Development of Model Reduction using Stability Equation and Cauer Continued Fraction Method

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1 Intenational Jounal of Electical and Compute Engineeing. ISSN Volume 5, Numbe (03), pp. -7 Intenational Reeach Publication Houe Development of Model Reduction uing Stability Equation and Caue Continued Faction Method Haenda Singh and V.R. Singh Reeach Schola, Depatment of Electical Engineeing, Mewa Univeity, Ganga, Chittogah (Rajathan) Diecto, PDM College of Engineeing, Bahadugah (Hayana) Abtact A mixed method i peented fo ode eduction of linea time invaiant ytem fo ingle input ingle output ytem. Numeato of polynomial i deived fom Caue econd fom of continued faction and educed denominato i obtained fom tability equation method. Reduced ode model etain the teady tate value and tability of oiginal ytem. Keywod: - Model eduction, Caue econd fom (continued faction expanion), Stability equation method. I.INTRODUCTION The deign and analyi of high ode ytem become complex and tediou. Numeou method have been developed fo appoximation of oiginal ytem of linea time invaiant ytem fo ingle input and ingle output (SISO).Routh appoximation method i good fo tability and implicity () but it i found that thi method i unable to poduce good educed model..seveal method have been popoed baed on pade appoximation, () continued faction expanion (3-5) and eo minimization technique (6-7) and time moment matching etc.many pape have been eviewed. Thee method contain advantage and diadvantage. Continued faction method i ued a poweful tool fo educing the oiginal ytem. Seveal method have been made in model eduction a pade appoximation, Routh appoximation (9).Thi pape deal with both caue econd fom of continued faction and tability equation method..pade appoximant method i one of educed technique baed on matching of time moment of educed ode model and oiginal ytem. thi technique i vey imple in computation and fitting of initial time moment

2 Haenda Singh and V.R. Singh and teady tate value of output of educed ode model and oiginal ytem in fom of α i t i.a known dawback of thi method i untable educed ode model aie fom oiginal table ytem, wa., impoved by uing thi method. Pal and Pathathy and Jayaimha (4) combine thi method with continued faction method to etail tability in educed ode model. II. Method Ued. Statement of Appoximation Method. Suppoe nth ode SISO ytem i given by tanfe function N( ) B B S B S... B S H() = D( ) B B S B S... B S n 3 n n 3 n H() = n j n j B j B j j j () Whee B,B B n and B,B..B n+ ae contant. Reduced ode model i given by N ( ) D D D... R( ) D ( ) D D D R( ) j j D j D j j j (). Ode Reduction Method. Thi method conit of two type Step Fit numeato of educed model i fomulated in fom of continued faction Expanion () H ( ) = h h / h3 h4 / (3)

3 Development of Model Reduction uing Stability Equation 3 The quotient h i (i =,, 3 ) ae detemined by uing Routh algoithm (). B i,j = B i, j + h i B i, j + i = 3, 4.. j =, h i Bi, j, i=,, Bi j B i+,j 0 (4) Eqn. () give the econd caue econd fom of continued faction expanion of oiginal ytem of H() and the fit quotient ae etained and tuncated. The Continued faction i inveted to obtain the th ode of educed ode model R(). Step Detemination of the denominato coefficient of educed ode model:- Fo table oiginal ytem H(), denominato of H() i epaated in even pat and odd pat, a tability equation n De ( ) c ( ) (5) 00 i zi n Do( ) c ( ) 0 i pi whee n and n ae intege pat of k/ and k / epectively and z < p < z < p by neglecting facto with lage magnitude of z i and p i in eqn(5).stability equation fo th ode of educed ode model i obtained. a n3 De ( ) c00 ( ) z i n4 D ( ) c ( ) o 0 i pi i (6) Whee n 3 and n 4 ae intege pat of / and

4 4 Haenda Singh and V.R. Singh / / D ( ) c ( ) c ( ) (7) 00 0 i zi i pi Combing thee even and odd pat of educed ode denominato and ecipocating, we get D () = D e () + D o () D () i obtained and i given by a D ( ) j D, j (8) j Coefficient D,j in eqn. (8) ae matched with h i in eqn. (4) to detemine educed ode numeato N (), by applying the following evee Routh algoithm B i, j B i, j h i B, ( B, B ) i j i j i, j i, =, ( ) j =, ( ) &B, + = The educed ode model (ROM) i obtained a N ( ) R( ) D ( ) Step 4:- Thee i teady tate eo between the output of oiginal and educed ode ytem. To avoid teady tate eo we match the teady tate epone by the following elationhip to obtain coection facto k a contant. B B D k (9) D III. Reult: - Numeical Analyi. Let tanfe function of 5 th ode ytem (4) i given a H ( ) (0)

5 Development of Model Reduction uing Stability Equation 5 fom above oiginal tanfe function, econd ode educed model i to be obtained Step :- The quotient h i fo i =,, 3. ae detemined uing Routh algoithm a h = 0.56, h =.9 h 3 = 0.87 h 4 =.7 Step :- Sepaating the denominato of the above high ode ytem (HOS) in even and odd pat, we get tability equation a D e () = ( )( ) D o () = ( )( ) by dicading the facto with lage magnitude of z i and p i in D e () and D o () epectively. The tability equation fo educed model of econd ode i given by De ( ) 40( ) & Do ( ) Combing thee educed tability equation, we get denominato of educed nd ode model a D () = Step 3:- By matching the quotient of h i of caue econd fom with the coefficient of educed denominato and uing evee Routh algoithm to get N = The tanfe function fo educed ode model (ROM) with nd be expeed a R ( ) ode can

6 6 Haenda Singh and V.R. Singh Step 4:- In thi numeical ection, thee i no teady tate eo between tep epone of oiginal ode and educed model, hence k = and the final model eduction emain unchanged. The oiginal and educed ode model (ROM) i compaed by tep epone plot hown in fig.. Figue IV. Dicuion: In popoed model eduction method algoithm ued which combine to take both advantage of tability equation method to deive the denominato of polynomial and the caue econd fom to detemine the numeato of polynomial.it i obeved that popoed method peeve the teady tate and tability in educed ode model. Thi algoithm ha been implemented in metlab 7.0. Steady tate and tability avoid any eo in initial and final value of epone of oiginal and educed ode ytem. Thi method i extended fo multivaiable ytem. Thee method ae eaily contibution to model eduction method due to flexibility of algoithm. Thi method i cuioity method to develop educed ode model fo appoximation of oiginal ytem. Refeence [] Hutton M F and Fiedland Routh appoximation fo educed ode of linea time invaiant ytem. IEEE 975, AC-0,pp

7 Development of Model Reduction uing Stability Equation 7 [] Y Shamah table educed ode model uing pade type appoximation. IEEE TRANS.on automatic contol. Vol 9, 974, pp [3] Chuang SC application of continued faction method fo modeling tanfe function to give moe accuate initial tanient epone. Electonic, Lett., 970, 6, pp [4] T, N LUCAS Continued faction algoithm fo biaed model eduction. Electonic Lette, 9 th June 983, Vol 9, No [5] R Paad and J Pal A table eduction of linea ytem by continued faction. J Int.Eng.India, IE (I) Jounal-EL, Vol 7, pp3-6 Octobe 99. [6] S.Mukhejee and R.N Miha ode eduction of linea ytem uing an eo minimization technique. Jounal of Fanklin Int, Vol.33, No,pp. 3-3,987. [7] K. Rameh, A. Nimal Kuma and G. guuamy ode eduction by eo minimization technique. Poceeding of the 008 intenational confeence on computing, communication and netwok(icccn 008), /08,008IEEE [8] M F Hutton, B. fiendland ROUTH appoximation fo educing ode of linea time vaying ytem IEEE tan.automatic contol,vol-44,no 9,pp [9] T C Chen, CY Chang and KW Han Stable educed ode Pade appoximant uing tability equation method. Electonic Lette.Vol 6,No. 9,pp [0] Jayanta pal ytem eduction by a mixed method.tanaction on Automatic contol, vol.ac-5, No 5, Octobe 980. [] J. Pal Impoved pade appoximant uing tability equation method Electonic Lette, vol 9, No, pp may, 983. [] Chen C F and Sheih LS Novel appoach to linea model implification. int.j.contol, 968, No 8, pp [3] R. Pathaathy and K.N Jayaimha ytem eduction uing tability equation and modified caue continued faction method. Poceeding IEEE, vol 70, No0,pp Octobe98. [4] Luca. T N Biaed model eduction by facto diviion method Electonic Lette, 984, 0 pp [5] Bai-Wu Wan Linea model eduction uing Mihailov citeion and Pade appoximation technique Int.J.Contol,Vol. 33,pp ,98.

8 8 Haenda Singh and V.R. Singh