Identification of Linear Partial Difference Equations with Constant Coefficients

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1 J. Basc. Appl. Sc. Res., 3(1)6-66, , TextRoad Publcaton ISSN Journal of Basc and Appled Scentfc Research Identfcaton of Lnear Partal Dfference Equatons wth Constant Coeffcents Mohsen Shaferad, Masoud Shafee*, Mehrdad Abed Electrcal Department, Amrkabr Unversty of Technology, Tehran, Iran ABSTRACT In ths paper, an algorthm based on the structural model s developed for parameter estmaton of a lnear two dmensonal partal dfference equaton. Identfcaton of two dmensonal partal dfference equatons paves the way for dentfyng equatons wth hgher dmensons. The proposed algorthm s based on two man parts, nstrumental varable method and pre-flterng technque. Based on the analyss results, 2DPIV algorthm s presented. Fnally, the performance of the proposed algorthm s evaluated by a smulaton example. KEYWORDS: Parameter Estmaton, Partal Dfference Equatons, Structural Model, Instrumental Varable, Transfer Functon. I. INTRODUCTION The parameter estmaton of two-dmensonal partal dfference equatons s an mportant problem that has applcatons n many branches of scence such control, envronmental scence, mage processng, aerospace, etc., [2], [3]. In recent years, there has been ncreasng nterest n the use of transfer functon (TF) models n dentfcaton problems. TF modelng s approprate for modelng of lnear systems. TF models have been wdely used for dentfcaton and parameter estmaton problems. There exst a number of approaches for the parameter estmaton of partal dfference equatons [4]. Durng the last decades, several methods have been proposed for parameter estmaton of ordnary dfference equatons, but parameter partal dfference equaton dentfcaton has not been studed as much as one dmensonal system dentfcaton []. The authors In [4] have been proposed some methods to determne the model order of two dmensonal ARMA that can be descrbed as a lnear partal dfference equaton. Hangbn et al. present a two dmensonal system dentfcaton scheme that makes use of ampltude estmaton [6]. There have been some other algorthms reported for the estmaton of parameters n two dmensonal systems, such as lnear predcton based methods [7] [8], least-squares (LS) methods [9] [11], and maxmum-lkelhood (ML) methods [1], [12] [3]. Instrumental Varable approaches to system dentfcaton have been developed snce 197s [13]. The IV approach to least squares parameter estmaton has ts foundatons n classcal statstcal estmaton theory, where t represents one approach to the problem of estmatng structural model parameters [14]. In the structural model, the basc relatonshp between the parameters s n the normal estmaton equaton form but the elements of varable vector are not exactly known and can only be observed n error. The frst applcaton of the IV method n the process dentfcaton was by Joseph et al. []. Although they dd not refer to t by name, Joseph et al. suggested an IV procedure for dentfyng the parameters of a process descrbed by an ordnary dfference equaton model. For dentfcaton of partal dfference equatons, the IV method has not been used so far. In ths paper, SRIV algorthm [16] s evaluated and extended to the two dmensonal extended algorthm for two dmensonal partal dfference equaton. Young et al. proposed SRIV method to estmate the parameters of system model n one dmensonal case [16]. The pre-flerng technque n our algorthm, presents a consstent estmaton that can be mnmum varance for whte nose. The order of partal dfference equaton s assumed to be known and the structure s fxed. Note that the paper does not attempt to determne the order of partal dfference equaton and the dentfcaton problem s to estmate the unknown parameter of system. Ths problem s known as parameter estmaton. The paper s organzed as follows. Secton II presents the modelng and problem formulaton needed by the proposed two dmensonal pre-fltered IV algorthms that s presented n secton III. In secton III, the general one dmensonal IV method s extended to two dmensonal IV method and then an algorthm to estmate the parameters of two dmensonal partal dfference equaton, usng the proposed two dmensonal IV method s presented. Smulaton results to support the effcency of the proposed algorthm are presented n secton IV. Fnally, we draw some conclusons. *Correspondng Author: Masoud Shafee, Electrcal Department, Amrkabr Unversty of Technology, Tehran, Iran Emal: mshafee@aut.ac.r 6

2 Shafee and Shaferad, 213 II. Problem Formulaton In ths secton a two dmensonal partal dfference equaton s formulated and then, descrbed n transfer functon (TF) form. It s assumed that the partal dfference equaton under estmaton s a lnear equaton wth constant coeffcents. The nput r and the output x are related by the followng lnear two dmensonal partal dfference equaton: α x(k, l ) = β r(k, l ) (1) where the constant coeffcents α s and β s n (1), are the parameters to be estmated. n, n are the hghest dfference order of x wth respect to the varables k and l, respectvely. Also m, m are the hghest dfference order of r wth respect to the varables k and l, respectvely. The set {(n, n ), (m, m )} s called as the order of equaton. Here, we suppose n max(m, m ) and n max(m, m ). In TF terms, the above lnear two dmensonal partal dfference equaton can be wrtten as the followng TF: x(z, z ) = (, ) (, ) r(z, z ) (2) where D(z, z ) = 1 + α z + + α z + α z + + α z z N(z, z ) = β + β z + + β z + β z + + β z z Operator z s partal dfference operator,.e. z z f(z, z ) s the two dmensonal z-transform of f(k, l ). The output of system, x, s corrupted by addtve measurement nose e. We consder the whte nose, n ths paper. The measurement y of x, can be wrtten as the followng observaton equaton: y(k, l) = x(k, l) + e(k, l) (3) where e(k, l) s the two dmensonal whte nose wth zero mean and fnte and constant varance. By consderng (2), the observaton equaton (3) can be wrtten as the followng two dmensonal TF Model: y(z, z ) = (, ) (, ) r(z, z ) + e(z, z ) (4) We consder the unknown parameter vector θ, n terms of the parameters n the TF polynomals D(z, z ) and N(z, z ) as: θ = α α α α β β β β β T () The partal dfference equaton (1) can be formulated as the followng vector form: x(k, l) = φ (k, l)θ (6) such that φ (k, l) = [ x(k, l 1)... x(k n, l n ) r(k, l)... r(k m, l m ) ] (7) The number of measured samples are N N,.e. (k, l) = (T, T ) such that (1 N, 1 N ) and, N. Now, the parameter estmaton problem posed by the TF model (4) s to estmate unknown parameter vector θ based on N N unformly sampled nput-output data set {r(t, T ); y(t, T )} such that (1 N, 1 N ) and, N. It should be noted that the order of the partal dfference equaton,.e. set {(n, n ), (m, m )} s known a pror. III. Parameter Estmaton Accordng to the modelng and formulatons stated above, n ths secton the parameter estmaton problem s analyzed. At frst, the data are pre-fltered by a sutable flter and then, by developng a two dmensonal IV algorthm, the unknown parameters are yelded. A. Data Pre-flterng In ths sub-secton, the predcton error mnmzaton (PEM) approach s used. Mnmzaton of a least squares crteron functon n ε, provdes the bass for optmal parameter estmaton. Under the Gaussan normalty assumptons on e(k, k ), a sutable error functon can be gven by, ε(z, z ) = y(z, z ) (, ) r(z (, ), z ) (8) whch can be wrtten as ε(z, z ) = (D(z (, ), z )y(z, z ) N(z, z )r(z, z )) (9) The flter p can be defned as 66

3 J. Basc. Appl. Sc. Res., 3(1)6-66, 213 p(z, z ) = (, ) Ths flter s nverse of characterstc equaton of system.e. 1 D(z, z ). Ths flter has a physcal meanng. Indeed, t attenuates all sgnals outsde the pass band of system, ncludng hgh frequency nose. By nsertng flter p nto brackets n (9) and pre-flterng y and r by p, (9) takes the followng form: ε(z, z ) = D(z, z )y (z, z ) N(z, z )r (z, z ) (11) The subscrpts p denote that the varable has been pre-fltered by flter p. As a result, the estmaton equaton can be wrtten as: y (k, l) = φ (k, l)θ + e(k, l) (12) where φ (k, l) = y (k, l 1)... y (k n, l n ) r (k, l)... r (k m, l m ) (13) B. Two Dmensonal IV Approach Now, we present the two dmensonal IV estmaton method. Ths method s extended based on the tradtonal one dmensonal IV approach. Gven the estmaton equaton (12), the two dmensonal IV normal estmaton equatons are obtaned as: φ (k, l)φ (k, l) θ φ (k, l)y (k, l) = (14) It should be noted agan that (k, l) = (T, T ). Optmzaton wth respect to θ yelds a two dmensonal IV soluton θ: θ = argmn θ φ (k, l)φ (k, l) θ φ (k, l)y (k, l) θ = φ (k, l)φ (k, l) φ (k, l)y (k, l) () In these equatons, φ s the IV vector defned as follows: φ (k, k ) = x (k, l 1)... x (k n, l n ) r (k, l)... r (k m, l m ) (16) such that x(z, z ) = (, ) r(z (, ), z ) (17) Tll now, two technques pre-flterng and IV method have been gven whch are the bases for the man algorthm. C. Two Dmensonal Pre-fltered IV (2DPIV) Algorthm In ths secton, we summarze the results as pre-fltered IV (PIV) algorthm to estmate the parameters of a partal dfference equaton. 2DPIV Algorthm: Step 1. Make an ntal estmaton by usng a dscrete two dmensonal estmaton algorthm or based on pror knowledge of system. Step 2. Generate the IV varables from (17) wth the ad of estmaton n last teraton (for frst teraton, use ntal estmaton n Step 1). Step 3. Pre-flter the nput and output sgnals accordng to (16). Step 4. Estmate the system parameters accordng to sub-secton B. Step. Check the convergence condton. If t s not satsfed, go to Step 2. End Let us conclude ths secton wth some remarks: Remark.1: The convergence condton n Step can be descrbed as: The covarance matrx P assocated wth estmaton θ, converges to zero, as the number of samples goes nfnty. The covarance matrx P s defned as follow: P(k, l) = φ (k, l)φ (k, l) (18) Remark.2: Applyng IV method and pre-flterng technque n mult dmensonal parameter estmaton s a new dea and have not been fully nvestgated. Remark.3: A necessary condton for dentfably of system s stablty of the TF model,.e. the roots of D(z, z ) should le nsde the unt crcle. Ths condton must be observed n Step 1 of the 2DPIV algorthm, too. IV. Smulaton Results In the followng example, the 2DPIV algorthm s appled to 2 samples of smulated data generated by the followng TF model: (1) 67

4 Shafee and Shaferad, 213 x(k, l) +.6x(k, l ) = 3r(k, l) The nput sgnal r s zero mean whte nose wth varance 1. Also e s zero mean whte nose wth varance.. Here, N = and N =. The observaton y s shown n fgure y fgure 1. Observaton of system In above partal dfference equaton, α =.6, β = 3. Therefore the system parameters vector () are θ = [α β ] = [.6 3 ]. By applyng 2DPIV algorthm and consderng ntal estmaton as θ = [1 ], the estmaton of system parameters vector after 2 teratons has been obtaned as θ = a, b, = [ ]. The ratonal error between θ and ts real value θ s about [.3%.6% ]. The error between the output x and ts estmaton x s shown n fgure 2. The maxmum error s.73. It s obvous that the estmaton error s too low error of x fgure Dfference between output x and ts estmaton x 68

5 J. Basc. Appl. Sc. Res., 3(1)6-66, 213 Now we consder the output of system wth two dmensonal step nput,.e. r(k, l) = 1. Fgure 3 shows the estmated output of system and the error between the real and estmated step response of system s shown n fgure estmated step response fgure 3. 4 Estmated output x x error of estmaton n step response fgure Dfference between the real and estmated step response of system From fgure 4 we can see that the amount of error s very low. 69

6 Shafee and Shaferad, 213 V. Concluson In ths paper, pre-fltered IV (PIV) algorthm was presented to estmate the parameters of two dmensonal partal dfference equatons. The algorthm was gven based on two technques: pre-flterng and IV method. An example was put worth to demonstrate the effectveness of the proposed approach. There are many new topcs n dentfcaton the partal dfference equaton that can be consdered as future works such as model order determnaton and consderaton the colored nose n analyss. REFERENCES [1] J. L and P. Stoca, An Adaptve Flterng Approach Spectral Estmaton and SAR Imagng, IEEE Trans. Sgnal Processng, vol. 44, no. 6, pp , [2] H. Kaufman, J. W. Woods, S. Dravda and M. Tekalp, Estmaton and Identfcaton of Two-Dmensonal Images, IEEE Trans. Automatc Control, vol. AC-28, no. 7, pp. 7-76, [3] K. B. Eom, 2-D movng average models for texture synthess and analyss, IEEE Trans. Image Processng, vol. 7, pp , [4] M. S. Sadabad, M. Shafee and M. Karrar, Two-Dmensonal ARMA Model Order Determnaton, ISA Transactons-Elsever, vol. 48, no. 3, pp , 29. [] G. A. Perez and J. M. V. Fernandez, Prewhtenng-based estmaton n partal lnear regresson models: a comparatve study, REVSTAT: Statstcal Journal, vol. 7, pp. 37-4, 29. [6] H. L, W. Sun, P. Stoca and J. L, Two-Dmensonal System Identfcaton Usng Ampltude Estmaton, IEEE Sgnal Processng Letters, vol. 9, no. 2, 22. [7] D. E. Dudgeon and R. M. Mersereau, Multdmensonal Dgtal Sgnal Processng, Englewood Clffs, NJ: Prentce- Hall, [8] S. R. Parker and A. H. Kayran, Lattce parameter autoregressve modelng of two-dmensonal felds Part I: The quarter-plane case, IEEE Trans. Acoust., Speech, Sgnal Processng, vol. ASSP-32, pp , [9] R. L. Kashyap, R. Chellappa, and A. Khotanzad, Texture classfcaton usng features derved from random feld models, Pattern Recognt. Lett., vol. 1, no. 1, pp. 43-, [1] R. L. Kashyap and R. Chellappa, Estmaton and choce of neghbors n spatal-nteracton models of mages, IEEE Trans. Inform. Theory, vol. IT-29, pp. 8-72, [11] R. Chellappa and S. Chatteree, Classfcaton of texture usng Gaussan Markov random felds, IEEE Trans. Acoust, Speech, Sgnal Processng, vol. ASSP-33, pp , 198. [12] G. Sharma and R. Chellappa, Two-dmensonal spectrum estmaton usng non causal autoregressve models, IEEE Trans. Inform. Theory, vol. IT-32, pp , [13] R. E. Kopp and R. J. Ortord, Lnear Regresson Appled to System Identfcaton for Adaptve Control systems, Aaa Journal 1, pp , [14] J. Durbn, Errors n Varables, Rev. Int. Statst. Inst. vol. 22, pp , 194. [] P. Joseph, J. Lews and J. Tou, Plant Identfcaton n the Presence of Dsturbances and Applcaton to Dgtal Adaptve Systems, Aee Trans, [16] P. C. Young and A. J. Jakeman, Refned nstrumental varable methods of tme-seres analyss: Part I, SISO systems, Internatonal Journal of Control, vol.29, pp. 1-3,