratio of two polynomial matrices which are • Right Matrix Fraction Description (RMFD) G ( q − 1 ) = C ( q − 1 ) D − 1 ( q − 1 ) (2) • Left Matrix Fraction Description (LMFD) G ( q − 1 ) = A − 1 ( q − 1 ) B ( q − 1 ) (3) Where, the matrix polynomials A −1 ( q −1 ) , B ( q −1 ) , C ( q −1 ) and D ( q −1 ) have the following structures:  A( q −1 ) = I p + A1 q −1 + ⋯ + Ana q − na  −1 −1 − nb  B( q ) = B0 + B1q + ⋯ + Bnb q (4)  −1 −1 − nc C ( q ) = C0 + C1q + ⋯ + Cnc q  −1 −1 − nd  D(q ) = I m + D1q + ⋯ + Dnd q And the matrix coefficients have the following dimensions: Ai ∈ R p × p , Bi ∈ R p × m , C i ∈ R m × p And D i ∈ R m × m 3.

Extended Least Squares Based On Left MFD

A MIMO ARX (Autoregressive with Exogenous Excitation) model given by: A ( q −1 ) y[ k ] = B ( q −1 )u[ k ] + e[ k ] (5) Can be written in Left MFD form as:

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y[ k ] = A ( q − 1 ) −1 B ( q − 1 ) u [ k ] + A ( q −1 ) −1 e[ k ] (6)

Where u [ k ] ∈ R m and B i ∈ R p × m are inputs and outputs vectors of the system respectively, while e[ k ] ∈ R p is a white-noise signal and the polynomial matrices A −1 ( q −1 ) , B ( q −1 ) have the following structures: A ( q − 1 ) = I p + A1 q − 1 + ⋯ + A na q − n a (7) B ( q −1 ) = B1 q −1 + ⋯ + B nb q − nb (8)

The objective is to identify the matrix coefficients Ai ∈ R p × p and B i ∈ R p × m of the matrix polynomials C − 1 ( q − 1 ) = I p and B ( q −1 ) assuming in this case C − 1 ( q − 1 ) = I p . Taking the transpose of equation (5) and expanding A −1 ( q −1 ) and B ( q −1 ) yield:

eT [k ] = ( yT [k ] + yT [k − 1] A1T ⋯ + yT [k − na] Ana T ) − ( u T [k − 1]B1T ⋯ − u T [k − nb]BnbT ) T

T

T

(9)

e [k ] = y [k ] − ϕ [k ]θ Where ϕ T [ k ] = [ − y T [ k − 1] ... − y T [ k − n a ], u T [ k − 1] ... u T [ k − nb ]]

θ =  A1T ⋯ AnaT , B1T ⋯ BnbT 

T

The estimator based on least squares is given as follows: θˆls = Φ T Φ − 1 Φ T Y (10) Where, Y and

Φ are functions of y and u,

Hence

 yT [n + 1,:]    − y T [n,:]  ⋯  − yT [n − na + 1,:]         ⋮ Y = ⋮ ⋮  , Φ = Φ y ⋮ Φu , Φ y =    ⋯    (11)   − y T [ M − 1,:]  ⋯  − yT [ M − n ,:]    y T [ M ,:]  a      

(

)

  u T [ n,:]  ⋯  u T [ n − nb + 1,:]       Φu =   ⋮ ⋮  ⋯    (12)   u T [ M − 1,:]  ⋯  u T [ M − n ,:]   b    

With, n a and M is the number of (Inputs/Outputs) data. 4.

Model Validation

We have been using two criteria of validations in this work to choice the good model[22,23,24,25].: 4.1 AIC Is the Akaike Information Criterion for an estimated model

AIC = Ln(V ) +

2d (15) N

Where, V is the loss function, dis number of estimated parameters, N is the number of values in the estimation data set.

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B. Nail et al.: Parametric Identification of Multivariable Industrial System Using Left Matrix Fraction Description

4.2 FPE  1+ FPE = V   1 − 

d N d N

   (16)  

Relation between AIC and FPE given by:

AIC ≈ Lg ( FPE ) (17) In order to select the best model structure, we must have to test a number of different ones and compare the resulting models on the evaluation of different criteria assessing the model structures under study. AICs and FPEs uses a Criterion with complexity terms obtained by supplementing the cost function with some extra term penalizing for model complexity. Then the model structure giving the smallest value of this criterion is selected. The criterions takes the forms as shown in the preceding equations (15), (16).

Figure 1 Schematic Bloc Diagram of Gas Turbo-compressor with Inputs/Outputs Model Identification.

5

Application and Results

The system under consideration is power plant Turbo-compressor composed of two inputs and two outputs, its schematic diagram is shown in figure 1, the length of the real data used in this identification is 604 samples taken in period of 1208 hours, the data treated and filtered to be valid

5.1 Validation and Numerical Model The model that have been select must have order of even numbers (2, 4...) because each matrix block $q$ have a dimension of 2×2. The table shown below clarify the change of the orders $n$ until we stop at the best one.

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TABLE I: Turbo-Compressor System model orders AICs andFPEs q

n

AIC

FPE

1

2

1.7200

2

4

0.7445

2.1024

3

6

0.1789

1.1923

4

8

0.1319

1.1350

5

10

0.1251

1.1242

6

12

0.1420

1.1395

5.5826

The best model having both minimums AICs FPEs and no Pole/Zero cancellation is the one having order n=10 with q=5 blocks.The system write in state space Block Observable canonical Form as follow:

Z  I A = Z  Z Z 

Z Z

Z Z

Z Z

I

Z

Z

Z

I

Z

Z

Z

I

− A5   B5  Z       − A4   B4  Z  T − A3  , B =  B3  , C =  Z  , D = Z , I =  1      0 − A2   B2  Z   I ′ B  − A1     1

0 0  Z = 1 0

0  0

Yield,

0  0 1  0 0 A= 0 0  0  0 0 

0 0 0 0 0 0 0 −0.0152 0.0185   0.0245 0.0262  0 0      0 0 0 0 0 0 0 0.0389 0.0436   0.0686 0.0045  0 0  0.0219 0.0425  0 0 0 0 0 0 0 0 0 0.0666 0.0696       1 0 0 0 0 0 0 0.0881 0.1080   0.0199 0.0710  0 0  0.0820 0.2317  0 0 0 1 0 0 0 0 0 0.0959 0.0784   , B=  , CT =   0 0 1 0 0 0 0 0.0508 0.0701   0.1615 0.1233  0 0  0.1762 0.0941  0 0 0 0 0 1 0 0 0 −0.0171 0.0319        0.3274 0.1871  0 0 0 0 0 0 1 0 0 0.1294 0.0158       0 0 0 0 0 1 0 −0.0052 0.0145   0.0857 0.2576  1 0  −0.0801 −0.0413 0 1 0 0 0 0 0 0 1 0.0699 0.0038     

4.2 Inputs Outputs and Error Signals The next figures represents: The real and the estimated outputs (Refoulement Pressure/Temperature) signals, the control (Aspiration Temperature/Pressure) signals and the error between the real and the estimated (Refoulement Pressure/Temperature) signals.

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B. Nail et al.: Parametric Identification of Multivariable Industrial System Using Left Matrix Fraction Description

Refoulement Pressure Kg/cm2

120 100 80 60

REAL MLS

40 20

400 Hours 0 0

200

400

600

Time(2*h) Figure. 1 Outputs Signals of Real and Estimate Refoulement Pressure

Refoulement Temperature C°

140 120 100 80 REAL

60

MLS

40 20

400 Hours 0 0

200 400 Time(2*h)

600

Figure 2Outputs Signals of Real and Estimate Refoulement Temperature

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Aspiration Temperature C°

70 65 60 55 50

400 Hours 45 0

200

400

600

Time(2*h) Figure 3 Input Signal of Aspiration Temperature

Aspiration Pressure Kg/cm2

70 65 60 55 50

400 Hours 45 40 0

200

400

600

Time(2*h) Figure 4 Input Signal of Aspiration Pressure

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B. Nail et al.: Parametric Identification of Multivariable Industrial System Using Left Matrix Fraction Description

4

Error

2

0

-2

-4 0

200

400

600

Time(s) Figure 5Error Signal of Real and Estimate Refoulement Pressure

4 3

400 Hours

Error

2 1 0 -1 -2 -3 0

200

400

600

Time(2*h)

Figure 6 Error Signal of Real and Estimate Refoulement Temperature

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6. CONCLUSIONS

To the best of our knowledge, this is the first application of theory of Left Matrix Fraction Description (LMFD) inidentification of large scale parametric model of Gas TurbineTurboCompressor System based on the real inputs outputsdata, the choice of MIMO least square (MLS) estimator isnot arbitrarily it was selected for many raisons among themthe reliability of this estimator (unbiased estimator) comparedwith the recursive estimators give the same result , the simplestmathematical structure for implementation and the main one isthe richest of data in frequency, this last one has facilitated usthe choice of the estimator. For the validation of the obtaineddynamical model, we restrict ourselves to use two criteria (AICand FPE) dedicated for the choice of the best model orderand with no Pole/Zero cancellation (left coprime). The furtherworks have been dedicated to applied many advanced controltheory on this dynamical model.

REFERENCES

[1] AbdelhafidBenyounes, Ahmed Hafaifa&MouloudGuemana, Gas turbinemodelling based on fuzzy clustering algorithm using experimentaldata. Journal of Applied Artificial Intelligence, Taylor & Francis, 2016,vol. no., pp. [2] AhmedHafaifa, GuemanaMouloud, and BelhadefRachid, Fuzzy Modelingand Control of Centrifugal Compressor Used in Gas PipelinesSystems. Multiphysics Modelling and Simulation for Systems Design andMonitoring, Book Chapter, Applied Condition Monitoring, 2015, vol. 2,pp.379-389. [3] AbdelhafidBenyounes, Ahmed Hafaifa and MouloudGuemana, Gasturbine parameters modelling based on fuzzy logic and artificial neuralnetworks: Solar TITAN 130N investigation. 5me Symposium de lAssociationAlgrienne de lIndustrie du Gaz AIG Sonatrach, 16-17 Fvrier2016, Alger. [4] F. Laaouad, H. Fekhar, A. HafaifaModeling Centrifugal CompressorUsing Fuzzy Logic. XVI. International Conference on Thermodynamic.Saint Petersbourg. Russie. Juillet 2002. [5] F. Laaouad, H. Fekhar, A. Hafaifa Adaptive control of centrifugalcompressors. 15th International Congress of Chemical and ProcessEngineering. Prague. Tchequie. August 2002. [6] Akroum, M., Hariche K., An optimal instrumental variable identificationmethod for LMFD models, Studies in informatics and control, Vol 17,No.4, December 2008. [7] Akroum, M., and Hariche K., Extending the SRIV algorithm to LMFDmodels, Journal of electrical engineering and technology, Korean Instituteof electrical engineers, Vol. 4, No.1, march 2009. [8] Akroum, M., and Hariche K., A matrix fraction description-basedidentification algorithm for MIMO ARMAX models, submitted to Asianjournal of control, 2008 [9] Al-Muthairi, N.F., S.Bingulac and M.Zribi, Identification of discretetimeMIMO systems using a class of observable canonical-form, IEEProc-Control Theory, 2002. [10] de Callafon, R.A., de Roover, D. & Van den Hof, P.M.J. 1996. Multivariableleast squares frequency domain identification using polynomialmatrix fraction descriptions. Proceedings of the 35th IEEE, Decisionand Control 2: 2030-2035 [11] Akroum, M., and Hariche K., Right matrix fraction description modelsidentification using constrained PEM, Kuwait J. Sci. & Eng.37 (1B) pp.69-88, 2010. [12] Yucai Zhu and Ton Back., Identification of Multivariable IndustrialProcesses Springer-Verlag London 1993. [13] Bastogne, T. et al., Multivariable identification of winding process bysubspace methods for a tension control, Control engineering practice ,Vol. 6, No.9, pp. 1077-1088, 1998. [14] Bastogne, T. et al., Multivariable identification of winding process bysubspace methods for a tension control, Control engineering practice,Vol. 6, No.9, pp. 1077-1088, 1998. [15] R. Johansson, Multivariable system identification via continued-fractionapproximation, IEEE Transactions on Automatic Control, 40(3), 507-512, 1995. [16] R. Guidorzi, Canonical structures in the identification of multivariablesystems, Automatica, pages 361374,1975. [17] T. Kailath, W. Li, Linear Systems,Prentice Hall, 1980. [18] L. S. Shieh and Y. T. Tsay, Transformation of a class of multivariablecontrol systems to Block companion forms,IEEETruns. Autonwt. Control27: 199-203 (1982). [19] L. S. Shieh and Y. T. Tsay, Block modal matrices and their applicationsto multivariable control systems,IEE Proc. D Control Theory Appl. 2:41-48(1982). [20] C. T. Chen, Linear System Theory and Design,Holt, Reinhart andWinston, 1984.

229

B. Nail et al.: Parametric Identification of Multivariable Industrial System Using Left Matrix Fraction Description

[21] Belkacem Bekhiti, A. Dahimene, B. Nail, K. Hariche, and A.Hamadouche, On Block Roots of Matrix Polynomials Based MIMO Control System Design , 4-th IEEE International Conference on Electrical Engineering, ICEE, Boumerdes (2015). [22] Ljung, Lennart .Torsten Soderstrom, Theory and practice of recursive identification, MIT press Cambridg, Massachusetts, London, England1987. [23] Ljung, L., Glad, T., Modeling of dynamic systems (Prentice Hall), 1994. [24] Ljung, L., System Identification : Theory for the user (Prentice Hall ),1999. [25] Ljung, L., System Identification toolbox for Matlab, Version, 6.1.2(www.mathworks.com), 2005. [26] I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, 1982. [27] J. E. Dennis, J. F. Traub, R. P. Weber, The Algebraic Theory of Matrix Polynomials, SIAM J. Numer. Analy., vol 13, pp.831-845, 1976. [28] I. Gohberg, P. Lancaster, L. Rodman, Spectral Analysis of Matrix Polynomials: I. Canonical Forms and Divisors, Linear Algebra Appl.,20, pp. 1-44, 1978. [29] I. Gohberg, P. Lancaster, L. Rodman, Spectral Analysis of Matrix Polynomials:II.Theresolvent Form and Spectral Divisors, Linear Algebra Appl., 21, 1978. [30] Bachir Nail, Abdellah Kouzou, Ahmed Hafaifa, Belkacem Bekhiti, Stabilizing Linear Multivariable Gas Turbine Model Via Sliding Mode Control, International Conference on Engineering, Technologies andSystems TECHSYS 2016, Technical University Sofia, Plovdiv branch26-28 May, Plovdiv,Bulgaria (2016).

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