ORDER REDUCTION USING POLE CLUSTERING AND FACTOR DIVISION METHOD

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1 Author Nme et. l. / Interntionl Journl of New Technologies in Science nd Engineering Vol., Issue., 7, ISSN 9-78 ORDER REDUCTION USING POLE CLUSTERING AND FACTOR DIVISION METHOD A Chinn Nidu* G Dileep** G Jyothirmi*** M S Venkt Lkshmi**** Deprtment of EEE Avnthi Institute of Engineering & Technology, Nrsiptnm niduc96@gmil.com *, jyothismiles7@gmil.com **, dileepgorli9@gmil.com ***, msv.lkshmi@gmil.com **** Astrct: In this pper method is proposed for finding stle reduced order models of single-input-single-output lrge scle systems using Fctor division lgorithm nd the clustering technique. The denomintor polynomil of the reduced order model with respect to originl model is determined y forming the clusters of the numertor polynomil with respect to originl model re otined y using the Fctor division lgorithm. The mixed methods re simple nd gurntee the stility of the reduced model if the originl system is stle. The methodology of the proposed methods illustrted with the help of exmples from literture Keywords: Pole Clustering, Order reduction, Fctor Division, Trnsfer Function I. INTRODUCTION A gret numer of prolems re rought out y the present dy technology nd societl nd environmentl process which re highly complex nd lrge in dimensions nd stochstic in nture. There hs een no ccepted definition for wht constitute lrge scle system. Mny viewpoints hve een presented on this issue. One view point hs een tht system is considered lrge scle if it cn e decoupled or prtitioned in to numer of interconnected susystems or smll scle systems for either computtionl or prcticl resons. Another view points is tht system is lrge scle when its dimensions re so lrge tht conventionl techniques of modeling, nlysis, control, design nd computtionl fil to give resonle solutions with resonle computtionl efforts In other words system is lrge when it requires more thn one controller. Since the erly 95s,when clssicl control theory ws eing estlished, engineers hve devised severl procedures, oth within the clssicl nd modern control contexts, which nlyze or design given system. These procedures cn e summrized s follows.. Modelling procedures which consist of differentil equtions, input-output trnsfer functions nd stte spce formultions.. Behviorl procedures of systems such s controllility, oservility nd stility tests nd ppliction of such criteri s Routh-Hurwitz, Nyquist, Lypunov s second method etc.,. Control procedures such s series compenstion, pole plcement, optiml control etc., The underlying ssumption for ll such control nd system procedures hs een centrlity i.e., ll the clcultions sed upon system informtion nd the informtion itself re loclized t given center, very often geogrphicl position. Necessity of Model Reduction: In prcticl situtions most of the systems re of very high order so, their exct nlysis nd design ecome oth tedious nd costly. In order to perform simultion nlysis or control design on those higher order models one will fce mny difficulties. It is desirle to represent those physicl systems with reduced order model tht will resemle the originl model in time nd frequency domin. Aville 59

2 Author Nme et. l. / Interntionl Journl of New Technologies in Science nd Engineering Vol., Issue., 7, ISSN 9-78 II. CONVENTIONAL TYPE OF REDUCTION METHODS These re mny methods ville in literture for the reduction of higher order liner continuous time systems. The numertor nd denomintor cn e solved y two different method. Some of the fmilir methods ville for the reduction of higher order continuous time system nd studies for the thesis work re Aim of this Work:. Moment mtching method,. Continued frction method,. Stility eqution method,. Routh pproximtion method, 5. Interpoltion method, 6. Pde pproximtion method. To study some importnt nd recently developed model reduction technique ville in literture for reduction of high order SISO continuous time system. To consider nd verify the some of the existing methods of model reduction techniques ville in the literture for the high order SISO continuous time systems. A mixed method is considered for reduction of higher order SISO time system using pde pproximtion nd stndrd chrcteristics of originl higher order system which overcome the drwck of existing methods ville in the literture. III. MODEL REDUCTION METHODS FOR HIGHER ORDER CONTINOIUS TIME SYSTEMS. ROUTH APPROXIMATION METHOD (RAM): REDUCTION PROCEDURE: Consider the trnsfer function of nth order originl systems s: G(s) The ove eqution cn e written in the following cnonicl form G(s)= G(s)= Where i=,.nnd, k=,.n re determined y the continued frction Aville 6

3 Author Nme et. l. / Interntionl Journl of New Technologies in Science nd Engineering Vol., Issue., 7, ISSN 9-78 Aville 6 And Equtions re clled lph-et expnsions of G(s). Prmeters cn e otined y the following tles. -Tle: Tle: The lph prmeters re defined s i =, i=,,..n The et prmeters re defined s

4 Author Nme et. l. / Interntionl Journl of New Technologies in Science nd Engineering Vol., Issue., 7, ISSN 9-78 i =, i=,,.. The numertor nd denomintor of the k th order reduced model follows. Reduced Order Numertor: ( ( s)) R k cn e expressed in s re s The generl expression for the numertor is Where k=,,.nd Reduced Order Denomintor: Where k=,,. And The reduced model preserves high frequency chrcteristics nd for control ppliction it is preferle to use reciprocl trnsfer function defined y G(s)= G Algorithm: Determine the reciprocl trnsformtion of the given originl system G ˆ( s ).. Construct α-β tles corresponding to G ˆ ( s ) from the tle- nd tle- respectivel. Find out numertor nd denomintor for kth order reduced model y using equtions., Where k=,, nd. Constitute reduced order model s. Any reciprocl trnsformtion on Rk(s) nd to otin Rk(s) NUMERICAL EXAMPLE: Replce y Aville 6

5 Author Nme et. l. / Interntionl Journl of New Technologies in Science nd Engineering Vol., Issue., 7, ISSN 9-78 From method, we cn find. MIXED MATHEMATICAL METHOD: REDUCTION PROCEDURE: Numertor is solved y using the pde pproximtion nd stndrd chrcteristics of higher order eqution. Consider function From this equtions we get c,c,c,... vlues. Reducing the denomintor y utilizng the chrcteristics of the system, like dmping rtio (ξ), undmped nturl frequency of oscilltions (ωn) etc. For n periodic or lmost periodic system, ξ =.99, numer of oscilltions efore the system settles = Since, ωn= /ξ*ts Reduced denomintor: D(s) = s + ξωns + ωn By using the reduced denomintor coefficients,,. We re clculting, vlues The normlized reduced order G(s)= Aville 6

6 Author Nme et. l. / Interntionl Journl of New Technologies in Science nd Engineering Vol., Issue., 7, ISSN 9-78 NUMERICAL EXAMPLE: Form the procedure we get, C= nd C=-.6 The reduced system =.5 Step Response.5 Amplitude System: sys Settling Time (sec): Time (sec) Figure : Step response of originl higher order system In these ξ =.99 From the ove figure settling time is Ts=.8, ω n=88 The reduced denomintor is here =.7, =6596 The reduced order trnsfer function is POLE CLUSTERING METHOD: Let the trnsfer function of the originl high order liner dynmic SISO system of order n e: nd Let the corresponding rth order reduced model is synthesized s: Aville 6

7 Author Nme et. l. / Interntionl Journl of New Technologies in Science nd Engineering Vol., Issue., 7, ISSN 9-78 = Further, the method consists of the following steps. Step : Determintion of the reduced order denomintor polynomil with n improved pole clustering technique: Clculte the n numer of poles from the given higher order system denomintor polynomil. The numer of cluster centres to e clculted is equl to the order of the reduced system. The poles re distriuted in to the cluster centre for the clcultion such tht none of the repeted poles present in the sme cluster centre. Minimum numer of poles distriuted per ech cluster centre is t lest one. There is no limittion for the mximum numer poles per cluster centre. Let k numer of poles ville in cluster centre:. The poles re rrnged in mnner such tht. The cluster center for the reduced order model cn e otined y using the following procedure.. Let k numer of poles ville re.. Set L=;. Set L=L+. Set Pole Cluster Centre 5. Check for L=K. if yes, then the finl cluster center is nd termintes the process. Otherwise proceed on to next step. 6. Check for L=K. if no, then go to step5. Otherwise go the next step. 7. Finl cluster center is On clculting the cluster center vlues, we hve following s in. Cse : All the denomintor poles re rel: The corresponding reduced order denomintor polynomil cn e otined s, Where re the improved cluster vlues required to otin the reduced order denomintor polynomil of order r Cse : All the poles re complex: Let t = ( ) pirs of complex conjugte poles in L th cluster e, [( Aville 65

8 Where, Author Nme et. l. / Interntionl Journl of New Technologies in Science nd Engineering Vol., Issue., 7, ISSN 9-78.pply the proposed lgorithm individully for rel nd imginry prts to otin the respective improved cluster centers. The improved cluster is the form of.where, is the improved pole cluster vlues otined for rel nd imginry prts respectively. The corresponding reduced order denomintor polynomil cn e otined s, Cse : ).Where j=r If some poles re rel nd some poles re complex in nture,pplying n improved clustering lgorithm seprtely for rel nd complex terms.finlly otined improved cluster centers re comined together to get the reduced order denomintor polynomil. FACTOR DIVISION METHOD: Determintion of the numertor of k th order reduced model using fctor division lgorithm. After otining the reduced denomintor, the numertor of the reduced model is determined s follows. Where N(s)= is reduced order denomintor There re two pproches for determining of numertor of reduced order model.. By performing the product of N8(s) nd Dk(s) s the first row of fctor division lgorithm nd D8(s) s the second row up to sk- terms re needed in oth rows.. By expressing N8(s)Dk(s)/D8(s) s N8(s)/[D8(s)/Dk(s)] nd using fctor division lgorithm twice ;the first time to find the term up to sk- in the expression of D8(s)/Dk(s) (i.e. put D8(s) in the first row nd Dk(s) in the second row, using only terms up to s^k-), nd second time with N8(s) in the first row nd expnsion [D8(s)/Dk(s)] in the second row. Therefore the numertor Nk(s) of the reduced order model will e the series expnsion of Aout s= up to term of order s^k-. This is esily otined y modifying the movement generting.which uses the fmilir routh recurrence formul to generte the third, fifth, seventh etc rows s Aville 66

9 Author Nme et. l. / Interntionl Journl of New Technologies in Science nd Engineering Vol., Issue., 7, ISSN Where... NUMERICAL EXAMPLE Therefore the numertor is given y the poles re: -, -, -, -, -5, -6, -7, -8 Let the nd order reduced model is required to e relized, for this purpose only two rel clusters re required. Let the first nd second cluster consists the poles (-, -, -, -) nd (-5,-6,-7,-8) respectively. The modified cluster centers re computed s: For (-,-,-,-) Set L=; C=-.9 Set L=L+, the Modified Cluster center is C=-. Set L=L+, the Modified Cluster center is C=-. Set L=L+, the Modified Cluster center is C=-.6(finl Cluster) For -5,-6,-7,-8) Set L=; C=-6. Set L=L+, the Modified Cluster center is C= Set L=L+, the Modified Cluster center is C=-5.7 Set L=L+, the Modified Cluster center is C=-5.(finl Cluster) The denomintor polynomil D(s) is otined s; Aville 67

10 Author Nme et. l. / Interntionl Journl of New Technologies in Science nd Engineering Vol., Issue., 7, ISSN 9-78 Determintion of the Numertor of Kth order reduced model using Fctor Division Algorithm Where Dk(s) is reduced order denomintor The Reduced Model is given s: IV. RESULTS Step Response Of Higher & Equivlent Reduced Order Systems for the following Exmple Let us consider higher order eqution s Reduction Method Reduced Order System Pole Clustering Method Routh Approximtion Method ( Mixed Mthemticl Method Tle : Equivlent Reduced Order Systems for Exmple Time Domin Specifictions Of Originl And Reduced Order Systems for Exmple Time Domin Originl Routh Approximtion Pole Mixed Specifictions System Method ( Clustering Method Pek mplitude Overshoot Pek time Settling time Rise time Finl vlue Tle : Comprison of Time Domin Specifictions of Higher & Reduced Order Systems for Exmple Aville 68

, Issue., 7, ISSN 9-78.5 Step Response.5 Amplitude.

System & Equivlent Reduced Order Systems for Exmple Figure : Vrition in

Different Methods Figure : Vrition in Pek Time with Different Methods

11 Author Nme et. l. / Interntionl Journl of New Technologies in Science nd Engineering Vol., Issue., 7, ISSN Step Response.5 Amplitude Time (sec) Figure : Comprison of Step Response of Higher Order System & Equivlent Reduced Order Systems for Exmple Figure : Vrition in Rise Time with Different Methods Figure : Vrition in Settling Time with Different Methods Figure : Vrition in Pek Time with Different Methods Figure 5: Vrition in Pek Amplitude with Different Methods Figure 6: Vrition in Pek Overshoot with Different Methods Aville 69

12 Author Nme et. l. / Interntionl Journl of New Technologies in Science nd Engineering Vol., Issue., 7, ISSN 9-78 From the ove Tles y compring time domin specifictions of reduced order systems with higher order system, definitely reduced order system hs more dvntges in terms sme finl vlue, less pek overshoot, less settling time, less pek mplitude nd fster rising time. Wheres in reduced order systems with different reduction techniques hving mixed dvntges. From the tle, oserved tht Routh Approximtion nd Mixed method hs less pek mplitude compred to remining methods. Pole Clustering method hs fster rise time when compred to originl nd remining methods. Pole Clustering method hs less settling time compred to remining methods ut hving higher pek overshoot when compred to other Method. So from the ove oservtions, concluded tht Pole Clustering Method hs mixed dvntges like less settling time, pek time nd rise time compred to ll the methods nd Originl higher order system. V. CONCLUSION From the ove results we re proposed n order reduction method for the liner single-input-single-output higher order systems. The determintion of the polynomil of the reduced model is done y using the pole clustering method while the numertor coefficients re computed y fctor division method. The merits of proposed method re stle, simplicity, efficient nd computer oriented. The proposed lgorithm hs een explined with n exmple tken from the literture. The step responses of the originl nd reduced system of second order re shown in the Fig (-6). A comprison of proposed method with the other well known order reduction methods in the literture is shown in the Tle from which we cn concluded tht proposed method is comprle in qulity. REFERENCES [] B.C.Moore, Principl component nlysis in liner systems: model reduction, IEEE [] Y. Shmsh, Singlevrile system vi modl methods nd Pde Approximtion, IEEE trnsctions, [] J. Pl, System reduction y mixed method, IEEE trnsctions, utomtic control, 98, vol. 5, issue 5, pp [] Y. Shmsh, The viility of nlyticl methods for the reduction of single vrile systems, IEEE proceedings, 98, vol. [5] S.A. Mrshll, The design of reduced order systems, interntionl journl of control, 98, vol., issue, pp [6] P.O. Gutmn, C.F. Mnnerfelt, P. Molnder, Contriutions to the model reduction prolem, IEEE trnsctions, utomtic control, 98, vol. 7, issue, pp [7] G. Prmr, S. Mukherjee, R. Prsd system reduction using fctor division lgorithm nd eigen spectrum nlysis interntionl journl of pplied mthemtics modelling, 7, vol., pp [8] T. C. Chen, C. Y. Chng nd K. W. Hn, Model reduction using stility eqution method nd the continued frction method interntionl journl of control, 98, vol., issue, pp. 8-9 [9] R. Prsd, S. P. Shrm nd A. K. Mittl, Liner model reduction using the dvntges of Mihilov criterion nd fctor division, j. inst. eng. Indi ie(i),, vol. 8, pp. 7-. [] R. Prsd, j. pl,. k. pnt, single vrile system reduction using model methods nd Pde type pproximtion, j. inst. eng. Indi pt el, 998, vol. 79, pp [] R. Prsd, Pde type model order reduction for single vrile system using Routh Approximtion, computers nd electricl engineering,, vol. 6, pp Aville 7

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response