EL2520 Control Theory and Practice
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1 oals EL252 Cotol Theoy ad Pactice Lecte 2: The closed-loop system Mikael Johasso School of Electical Egieeig KTH, Stockholm, Sede Afte this lecte, yo shold: Ko that the closed-loop is chaacteied by 6 tasfe fctios Dageos to desig fo oly oe Cacellatios ad the cocept of iteal stability Detemie, aalye ad desig desied sesitivity fctios Sesitivity fctio fo distbace ejectio Complemetay sesitivity fctio fo obst stability Udestad limitatios ad coflicts, elatio to stability magis Mateial: cose book Chapte 6. Cotets The closed-loop system. The closed-loop system 2. The cotol poblem ad six cetal tasfe fctios 3. The sesitivity fctio ad distbace ejectio 4. The complemetay sesitivity ad obst stability 5. The closed-loop tasfe fctio ad efeece folloig F Cotolle: feedback F y, feedfoad F Distbaces:, dive system fom desied state Measemet oise: copts ifomatio abot Aim: fid cotolle sch that follos, ith limited se of
2 The desig poblem Relatio betee sigals Desig poblem: fid a cotolle that F a) Redces the effect of load distbaces b) Does ot iject too mch measemet oise ito the system c) Makes the closed loop isesitive to pocess vaiatios d) Makes the otpt follo commad sigals Ofte coveiet ith to-degee of feedom cotolle (sepaate tasmissio fom yà ad fom à ) Use feedback to deal ith a,b,c; se feedfoad to deal ith d. Similaly, e fid Closed-loop fo SISO (s) chaacteied by six tasfe fctios Tasfe fctios ad obsevatios A aig! Idividal time esposes may look good ->.8 Otpt Obsevatio: eed to look at all! May tadeoffs (e.g. S+T=) bt yo eed to veify that all tasfe fctios ae as desied! 2
3 Fo esposes What s goig o? -> -> Pocess:.8 8 Otpt.6.4 Otpt 6 4 Cotolle:.2 2 Tasfe fctios: > ->.8.8 Otpt.6.4 Otpt Iteal stability Sesitivity fctios F Sesitivity ad complemetay sesitivity ae paticlaly impotat: S detemies sppessio of load distbaces, T detemies obstess to oise ad modelled dyamics Both coected to classical stability magis (gai, phase magi) Defiitio. The closed loop system above is iteally stable if it is ipt-otpt stable fom to all otpts. Fist tade-off: S+T= - ca t make both eo at the same time. Theoem. If is SISO, the closed-loop system is iteally stable stable if ad oly if ae stable 3
4 Distbace ejectio Nyqist cve itepetatio The tasfe fctio fom to i ope loop is hile the closed-loop cote-pat is is ivese distace fom Nyqist cve to - Distbace atteatio at feqecies hee Nyqist cve is iside it cicle ceteed at the - poit. Ths S qatifies the distbace atteatio de to closed-loop cotol. Distbaces at feqecies ith S(i!) amplified by feedback! Obsevatio: ca t avoid cicle if pole excess 2, mst amplify distbaces at some feqecies (moe ext lecte!) Maximm sesitivity ad M s -cicles Specificatio : loop gai otside cicle ith adis M s - Sesitivity shapig Obsevatios: Ca t atteate distbaces at all feqecies (if pole excess 2) Need to limit S(i!) at feqecies ith sigificat distbaces Reasoable desig specificatio Fobidde aea Reasoable vales:.2 Ms 2 (picte shos M s =2) 4
5 Sesitivity to cetaities Robst stability Sesitivity of hat, ad to hat? Sesitivity of closed-loop tasfe fctio to model cetaities The espose of to (assmig o distbaces) is If thee is a model eo, so that the te system is Ucetaities also affect stability of closed-loop system. Assme te system is give by F the the te espose to is What liea ca be toleated ithot jeopadiig stability? Robst stability via small gai theoem Robst stability via small gai theoem Assme all exogeos ipts (,,, ) to be eo, e-ite -T Note that Assme stable ad omial-system T iteally stable. If the the above system (ad, hece, the oigial system) is ipt-otpt stable. Poof. Small-gai theoem kt k apple 5
6 Sesitivity shapig Extesio: shapig feqecy esposes Natal desig citeio: make se that both the sesitivity S ad the complemetay sesitivity T avoid fobidde aeas Fobidde aea Fobidde aea Ca shape all elevat tasfe fctios (i the gag of six ) ksw S k apple ktw T k apple. ksf W SF k apple This is the topic of Compte Execise b! o ksw S k apple ktw T k apple Complemetay sesitivity i Nyqist cve Closed loop tasfe fctio ad tackig Costait o complemetay sesitivity kt k apple M t also yields cicles that shold be avoided by the Nyqist cve. Refeece folloig detemied by closed-loop tasfe fctio M P c B 3 db Cicles ceteed at ith adis! B 2 Feqecy (ad/s) Desig citeio: choose F so that M p ad eqal desied vales Note: potetial coflict ith S, cotol sigal limitatios! B 6
7 Steady-state eos Smmay Step i efeece sigal If F = F y, the ad eqies Te if, i.e. if itegato i F y o. Tackig a amp sigal eqies to itegatos, etc. Pefect sppessio of distbaces teated aalogosly. Closed-loop system chaacteied by 6 tasfe fctios Need to coside all! Sesitivity ad complemetay especially impotat S: distbace atteatio, pefomace sesitivity T: oise atteatio, obst stability Close elatioship ith classical stability magis Cotol system desig via sesitivity shapig Coflicts ad limitatios S+T= S(i!) fo some! (distbace amplificatio!) Mch moe ext lecte! 7
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