Appropriate Realisation of MIMO Gain-Scheduled Controllers. D.J.Leith W.E. Leithead

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1 Appopte Relsto of MIMO GScheduled Cotolles D.J.Leth W.E. Lethed Deptet of Electoc & Electcl Egeeg, Uvest of Stthclde, GLASGOW G QE, U.K. Astct The dc chctestcs of cotolle desged the gschedulg ppoch c e stogl depedet o the elsto dopted; tht s, the e hch the locl le cotolle desgs e coed to ot olocl cotolle. The pupose of the peset ppe s to vestgte the choce of ppopte elstos fo geel MIMO gscheduled cotolles. A eteded locl le equvlece codto fo MIMO gscheduled ole cotolles s poposed hch ses the cotolle olet. It s sho tht, th fe eceptos, t s possle to else ll gscheduled cotolles s ole cotolles stsfg the eteded locl le equvlece codto d equg the cotolle to do so s ot t ll estctve.. Itoducto Gschedulg cotol hs log pedgee d, hvg ogted flght cotol, s o del eploed vet of pplctos hee hgh pefoce hs to e cheved ove od opetg evelope. Tdtoll, gscheduled cotolle s djusted th efeece to slolvg etell esued qutt hch, soe sese, cptues the ole ehvou of the plt Asto & Wttek 989. Hoeve, the te gscheduled s lso del ppled to ecopss od ge of cotolles cludg those hee the schedulg vle ves pdl Sh & Aths 99, hee tel stte of the cotolle s eploed to plctl schedule the cotolle Rugh 99 d hee the plt s essetll fed ut the cotol ojectves v Leth & Lethed 994, 996. I these ltte pplctos, the cotolles e stogl ole d the dc chctestcs, geel, e lttle elto to those of slolvg eogeousl gscheduled cotolles. Nevetheless, the cotolles e lked the desg ppoch eploed hee ole cotolle s costucted cotuousl tepoltg, soe e, etee the ees of fl of le cotolles. Ech le cotolle s, tpcll, ssocted th specfc equlu opetg pot of the plt d s desged to esue tht, locll to the equlu opetg pot, the pefoce equeets e et. The estece of set of equlu opetg pots hch ecopss the evelope of plt opeto s cetl to ost gschedulg geets d t s suffcet to estct cosdeto to sgle, solted, equlu opetg pot. eplog sees epso lesto hch, locll to the equlu opetg pot, hs sl dcs to the plt, le techques e used to esolve ths locl desg tsk. Cotut s, theefoe, ted th estlshed le desg techques fo hch cosdele od of epeece hs ee ccuulted. It s ko tht the dc chctestcs of cotolle desged the gschedulg ppoch c e stogl depedet o the elsto dopted; tht s, the e hch the locl le cotolles e coed to ot olocl cotolle see, fo eple, Lethed et l. 99, Leth & Lethed 996. Ideed, the pefoce poveet, copso to le cotol, cheved eplog gscheduled cotolle c e etel lost he ppopte cotolle elsto s dopted Leth & Lethed 996. Hoeve, ltc suppot fo the choce of ppopte cotolle elsto s the pool developed. Lece & Rugh 995 popose tht the cotolle elsto should e estcted to esue tht, t ech equlu opetg pot, the dcs of the cotolle sees epso lesto tches the coespodg ee of the fl of le cotolles. A sl ppoch s lso dvocted, oe estcted cotet, Ke et l Hoeve, the esults e of lttle utlt Leth & Lethed 996, 997. Leth & Lethed 996 popose, sted, tht the cotolle elsto should stsf soclled eteded locl le equvlece codto d deve geec elstos hch stsf ths codto fo od clss of SISO cotolles. Eplog elstc eple fo d tue egulto, t s deostted tht elstos hch stsf the eteded locl le equvlece codto povde susttl poveet pefoce copso th elstos hch do ot stsf ths codto cludg those hch stsf the locl le equvlece codto of Lece & Rugh 995. The ojectve of the peset ppe s to futhe develop the ppoch of Leth & Lethed 996 d, ptcul, eted t to ecopss the choce of ppopte elsto fo geel ultput ultoutput gscheduled

2 cotolles. The ppe s ogsed s follos. I secto, soe pel detls e peseted d, secto, cte fo the selecto of ppopte elstos e dscussed. I secto 4, vous pctcl ssues e ddessed d elstos, stsfg the eteded locl le equvlece cteo deved secto, e vestgted. The coclusos e sused secto 5.. Gschedulg desg The tdtol eogeousl gscheduled cotolle, hch s djusted th efeece to etell esued vle, t, hs the fo At t, Ct Dt The dc popetes chge th t ut, povded tht the te of chge s ot too pd, the the dc popetes of the tevg cotolle,, e sl to those of the le cotolles oted feezg the vlue of ; tht s, the ole cotolle hets the dc popetes of the fl of le cotolles see, fo eple, Desoe 969. It s oted tht thee e o dect estctos o the stte,, o the put,. The ol estcto s o the te of chge of the schedulg vle; deed, he the schedulg vle s costt, the cotolle s le. Hoeve, ttes e ot so sple he the schedulg vle s fucto of tel quttes. I ths ppe, cosdeto s estcted to the ltte.. Le sthess I the gschedulg desg ppoch, the usull ole plt dcs e ppoted, locll to specfc equlu opetg pot, sees epso lesto. A le tevt cotolle s the desged hch esues ppopte closedloop pefoce he eploed th the plt lesto. Ths pocess s epeted fo set of equlu opetg pots, coveg the evelope of opeto, hlst esug tht the le cotolle desgs hve coptle stuctues; fo eple, he soothl gscheduled cotolle s equed, the le cotolle desgs e selected to pet sooth tepolto, soe ppopte e, etee the desgs. At equlu opetg pot of the plt, ll of the plt puts, cludg the cotolle output R p, d ll of the plt outputs, cludg the cotolle put R, e costt. It s ssued tht the equlu opetg pots of the plt e ll petesed qutt, π R q. Cosequetl, oth the equlu opetg pots of the gscheduled cotolle d the locl le tevt cotolles lso e petesed π. Hece, thee ests fl, o π, o π, of costt cotolle puts d outputs d fl of le tevt cotolles, δ A δ δ, δ C δ D δ π π π π π π π π π π δ π oπ, o π δ π coespodg to the set of plt equlu opetg pots. Clel, s ot uque d the sttes, δ π, e suject to osgul le tsfoto hch e dffeet fo ech equlu opetg pot thout chgg the dcs. I ddto, t s ssued tht the degees of the put, output d stte e ot depedet o π d tht the tces, A, π, π C d π D π e cotuous th espect to π.. Nole elsto I ddto to the sthess of fl of le cotolles, the gschedulg desg ppoch eques the deteto of sutle ole cotolle; tht s, the deteto of ppopte elsto. It s ths step hch eceves cosdeto the peset ppe. Nole gscheduled cotolle elstos of the fo, F,, G, e cosdeed, hee F, d G, e cotuous d dffeetle, d R deotes the stte of the cotolle. The set of equlu opetg pots of the ole cotolle,, cossts of those pots, o, o, o, fo hch F o, o, o G o, o 4 Let Φ:R R deote the spce cosstg of the uo of the stte,, th the put,. The locus of the set of equlu opetg pots of the ole cotolle, o, o, fos sufce Φ d the espose of the cotolle to geel tevg put, t, s depcted tjecto Φ. Assue tht 4 s sutl vetle so tht the equlu vlues of the stte, o, e deteed fo the equlu vlues of the cotolle put, o, d output, o. Ths ssupto esues tht the equlu chctestcs e depedet of the choce of cotolle stte. Fo cosstec th secto., the set of equlu opetg pots, o, o, ust e the se s the fl o π, o π d the sufce of equlu opetg pots Φ ust e of deso q. The qutt, π, povdes de fo the equlu opetg pots hch s depedet

3 of the elsto hees the pots o, o theselves, d the ppg fo π to o, o d the vese ppg fo o, o to π, e depedt o the choce of elsto,. The schedulg vle s fucto, : o, o R q, hch s soophs of the vese ppg fo o, o to the petesto of the equlu opetg pots. Hece, the set of equlu opetg pots fo the ole cotolle,, c e petesed th espect to ; el, o, o, o. The do of s usull ot cofed to the set of equlu opetg pots, o, o : t geel opetg pot,,, :, R q such tht, o, o fo soe o, o. The schedulg vle, thee, dctes soe ee of the le fl,, t te, egdless of hethe the sste s equlu. I ths ppe, t s ssued tht s cotuous fucto; tht s, the dscusso hee ecludes osooth schedulg such s stchg. Tpcll, the selecto of ppopte schedulg vle s sed o phscl sght. The ole cotolle,, s equed to hve, soe sese, sl dc popetes to the ees of the le fl,, fo the dest possle ge of opetg codtos. Itll, cosde the cse hee the cotolle s opetg locll to specfc equlu opetg pot, o o, o o. Thee ests clss, C, of puts d tl codtos fo the stte, fo hch the ole cotolle d the ssocted ee of the le fl e judged to hve sl dc ehvou. The defto of the cte, hch the slt of dc ehvou s ssessed, s depedet o the cotet, hch the equeet fo gscheduled cotolle ses, d o the tue of the cotolles. Let the clss, C, e the uo ove the o of the clsses C. No cosde the cse hee the cotolle s ot cofed to opetg locll to soe specfc equlu opetg pot. Thee ests clss, C N, of puts d tl codtos fo the stte such tht the espose of the ole cotolle, hlst the vct of ptcul equlu opetg pot, s judged, soe cotetull ppopte sese, to hve sl dc ehvou to the coespodg ee of the fl of le cotolles. The clsses, C d C N, e oept sce the cot the equlu vlues o o d o o. The deftos of the cte, hch the slt of dc ehvou s ssessed the to cses, e ssued to e such tht, fo sequece of ole elstos fo fl of le cotolles, the ted s fo the eeshp of the clss C N to cese s the eeshp of the clss C ceses. Ths ssupto s ot utul sce the eeshp of the clsses s eflecto of the stegth of the cotolle olet hteve es t s defed. It should e oted tht the eltoshp etee the equlu put d output of the cotolle, o d o, c hve cosdele pct o the ole cotolle desg. Whlst ths ssue s lgel eglected the ltetue, t s cle tht stog estcto s posed o the clss of llole ole cotolles; fo eple, the fuctos, F, d G,, e ot depedet sce, F o, o, G o, o o ust e jotl stsfed d, he F, s vetle so tht o s deteed o, t follos tht the output fucto, G,, s, essetll, copletel specfed the equlu put/output eltoshp d the choce of F,. It s possle to stsf the eltoshp etee the equlu put d output, hlst tg desg feedo, cludg eplct fuctos fo o d o th the cotolle Sh & Aths 99, Lece & Rugh 995. Hoeve, the devto of the eplct fuctos s eteel oeous d qute udesle ost pplctos. Uless othese stted, t s ssued tht the cotolles dscussed elo e coptle th the equlu put/output codtos. ο ο. Pevous ppoches Pehps the ost coo ppoch to elsg gscheduled cotolle s to spl susttute schedulg vle,,, fo π the fl of locl le cotolles,, to ot ole cotolle. Whe the schedulg vle,, ves suffcetl slol tvll, he the te of vto s zeo, the dc chctestcs of the le fl e heted the ole cotolle. Hoeve, the ole cotolle the schedulg vle,, ves th the put d/o the stte hlst the locl le cotolle,, t s fed. Hece, geel, the sees epso lesto of the ole cotolle out equlu opetg pot, t hch the schedulg vle hs the vlue o, cots tes, elted to the petutos, δ, the schedulg vle, ot peset the le fl, hee δ equls o. Cosequetl, fo the ole cotolle to hve sl dcs to the le fl, t s ecess to pose costt o the vto of ; el, tht the fst ode petutos δ, d so δ d δ, ust e eglgle. Ths costt pples eve he the stte s cofed to sll eghouhood out sgle equlu opetg pot hch s clel ot po ecess. Hece, fo ths clss of ole cotolle the costts posed o see uecessl stct. As ltetve, Lece & Rugh 995 d lso Ke et l. 995, oe estctve cotet popose tht the ole cotolle should e selected to esue tht ts sees epso lesto t ech equlu opetg pot coespods to the ppopte ee of the le fl,. I othe ods, th the

4 eghouhood, out specfc equlu opetg pot, fo hch the secod ode petutos δ d δ e eglgle, the ole cotolle s equed to hve sl dcs to the ppopte ee of the le fl. The ole cotolle, theefoe, hs the equed dc ehvou, locll to ech equlu opetg pot, fo less stct costt o δ d δ th the pevous ppoch.. Selecto cte fo the elsto of gscheduled cotolles It s clel ttctve to eque tht the cotolles e cosstet th the ppoch of Lece & Rugh 995 sce the sees epso lesto of the cotolle, t ech equlu opetg pot, the coespods to the ppopte ee of the le fl,. Hoeve, thee est ftel ole cotolles stsfg the locl le equvlece codto of Lece & Rugh fo ptcul fl of le cotolles. The sze of the clsses, C, of puts d tl codtos v getl fo dffeet choces of cotolle elsto d, ο fct, e eltvel lge fo soe choces ut vshgl sll fo othes. Sce the sze of the clsses of vld puts d tl codtos s vle d e uecessl estcted the choce of elsto dopted, the equeet of locl le equvlece s, tself, dequte gude to the choce of ll ole cotolle elsto; tht s, elsto fo hch the estctos o the puts d the tl vlues of the sttes e l Leth & Lethed 997. A cteo s, theefoe, equed hch povdes gudce to those elstos stsfg locl le equvlece tht do ot uecessl dsh the clss of vld puts d tl codtos.. Eteded locl le equvlece The etet to hch the dcs v ove the locus of equlu opetg pots, eg defed the fl of le cotolles,, s clel heet to the gscheduled cotolle. Hoeve, the etet to hch the dcs v s the opetg pot s dsplced fo the locus of equlu opetg pots s, cotst, depedet o the cotolle elsto. Restctos e posed o the puts d tl codtos oth spects of the cotolle olet. The estcto posed the fst spect s soeht sl to tht ecouteed eogeousl gscheduled cotolles; oughl spekg, the puts d tl codtos e estcted to pevet the locus of equlu opetg pots fo eg tvesed too pdl thee pevetg the dc popetes, hch v ove the locus of equlu opetg pots, fo vg too pdl. Whe the puts d tl codtos e such tht the vto the schedulg vle s sll d cosdeto e cofed to sgle ee of the le fl, ths estcto does ot ppl. Hoeve, the secod spect stll poses estcto o the puts d tl codtos. The estcto posed the secod spect s, evdetl, ddtol to d, soe sese, depedet of the fst. The ole cotolle,, e efoulted s, δ F o o, o o δ F o o, o o δε F 5 δ G o o, o o δ G o o, o o δε G 5 δ o o, o o δ 5c hee, ε F F, F o o, o o δ F o o, o o δ 5d ε G G, G o, o G o o, o o δ G o o, o o δ 5e o o δ 5f fo o. The locl le equvlece equeet s stsfed t equlu opetg pot, t hch hs the vlue, o, povded the devtves F, F, G d G tch, th osgul tsfoto of the sttes, the tces, A, π, π C d π D π, of the ppopte ee of the fl of le cotolles. Whe ssessg hethe the dc ehvou of the ole cotolle,, s locll sl to ptcul ee of the fl of le cotolles, t tull e ssued tht the cte eploed poses estcto o the gtude of the esduls, ε F d ε G. I geel, the esduls e depedet o eve eleet of δ d δ d, theefoe, estcto s posed o the gtude of eve eleet. Hoeve, of these estctos e ot po ecess. The heet ole chcte of the gscheduled cotolle s eoded the vto etee the ees of the fl of le cotolles,. It follos tht the gscheduled cotolle s heetl ole th espect to the schedulg vle,, d ptcul ee of the fl of le cotolles s heetl ol vld he the petuto, δ, s estcted to soe ge hch s depedet o the popetes of the fl of le cotolles. To se the estctos o the petutos the eleets of,, t s, theefoe, qute tul to eque tht the esduls deped puel o δ loe; tht s, o d o othe cotos of the eleet. Hece, the olet ust puel e fucto of. Moeove, sce ll peteses the fl of le cotolles, t s cle tht ths s the ekest fuctol depedece of the esduls possle.

5 Whe the cotolle olet s puel fucto of, the ole cotolle,, ust e of the fo, A f, C D g 6 hee A,, C, D e costt tces, f d g e dffeetle ole fuctos, d, e fuctos of loe. The schedulg vle,, R q, equls the costt vlue, o, upo sufce of codeso q Φ d d e costt ove ech sufce. Hece, the ol to ech sufce s detcl t eve pot o the sufce d ech sufce s, theefoe, ffe. Moeove, to esue tht s uque fucto of d, these sufces ust e pllel fo ll. Cosequetl, t e ssued, thout loss of geelt, tht d e costt d s le coto of the eleets of the stte d put. The cotolle dcs e le those cotos of the eleets of, tht e lel depedet of. The lesto of the ole cotolle, 6, t specfc equlu opetg pot, o o, o o, s, δ A f o δ f o δ 7 δ C g o δ D g o δ 7 Uecess estctos o the eeshp of the clsses, C ο, of puts d tl codtos fo the stte, fo hch the lestos e locll vld, e voded the elsto, 6; ths sese, the sze of the clsses, C ο d so C, s sed. Sce the eeshp of the clss, C N, teds to cese s the eeshp of the clss, C, ceses, eeshp of C N s, geel, gete fo cotolle th the elsto 6,.e. stsfg the soclled eteded locl le equvlece codto, th fo cotolle stsfg the locl le equvlece codto of Lece & Rugh 995. The estcto posed o the clss of puts d tl codtos the etet, to hch the dcs v s the opetg pot s dsplced fo the locus of equlu opetg pots, s copletel eled fo the elstos stsfg the eteded locl le equvlece codto. Ol the heet estcto posed o the clss the etet to hch the dcs v ove the locus of equlu opetg pots es plce. The foulto, 6, s equvlet, SISO cotet, to the eteded locl le equvlece codto poposed Leth & Lethed 994, Lesto t oequlu opetg pots Fo cosstec th the estcto o the gtude of the esduls posed the equeet tht the dc ehvou of the ole cotolle,, s locll sl to ptcul ee of the fl of le cotolles, the stte, δ, of the ole cotolle, 5, d the put, δ, ust elog to soe eghouhood of o o, o o th hch the e suffcetl sll tht ε F d ε G e eglgle. Altetvel, the stte of the le cotolle, oted fo 5 settg ε F d ε G to zeo, d the put could e equed to elog to the eghouhood. Although the clss, fo hch ths ltte equeet s et, s ot detcl to C, t s ot susttll dffeet. Whe copg the eeshp of the clss of puts d tl codtos fo dffeet elstos, ths ltetve equeet hs the dvtge tht the stte does ot chge th the elsto sce the lestos e the se. The stuto fo cotolle stsfg the locl le equvlece codto of Lece & Rugh 995 s llustted fgue fo SISO fstode cotolle hlst the stuto fo cotolle stsfg the eteded locl le equvlece codto, s llustted fgue : the shded egos otoll dcte the eghouhoods th hch the δ d δ e suffcetl sll tht ε F d ε G e eglgle d ptcul lesto s vld. It should e oted tht the eghouhoods fo the elsto, 6, e fte etet, hvg o po estcto the dectos hch s costt. Iespectve of dffeece sclg of the sttes, the eghouhoods fgue e sgfctl lge th the eghouhoods fgue. The ole cotolle, 6, c e lesed t opetg pot, cludg oequlu opetg pots, fo hch the vlue of s th ts do; cses t opetg pot the spce, Φ. A pot, fo hch hs the vlue o, s the eghouhood ssocted th the equlu opetg pot fo tht vlue of o. Hece, t oequlu opetg pot the ole cotolle c e lesed ssoctg t th the le cotolle t the equlu pot fo hch the vlue of s the se; tht s, t pot fo hch the schedulg vle hs the vlue o, the lesto of 6 s 7. Note, the lesto s ot oted petug the sste out the oequlu opetg pot d eglectg the hoogeeous te, lthough tht ould esult sl descpto fo those sstes stsfg the eteded locl le equvlece codto. Ths lesto s vld eghouhood of the oequlu opetg pot hch s coted th the eghouhood ssocted th the coespodg equlu opetg pot. Hece, the eteded locl le equvlece codto c e tepeted s equg tht the lesto of the ole cotolle coespods to the ppopte ee of the le fl t ll opetg pots d ot just the equlu opetg pots s equed the locl le equvlece codto of Lece & Rugh 995. ο

6 . Eple The foegog pots e llustted the follog eple. Suppose tht the fl of secodode SISO le cotolles, δ A δ δ, δ C δ D π δ 8 π π π hee δ [δ δ ] T,, C [ g ], D π Relsto A Cosde the ole cotolle oted spl eplcg g π 8 the fucto g s depcted fgue. Such ppoch s del eploed to costuct gscheduled cotolles. The dcs of the ole cotolle e desced A f,, C D g, hee [ ] T A,, C [, ] D, f,, g, g g d s soluto to the plct ole equto, g c Fo ssuptos d, the stght les of costt, c, cove the coplete, ple. Hece, thee ests t lest oe vlue of stsfg c fo ll d. Whe oe th oe vlue ests, the ppopte oe s deteed cotut. The tl codto fo the stte,, ust e ccoped ppopte, cotet depedet, choce of. The locus of equlu opetg pots s the set of pots, o /g o, o,, < o <, d the schedulg vle,, s costt o the ples g o. Locll to equlu opetg pot, t hch the ol vlue of s o, δ, o δ, o δ, o δ d the ole cotolle,, c e efoulted s A π g π π π 8 π hs ee desged, hee g π ves s cotuous fucto of the equlu opetg pot d s ozeo. The tsfe fucto of the ee of the fl ssocted th π equl to π o s, s Ys g πo Rs 9 ss hee Ys, Rs e the Lplce tsfos of, espectvel, δt d δt. The vlues of g π e ssued to e fte d ethe stctl postve o stctl egtve; tht s, thee est d such tht < g π < o < g π < The output,, s selected s the schedulg vle o the ssupto tht the equlu opetg pots of the cotolle c e petesed d tht thee ests cotuous fucto, g : R {g π }. Og to the pue tegl cto peset the ees of the fl, 8, the put to the cotolle ust e zeo t equlu opetg pot d the locus of equlu opetg pots cot, theefoe, e petesed the put. δ ε g o δ δ 4 δ [g o ]δ ε 4 hee ε [ g o δg o o δ ] T, ε g o δg o o δ 4c The le sste oted settg ε d ε to zeo 4 s the se s 8 s equed. Fo the lesto to e locll vld, δ d δ ust e suffcetl sll tht ε d ε e eglgle. It should e oted tht, s o ves d the gtude of o ceses, the eeshp of the clss, C, pdl dshes utl t ecoes, tes of the cotet, sgfct. Ths s cosequece of the elsto ot stsfg the locl le equvlece codto, th ο

7 the esult tht fst ode petutos δ e peset the epessos fo ε d ε. Uless the opetol evelope of the cotolle s cofed to the vct of the equlu opetg pots fo hch the gtude of o s estcted, ε d ε c e tl lge fo soe o fo ozeo vlues of δ. Relsto A ltetve ole cotolle, hch eplos veloct foulto sl to tht poposed Ke et l. 995 d th the tegleo fo dscussed Lece & Rugh 995, s sho fgue. The dcs of the cotolle e desced A f,, C D g, 5 hee [ ] T g,, [ ],,,, g, A C D f 5 g The locus of equlu opetg pots s the set of pots,, o,, < o <, d the schedulg vle,, s costt o the ples o. Locll to equlu opetg pot t hch the ol vlue of s o, δ, δ, δ, o δ 6 d the ole dcs, 5, c e efoulted s g o δ δ g o δ ε 7 δ [ ] δ 7 hee ε [g o δ g o δ g o δ g o δ] T 7c The le sste oted settg ε to zeo 7 s the se s 8 s equed. Fo the lesto to e vld, δ d δ ust e suffcetl sll tht ε s eglgle. It should e oted tht, s o ves, the eeshp of the clss, C, es sgfct. Ths s cosequece of the ole cotolle stsfg the locl le equvlece ο codto. Hece, geel, the clsses, C ο, fo elsto, e lge th the clsses, C ο, fo elsto A. Clel, elsto s oe ekl ole th elsto A. Hoeve, t s evdet tht the cotolle of fgue does ot stsf the eteded locl le equvlece codto. Relsto C No, cosde the ole cotolle sho fgue c. Its dcs e desced the dffeetl equto A c c f c,, C c D c g c, 8 hee [ ] T, Ac, c, Cc [ ], Dc, fc,, gc, g 8 The elsto, 8, does ot eplctl hve the eteded locl le equvlece codto fo, 6. Hoeve, the cotolle dcs, 8, e efoulted s A f,, C D g, 9 hee [ ] T, ˆ A, ˆ c G gs ds c c c c c c, Cˆ c, Dˆ c c, fˆ,, ĝ, Gˆ c hch eplctl stsfes the eteded locl le equvlece codto. ssupto, G ests sce G s ootoc d, theefoe, vetle. The locus of equlu opetg pots s the set of pots, o,, < o <, d the schedulg vle,, s costt o the ples G o. Locll to equlu opetg pot t hch the ol vlue of s o, δ, δ, o δ, o δ d the ole dcs, 9, c e efoulted s, c 9 9c

8 δ ˆ δ δ g δ ε δ [ o ] hee ε g o [G o δ G o G o δ] c The le sste oted settg ε to zeo s the se s 8 s equed. Fo the lesto to e vld, ol δ ust e suffcetl sll tht ε s eglgle. Ths s cosequece of the ole cotolle elsto C stsfg the eteded locl le equvlece codto. I geel, the eeshp of the clsses, C, fo the elsto C e lge th the clsses, C ο, fo elsto sce the gtude of oth the put, δ, d the output, δ, e estcted the ltte ut ol the gtude of the output, δ, s estcted the foe th o p o estcto o the put, δ. Clel, elsto C s oe ekl ole th elsto. The eghouhoods ssocted th the equlu opetg pots cove the ete, ẍ, spce. ο Whe copg the thee elstos, the clsses, C ο, e suffcetl dffeet tht thee s o eed to ke the stte spce fos of the le cotolles detcl tsfog the sttes. The foegog sple eple does ot dectl cosde the pefoce eefts of doptg gscheduled cotolle tht stsfes the eteded locl le equvlece codto d hch, thee, hs ppopte dcs fo the dest clss of puts d tl codtos. Hoeve, these eefts, hch ust, s usul, e cofed lss d/o sulto studes, c e qute cosdele. Fo eple, the cotet of d tue egulto, Leth & Lethed 996 oseve tht ppoptel elsed gscheduled cotolle cheves susttl poveet pefoce copso to elldesged le cotolle. Hoeve, ths pefoce g s effectvel lost he gscheduled cotolle elstos e eploed hch stsf ol locl le equvlece out the equlu opetg pots the th the eteded locl le equvlece codto. Cosequetl, the tue of the elsto dopted pls cetl ole ttg the equed pefoce. The cotet of d tue egulto s ptcull ell suted to ssessg the effectveess of the ole cotolle tht, eve though the plt s fed, the cotolle s stogl scheduled; tht s, the olet s stog d the schedulg vle cot po e ssued to e slol vg.. Relto of estcto o tl codtos I secto., ll ole cotolle elstos e defed; el, those stsfg eteded locl le equvlece codto, 6. Fo these elstos, the estcto posed o the clss of puts d tl codtos of the stte the etet, to hch the dcs v s the opetg pot s dsplced fo the locus of equlu opetg pots, s copletel eled. Ol the estcto posed o the clss of puts d tl codtos the etet, to hch the dcs v ove the locus of equlu opetg pots, es. The ltte s essetll estcto o the te of vto of the schedulg vle 6. The espose of the sste, 6, d so the te of vto of the schedulg vle depeds o the put d tl codtos of the stte. I geel, the depedece o the tl codtos ecoes stoge s the dsplceet of the tl codtos fo the locus of equlu opetg pots ceses. Hoeve, ol the depedece of the te of vto of the schedulg vle o the put s ell heet to the ole cotolle d fo soe choces of elsto the depedece o the tl codtos s ekeed. Cosde the stuto he the ozeo os of e o egevectos of A 6. Ude these ccustces A f Λ f Λ hee Λ s dgol t, the ozeo dgol eleets of hch e egevlues of A. A estcto o the te of vto of the schedulg vle,, poses estcto o the put,, d the tl codtos of the schedulg vle. Thee e o estctos o those le cotos of the stte hch e lel depedet of the schedulg vle. Eple Cosde the ole cotolle A f,, C D g, hee [ ] T d th f A,, [, ] D d,,,, df f d C f df f g 4 The tsfe fuctos fo the fl of le cotolles, oted lesg the ole cotolle, e

9 ds s f s s f d the dc ehvou of the schedulg vle,, s desced f 5 6 Of ptcul teest e those elstos, stsfg the eteded locl le equvlece codto, fo hch the schedulg vle depeds solel o the put; tht s, fo hch s zeo. Wth these elstos, thee e o estctos o the tl codtos of the stte: estcto s posed the etet, to hch the dcs v ove the locus of equlu opetg pots, o the clss of puts ol d ot o the tl codtos. Whe the put s costt, the schedulg vle s costt d the dcs of the cotolle e puel le. The stuto s ectl logous to the eogeousl gscheduled cotolle,..4 Equvlet fles of le cotolles I secto., elstos stsfg the eteded locl le equvlece codtos e defed th efeece to the fo of the ole cotolle. Equvletl, the c e defed th espect to the fo of the fl of le cotolles. Assocted th ech fl of le cotolles,, s the fl of tces A π π Σ π 7 Cπ D π hee the deso of π s q. Cosde the dffeece t etee fed ee of the fl, Σπ o, d othe ee, Σπ. S π S π π Σπ o Σπ 8 I I hee Sπ R s osgul d coespods to stte tsfoto th espect to hch the dcs of the le cotolle t π e uchged. Povded tht the k of ll π, fo soe Sπ, s q d, ddto, the os of ll π elog to qdesol suspce of R p q, the thee ests fˆ π d ĝ π such tht fˆ π π [ M N] 9 gˆ π hee [M N] s costt t, hs k q d sps the qdesol suspce. It follos tht A fˆ π M fˆ π N Σ π C gˆ π M D gˆ π N hee A Σ π o C D Assue π s elted to the qdesol schedulg vle,, the oetooe ppg πh d tht thee est solutos, f d g, of, espectvel, f fˆ h, g gˆ h, The ltte ssupto s stsfed he f f, g g, j, hee ι j j j j deotes the th eleet of. Whe the fl of cotolles s deved soothl tepoltg etee fte ue of cotolle desgs, ths equeet s ot estctve. The usul gschedulg desg ppoch s to desg le cotolles fo ue of dstct equlu opetg pots spg the opetg evelope. Moeove, the deso the schedulg vle,, s tpcll qute sll. Hece, the codto,, s edl stsfed. The fl of le cotolles,, c e elsed ole cotolle, 6, fo hch M N povded tht thee ests, fo ll vlues of, soluto of

10 o g f N M D I C A o o 4 such tht o, o o, o he. Futheoe, the o, o, o e the equlu opetg pots of the ole cotolle. It should e oted tht the estece of soluto of 4 does ot deped o the choce of h. Whe the t o the lefthd sde of 4 s full k, the estece of soluto egdless of the choce of h s cle. Whe the t s ot full k, soluto of 4 ests povded, fo soe costt tces X, Y d Z, depedet o A,, C, D, M d N d ot ll zeo, Xf Yg Z 5 fo ll. The tl codtos fo ust e chose such tht 5 s stsfed fo soe o ; fo eple, choose o d the tl codtos fo 5 to e f g. It follos tht 5 s stsfed povded X f Y g Z 6 hch s equvlet to ˆ ˆ π π Z Yg Xf 7 Sce 7 d the choce of the tl codtos fo 5 e depedet of h, the estece of soluto of 4 does ot deped o the choce of h. Dffeet h coespod to the dffeet soophss possle he defg the schedulg vle. Clel, ot ll fles of le cotolles,, c e elsed ole cotolle, 6, stsfg the eteded locl le equvlece codto. Hece, eteded locl le equvlece ght ppe to e the stog codto..5 Utlt of elstos Sce eteded locl le equvlece ppes to e the stog codto, cotolles stsfg t ght e epected to e of lted utlt. It s, theefoe, ecess to detee hethe equg the gscheduled cotolle to e ee of the clss of ole cotolles, hch stsf the eteded locl le equvlece codto, s ovel estctve. Cosde fl of put sgleoutput le cotolles. The tsfe fucto eltg the output, t, d the th put, t, s s s... s Ys d R s 8 s s... s hee Ys d R s e, espectvel, the Lplce tsfos of t d t, d t s ssued tht Ys/R s Ys/R j s j The coeffcet, d, s supefluous d could e soed to the coeffcets, j, thout loss of geelt. Accodgl, ssug d s ut, t s evdet tht ech tsfe fucto s copletel specfed ts ueto coeffcets, j, d deoto coeffcets,. Sce the tsfe fuctos hve, thout loss of geelt, coo deoto, the putoutput chctestcs of the th ode put sgleoutput cotolles e specfed coeffcets totl. A coespodg l sttespce epesetto s 9 [ ] [ ] d d 9

11 hee, og to the ptcul choce of sttes, the tsfe fucto coeffcets ppe stghtfod e the sttespce tces. Of couse othe, l, sttespce epesetto stsfg the putoutput eltoshps, 8, e deved fo 9 sutle stte tsfoto. It follos fo 7 tht fl of le cotolles coespodg to ole cotolle stsfg the eteded locl le equvlece codto hs, t ost, q degees of feedo. I ll ut the eceptol cse he q ote, q, q< d ot ll of the coeffcets of 8 c e feel ssged; tht s, ol suset of q coeffcets 8 c e scheduled depedetl suject to stsfg, th the eg coeffcets ethe depedet o ths suset o costt. The dett of the costt coeffcets, d the eltoshps etee the ede, deped o the defto, tes of le cotos of the eleets of the stte d put, of. Nevetheless, th fe eceptos, ll gschedulg equeets c e ccoodted sce, th the pelt of ol elsto, the ode,, of the cotolle e cesed to povde t degee of flelt. A fl of le cotolles, equvlet to the ole cotolle stsfg the eteded locl le equvlece codto, e costucted to ppote closel fl of pevousl desged, pehps loe ode, le cotolles. Altetvel, dug sthess, the le cotolles c e dectl desged to hve the equed ptte of coeffcet schedulg depedece. Eple The fl of le cotolles, desged fo d tue Leth & Lethed 996, cosst of fed copoet Ks Cs/s 4 hee s.7s.8s 7.59s 68.6s s 4.4s s Cs s.s.7s s s s 4.4s 8s 46.6s 65.8s 9 togethe th copoet, hch ves th the equlu opetg pot,..47s.s Cπ s g π s πs π hee π peteses the equlu opetg pots. The choce of π s, of couse, ot uque. Selectg π to e elted to the d speed, V, π.44v.95v4.97 4d the vlue of π coespods to the ptch gle of the tue oto ldes, t the equlu opetg pots, d π.π.75π.75, π.6π58.4, gπ.8π.98 4e I 4c ll the poles v th the opetg pot hle the zeoes e costt. Suppose t s equed, sted, tht the zeoes v hle the poles e costt. The fl of le cotolles, 4, c e edesged such tht K sc s/s 4 hee C s.7s.8s s 4.4s s s 4 s.s.7s s 4.4s 8s 46.6s 65.8s 9 d 4 C π's π' s π' s 4 π' s 5 π' π s 4c 4 s 5.8s 88s 65s 4 Selectg π to e elted to the d speed, V, π.7v 4.7V 9.V V44.7.4π 88.5π.7 4d the vlue of π coespods to the put to C π s, t the equlu opetg pots, d π π.5 π.4, π.4 5 π.68 π.57, π.78 4 π.5 π 46., 4 π π.7π 676.6, 4e 5 π.8 π.9π 95. The equed schedulg of tsfe fucto coeffcets hs ee cheved. As ght e epected, copso to the ode of C π s, the ode of C π s s cesed fo to 4. Hoeve, the ovell ode of the cotolle s ot cesed sce, copso to the ode of Cs, the ode of C s s decesed fo to 8. The ode plots of the to fles of le cotolles, C π scs d C π sc s, e effectvel dstgushle t equvlet equlu opetg pots. Although the ol otvto, ths cotet, fo covetg fl of le cotolles fo oe fo to othe ould e to chge the fl fo oe ot stsfg the eteded locl le equvlece codto to oe stsfg ths codto, o ths occso oth C π s, th the schedulg vle ts output, d C π s, th the schedulg vle ts put, e equvlet to ole cotolles stsfg the eteded locl le equvlece codto. Ths eple, thee, lso llusttes the depedece, o the defto of the schedulg vle, of the 4 4c

12 e hch the le cotolles tsfe fucto coeffcets e elted to the schedulg vle; tht s, the dett of the costt coeffcets d the tedepedece of the ede. The ove guets c clel e eteded to the MIMO cse. Hece, geel, estcto o the desg of gscheduled cotolles, due to doptg elsto stsfg the eteded locl le equvlece codto, c e epected to e the ek. The ppet stegth of the eteded locl le equvlece codto s deceptve, let, t the epese of doptg elstos of, pehps, ol degee. 4 Pctcl elstos I secto, the popetes of ptcul ole cotolles, el those stsfg the eteded locl le equvlece codto, e vestgted. The e ll ole ut suffcetl flele to ct s elstos fo ll gscheduled fles of le cotolles. Hoeve, sevel ssues cetl to the pctcl desg d elsto of gscheduled cotolles e ot ddessed: fstl, desgg the fl of le cotolles, fo hch the coeffcets of the tsfe fuctos e ot depedet of ech othe ut ust stsf cet eltoshps; secodl, deteg the ole fuctos of the schedulg vle the ole cotolle, the ptl devtves of hch e elted to the le cotolles tsfe fucto coeffcets; thdl, esug tht the cotolle put d output hve ppopte vlues t the equlu opetg pots. These ssues e dscussed elo. 4. Septo of coeffcets The fst ssue, el the tedepedece of the le cotolles tsfe fucto coeffcets, s esolved septg suset of the coeffcets, ll of hch e essetll feel scheduled, th the eg coeffcets costt. Although ll of the pevousl cosdeed eples, hvg ee chose fo splct d clt, do cofo to ths esoluto of the tedepedece of the le cotolles tsfe fucto coeffcets, t s cle fo 7 tht, geel, the fl of le cotolles eed ot. Whethe the septo of the coeffcets occus s depedet o the defto of the schedulg vle. It s suffcet to vestgte put sgleoutput cotolles sce geel MIMO cotolle e oted sutl cog sevel ultpleput sgleoutput cotolles. Cosde the put sgleoutput ole cotolle elsto, depcted fgue 4, [ ] 4 fo soe fuctos j, j,,.., hee [ ] T. Povded tht the locus of equlu opetg pots c e petesed the put,, the cotolle, 4, hs the fo equed 6. Hece, ths cotolle elsto stsfes the eteded locl le equvlece codto th the schedulg vle spl the put,. Moeove, t equlu opetg pot, the lesto of 4 d ts tsfe fuctos hve the fos, 9 d 8, espectvel, th j j o,,,..,, j,,.., 4 Adopto of the cotolle elsto of fgue 4 eles the eteded locl le equvlece codto to e stsfed fo fles of put sgleoutput le cotolles, fo hch the cotolle zeoes e scheduled essetll depedetl th espect to the put,, ut the cotolle poles e fed. Hece, the equeet to septe suset of coeffcets, those of the uetos of the tsfe fuctos ths elsto, ll of hch c e feel scheduled, th the eg coeffcets, those of the deoto, costt, s et. It should e oted tht the le cotolles ssocted th 4 hve the u possle schedulg degees of feedo, q. Altetvel, cosde the put sgleoutput ole cotolle elsto, depcted fgue 5, Gz, ˆ z, ˆ z, ˆ z, ˆ z, 44

13 fo soe fuctos G, d j,, j,, hee, [ ] T d z. Povded tht the locus of equlu opetg pots c e petesed put,, togethe th the output,, the the cotolle, 44, hs the fo equed 6. Hece, ths cotolle elsto stsfes the eteded locl le equvlece codto th the schedulg vle, T T. Moeove, t equlu opetg pot, the lesto of 44 d ts tsfe fuctos hve the fos 9 d 8, espectvel, th j z j z o, o, j,,.., d z Gz o ; o,,,..; j j z o, o,,,.., j,.. 45 Wthout loss of geelt, t e ssued tht the devtve of G s ethe stctl postve o stctl egtve. It follos tht G s ootoc d G ests. Hece, the ole cotolle, 44, e tepeted s eg scheduled o the output, let plctl, d the eleets of the put,,. Adopto of the cotolle elsto of fgue 5 eles the eteded locl le equvlece codto to e stsfed fo fles of put sgleoutput le cotolles, fo hch the cotolle poles e scheduled essetll depedetl th espect to the output,, d the eleets of the put,,. I ddto, othe th the tsfe fucto eltg to, fo hch the zeoes e fed, the zeoes of the tsfe fuctos eltg the puts dvdull to the output e lso scheduled essetll depedetl th espect to the output,, d the eleets of the put,,. Hece, the equeet to septe suset of coeffcets, ll of hch c e essetll feel scheduled th the eg coeffcets costt, s et fo ths elsto, th the suset cosstg of ll the tsfe fucto coeffcets ecept those of the ueto of the tsfe fucto eltg to. The le cotolles ssocted th 44 hve the u possle schedulg degees of feedo, q. Hoeve, ol q coeffcets e scheduled th espect to oth d,, th q coeffcets, el,,, scheduled th espect to, loe d oe coeffcet, el d, scheduled th espect to loe. The elsto, 44, s equvlet, SISO cotet, to the elsto stsfg the eteded locl le equvlece codto desced Leth & Lethed 996. Eple Wth efeece to the d tue cotolles desced the Eple secto.5, the tsfe fucto ueto coeffcets of the fl of le cotolles, 4, e scheduled hlst the deoto coeffcets e costt. It s ppopte, theefoe, to detf the cotolles, 4 th the elsto, 4, speclsed to SISO sste. I othe ods, the schedulg vle c e detfed s the put to the ole cotolle equvlet to the C π s, el,, d the ole cotolle, deoted C π, stsfg the eteded locl le equvlece codto, s , hee , , , , O the othe hd, the tsfe fucto deoto coeffcets, togethe th the ovell g, of the fl of le cotolles, 4, e scheduled hlst the ueto coeffcets e costt. It s ppopte, theefoe, to detf the cotolle, 4 th the elsto, 44, speclsed to SISO sste. I othe ods, the schedulg vle c e detfed s the output fo the ole cotolle equvlet to the C π s, el, d the ole cotolle, deoted C π, stsfg the eteded locl le equvlece codto, s z., Gz z 47 hee z.47, Gz.59e.8z z.557e.76z.96e.8z.6z, z 4.67e.8z 5.4z 47 Notce tht hlst the tsfe fucto coeffcets of the le fl, 4, e petesed the cotolle output, the ssocted ole fuctos eploed the elsto, 47, e epessed tes of z. The elsto s eplctl scheduled th espect to z ut, sce Gz s vetle, plctl scheduled th espect to the output,. The depedece of the fo of the fl of le cotolles o the defto of the schedulg vle s ve evdet the foegog. It s ot dffcult to devse othe elstos, coespodg to dffeet deftos of the schedulg vle, hch cheve the septo of schedulg depedet coeffcets othe s; fo eple, the

14 Eple of secto.. Whe, s secto.4, the ees of the fl of le cotolles e defed geel sttespce fo, the equvlece to ole cotolle stsfg the eteded locl le equvlece codto s ot ovous. I cotst, he, s ove, the ees e defed tsfe fucto fo o, equvletl, cocl sttespce fo hch s eplctl petesed the tsfe fucto coeffcets, the equvlece s ovous due to the stog coespodece etee the fos of the ole cotolle d the ees of the fl of le cotolles. I ptcul, the equvlece d the ppopte defto of the schedulg vle e esl detfed fo those fles ehtg septo of the schedulg depedet coeffcets. Ths ese of detfcto getl sssts the desg of le cotolles to e elsed s gscheduled ole cotolle stsfg the eteded locl le equvlece codto fo ptcul defto of the schedulg vle. Covesel, t s cle tht the elsto of the gscheduled cotolle c e fcltted the judcous choce of ppopte schedulg vle. I geel, thee e le cotos of the eleets of o, o, o, hch petese the equlu opetg pots; tht s, thee e possle choces of schedulg vle. Choosg the schedulg vle, s ove, to fcltte the elsto of the ole gscheduled cotolle ght ppe to coflct th the oe usul ppoch of choosg the schedulg vle o the ss of phscl sght hee soe cotos of the eleets of the put, stte d output, epesetg the vles th espect to hch the plt dcs o the cotol ojectves see Leth & Lethed 996 v, e chose. Fo the d tue of the Eple secto.5, the cotolle ould dell e scheduled th espect to the d speed. Ufotutel, esueet of the d speed epeeced the tue s possle Leth & Lethed 996. Hoeve, povded the cotolle s effectve, the ptch gle of the oto ldes cts s good estte of the d speed. Moeove, sce the ddth of the lde ptch ctuto s, tpcll, lge copso to the ddth of the closedloop cotolled sste, the ptch ded to the ctuto,.e. the cotolle output, lso cts s good estte of the d speed. Accodgl, sutle cotolle fo the d tue s KsC π, th Ks postoed efoe C π, d C π elsed s 47 so tht the schedulg vle s the output of the cotolle. Nevetheless, t s ol the te of vto of schedulg tht ell ttes d schedulg vle, the spect of hch hs sl ddth, ould e equll sutle. Fo the d tue, desg, the put d output of the C π s hve sl ddths. Hece, equll sutle cotolle s K sc π, th K s postoed efoe C π, d C π elsed s 46. Cosequetl, thee s o coflct etee choosg the schedulg vle to fcltte the elsto of the ole gschedulg cotolle d choosg t o the ss of phscl sght. 4.. Utlt evsted Tkg to ccout the ue of tsfe fucto coeffcets tht c e feel scheduled el, out of fo the elstos vestgted ove, the ge of possle choces of the schedulg vle equvletl, the fos of the fl of le cotolles d the lt to desg/edesg le cotolles to hve specfc fo see Eple secto.5, the, eve he estcted to those chevg septo of the schedulg depedet coeffcets, the flelt of elstos stsfg the eteded locl le equvlece codto s ppet. Clel, t s possle to else, th fe eceptos, ll gscheduled cotolles s ole cotolles stsfg the eteded locl le equvlece codto d equg the cotolle to e so elsed s ot t ll estctve. 4. Veloctsed elstos The secod ssue, el the equeet to detee the ole fuctos of the schedulg vle the ole cotolle, s esolved doptg elstos fo hch le cotolles e oted feezg the schedulg vle t ts cuet vlue. Iplctl, ths pocedue eques the ole cotolle to e lesed t opetg pot, fo hch the vlue of the schedulg vle s th ts do, d ot just t the equlu opetg pots. Hoeve, ths s possle fo ole cotolles stsfg the eteded locl le equvlece codto s dscussed secto... The ole cotolle, 6, c e efoulted, dffeettg, s A f f 48 C g D g 48 48c hee. Dcll, 48, th ppopte tl codtos, el, A f, 49 d 6 e equvlet. Ideed, he A f 5 C D g 5

15 s vetle such tht e epessed s fucto of, d, ths efoulto s equvlet to lgec stte tsfoto 48c c the e eplced dect evluto of. Povded tht, equed s the put, d pehps, he equed the schedulg vle, e vlle d povded tht uted costt dsplceet of the stte o output, due to the dffeetto of the put folloed the tegto to detee the output fo, s voded, the 48 s vle ltetve epesetto fo 6. Copg the le cotolle, oted settg to o 48, to the le cotolle, 7, t s evdet tht the tsfe fuctos eltg to e the se. Hece, fo the veloctsed elsto, 48, the le cotolles e oted spl feezg the schedulg vle s equed. The equvlet veloctsed elsto fo the elsto, 4, depcted fgue 6 th β [ ],,.., s 5 [ ] [ ] 5 hee j j,,,..,, j,,.., 5c Whe s ozeo, ths efoulto s equvlet to lgec ole stte tsfoto. The stog coespodece to the foze schedulg vle le cotolles s evdet fo copg 5 to 9. The equvlet veloctsed epesetto, fo the elsto, 44, depcted fgue 7 th β [ ] ˆ ˆ d β, [ ] ˆ, ˆ,,,.., s ˆ, ˆ, ˆ, ˆ, ˆ, ˆ, ˆ, ˆ, ˆ, ˆ, ˆ, ˆ, ˆ, ˆ, ˆ, ˆ, ˆ ˆ ˆ ˆ 5 [ ] [ ] ˆ d d 5 hee j, ˆ z j G, ˆ, j,,..,; ˆ ˆ,,, ; ˆ, G ˆ, j j,,,.., j,,..; d z GG 5c Whe, ˆ s vetle fo ll ˆ, ths epesetto s equvlet to lgec ole stte tsfoto of 44. The schedulg th espect to the output s o loge plct ut s o eplct vtue of the vetlt of G. The stog coespodece to the foze schedulg vle le cotolles s g evdet fo copg 5 to 9.

16 Eple Retug to the d tue Eple secto.5, the fst elsto, 46, c e ecst s the veloctsed elsto, , hee , , , , Sll, the secod elsto, 48, c e ecst s.47., d d hee ,.658.4, d The coespodece to the ogl fles of le cotolles, 4 d 4 espectvel, s cle; ptcul, cope 5 to 4e d 54 to 4e. Whlst, geel, the efoulto to veloctsed elstos ceses the cotolle ode, the elstos of fgues 5 d 6 hve the dstct dvtge of eg dectl elted to the ees of the fl of le cotolles. The ssocted fl of le cotolles fo the elsto, 5, o cossts spl of 9 o, equvletl, the tsfe fuctos, 8, th d d,,,.., ; j,,..,. The ssocted fl fo the j j o elsto, 5, cossts of 9 o, equvletl, 8, th dd o, j j o, ˆ o, ˆ o, j j o, ˆ o,,..,j, Itegl Acto The thd ssue, el ecoclg the equlu vlues of the cotolle put d output, s esolved cludg tegl cto the cotolle. As oted secto., s put to the plt d s output fo the plt, the vlues of hch, t equlu opetg pot of the plt, e o π, o π th o π depedet o o π v the plt. Hoeve, sce s the cotolle put d s the cotolle output, o π, o π ust lso e equlu opetg pot of the cotolle th o π depedet o o π v the cotolle. Requg cosstec poses, geel, stog estcto o the llole ole cotolles. Fotutel, the cotol desg tsk c usull e defed such tht the cotolle cts upo eo hch s, tpcll, equed to e zeo o sll he equlu. Sce, ode to eet pefoce equeets, cotolles ofte possess tegl cto, hch esues tht the put ust e zeo equlu, the foe cse s fequetl ecouteed. I the ltte cse, the lck of pecso defg ht s et sll fequetl eles the equlu vlue of the put,, to e edefed to e zeo; tht s, the chge duced the equlu opetg pot ths edefto s tpcll suffcetl sll tht t s tel. Whe the put to the cotolle s zeo equlu, the o π e o loge depedet o the o π, ut c hve vlue. Hece, the cosstec equeet o the equlu vlues of the put d output of the cotolle s ot estctve Sh 988, the oe estcted cotet, ke sl osevto egdg the ole of tegl cto gscheduled cotolles. 4..Veloctsed elstos evsted It s oted secto 4. tht, fo veloctsed elsto to e vle, uted costt dsplceet of the stte o output, due to the dffeetto of the put folloed the tegto to detee the output fo, ust e voded. Ths c, lso, e cheved the cluso of tegl cto the cotolle. Cosde the sple ole cotolle cosstg of sgle ole g togethe th tegl cto s depcted fgue 8. The ole copoet G 55

17 tself costtutes ole cotolle stsfg the eteded locl le equvlece codto. The schedulg vle c e cosdeed to e ethe the put,, o, he the vese of G ests, plctl the output,. Postog the tegto efoe the ole g edtel kes oth d vlle. The ole cotolle, th s the schedulg vle, c e ecst s the veloctsed elsto, depcted fgue 8, G 56 The ole cotolle, th s the schedulg vle, c e ecst s the veloctsed elsto, depcted fgue 8c, g, g GG 57 I oth fgues 8 d 8c, the dffeetto d tegto, ssocted th the veloctsed elsto, do ot ppe eplctl ut eel cuse epostog of the pue tegto fo efoe to fte the ole eleet. Note tht the elsto, 56, the schedulg vle,, ust e deteed tegtg, ut the elsto, 57, the schedulg vle,, s dectl vlle. Sce the closedloop sste s stle, eo the tl codto fo pdl evolves out. The elsto, depcted fgue 8c, s the sple d effectve es of elsg sste cosstg of sgle ole g togethe th tegl cto. It s del pplcle; fo eple, d tue cotol to cte fo the vto the eodcs th d speed, see Lethed et l. 99. A soeht oe goous eplto fo the ekle effectveess of ths ppoch to schedulg the cotolle, he ppled to d tues, s peseted Leth & Lethed 997. The eso fo doptg the veloctsed elstos s to vod hvg to detee the ole fuctos of the schedulg vle the ole cotolle fo the eltoshps of the ptl devtves to the le cotolles tsfe fucto coeffcets. Ufotutel, th the toducto of tegl cto, the veloctsed elsto o loge esolves ths ssue sce the equlu opetg pots cot e petesed the put, o. Cosde the sple ole cotolle, 55, thout tegl cto d ts veloctsed elsto th the schedulg vle, 56. The equlu opetg pots, o, o, e petesed o o o. Hece, he desgg the le cotolles the gs t the equlu opetg pots, t s possle to dectl petese the o d so to dectl detee G. Hoeve, he tegl cto s cluded d tll postoed, s ove, efoe the ole eleet, the equlu vlues of the put, o, e o loge elted to the plt put ut e elted to the cotolle output, o, v the ole cotolle tself. Hece, he desgg the le cotolles, t s ot possle to dectl petese the gs o. Isted, the gs ust fst e petesed o, s g o. Fo 55 d 57, t the follos tht G gg 58 th G o ds, gs g s 59 gs d tegto of the coeffcets, ths cse the vese of the g, cot e voded. Of couse, the to elstos, 4/5 d 44/5, c e teted sl e to the ove sple ole cotolle cosstg of sgle ole g togethe th tegl cto. Adopto of the veloctsed elstos, 5 d 5, eel cuses the epostog of the tegl cto fo efoe to fte the ole cotolle. Ecept fo elsto 5, he speclsed to the SISO cse, the schedulg vle stll eques to e deteed tegtg d the veses of the cotolle gs stll eque to e tegted to ot dect petesto of the le cotolles tes of the put,. Eple Retug to the d tue Eple secto.5, the fed copoet, Ks o K s, hch s postoed efoe the gscheduled copoet, C π o C π, cludes tegl cto. Hece, the fst veloctsed elsto, 5, c e efoulted s depcted fgue 9 th the coeffcets,..5, defed 5. The petesto of the, 5, d equvletl the π, 4e, e oted sl pocedue to tht desced ove fo the sple ole cotolle cosstg of ole g togethe th tegl cto. The eltoshp etee the o d o s o.4 o 88.5 o.7. The schedulg vle,, s oted tegtg. A stle ek feedck fo the cotolle output s eploed to esue tht eo the tl codto fo evolves out. The secod veloctsed elsto, 54, c e efoulted s depcted fgue 9, th the coeffcets, d d defed 54. Ths elsto s the oe epoted Leth & Lethed 996. oth of these elstos, fgue 9 d fgue 9, hve ee pleeted o detled ole sulto of the d tue d the pefoce ssessed: the e foud to e equll effectve. It s ell ko epcll, lthough el epoted the ltetue Lece & Rugh 995, tht t s eefcl, hee possle, to posto pue tegto t the output of gscheduled cotolle. The foegog dscusso povdes ltc suppot fo ths heustc ule.

18 4.4 Dect elstos I sectos 4.4., sevel ssues elted to the pctcl desg of ole cotolles stsfg the eteded locl le equvlece codtos e vestgted d vous s of esolvg the deteed. Of couse, ptcul pplcto, the ost ppopte ethods fo the cotet should e dopted. Hoeve, elsto hch utotcll esolves ll of these ssues ould e useful. Just such elsto fo MIMO cotolles s developed elo. The geel stuctue of the elsto s depcted fgue. The copoets C po d C p e fed, th ll of the scheduled copoets C, d the cotolle s ssued to clude tegl cto postoed s sho eplctl fgue. Wthout loss of geelt, C po d C e ssued sque; tht s, osqueess s copoted to C p. ecuse the put to the cotolle s zeo t the equlu opetg pots, the u ue of schedulg vles hch c e ccoodted the elsto of fgue s p, the degee of the output; tht s, the degee of sce C po s sque. The copoet C po s chose such tht the petesto of the equlu opetg pots the eleets of s coptle, s tul e s possle, th the schedulg equeets. A ecess osgul tsfoto of the outputs s copoted to C po. A futhe equeet o the ssget of copoets of the cotolle to C p d C po s to esue tht, s dscussed secto 4., the spect of the schedulg vle hs sutle ddth. Fo the d tue Eple secto.5, t s ppopte to schedule the cotolle th espect to the ptch gle of the oto ldes. Whe the ddth of the ctuto s lge, sutle ltetve to the ptch gle s the ptch ded to the ctuto; tht s, the cotolle output. Hoeve, fo soe d tues, ptcull lgescle ches, the ddth of the ctuto c e ve lo see, fo eple, Leth & Lethed 997 d the efeeces thee. Cosequetl, the te of vto of the ptch ded c e sgfctl fste th the ptch gle tself. The eed, dopted Leth & Lethed 997 let, oe goousl justfed, s to choose C po to e vese o pseudovese, A, of the ctuto dcs. Whe ths eed s ppled to the sple cotolle of fgue 8c, t s eded s sho fgue 8d. The tegl cto s eploted pettg A to e sple led te, th tsfe fucto s/, sce the coed tsfe fucto s pope. The ole copoet, C, of the cotolle s scheduled th espect to the put,. Wth ths choce of schedulg vle, the elsto, stsfg the eteded locl le equvlece codto, fo C d the coespodg fl of le cotolles hve the ost coveet fo, see secto 4., d the estctos o the clss of vld puts d tl codtos s educed to solute u, see secto.. Itepetg C to cosst of p ultput sgleoutput ole cotolles, the elsto fo ech cofos to tht of fgue 4. Fo ech of the p ultput sgleoutput ole cotolles, let the vle, z, e the scl fucto,, of the put,. To dstgush ths eltoshp fo the dffeet ultput sgleoutput cotolles, de k s used; fo the cotolle eltg to the k th eleet of, z k k. Let z e the vecto hose eleets e the z k, d e the vecto hose eleets e the k, the z 6 Assue tht the vese,, of ests. The cotolle of fgue c the e efoulted s sho fgue. The ole cotolle, C *, cossts of p ultput sgleoutput ole cotolles th the elsto depcted fgue, hee, fo the k th ultput sgleoutput ole cotolle, c s the vecto fo hch the k th eleet s ut, ll othe eleets eg zeo, d ˆ k,,... It should oted tht, due to the defto of, the eleets of z e equl to the scled eleets of the t equlu opetg pot. The equvlet veloctsed elsto fo the ultput sgle output ole cotolle of fgue s 5 th the coeffcets defed z ˆ z,,.., j,.., c,,.., 6 j z j hee c s the th eleet of c. Ths elsto s depcted fgue, hch coespods to fgue 6, th ˆ β z [ z... z],,... I fgue, the feedck loops hve ll ee shfted oe tegto to the ght. It should e oted tht z s vlle tell to the elsto depcted fgue d, theefoe, ol z s eeded s put to C *. Sce the closedloop sste ould e stle, eo the tl codtos fo the tegto of z pdl evolves out. The ole cotolle, cosstg of the fucto,, togethe th the tegl cto, c lso e efoulted sl e to the sple scl cse cosdeed secto 4. s the veloctsed elsto depcted fgue, hee ˆ z [ˆ z] [ z]. Sce z s ot equed s put to C *, the tegl cto the j j veloctsed elsto c e otted he the coplete cotolle hs the elsto depcted fgue c, C * eg elsed s fgue. Moeove, sce z hs ee chose so tht, t equlu opetg pot, the