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2 Chaptr 7 A Coputational Approach to Vibration Control o Vhicl Engin-Body Systs Haid Rza Karii Additional inoration is availabl at th nd o th chaptr Introduction In rcnt yars, th nois and vibration o cars hav bco incrasingly iportant [, 3, 9, 3, 35]. A ajor coort aspct is th transission o ngin-inducd vibrations through powrtrain ounts into th chassis (s Figur ). Engin and powrtrain ounts ar usually dsignd according to critria that incorporat a trad-o btwn th isolation o th ngin ro th chassis and th rstriction o ngin ovnts. h ngin ount is an icint passiv ans to isolat th car chassis structur ro th ngin vibration. Howvr, th passiv ans or isolation is icint only in th high rquncy rang. Howvr th vibration disturbanc gnratd by th ngin occurs ainly in th low rquncy rang [8, 9, 3, 3]. hs vibrations ar rsult o th ul xplosion in th cylindr and th rotation o th dirnt parts o th ngin (s Figur ). In ordr to attnuat th low rquncy disturbancs o th ngin vibration whil kping th spac and pric constant, activ vibration ans ar ncssary. A varity o control tchniqus, such as Proportional-Intgral-Drivativ (PID) or Lad-Lag copnsation, Linar Quadratic Gaussian (LQG), H, H, -synthsis and dorward control hav bn usd in activ vibration systs [, 3, 4,,, 5, 4, 6, 3, 3, 34, 35]. h ain charactristic o dorward control is that inoration about th disturbanc sourc is availabl and is usually ralisd with th Filtrd-X Last-Man-Squars (Fx-LMS) algoriths. Howvr, th disturbanc sourc is assud to b unknown in dback control, thn dirnt stratgis o dback control or vibration attnuation o unknown disturbanc xist ranging ro classical thods to a or advancd thods. Rcntly, th proranc rsult obtaind by H dback controllr with th rsult obtaind by dorward controllr using Fx-LMS algoriths or vhicl ngin-body vibration syst was copard in [3, 35]. On th othr hand, wavlt thory is a rlativly nw and an rging ara in athatical rsarch []. It has bn applid in a wid rang o nginring disciplins such as signal Karii, licns Inch. his is an opn accss chaptr distributd undr th trs o th Crativ Coons Attribution Licns ( which prits unrstrictd us, distribution, and rproduction in any diu, providd th original work is proprly citd.

3 . 58 Advancs on Analysis and Control o Vibrations hory and Applications procssing, pattrn rcognition and coputational graphics. Rcntly, so o th attpts ar ad in solving surac intgral quations, iproving th init dirnc ti doain thod, solving linar dirntial quations and nonlinar partial dirntial quations and odlling nonlinar siconductor dvics [5, 6, 7, 3, 6, 7, 8,, 7]. chassis subra ngin ount point Figur. Front axis o AUDI A 8 ro [, 3] (Wrkbild Audi AG). Z: Bounc X Engin Block Chassis O Pitch Y Piston Crank Engin Mount Figur. Chassis xcitd by th ngin vibration.

4 A Coputational Approach to Vibration Control o Vhicl Engin-Body Systs 59 Orthogonal unctions lik Haar wavlts (HWs) [3, 6], Walsh unctions [7], block puls unctions [7], Lagurr polynoials [4], Lgndr polynoials [5], Chbyshv unctions [] and Fourir sris [8], otn usd to rprsnt an arbitrary ti unctions, hav rcivd considrabl attntion in daling with various probls o dynaic systs. h ain charactristic o this tchniqu is that it rducs ths probls to thos o solving a syst o algbraic quations or th solution o probls dscribd by dirntial quations, such as analysis o linar ti-invariant, ti-varying systs, odl rduction, optial control and syst idntiication. hus, th solution, idntiication and optiisation procdur ar ithr gratly rducd or uch sipliid accordingly. h availabl sts o orthogonal unctions can b dividd into thr classs such as picwis constant basis unctions (PCBFs) lik HWs, Walsh unctions and block puls unctions; orthogonal polynoials lik Lagurr, Lgndr and Chbyshv as wll as sin-cosin unctions in Fourir sris []. In th prsnt papr, w, or th irst ti, introduc a coputational solution to th initti robust optial control probl o th vhicl ngin-body vibration syst basd on HWs. o this ai, athatical odl o th ngin-body vibration structur is prsntd such th actuators and snsors usd to invstigat th robust optial control ar slctd to b collocatd. Morovr, th proprtis o HWs, Haar wavlt intgral oprational atrix and Haar wavlt product oprational atrix ar givn and ar utilizd to provid a systatic coputational rawork to ind th approxiatd robust optial trajctory and init-ti H control o th vhicl ngin-body vibration syst with rspct to a H proranc by solving only th linar algbraic quations instad o solving th dirntial quations. On o th ain advantags is solving linar algbraic quations instad o solving nonlinar dirntial Riccati quation to optiiz th control probl o th vhicl ngin-body vibration syst. W donstrat th applicability o th tchniqu by th siulation rsults. h rst o this papr is organizd as allows. Sction introducs proprtis o th HWs. Mathatical odl o th ngin-body vibration structur is statd in Sction 3. Algbraic solution o th ngin-body syst is givn in Sction 4 and Haar wavlt-basd optial trajctoris and robust optial control ar prsntd in Sctions 5 and 6, rspctivly. Siulation rsults o th robust optial control o th vhicl ngin-body vibration syst ar shown in Sction 7 and inally th conclusion is discussd. h notations usd throughout th papr ar airly standard. h atrics I r, r and r s ar th idntity atrix with dinsion r r and th zro atrics with dinsions r r and r s, rspctivly. h sybol and tr( A ) dnot Kronckr product and trac o th atrix A, rspctivly. Also, oprator vc( X ) dnots th vctor obtaind by putting atrix X into on colun. Finally, givn a signal xt (), xt () dnots th L nor o xt (); i.., xt () xt () xt () dt.

5 6 Advancs on Analysis and Control o Vibrations hory and Applications. Proprtis o Haar Wavlts Proprtis o HWs, which will b usd in th nxt sctions, ar introducd in this sction... Haar Wavlts (HWs) h oldst and ost basic o th wavlt systs is nad Haar wavlt that is a group o squar wavs with agnitud o.. in th intrval, [6]. In othr words, th HWs ar dind on th intrval, as () t, t,,, or t,, () t, or t,, and () ( j i t t k ) or i and w writ i j k or j and k j. W can asily s that th () t and () t ar copactly supportd, thy giv a local dscription, at dirnt scals j, o th considrd unction. ().. Function approxiation h init sris rprsntation o any squar intgrabl unction y() t in trs o an orthogonal basis in th intrval,, naly yˆ( t ), is givn by y ˆ () t a i i(): t a () t () i whr a: a a a and (): () () () t t t t or j and th Haar coicints a i ar dtrind to iniiz th an intgral squar rror ( y ( t ) a ( t )) dt and ar givn by Rark. h approxiation rror, j a y ( t ) ( t ) dt (3) i i ( ): ( ) ˆ y y t y ( t ), is dpnding on th rsolution and is approaching zro by incrasing paratr o th rsolution. h atrix H can b dind as H ( t), ( t),, ( t ) (4)

6 A Coputational Approach to Vibration Control o Vhicl Engin-Body Systs 6 whr i t i and using (), w gt i h intgration o th vctor ( ) can b approxiatd by ˆ ˆ ˆ y ( t ) y ( t ) y ( t ) a H. (5) t whr th atrix t () t dt P () t (6) t t P ( ) d, ( t) ( r) dr ( t) dt rprsnts th intgral oprator atrix or PCBFs on th intrval, at th rsolution. For HWs, th squar atrix P satisis th ollowing rcursiv orula [3]: P P H H (7) with P and H H () diag r whr th atrix H dind in (4) and also th vctor r is rprsntd by r: (,,,,4,4,4,4,,( ),( ),,( )) ( ) lnts or. For xapl, at rsolution scal j 3, th atrics H 8 and P 8 ar rprsntd as and H 8 ( t) ( t) ( t7) ( t) ( t) ( t7) ( t) ( t) ( t7) 3( t) 3( t) 3( t7) 4( t) 4( t) 4( t7), 5( t) 5( t) 5( t7) 6( t) 6( t) 6( t7) 7( t) 7( t) 7( t7)

7 6 Advancs on Analysis and Control o Vibrations hory and Applications P 8 4H H 4H H4 6P4 H H 6 H , 64 or urthr inoration s [3, 5]. H 4.3. h product oprational atrix In th study o ti-varying stat-dlayd systs, it is usually ncssary to valuat th product o two Haar unction vctors [3]. Lt us din whr R ( t ) satisis th ollowing rcursiv orula R (): t () t () t (8) R () t R() t H diag( ()) b t ( H diag( b( t))) diag( H a( t)) (9) with R() t () t () t and a(): t (), t (), t, () t () t b(): t (), t (), t, () t. () Morovr, th ollowing rlation is iportant or solving optial control probl o tivarying stat-dlayd syst: R () t a a () t ()

8 A Coputational Approach to Vibration Control o Vhicl Engin-Body Systs 63 whr a a and a H diag( a ) b a diag( ab) H diag( a ) ah () with a : a a, a,, a a ( ) t ab : a(), t a (), t, a () t. (3) l L Figur 3. h sktch o ngin-body vibration syst 3. Mathatical odl dscription A schatic o th vhicl ngin-body vibration structur is shown in Figur 3. h actuator and snsor usd to this control rawork ar slctd to b collocatd, sinc this arrangnt is idal to nsur th stability o th closd loop syst or a slightly dapd structur [6]. In our study, only th bounc and pitch vibrations in th ngin and body ar considrd [35]. h ngin with ass M and inrtia ont I is ountd in th body by th ngin ounts k and c. h ront ount is th activ ount, th output orc o

9 64 Advancs on Analysis and Control o Vibrations hory and Applications which can b controlld by an lctric signal. h activ ount consists o a ain chabr whr an oscillating ass (inrtia ass) is oving up and down. h inrtia ass is drivn by an lctro-agntic orc gnratd by a agntic coil which is controlld by th input currnt. h vhicl body with ass M b and inrtia ont I b is supportd by ront and rar tirs, ach o which is odld as a syst consisting o a spring k b and a daping dvic c b. hror, a our dgr-o-rdo vibration suspnsion odl shown in Figur 3 can b dscribd by th ollowing quations Mx cx kx cx kx ( Llcx ) 4 ( Llk ) x4 t () d() t M bx ( c cb) x ( k kb) x cx kx ( Ll) cx 4 ( Ll) kx4 ( t) I x3 lcx 3 lkx 3 lcx 4 lkx 4 lt ( ) I b x4 (( L ( Ll)) c L cb) x 4 (( L ( L)) l k L kb) x4 l cx 3 lkx 3 lcx lkx ( Llcx ) ( Llkx ) Lt ( ) (4) whr th stats x (), t x (), t x 3 () t and x 4 () t ar th bouncs and pitchs o th ngin and body, rspctivly, whr displacnt o th chassis ( x( t)) is usually takn as an output. Input orc, () t, is usd as th activ orc to copnsat th vibration transittd to vhicl body. Morovr, ngin disturbanc d( t ) can b th xcitation, gnratd by th otion up/down o th dirnt parts insid th ngin; h syst Eq. (4) can b rprsntd in th ollowing stat-spac or Mx () t Cx () t Kx() t B () t Bdd(), t t, C x() t zt () C xt () C3 () t (5) 4 whr xt () is th stat; () t is th control input; d( t) is th disturbanc input which blongs to L [, ) ; and zt () is th controlld output with C, C and C 3 is a positiv scalar. h stat-spac atrics ar also dind as M Mb M I Ib c c ( Ll) c c ( c c ) ( Ll) c, b, C lc lc lc ( Ll) c l c

10 A Coputational Approach to Vibration Control o Vhicl Engin-Body Systs 65 B, l L B d, k k ( Ll) k k ( k kb) ( Ll) k K lk lk. lk ( Ll) k l k ( L ( L l) ) k L k b aking displacnt o th chassis ( x ( t )) as an output thn a coparison o th displacnt rspons rspct to th input orc () t and th xtrnal disturbanc d( t ) in th rquncy rang up to KHz is dpictd in Figur 4a) and 4b). hr rlvant ods occur around th rquncis, 5 and 9 Hz, rspctivly, which rprsnt th dynaics o th ain dgrs o rdo (DOFs) o th syst. 4. Algbraic solution o syst quations In this sction, w study th probl o solving th scond-ordr dirntial quations o th ngin-body syst (4) in trs o th input control and xognous disturbanc using HWs and dvlop appropriat algbraic quations. Basd on HWs dinition on th intrval ti,, w nd to rscal th init ti intrval, into, by considring t ; noralizing th syst Eq. (5) with th ti scal would b as ollows Mx ( ) Cx ( ) Kx( ) B ( ) B d ( ) (6) d Now by intgrating th syst abov in an intrval,, w obtain d M( x( ) x()) C x( ) d K x( ) dd B ( ) dd B d ( ) dd ( Mx () Cx()) d. By using th Haar wavlt xpansion (), w xprss th solution o Eq. (5), input orc ( ) and ngin disturbanc d ( ) in trs o HWs in th ors (7) x( ) X ( ), (8) ( ) F ( ), (9)

11 66 Advancs on Analysis and Control o Vibrations hory and Applications 4 d ( ) D ( ), () whr X, F and D dnot th wavlt coicints o x( ), ( ) and d( ), rspctivly. h initial conditions o x () and x () ar also rprsntd by x() X ( ) and x 4 () X ( ), whr th atrics { X, X} ar dind, rspctivly, as X : x() 4 4 ( ) X : x () 4 4 ( ) () () hror, using th wavlt xpansions (8)-(), th rlation (7) bcos d M( XX ) CXP KXP B FP B D P ( MX CX ) P (3) For calculating th atrix X, w apply th oprator vc (.) to Eq. (3) and according to th proprty o th Kronckr product, i.. vc( ABC) ( C A) vc( B), w hav: ( I M)( vc( X) vc( X )) ( P C) vc( X) ( P K) vc( X) P B vc F P Bd vc D ( ) ( ) ( ) ( ) P C vc X P M vc X ( ) ( ) ( ) ( ). (4) Solving Eq. (4) or vc( X ) lads to vc( X) vc( F) vc( D ) vc( X ) vc( X ) (5) 3 4 whr th atrics 4 44 {, } and {, } ar dind as 3 4 ( ( P C) ( P K) I M) ( P B) ( ( P C) ( P K) I M) ( P Bd) 3 ( ( P C) ( P K) I M) ( I M P C) 4 ( ( P C) ( P K) I M) ( P M). (6) Consquntly, using (5) and (6) and th proprtis o th Kronckr product, th solution o syst (5) is x ( ) ( ( ) I ) vc ( X ) (7) 4

12 A Coputational Approach to Vibration Control o Vhicl Engin-Body Systs 67 and it is also clar that to ind th approxiatd solution o th syst, w hav to calculat th invrs o th atrix ( P C) ( P K) I M with dinsion 44 only onc. 3 (a) Displacnt (db) (Hz) - Frquncy 3 3 (b) Displacnt (db) (Hz) - Frquncy 3 Figur 4. Displacnt o th chassis rspct to () t (a) and d () t (b).

13 68 Advancs on Analysis and Control o Vibrations hory and Applications 5. Optial control dsign h control objctiv is to ind th optial control () t with rspct to a quadratic cost unctional approxiatly such acts as th activ orc to copnsat th vibration transittd to vhicl body. h quadratic cost unctional wights th stats and thir drivativs with rspct to ti in th cost unction as ollows: ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ) (8) J x S x x S x x t Q x t x t Q x t R t dt whr S :4 4, S :4 4, Q :4 4 and Q :4 4 ar positiv-dinit atrics and R is a positiv scalar. W can rwrit th cost unction (8) as ollows: x() x( ) [ () ()] ([ ( ) J x x S x x ( )] Q R ( ) ) d x() x. (9) ( ) whr S diag( S, S) and Q diag( Q, Q ) with th ti scal t. Fro (5) and th rlation x ( ) X ( ), whr X:4 dnots th wavlt coicints o x ( ) atr its xpansion in trs o HFs, w rad x( ) X ( ): Xaug( ) x ( ) X (3) X whr Xaug X and vc( Xaug) vc ( X) vc ( X) (3) Rark. By substituting x ( ) X ( ) into x( ) x() x ( t) dt, w hav: X ( ) X ( ) X ( ) d, (3) and using (4), w rad XX XP. hn, by applying th oprator o vc (.) and according to th proprtis o Kronckr product in Appndix A, w obtain vc( X) vc( X ) ( P I ) vc( X) (33) n

14 A Coputational Approach to Vibration Control o Vhicl Engin-Body Systs 69 By substituting th dinition (3) in (33) and using th proprtis o th oprator tr (.) in Appndix A, th cost unction (8) is givn by ( ( ) ( ) J vc Xaug vc Xaug vc ( F ) vc ( F )) (34) whr th atrics :8 8 and : ar dind as M S ( M Q ) and R M, rspctivly, and th atrics M : and M : ar dind as M : ( ) ( ) d and M : () (), rspctivly. i i It is clar that th cost unction o J (.) is a unction o i, thn or inding th optial control law, which iniizs th cost unctional J (.), th ollowing ncssary condition should b satisid J vc( F) (35) By considring vc( X aug), which is a unction o vc( F ), and using th proprtis o drivativs o innr product o Kronckr product in Appndix A, w ind J [ ( P 4)] ( ) ( ) I vc Xaug vc F vc( F) (36) hn th wavlt coicints o th optial control law will b in vctor or as 4 aug vc( F) [ ( P I )] vc( X ) (37) Consquntly, th optial vctors o vc( X ) and vc( F ) ar ound, rspctivly, in th ollowing ors I vc( X) ( I ( [ ( P I )] ) ( vc( D ) ( 4 [ ( P 4)] 3) ( ) 4 ( )), I vc X vc X ( P 4) I ( P I4) (38) and

15 7 Advancs on Analysis and Control o Vibrations hory and Applications I 4 vc( F) [ ( P 4)] { I ( P 4) I I ( I [ ( P I )] ) ( P I4) 4 ( vc( D) ( [ ( P 4)] 3) ( ) I vc X ( P 4) I 4 4vc( X)) vc( X)}. ( P 4) I (39) Finally, th Haar unction-basd optial trajctoris and optial control ar obtaind approxiatly ro Eq. (7) and () t () t vc( F). 6. Robust optial control dsign In this sction, an optial stat dback controllr is to b dtrind coputationally such that th ollowing rquirnts ar satisid: i. th closd-loop syst is asyptotically stabl; ii. undr zro initial condition, th closd-loop syst satisis non-zro d( t) [, ) whr is a prscribd scalar. zt () d() t or any h control objctiv is to ind th approxiatd robust optial control () t with H proranc such () t acts as th activ orc to copnsat th vibration transittd to vhicl body, i.. guarants dsird L gain proranc. Nxt, w shall stablish th H proranc o th syst (5) undr zro initial condition. o this nd, w introduc J x ( ) S x( ) x ( ) S x( ) ( z ( t) z( t) d ( t)) dt. (4) It is wll known that a suicint condition or achiving robust disturbanc attnuation is that th inquality J or vry d () t L[, ) [33, 36]. hror, w will stablish conditions undr which In Sup J( vc( F), vc( D )) (4) vc( F) vc( D ) Fro (5), th Eq. (4) can b rprsntd as

16 A Coputational Approach to Vibration Control o Vhicl Engin-Body Systs 7 x() J ( x () x ()) S () x x( ) (( x ( ) x ( )) C C3 ( ) d ( )) d x ( ) (4) whr t, S diag( S, S) and C diag( C C, C C ). Using th rlation x ( ) X ( ), w rad x( ) X ( ): Xaug( ) x ( ) X (43) X whr Xaug X and vc( Xaug ) vc ( X) vc ( X) (44) Morovr, according to Rark in [8], th ollowing rlation is alrady satisid btwn vc( X ) and vc( X ) By using th dinition (44) in Eq. (45), w hav 4 vc ( X ) vc ( X ) ( P I ) vc ( X ) (45) ( ( )) ( ( ) ( 3 ) ( J tr M Xaug SX aug tr M Xaug CX aug tr C M F F tr M D D )) (46) Using th proprty o th Kronckr product, i.. tr( ABC) vc ( A ) ( Ip B) vc( C), ( A C)( DB) AD CB and vc( ABC) ( C A) vc( B), w can writ (4) as ( ( ) ( ) 3 ( ) ( ) J vc Xaug vc Xaug C vc F vc F vc ( D ) vc ( D )) (47) whr th atrics 8 8, ar dind as M S ( M C ) and M, rspctivly. It is asy to show that th worst-cas disturbanc in Eq. (47) occurs whn vc ( D) ( P 4) ( ): ( ) I vc Xaug dvc X aug (48)

17 7 Advancs on Analysis and Control o Vibrations hory and Applications By substituting Eq. (48) into Eq. (47) w obtain In Sup J ( vc ( F ), vc ( D )) In J ( vc ( F ), vc ( D )) (49) vc( F) vc( D ) vc( F) Miniizing th right-hand sid o Eq. (49) rsults in th algbraic rlation btwn wavlt coicints o th robust optial control and o th optial stat trajctoris in th ollowing closd or vc( F) C3 ( P 4) ( ) ( ) I d d vc X aug : vc( X ). aug (5) As a rsult w hav vc( F) vc( D ) aug d aug In Sup J( vc( F), vc( D )) vc ( X )( R ) vc( X ) (5) Consquntly, i thr xists positiv scalar to th atrix inquality d C3 d (5) thn inquality (4) is concludd. Fro th rlations abov w obtain th robust optial vctors o vc( X ) and vc( F ) atr so atrix calculations, rspctivly, in th ollowing ors d and I 4 vc( X) ( I4 ( ) d ) (( 3( d) ( P 4) I 4 ) vc( X) 4vc( X)), ( P 4) I I 4 I 4 vc( F) {( (( I4 ( ) d ) ( P 4) ( 4) I P I 4 4 ( 3( d) ) ) vc( X) ( P 4) ( 4) I P I I 4 I 4 ( I4 ( ) d ) 4vc( X))} ( P 4) I ( P 4) I (53) (54) Finally, th Haar wavlt-basd robust optial trajctoris and robust optial control ar obtaind approxiatly ro Eq. (7) and () t () t vc( F), rspctivly.

18 A Coputational Approach to Vibration Control o Vhicl Engin-Body Systs Nurical rsults In this sction th proposd coputational thodology is applid to th vhicl nginbody vibration syst (5) such th xognous disturbanc d( t ) is assud to b a Sin (.) unction at th rquncy o Hz. h syst paratrs, usd or th dsign and siulation ar givn in abls and in th Appndix B. abl 3 in th Appndix givs th pol-zro locations o 8 th ordr odl o th vhicl ngin-body vibration syst. It is clar that th vhicl ngin-body vibration syst is unstabl and has th noniniu phas proprty. h objctiv is to ind th approxiatd robust optial displacnt o th chassis and robust optial input orc with H proranc using HWs at th init ti intrval,. Morovr, th atrics 44 { S, S} and th vctors C, C and th scalar C 3 in th controlld output zt () in Eq. (5) ar chosn as S S 4, C [,,,], C [3,,,] and C3. Figur 5. Coparison o displacnt o th chassis ound by HWs at rsolution lvl j 5 (solid) and by analytical solution (dashd). o copar th approxiat solutions x () t and () t, ound by HWs, to th analytical solution ound by hor in th Appndix C, w choos th proranc bound and th rsolution lvl qual to 3.5 and 5, rspctivly, i and j 5. h ti curvs ound ar plottd in Figurs 5 and 6. It is clar that th ct o th ngin

19 74 Advancs on Analysis and Control o Vibrations hory and Applications disturbanc is attnuatd onto th displacnt o th chassis as th output as wll. In othr words, () t copnsats th vibration transittd to th chassis. Copar th Haar wavlt basd solutions to th continuous solutions using th dirntial Riccati quation, th approxiat solutions (53) and (54) dlivr both, robust control () t and stat trajctory xt () in on stp by solving linar algbraic quations instad o solving nonlinar dirntial Riccati quation, whil accuracy can asily b iprovd by incrasing th rsolution lvl j. Figur 6. Coparison o input orc ound by HWs at rsolution lvl j 5 (solid) and by analytical solution (dashd). 8. Conclusion his papr prsntd th odlling o ngin-body vibration structur to control o bounc and pitch vibrations using HWs. o this ai, th Haar wavlt-basd optial control or vibration rduction o th ngin-body syst was dvlopd coputationally. h Haar wavlt proprtis wr introducd and utilizd to ind th approxiat solutions o optial trajctoris and robust optial control by solving only algbraic quations instad o solving th Riccati dirntial quation. Nurical rsults wr prsntd to illustrat th advantag o th approach.

20 A Coputational Approach to Vibration Control o Vhicl Engin-Body Systs Appndix 9.. Appndix A A. So proprtis o Kronckr product Lt A : p q, Bq : r, C: r s and D: q t b ixd atrics, thn w hav: vc( ABC) ( C A) vc( B), tr( ABC) vc ( A )( I B) vc( C), tr( ABC) vc ( A )( I B) vc( C), ( AC)( DB) ADCB. A. Drivativs o innr products o Kronckr product Lt A : n n b ixd constants and x: n b a vctor o variabls. hn, th ollowing rsults can b stablishd: ( Ax) vc( A), x ( Ax) A, x ( x Ax) A x A x x A3. Chain rul or atrix drivativs using Kronckr product Lt b a p q atrix whos ntris ar a atrix unction o th lnts o Y: s t, whr Y is a unction o atrix : n. hat is, ( Y), whr Y ( X). h atrix o drivativs o with rspct to is givn by 9.. Appndix B vc ( Y) IpIn. vc( Y) Paratrs M b I 8 [ b k b abl. h vhicl body paratr. c L b b p p Valus [kg] kg ] [N/] 3 [N//s].5 []

21 76 Advancs on Analysis and Control o Vibrations hory and Applications Paratrs M Valus 5 [kg] I 8. [ kg ] k c L [N/] [N//s].5 [] abl. h ngin paratrs. Pols i i i i 6.9 Zros -6.3 i i i abl 3. Pol-zro locations o th 8 th -ordr odl Appndix C hor (Stat Fdback) [9]. Considr dynaical syst xt () Axt () But () Bwt () zt () Cxt () Dut () undr assuption ( A, B, C) is stabilizabl. For a givn, th dirntial Riccati quation X A XXAX B B B B XC C ( ) has a positiv si-dinit solution Xt () such that A ( BB BB ) Xt ( ) is asyptotically stabl. hn th control law ut () BXtxt () (): Kt () Xt () is stabilizing and satisis zt () wt (). Author dtails Haid Rza Karii Dpartnt o Enginring, Faculty o Enginring and Scinc, Univrsity o Agdr, Gristad, Norway

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