Mixed Method of Model Reduction for Uncertain Systems

Transcription

1 SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol 4 No June Mixed Method of Model Reduction for Uncertain Sytem N Selvaganean Abtract: A mixed method for reducing a higher order uncertain ytem to a table reduced order one i propoed Interval arithmetic i ued to contruct a generalized Routh table for determining the denominator polynomial of the reduced ytem The reduced numerator polynomial i obtained uing factor diviion method and the teady tate error i minimized uing gain correction factor The propoed method i illutrated uing a numerical example Keyword: Model Reduction Uncertain ytem Factor diviion Gain correction factor Introduction The analyi and deign of practical control ytem become complex when the order of the ytem increae Therefore to analyze uch ytem it i neceary to reduce it to a lower order ytem which i a ufficient repreentation of the higher order ytem In recent decade much effort ha been made in the field of model reduction for fixed ytem and everal method like: Aggregation method [] Pade approximation [] Routh approximation [] Moment matching technique [3] and L optimization technique [] have been propoed Among them Routh approximation technique ha been recognized a the mot powerful method becaue of it ability to yield table reduced model for table high-order ytem [] In general the practical ytem have uncertaintie about it parameter Thu practical ytem will have coefficient that may vary and it i repreented by interval Interval arithmetic uch a addition ubtraction multiplication and diviion are dicued in [3 5] In literature [ 3] the author preented model reduction technique for higher order uncertain ytem The limitation of the above method are dicued in [8] A generalized method for contructing the DrNSelvaganean ME PhD Lecturer Department of Electrical and Electronic Engineering Pondicherry Engineering College Pondicherry 54 India n_elvag@rediffmailcom

2 N Selvaganean Routh table of interval polynomial i propoed in [4] which overcome ome of the limitation of [ 3] In thi paper the method propoed in [4 4] i integrated with factor diviion method [] for obtaining the table reduced order model Alo a gain correction factor i ued to improve the teady tate characteritic for interval ytem Problem Formulation Conider a higher order linear SISO uncertain ytem repreented by the tranfer function a N() G () = D() n c c + c c + c c + + cn c n G () = () n d d + d d + d d + + dn dn where ci c i i= n and di d i i= n are the interval coefficient of higher order numerator and denominator polynomial repectively The correponding k th order reduced model i Nk () R () = Dk () k r r + r r + r r + + rk r k R () = () k b b + b b + b b + + bk bk where ri r i i= k and bi b i i= k are the interval coefficient of lower order numerator and denominator polynomial repectively The problem i to reduce the ytem of form () to the ytem of the form () uch that the lower order ytem mimic the higher order one a cloely a poible Determination of reduced denominator Routh table i contructed for the denominator of the higher order ytem uing the algorithm propoed by Dolgin [4 4] The reduced denominator i obtained by direct truncation of the element in Routh table Conider the denominator polynomial of a higher order ytem

3 Mixed Method of Model Reduction of Uncertain Sytem D d d d d d d d d Correponding generalized Routh table i n () = n n d d d d d d n n n n n n 4 n 4 = d d d d d d n dn dn dn 3 dn 3 dn 5 dn 5 = d d d d d d n d3 d 3 d3 d 3 d3 d 3 dn dn dn d n From the Table the lower order polynomial can be obtained Thi algorithm offer table reduced order model for a given table higher order ytem Determination of reduced numerator The n th order original ytem given in equation () i equated to the k th order reduced model with unknown parameter repreented by equation () Hence c c + c c + + c c n n n n d d + d d + + dn dn k r r + r r + + rk r k k b b + b b + + bk bk = Rewriting the equation (3) we obtain ( ) ( ) k + n + ( cn c ) ( ) n b k b k = r r d d + + ( r r d d + r r d d ) + k + n ( ) c c b b + c c b b + c c b b + + r r k k dn dn = (3) (4) 3

4 N Selvaganean Equating the coefficient of the correponding term in the equation (4) the following relation are obtained: ( r r d d ) = ( c c b b ) ( r r d d + r r d d ) = ( c c b b + c c b b ) (5) ( rk r k dn d n ) = ( cn c n bk b k ) Comparing the firt k relation of the equation (5) ie comparing the coefficient of k unknown coefficient ri r i i= k are determined To minimize the teady tate error the Zero are adjuted by multiplying the numerator polynomial with the gain correction factor η It can be calculated uing the relation G () η= R () = For interval ytem η i calculated after converting the interval coefficient of Gand ( ) R( ) into the fixed coefficient by taking their mean Thu the gain correction factor i c b η= () d r where c + c b + b d + d r + r c = b = d = and r = 3 Simulation Reult Conider the eventh order ytem given by the tranfer function N() G () = D() 5 [ ] [ ] [ ] [ 885 5] + [ ] + + [ ] + [ ] + [ ] D ( ) =

5 Mixed Method of Model Reduction of Uncertain Sytem 5 4 [ ] [ ] [ ] [ ] [ ] + [ 899] + [ ] N ( ) = To obtain denominator polynomial the generalized Routh table i contructed for D( ) [ 955 ] [ 5359 ] [ ] [ ] [ 8993 ] [ ] [ 5839 ] [ ] 5 [ 333 ] [ ] [ ] 4 [ 89 ] [ ] [ ] 3 [ 8889 ] [ ] [ 343 ] [ 594 ] [ ] [ 594 ] By direct truncation from above table the econd order denominator polynomial i D ( ) = [ ] [ ] [ ] To obtain econd order reduced numerator polynomial G () = R () [ 85545] + [ 899] + + [ 9] [ ] + [ ] + + [ 955] = r r + r r [ 343] + [ 88835] + [ 594] Rewriting 5 =

6 N Selvaganean ([ 85545] [ 83] ) + ([ 899] [ 594] [ 85545] [ 88835] ) ([ 9] [ 343 ]) = r r [ ] ( ) ( r r [ ] r r [ ] ) + ( r r [ 955 ]) Comparing the coefficient of and and [ ] [ ] r r = 999 r r = The reduced numerator polynomial i N ( ) =η ([ ] [ ]) The gain correction factor found uing () i η = 994 Thu the reduced numerator polynomial i N ( ) = [ ] + [ 53858] The econd order reduced ytem for the higher order ytem G ( ) i obtained a [ ] + [ 53858] R () = [ 343] + [ 88835] + [ 594] The γ table for D( ) formed by the algorithm propoed in [] i obtained a [ ] [ 5435 ] [ 8883 ] [ 8893 ] [ ] [ ] [ 5353 ] [ 955 ] [ ] [ ] [ 595 ] [ ] [ 398 ] [ 5 ] [ 39448] [ 433 ] The econd order ytem obtained by method [] i R ( ) = [ 84] + [ 53] + [ 583] + [ 8]

7 Mixed Method of Model Reduction of Uncertain Sytem It i noted that the lower bound of the interval entry d5 d 5 of the γ table i negative thu retricting the completion of the table Hence reduced-order interval polynomial of degree four or greater cannot be obtained by [] The propoed method guarantee tability for a table higher order ytem and thu any lower order model can be derived with good accuracy Alo it may be noted that the propoed method involve le mathematical complexity compared to algorithm [] where both the γ and δ table need to be formulated which increae the complexity and require a great deal of computational effort Fig and Fig how tep repone and frequency repone of higher and reduced order tranfer function From the repone it can be oberved that the reduced order ytem i table and it cloely matche with the original ytem at all frequencie (in Fig ) Moreover it i found that the propoed method and the method propoed in [] match almot cloely Fig - Step repone of the original and the reduced order model But ytem dynamic error alway exit when a higher order ytem i reduced The quantitative criteria for meauring the performance are choen a Integral Square Error (ISE) Integral Abolute Error (IAE) and Integral of Time weighted Abolute Error (ITAE) e t e t t e t ISE= d IAE = d ITAE = d ISE IAE account mainly for error at the beginning of the repone and to a lower degree for the teady tate deviation ITAE take account of the error at the beginning but alo emphaize the teady tate For the reduced model

8 N Selvaganean (propoed method and []) and the original ytem the Performance Indice (ISE IAE and ITAE) are tabulated in Table and are found to be matching cloely Table Performance Indice for higher order and reduced ytem for G( ) R ( ) R ( ) by [ ] ISE IAE ITAE Kharitonov polynomial were contructed for the reduced model and their robut tability wa verified (in Appendix I) Alo the dominant pole for the higher order ytem G() propoed reduced order ytem R() and R() by [] are ± 9i and 338 ± 954i repectively which are cloely matching Thu the tranient repone of the higher order ytem i not much affected by thi reduction method Fig - Comparion of magnitude and phae frequency repone 8

9 Mixed Method of Model Reduction of Uncertain Sytem 5 Concluion The mixed generalized Routh table and factor diviion method are propoed to interval polynomial for obtaining the table reduced order ytem The reduction of eventh order interval ytem to econd order interval ytem give excellent tep a well a frequency repone The propoed method i mathematically imple and give all poible table lower order model Appendix I Kharitonov Theorem An interval family of polynomial D( ) i robutly table if and only if the Kharitonov polynomial are table D ( ) = a + a+ a + a3 + a4 + a D ( ) = a + a+ a + a3 + a4 + a D ( ) = a + a+ a + a3 + a4 + a D = a + a + a + a + a + a + ( ) For interval polynomial the teting et i at mot four Kharitonov polynomial II Theorem (Anderon Jury and Manour) The teting et for an interval polynomial of invariant degree i + D for n =3 ( ) ( ) ( ) D + D ++ ( ) ( ) ( ) D + D ++ D + ( ) ( ) ( ) ( ) for n =4 for n =5 D + D ++ D + D for n >5 For n = and n = a neceary and ufficient condition for robut tability i poitive lower bound on the coefficient The denominator polynomial of the higher order ytem in Section 4 i 5 [ ] [ ] [ ] [ 885 5] + [ ] + + [ ] + [ ] + [ ] D ( ) =

10 N Selvaganean n = therefore the neceary and ufficient condition i all the four polynomial hould be table 3 ( ) D = Contructing Routh table for D ( ) ( ) D = Contructing Routh table for D ( ) ( ) D = Contructing Routh table for D ( )

11 Mixed Method of Model Reduction of Uncertain Sytem ( ) D = Contructing Routh table for D ( ) From the above Routh table it i clear that all the four Kharitonov polynomial are table Therefore from Kharitonov Stability Criterion D ( ) i table The denominator polynomial of the reduced lower order ytem in Section 4 i D ( ) = [ 343] + [ ] + [ 594] n = therefore a neceary and ufficient condition for robut tability i poitive lower bound on the coefficient D ( ) =

12 N Selvaganean Contructing Routh table for D ( ) From the above Routh table it i clear that D () i table Thu the propoed method guarantee the robut tability of reduced order ytem Reference [] M Aoki: Control of large-cale dynamic ytem by aggregation IEEE Tran on Automatic Control Vol AC-3 98 pp 4-53 [] B Bandyopadhyay A Upadhye O Imail: γ δ Routh approximation for interval ytem IEEE Tran Automat Control Vol 4 Aug 99 pp 3 [3] BBandyopadhyay et al: Routh Pade approximation for interval ytem IEEE Tran Automat ControlVol pp [4] Y Dolgin E Zeheb: On Routh-Pade Model Reduction of Interval ytem IEEE Tran Automat Control Vol 48 3 pp - [5] ED Popova: Extended interval arithmetic in IEEE Floating-Point Environment Interval Computation No pp -9 [] K Glover: All optimal Hankel-Norm approximation of linear multivariable ytem and their L error bound Int J Control Vol 39 No 984 pp 5-93 [] MF Hutton B Friedland: Routh approximation for reducing order of linear time-varying ytem IEEE Tran Automat Control Vol 95 pp [8] C Hwang SF Yang: Comment on the computation of interval Routh approximant IEEE Tran Autom Control Vol 44 No pp 8 8 [9] VL Kharitonov: Aymptotic tability of an equilibrium poition of a family of ytem of linear differential equation Differentzialnye Uravneniya Vol 4 98 pp 8 88 [] V Krihnamurthy V Sehadri: Model Reduction uing the Routh Stability Criterion IEEE Tran on Automatic Control Vol 3 No 4 98 [] TN Luca: Factor diviion A ueful algorithm in model reduction IEE Proc Vol 3 No 983 p 3-34 [] Y Shamah: Stable reduced order model uing Pade type approximation IEEE tran on Automatic Control Vol 9 94 pp 5- [3] NK Sinha B Kuzta: Modeling and identification of dynamic ytem 33-3 New York: Van Notrand Reinhold 983 [4] SF Yang: Comment on On Routh Pade Model Reduction of Interval Sytem and Dolgin Y Author reply IEEE Tran Automat Control Vol 5 No Feb 5 pp 3 5