Robust Controller Design Using Loop-Shaping and the Method of Inequalities

Transcription

1 Robu Conroller Degn Ung H Loo-Shang and he Mehod of Inequale J F Whdborne, I Polehwae and D-W Gu Conrol Syem Reearch Dearmen of Engneerng Unvery of Leceer Leceer LE 7RH UK Ocober 99, reved July 99, Ocober 99 Abrac A new aroach o robu conroller degn rooed. By ung lan weghng funcon a he degn arameer, he aroach combne he mehod of nequale wh robu ablzaon of normalzed corme facor decron of he weghed lan o degn exlcly for cloed-loo erformance and ably robune. A rocedure for he degn of robu wo degree-of-freedom conroller reened, and lluraed on a hgh-ury dllaon column examle.

2 Inroducon Seccaon for he erformance of feedback conrol yem are ofen exreed n erm of nequale whch need o be aed. Th fac reuled n he develomen of he mehod of nequale [], a degn mehod where he degn objecve are exreed exlcly a a e of algebrac nequale rereenng dered bound on a e of erformance ndce. For a ucceful degn, he nequale mu be aed. A earae develomen ha been he ue of H -omzaon n a varey of aroache o degn robu conrol yem. One uch aroach he loo-hang degn rocedure (LSDP) [, ]. Th aroach nvolve he robu ablzaon o addve erurbaon n he ene of H -norm of normalzed corme facor of a weghed lan. The weghed lan ngular value are haed by adjung he weghng funcon o gve a dered oen-loo hae whch gve good cloed-loo erformance wh ably robune. Ceran aec of he LSDP make uable o combne h aroach wh he mehod of nequale o degn drecly for boh cloed-loo erformance and ably robune. Th aer decrbe h new aroach, and ale he rooed mehod o he degn of a conrol yem for a hgh ury dllaon column, a lan whch ha receved conderable aenon n he leraure of lae [6,, 5, 6, 9]. Normalzed Corme Facorzaon À The lan model G = M~ N~, a normalzed lef corme facorzaon (NLCF) of G f ~ ~ Ã Ã M; N RH; here ex V; U RH uch ha MV ~ + NU ~ = I; and MM ~ ~ + NN ~ ~ = I Ã where for a real raonal funcon of, X denoe X ( À). Ung he noaon À A B G( ) = D + C( I À A) B = 4 5 () C D ; hen a hown n [] [ N~ M~ ] = 4 A + HC B + HD H À= À= À= R C R D R 5 ; () À a normalzed corme facorzaon of G where H = À( BD + ZC ) R, R = I + DD, and he marx Z (ARE) he unque ablzng oluon o he algebrac Rcca equaon À À À À ( A À BS D C) Z + Z( A À BS D C) À ZC R CZ + BS B = ; ()

3 where S = I + D D Á - N Á M u ~ ~ À N -? - M - y K Fgure : Robu ablzaon wh reec o corme facor uncerany A erurbed model G P dened a À P M N G = ( M~ + Á ) ( N~ + Á ) (4) where Á ; Á RH. To maxmze he cla of erurbed model dened by (4) uch M N ha he conguraon of Fgure able, we need o nd he conroller K whch ablze he nomnal cloed-loo yem and whch mnmze where " # K À À = ( I À GK) M ~ : (5) I Th he roblem of robu ablzaon of normalzed corme facor lan decron a nroduced n [5]. From he mall gan heorem, he cloed-loo yem wll reman able f k N M k À [ Á Á ] < : (6) The mnmum value of for all ablzng conroller K " # K À À = nf ( I À GK) M ~ K ablzng I : (7) I hown n [5] ha max = = ( + ( ZX )) : (8)

4 where ( Á) rereen he maxmum egenvalue, and X he unque ablzng max oluon of he ARE À À À À ( A À BS D C) X + X( A À BS D C) À XBS B X + C R C = : (9) A conroller whch acheve gven n [] by A + BF + ( Q ) ZC ( C + DF ) ( Q ) ZC K = 4 B X ÀD À À 5 ; () where F = ÀS ( D C + B X), and Q = ( ) I + XZ. À À From he above, he omum conroller ynhezed by he oluon of wo ARE', unlke mo H roblem, whch requre an erave earch on o nd he omum. In racce, o degn conrol yem ung normalzed corme facorzaon, he lan need o be weghed o mee cloed-loo erformance requremen. A degn rocedure ha been develoed [, ], known a he loo-hang degn rocedure (LSDP), o chooe he wegh by udyng he oen-loo ngular value of he lan, and augmenng he lan wh wegh o ha he weghed lan ha an oen-loo hae whch wll gve good cloed-loo erformance. The nomnal lan G augmened wh re- and o-comenaor W and W À reecvely, o ha he augmened lan G G = W GW. Ung he rocedure oulned earler, an omum feedback conroller K ynhezed whch robuly ablze he NLCF of G gven by ( N ~ ; M~ ) where G = M~ N~. The nal feedback conroller K hen conruced by mly combnng K wh he wegh o gve K = W K W : () Noe ha from [], " À # W K À À = nf ( I À GK) [ W GW ] K ablzng W : () Eenally, wh he LSDP, he wegh W and W are he degn arameer whch are choen boh o gve he augmened lan a `good' oen-loo hae and o enure ha no oo large. a degn ndcaor of he ucce of he loo-hang a well a a meaure of he robune of he ably roery. The Mehod of Inequale Performance eccaon for conrol yem are frequenly gven n erm of algebrac or funconal nequale, raher han n he mnmzaon of ome objecve funcon.

5 For examle, he yem may be requred o have a re-me of le han econd, a elng me of le han 5 econd and an overhoo of le han %. In uch cae, obvouly more logcal and convenen f he degn roblem exreed exlcly n erm of uch nequale. The mehod of nequale (MOI) [] a comuer-aded mul-objecve degn aroach, where dered erformance rereened by uch a e of algebrac nequale, and where he am of he degn o mulaneouly afy hee nequale. The degn roblem exreed a S q he q-dmenonal ace IR and he co-ordnae of h ace are S f g q The boundary of h e dened by ( ) = ". A on IR a oluon o he ( ) " for = : : : n; () where " are real number, P a real vecor ( ; ; : : : ; ) choen from a gven e P and are real funcon of. The funcon are erformance ndce, he comonen of rereen he degn arameer and " are choen by he degner and rereen he large olerable value of nequale n order ha an acceable degn reached. For conrol yem degn, he funcon \ n S = S S q. The am he afacon of he e of ( ) may be funconal of he yem e reone, for examle he re-me, overhoo or he negral abolue error, or funconal of he frequency reone, uch a he bandwdh. They can alo rereen meaure of he yem ably, uch a he maxmum real ar of he cloed-loo ole. Addonal nequale whch are from he hycal conran of he yem can alo be ncluded, o rerc for examle, he maxmum conrol gnal. In racce, he conran on he degn arameer whch dene he e P are alo ncluded n he nequaly e, e.g. o conran he oble value of ome of he degn arameer, or o lm he earch o able conroller only. Each nequaly ( ) " of he e of nequale () dene a e of on n q, o = : ( ) " : (4) e of nequale () f and only f le nde every e, = ; ; : : : ; n and hence nde he e S whch denoe he nerecon of all he e, S. S ; ; : : : ; = : (5) called he admble e and any on n S called an admble on denoed 4

6 The objecve hu o nd a on uch ha S. Such a on ae he e of nequale () and ad o be a oluon. In general, a on unle he ube roblem,.e. S q S a on n he ace IR. In ome cae, here no oluon o he no unque an emy e. I hen neceary o relax he boundare of ome of he nequale,.e. ncreae ome of he number ", unl an admble on ex. The acual oluon o he e of nequale () may be obaned by mean of numercal earch algorhm, uch a he movng boundare roce (MBP), deal of he MBP may be found n [7] and []. The rocedure for obanng a oluon neracve, n ha requre uervon and nervenon from he degner. The degner need o chooe he conguraon of he degn, whch deermne he dmenon of he degn arameer vecor, and nal value for he degn arameer. The rogre of he earch algorhm hould be monored, and, f a oluon no found, he degner may eher change he arng on, relax ome of he dered bound " or change he degn conguraon. Alernavely, f a oluon found ealy, o mrove he qualy of he degn, he bound could be ghened or addonal degn objecve could be ncluded n (). The degn roce hu a wo way roce, wh he MOI rovdng nformaon o he degner abou concng degn requremen, and he degner makng decon abou he `rade-o' beween degn requremen baed on h nformaon a well a on he degner' knowledge, exerence and nuon abou he arcular roblem. The degner can be uored n h role by varou grahcal dlay [4] whch rovde nformaon abou he rogre of he earch algorhm and abou he concng degn requremen. In ome revou alcaon of he MOI, he degn arameer ha arameerzed a conroller wh a arcular rucure. For examle, = ( ; ) could arameerze a PI conroller + =. Th ha mean ha he degner ha had o chooe he rucure of he conrol cheme and he order of he conroller. In general, he lower he dmenon of he degn vecor, he eaer for he numercal earch algorhm o nd a oluon, f one ex. Alhough h doe gve he degner ome exbly and lead o mle conroller, and of arcular value when he rucure of he conroller conraned n ome way, doe mean ha beer oluon may ex wh more comlcaed and hgher order conroller. A furher lmaon of ung he MOI n h way ha a ably on mu be locaed a a re-reque o earchng he arameer ace o mrove he ndex e []., h ue addreed n more deal n 5

7 4 Robu Degn Ung a Corme Facor Plan Decron wh he Mehod of Inequale Two aec of degn ung robu ablzaon of normalzed corme facor decron of he weghed lan make amenable o combne h aroach wh he MOI. Frly, unlke mo H -omzaon roblem, he H -omal conroller for he wegh- ed lan can be ynhezed from he oluon of ju wo ARE' and doe no requre me-conumng -eraon. Secondly, n he LSDP decrbed n Secon, he weghng funcon are choen by conderng he oen-loo reone of he weghed lan, o eecvely he wegh W and W are he degn arameer. Th mean ha he degn roblem can be formulaed a n he mehod of nequale, wh he weghng arameer ued a he degn arameer o afy ome e of cloed-loo erformance nequale. Such an aroach o he MOI overcome he lmaon o he MOI decrbed a he end of Secon. The degner doe no have o chooe he order or rucure of he conroller, bu nead chooe he rucure and order of he weghng funcon. Wh low-order weghng funcon, hgh order conroller can be ynhezed whch ofen lead o gncanly beer erformance or robune han f mle low order conroller were ued. Addonally, he roblem of ndng a ably on doe no ex becaue ably guaraneed hrough he oluon o he robu ablzaon roblem, rovded ha he weghng funcon do no caue underable ole/zero cancellaon. Imroved erformance for rackng yem may be obaned wh a degree-offreedom ( DOF) cheme. T done by ncludng a re-comenaor K on he reference nu, a hown n Fgure. The re-comenaor arameerzed wh a ub-e of he degn arameer; he conroller K he oluon o he weghed normalzed corme facor aroach already decrbed, wh he weghng funcon W and W arameerzed wh he remanng degn arameer. An analycal mehod of degnng a DOF conroller wh he LSDP decrbed n [6], h mehod can alo be combned wh he MOI [, 8]. The degn roblem now aed a follow: Problem For he yem of Fgure, nd a ( W; K ) uch ha ( W ) " ; (6) 6

8 ref K W G u - -y K W Fgure : The DOF cheme and ( W; K ) " for = : : : n; (7) where r and ( W; K ) are funconal of he DOF cloed-loo yem, ", " are real number rereenng dered bound on and reecvely, W = ( W ; W ) a ar of xed order weghng funcon wh real arameer w = ( w ; w ; : : : ; w ) and K a re-comenaor wh a xed rucure and order and wh real arameer = ( ; ; : : : ; ). " À # W K À À ( W ) = nf ( I À GK) [ W GW ] ; (8) K ablzng W q Degn Procedure A degn rocedure o olve he above roblem : ) Dene he lan G, and dene he funconal. ) Dene he value of " and " ) Dene he form and order of he weghng funcon W and W. Bound hould be laced on he value of w o enure ha W and W are able and mnmum hae o reven underable ole/zero cancellaon. The order of he weghng funcon, and hence he value of q, hould nally be mall. v) Dene he form and order of he re-comenaor K. Bound may be laced on he value of f dered. The order of he re-comenaor ranfer funcon, and hence he value of r, hould nally be mall. v) Dene nal value of w baed on he oen-loo frequency reone of he lan. Dene nal value of. 7

9 v) Imlemen he MBP n conjuncon wh (8) and () o nd a W and K whch afy nequale (6) and (7). If a oluon found, he degn afacory. If no oluon found, eher ncreae he order of he weghng funcon or re-comenaor, relax one or more of he dered bound ", or ry agan wh deren nal value of w and. v) Wh afacory weghng funcon W and W, a afacory feedback conroller obaned from (). 5 Examle : The Dllaon Column The rooed degn mehod ued o degn a conrol yem for he hgh ury dllaon column decrbed n [9]. The column condered n ju one of conguraon, he LV conguraon, for whch he followng model relevan : 878 À: 864 ke GD( ; k ; k ; ; ) = : 8 À: 96 À : 4 : 6 À 5 (9) where : 8 k ; k : and ;, and all me un are n mnue. The me delay and acuaor gan value ued n he nomnal model G are k = k = : and = = : 5. The me delay elemen aroxmaed by a r-order Pade aroxmaon. The degn eccaon are o degn a conroller whch guaranee for all : 8 k ; k : and ; : ) Cloed-loo ably. h ) The ouu reone o a e demand h( ) ae y ( ) : for all, y ( ) : 9 for all > and y( ) : 5 for all. h ) The ouu reone o a e demand h( ) ae y ( ) : 5 for all, y( ) : 5 for all >, y( ) : 7 for all and y( ) : 55 for all >. h v) The ouu reone o a e demand h( ) ae y ( ) : 5 for all, y ( ) : for all and y ( ) : 9 for all >. v) Zero eady ae error. v) The frequency reone of he cloed-loo ranfer funcon beween demand nu and lan nu gan lmed o 5 db and he uny gan cro over frequency of large ngular value hould be le han 5 rad/mn. k e 8

10 The r degn aem wa o ue he MOI o afy he erformance degn eccaon for he nomnal lan G ung he conguraon of Fgure. The degn crera were, from (6) and (7), where he recrbed bound for 5 6 j j ( W ) " ; () ( G; W; K ) " ; for = ; ; : : : ; ; () no be oo large [], and here aken a D! no xed, bu for ably robune, hould " = 5: : () The erformance funconal ( G ; W; K ) are dened n he Aendx, and he reecve recrbed bound are decded from he degn eccaon and are hown n Table (noe ha " n db and " n rad/ec). An negraor erm wa ncluded n W, whch enure ha he nal conroller ha negral acon and he eady ae eccaon are aed. To enure he eady ae error eccaon are me, he re-comenaor e o be he gan marx K = K () W () where K () W () = lm K ( ) W ( ): () Wh weghng funcon W = w ( + w ) =( + w ) I, and W = I ; he degn rocedure decrbed n Secon 4 wa mlemened n Malab on a Sun SPARCaon, and a degn ha uccefully aed nequale () and () obaned ealy. The erformance wa hen eed wh varou value of,, k and k, and he degn wa found o be no very robu. To oban robu erformance, he nex aem wa o afy he erformance degn eccaon for everal lan model each a an exreme of he arameer range. The degn crera were hence amended o ( W ) " ; (4) ( G ; W; K ) " ; for j = ; ; ; 4 ; = ; ; : : : ; ; (5) where lan G, j = ; ; ; 4 have acuaor me delay and gan hown n Table. Thee exreme lan model were choen becaue hey were judged o be he mo dcul o mulaneouly oban good erformance. Wh weghng funcon a above, a afacory degn wa no acheved, o he order of he weghng funcon W and W wa ncreaed o gve + w + w w5 W = 4 5 ; ( + w + w4) w 6 (6) 9

11 4 k k G : 8 : 8 G : 8 : G : : 8 G : : Table : Exreme lan G, j = ; ; ; 4 j and j + W w w 8 = 7 + w I : 9 To enure ha he weghng are able and mnmum hae, he followng nequale were ncluded n he nequaly e (5): q Re Àw + w À 4w < ; (8) q Re Àw + w 4w < ; (9) À D (7) w < ; () w < : () To aem o afy all he erformance eccaon, a nd degree-of-freedom nroduced by eng K o be a four ae dynamc re-comenaor. K ha a eady ae gan of K () W () o enure ha he eady ae error eccaon are me. Afer mlemenng he MBP, a degn ha uccefully aed nequale (4) and (5) wa ealy obaned. However, necon of he cloed-loo e reone howed a very large amoun of underhoo, o four addonal nequale o rerc he mnmum underhoo o À: were ncluded. The underhoo funconal, ( G ; W; K ), = ; : : : ; 6, are dened n he Aendx, and he recrbed bound for he underhoo funconal are " = : for = ; : : : ; 6. The degn crera were hence amended o ( W ) " ; () ( G ; W; K ) " ; for j = ; ; ; 4 ; = ; ; : : : ; 6 : () Afer he MBP had eraed for abou wo hour, he erformance hown n Table wa obaned wh he wegh ( + : Æ : 7 j) 4: W = 4 5 ; (4) ( + : 9 Æ : 74 j) : 9 À À

12 " ( G ) ( G ) ( G ) ( G ) 4 : : 6 : 6 : 7 : 998 À: 9 À: 84 À: 95 À: 85 À: 9 : 5 : 7 : 79 : 465 : 4 4 : 5 : 4 : 47 : 45 : 4 5 À: 5 À: 97 À: 98 À: 99 À: 4 6 : 7 : 6 : 6 : 6 : 6 7 À: 55 À: 59 À: 59 À: 597 À: : 5 : 94 : 445 : 4 : 64 9 : : : 7 : : 998 À: 9 À: 9 À: 899 À: 958 À: 956 5: 5: 5 5: 6 5: 4 49: 59 5: 47: 47: 47: 47: : : : 56 : : 7 4 : : 9 : 85 : 4 : 5 : : 9 : 8 : 75 : 6 6 : : : 7 : 8 : Table : Performance requremen and nal erformance and ( ) = 77 ( + : W : 5) I ; (5) ( + : 67) and re-comenaor À: 469 À: 74 À: 57 À4: 5 À: 89 À4: 586 À: 868 : 9 À: 8 K = K() W () : (6) À: 59 : 69 : 55 : : 69 À: 99 À: 7 : 8 : À: 79 À: 88 : 7 4: 96 À: 59 : 474 The reulng omal comenaor K had ae. All he e reone crera were aed exce for ( G ) and ( G ). The 5 db gan lm wa margnally exceeded by ( G ), ( G ) and ( G ). The e reone

13 of he 6 oble exreme lan, ung 5h order Pade aroxman for he me delay, À are hown n Fgure along wh he maxmum ngular value of ( I ÀKG) WK, he demand o lan nu ranfer funcon. The recrbed bound on he reone are alo hown n he lo. Over all he exreme lan, he overhoo, re-me and crocoulng n he mulaon are no wore han for he four exreme lan ued for he degn, however, h no he cae for he underhoo. To reduce he underhoo, more exreme lan could have been ncluded n he MOI, bu h would be a he exene of addonal comuaonal eor. The reul comare favorably wh oher degn for he ame roblem [6, 9]. I wa found ha he recrbed gan and bandwdh bound, " and ", were he mo gncan facor n rercng he erformance, f hee bound were ucenly ncreaed, all he erformance eccaon could be me 6 Concludng Remark The ue of numercal mehod o degn he wegh n an -omzaon roblem aear o be new. The rooed mehod combne he exbly of numercal omzaon-ye echnque wh analycal omzaon n an eecve and raccal manner, a demonraed by he degn of a conroller for a hgh ury dllaon column. The MOI neracve, hu rovdng exbly o he degner n formulang a realc roblem and n deermnng degn rade-o. Unlke he LSDP, cloed-loo erformance exlcly condered n he formulaon of he degn roblem, and can nclude boh me and frequency doman erformance ndce. However, wa found n racce ha he nal choce of weghng funcon arameer very moran n he ubequen rogre of he MBP, and he LSDP aroach wa een a ueful n choong he nal arameer and he weghng funcon rucure. The aroach uggeed here ha ome advanage over he uual mlemenaon of he MOI. In he uual mlemenaon, a earch conduced n a e of xed order conroller o ry and nd a feable on. In he aroach here, he earch rerced o conroller whch are already robuly able, hu he roblem of ndng a ably regon doe no ex. Oher mul-objecve numercal mehod ex whch may be ued o olve mlar formulaon of he roblem. Thee nclude he goal-aanmen mehod [, 4], he vecor erformance omzaon mehod [7, 8] and a cla of numercal convex omzaon echnque []. Oher mehod are ummarzed n [4]. H

14 The ue of numercal mehod o degn he weghng funcon arcularly ued o he NLCF aroach becaue no -eraon requred. The MOI can be combned wh H -omzaon mehod whch requre -eraon [, 8], bu he roce conderably lower. The ue of ub-omal conroller would eed u he roce, however he funconal are nnely dconnuou, o radonal earch echnque canno be ued. Genec algorhm can be ued o olve mulobjecve conrol roblem [], and, becaue genec algorhm do no requre he objecve funcon o be connuou, nvegaon are beng conduced no ung genec algorhm o degn he weghng funcon for ub-omal H roblem. The mlemenaon of he MOI uggeed n Secon 4 requre he choce of a nomnal lan. Anoher obly o nclude ome of he nomnal lan arameer a degn arameer, he earch algorhm would hen aem o chooe he `be' nomnal lan ou of a e of oble nomnal lan. In he examle choen here, he erformance wa evaluaed for a elecon of exreme lan model choen by he degner. The roblem of ecenly deermnng he worcae erformance over he range of lan ll ex, alhough more general meaure of erformance robune could be ncluded n he erformance e. Th would be of arcular morance when he range of lan erurbaon le well-known. Aendx - Cloed Loo Performance Funconal A e of cloed-loo erformance funconal f ( G ; W; K ), = ; ; : : : ; 6 g, are de- h h > 4 5 > 6 D ned baed on he degn eccaon gven n Secon 5. Funconal o are meaure of he e reone eccaon. Funconal,, and are meaure of he overhoo;,, and are meaure of he re-me, and and are meaure of he cro-coulng. Denong he ouu reone of he cloed loo yem wh a lan GD a a me o a reference e h demand h( ) by y ([ h h ] ; ); = ;, he e reone funconal are = max y ([ ] ; ); (7) = À mn y ([ ] ; ); (8) = max y ([ ] ; ); (9) = max y ([: 4 : 6] ; ); (4) = À mn y ([: 4 : 6] ; ); (4) = max y ([: 4 : 6] ; ); (4)

15 ! 7 > 8 9 > D À f g À D = À mn y ([: 4 : 6] ; ); (4) = max y ([ ] ; ); (44) = max y ([ ] ; ); (45) = À mn y ([ ] ; ): (46) The eady ae eccaon are aed auomacally by he ue of negral acon. From he gan requremen n he degn eccaon, he H -norm (n db) of he cloed-loo ranfer funcon beween he reference and he lan nu, À = u ( I ÀK( j! ) G ( j! )) W ( j! ) K ( j! ) : (47) From he bandwdh requremen n he degn eccaon, a dened (n rad/mn) = max! uch ha ( I K( j! ) G ( j! )) W ( j! ) K ( j! ) : (48) Four addonal erformance funconal are dened o rerc he underhoo = À mn y ([ ] ; ); (49) = À mn y ([ ] ; ); (5) = À mn y ([ ] ; ); (5) = À mn y ([ ] ; ): (5) Reference [] S. Boyd and C. Barra. Lnear Conroller Degn: Lm of Performance. Prence-Hall, Englewood Cl, N.J., 99. [] P.J. Flemng and A.P. Pahkevch. Alcaon of mul-objecve omzaon o comenaor degn for SISO conrol yem. Elecronc Leer, (5):58{59, 986. [] C.M. Foneca and P.J. Flemng. Mulobjecve genec algorhm. In IEE Colloquum on Genec Algorhm for Conrol Syem Engneerng, age 6/{6/5, London, England, 99. [4] F.W. Gembck. Performance and Senvy Omzaon: A Vecor-Index Aroach. PhD he, Cae Weern Reerve Unvery, Cleveland, Oho, 974. [5] K. Glover and D. McFarlane. Robu ablzaon of normalzed corme facor lan decron wh H -bounded uncerany. IEEE Tran. Auom. Conrol, AC-4(8):8{8,

16 [6] D.J. Hoyle, R.A. Hyde, and D.J.N. Lmebeer. An H aroach o wo degree of freedom degn. In Proc. h IEEE Conf. Decon Conr., age 579{58, Brghon, England, 99. [7] G. Kreelmeer and R Senhauer. Syemac conrol degn by omzng a vecor erformance ndex. In Proc. IFAC Sym. on Comuer-Aded Degn of Conrol Syem, age {7, Zurch, 979. [8] G. Kreelmeer and R Senhauer. Alcaon of vecor erformance omzaon o a robu conrol loo degn for a gher arcraf. In. J. Conrol, 7():5{ 84, 98. [9] D.J.N. Lmebeer. The eccaon and uroe of a conroller degn cae udy. In Proc. h IEEE Conf. Decon Conr., age 579{58, Brghon, England, 99. [] D.J.N. Lmebeer, E.M. Kaenally, and J.D. Perkn. On he degn of robu wo degree of freedom conroller. Auomaca, 9():57{68, 99. [] D. McFarlane and K. Glover. Robu Conroller Degn Ung Normalzed Corme Facor Plan Decron, volume 8 of Lec. Noe Conrol & Inf. Sc. Srnger- Verlag, Berln, 99. [] D. McFarlane and K. Glover. A loo hang degn rocedure ung H ynhe. IEEE Tran. Auom. Conrol, AC-7(6):759{769, 99. [] G. Murad, I. Polehwae, D.-W. Gu, and J.F. Whdborne. Robu conrol of a gla ube hang roce. In Neherland, 99. Proc. nd Euroean Conr. Conf., Gronngen, [4] W.Y. Ng. Ineracve Mul-Objecve Programmng a a Framework for Comuer- Aded Conrol Syem Degn, volume of Lec. Noe Conrol & Inf. Sc. Srnger-Verlag, Berln, 989. [5] I. Polehwae, J.-L. Ln, and D.-W. Gu. Robu conrol of a hgh ury dllaon column ung À K eraon. In Proc. h IEEE Conf. Decon Conr., age 586{59, Brghon, England, 99. [6] S. Skogead, M. Morar, and J.C. Doyle. Robu conrol of ll-condoned lan: Hgh-ury dllaon. IEEE Tran. Auom. Conrol, AC-():9{5, 988. [7] J.F. Whdborne and G.P. Lu. Crcal Conrol Syem: Alcaon. Reearch Sude Pre, Taunon, U.K., 99. Theory, Degn and 5

17 [8] J.F. Whdborne, G. Murad, I. Polehwae, and D.-W. Gu. Robu conrol of an unknown lan { he IFAC 9 benchmark. Leceer Unvery Engneerng Dearmen Reor 9-5, Leceer, U.K., 99. To aear n In. J. Conrol. [9] O. Yanv and I. Horowz. Ill-condoned lan : A cae udy. In Proc. h IEEE Conf. Decon Conr., age 596{6, Brghon, England, 99. [] V. Zakan and U. Al-Nab. Degn of dynamcal and conrol yem by he mehod of nequale. Proc. IEE, ():4{47, (a) me - mn (c) me - mn gan - db (b) me - mn (d) frequency - rad/mn h Fgure : Reone of y ( {) and y (- - -) of all exreme lan o (a) nu h( ), h h (b) nu ( ), (c) nu h( ) and (d) maxmum ngular value of all exreme : 4 h : 6 À lan for ( I À KG) WK. 6