Hammerstein Model Identification Based On Instrumental Variable and Least Square Methods

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1 Intenational Jounal of Emeging Tends & Technology in Compute Science (IJETTCS) Volume 2, Issue, Januay Febuay 23 ISSN Hammestein Model Identification Based On Instumental Vaiable and Least Squae Methods Dalvinde Mangal Assistant Pofesso, Electonics and Instumentation Depatment, TITS Bhiwani, MD Univesity Rohtak, Hayana, India. Abstact: In this pape the instumental vaiable and ecusive least squae algoithm fo identification of Hammestein model consisting of the cascade connection of the static non-lineaity followed by a LTI system, have been given. Hee the linea dynamic block is descibed by genealised othonomal basis and the static nonlineaity is fomulated by a elay with dead zone chaacteistics. To estimate the paametes, a pope switching function is incopoated. Theeafte, the ecusive least squae algoithm and Instument vaiable method has been used to obtain the optimal estimates of the paametes. It is shown that the convegence of the paametes was faste using instumental vaiable identification algoithm as compaed to least squae algoithm. Keywods: Relay with dead zone, Instumental vaiable, Othonomal basis, SVD, Hammestein model, Recusive least squae.. INTRODUCTION In the last decades a consideable amount of eseach has been deployed fo modelling, identification and contol of nonlinea systems which can be epesented by the blockoiented models [], i.e., by inte-connection of linea dynamic and nonlinea static subsystems. Among these models, two of the common model stuctues consist of a combination of two blocks esulting the Hammestein (nonlinea-linea) [2] and Wiene (linea-nonlinea) models [3, 4, 5]. These model stuctues have been used extensively to epesent nonlinea systems in a vaiety of pactical applications in the aeas of biological pocesses [2, 6, 7] signal pocessing, contol, communication and chemical pocesses. Diffeent appoaches have been used in the liteatue fo identification of Hammestein and Wiene models. In one appoach the taditional iteative algoithm has been poposed [8] whee the appopiate paameteization of the system pemits the pediction eo to be sepaately linea in each set of paametes chaacteizing the linea and nonlinea pats. The estimation is then caied out by minimizing altenatively with espect to each set of paametes, a quadatic citeion on the pediction eos. Late on the othe fom of algoithm has been intoduced based on least squae estimation (LS) and singula value decomposition (SVD) [9]. Hee the estimates ae consistent only fo the distubances being white noise o fo the noise-fee case which esults in numeical obustness unde weak assumptions on the pesistency of excitation of the inputs. In the ecent yeas thee has been a lot of eseach on how to intoduce a pioi infomation in the identification of black box LTI model stuctues. Choosing the poles of the bases close to the (appoximately known) system poles, the othonomal basis functions ae geneated []. It is shown that including a pioi knowledge of the system dynamics via the use of othonomal basis function, fo the epesentation of the linea pat of the Hammestein model, into the identification pocess [] has the advantage of educing the numbe of paametes to be. The othogonal functions can be consideed as genealizations of, e.g., the pulse functions, Laguee functions and Kautz functions [2], and give ise to an altenative seies expansion of ational tansfe functions. When the covaiates ae exogenous, the small-sample popeties of odinay least squae estimato can be deived in a staightfowad manne by calculating the moments of the estimato conditional on covaiates. When some of the covaiates ae endogenous, instumental vaiable estimation is implemented. So Instumental vaiable method (IVMs) can be seen as genealization of the least squae (LS) estimates. The main idea behind this is to modify the least squae estimate so that it becomes consistent fo an abitay distubance [3]. In this pape the compaative study of two algoithms, Instumental vaiable [4] and the ecusive least squae algoithms [5], fo paamete identification of nonlinea dynamic systems based on Hammestein models in the pesence of noise is given. 2. HAMMERSTEIN MODEL The Hammestein model consists of the cascade connection of a static nonlineaity followed by a LTI system as shown in figue. Figue Hammestein Model Volume 2, Issue Januay - Febuay 23 Page 44

2 Intenational Jounal of Emeging Tends & Technology in Compute Science (IJETTCS) Volume 2, Issue, Januay Febuay 23 ISSN whee epesents the model input, is model output without noise, is noisy model output, is intenal vaiable epesenting the output of nonlinea block and is noise. With (.7) the paamete vecto: 2. Desciption of static nonlinea block The static nonlinea block is epesented by elay nonlineaity with dead zone [6] chaacteistics in figue 2. The slopes and ae the paametes to be. Figue 2 Relay non-lineaity with dead zone. The output of the nonlinea block depends on the position of the input signal in egad to the points of discontinuities zeo. It is descibed in the following way: (.) A switching function is descibed by the following equation: the obsevation vecto: (.8) (.9) Afte having the bief mathematical model of non-linea and linea block of Hammestein system, the algoithm used fo estimation is discussed in next section. 3. HAMMERSTEIN MODEL ESTIMATION ALGORITHM Step-I a) Minimizing the quadatic citeion on the pediction eo given by: (.) b) The paametes of vecto ae using anyone of the following ecusive algoithms: Recusive least-squae algoithm [2] (.2) then the static nonlineaity output as: (.3) can be witten o (.) Instumental vaiable algoithm [7] 2.2 Fomulization of linea block of Hammestein system The linea block of the Hammestein system is epesented by: (.4) Hee defines the numbe of poles and gives the genealized othonomal basis function which can be epesented by the following equations. Step-II Fom the paamete vecto given as: (.2) (.5) Whee and epesents pole and its conjugate espectively. System output equation is: (.6) In matix fom it can be witten as: having (.3) and The paametes, and ae sepaated by using SVD of as follows: (.4) Volume 2, Issue Januay - Febuay 23 Page 45

3 Intenational Jounal of Emeging Tends & Technology in Compute Science (IJETTCS) Volume 2, Issue, Januay Febuay 23 ISSN Having dim = dim et dim = dim and the block is, in fact, the fist singula value of, the estimates can be witten as: h h (.5) (.6) Thus the paametes and discontinuities in nonlineaity ae. Algoithm given by Eq.. to.6 is implemented on Hammestein model given by Figue fo both the identification schemes. paametes SIMULATION RESULTS -5 The Hammestein model given by Figue using elay with dead zone non-lineaity chaacteistic is identified using Recusive Least-Squae and Instumental Vaiable algoithms. The linea dynamic block is modeled as: Figue 3a Estimated paametes by IV method fo linea block of Hammestein system.5 (.7) The paametes of the discontinuities fo all the nonlineaities ae taken as [2]: The additive noise hee: given by Eq. 4.9 is ewitten (.9) whee is the white Gaussian noise with vaiance., and is also the white noise with vaiance. is a andom vaiable and its distibution is unifom within (, ). The input to the non-linea block is given though the filte whose tansfe function is given by Eq..2 given as: paametes (.8) R R Figue 4 Estimated paametes by RLS method fo nonlinea block of Hammestein system (.2) h h Relay non-lineaity with dead zone 2 5 paametes The paametes of static non-linea block as well as dynamic linea block ae shown fom figue 3 to 4 and 3a to 4a espectively using IV and RLS algoithms descibed above. -5 R R2.5 paametes Figue 4a Estimated paametes by RLS method fo linea block of Hammestein system Figue 3 Estimated paametes by IV method fo nonlinea block of Hammestein system Volume 2, Issue Januay - Febuay 23 Page 46

4 Intenational Jounal of Emeging Tends & Technology in Compute Science (IJETTCS) Volume 2, Issue, Januay Febuay 23 ISSN Based on the esults fom figue 3 to 4 and 3a to 4a the values of the paametes, fo elay with dead zone non-lineaity as well as the linea block, given by IV and RLS methods shows that the convegence of the paametes is faste in case of IV identified model which ae quantitatively shown in Tables and 2 espectively. Wheeas the eos in the paametes fo static non-linea blocks and thei espective dynamic linea block ae gaphically shown in Figue. 5 and 6 espectively and it has been obseved that these eos ae vey small. Table : Eos in paametes of IV and RLS identification of non-linea block fo elay with dead zone non-lineaity Table 2: Eos in paametes of IV and RLS identification of linea block fo elay with dead zone non-lineaity IV RLS h h tue paamete tue paamete IV paamete paamete tue paamete tue paamete RLS paamete Figue 5 Eos in tue and values of paametes fo non-linea block estima ted paam ete R R Figue 6 Eos in tue and values of paametes fo linea block 5. CONCLUSION The least squae and instumental vaiable algoithm fo identification of Hammestein model consisting of the cascade connection of the static non-lineaity followed by a LTI system, have been analyzed with the help of gaphical esults and ba gaph fo the convegence of system paametes. It has been obseved that the convegence of paametes is faste fo instumental vaiable identification algoithm as compaed to ecusive least squae method. The eos in paametes of non-linea block R= and R2= ae minimum when the Hammestein system having elay with dead zone non-lineaity, is identified using Instumental Vaiable algoithm. Wheeas the eos in paametes of linea block, h= and h=, ae minimum when the Hammestein system paametes ae using Recusive Least Squae algoithm. So it is concluded that RLS and IV methods both ae equally good fo minimizing the eos in paametes of linea as well as non-linea block of Hammestein system. REFERENCES [] Peason R. and M. Pottmann, 2, Gay-box identification of black-oiented nonlinea models. Jounal of pocess Contol,, [2] Eskinat E., Johnson, S., Luyben, W., Use of Hammestein models in identification of nonlinea systems, AICHE Jounal, 37(2), pp , 99. [3] Geblicki W., 994, Nonpaametic identification of Wiene systems by othogonal seies. IEEE Tansaction on Automatic Contol, 39(), [4] Wigen T., 993, Recusive pediction eo identification using the nonlinea Wiene model. Automatica, 29(4), -25. Volume 2, Issue Januay - Febuay 23 Page 47

5 Intenational Jounal of Emeging Tends & Technology in Compute Science (IJETTCS) Volume 2, Issue, Januay Febuay 23 ISSN [5] Wigen T., 994, Convegence analysis of ecusive identification algoithms based on the nonlinea Wiene model. IEEE Tansaction on Automatic Contol, 39(), [6] Koenbeg M., 978, Identification of biological cascades of linea and static nonlinea systems. In Poceedings of the 6 th Midwest Symposium on Cicuit Theoy, [7] Fuzetti K., A. Palazoglu and K. McDonald, 997, Nonlinea model pedictive contol using Hammestein models. Jounal of Pocess Contol, 7(), 3-4. [8] Naenda K. and P. Gallman, 966, An iteative method fo the identification of nonlinea systems using a Hammestein model. IEEE Tansaction on Automatic Contol AC-, [9] Bai E.W., 998, An optimal two-stage identification algoithm fo Hammestein-Wiene nonlinea systems. Automatica, 34(3), [] Gómez J. C., 998, Analysis of dynamic system identification using ational othonomal bases. PhD Thesis, The Univesity of New Castle, Austalia, PS file at [] Nalbantoğlu V., Boko J., Balas G. and Gaspa P., 23, System identification with genealized othonomal basis functions: an application to flexible stuctues. Contol Engineeing Pactice,, [2] Pete S.C., Heubege, Paul M.J., Van Den Hof and Okko H. Bosga, 995, A genealized othonomal basis fo linea dynamical systems. IEEE Tansactions on Automatic Contol, 4(3), [3] Södestöm T. and Stoica P., 22, Instumental vaiable methods fo system identification. Cicuits Systems Signal Pocessing, 2(), -9. [4] Holge Boman and Ǻke Andesson, 996, Instumental vaiables and pediction eo like second ode ecusive algoithms. IEEE Tansactions, [5] Elleuch K. and Chaai A., Modeling and estimation of Hammestein system with peload nonlineaity. UCA Unit of Automatic Contol National Engineeing School of Sfax, B. P: 73, 338 Sfax, Tunisia. [6] Bai E.W., 22, Identification of linea systems with had input nonlineaities of known stuctue. Automatica, 34, [7] Boman, H., Andesson A, Instumental vaiable (IV) and pediction eo like second ode ecusive algoithms, IEEE tansactions, Depatment of Applied Electonics Chalmes, Univesity of Technology, S , Gotebog, Sweden, pp , 996. Volume 2, Issue Januay - Febuay 23 Page 48