ELG456 Desig of Stte Vrible Feedbck Systems This chpter dels with the desig of cotrollers utilizig stte feedbck We will cosider three mjor subjects: Cotrollbility d observbility d the the procedure for determiig optiml cotrol system Ackerm s formul c be used to determie the stte vrible feedbck gi mtri to plce the system poles t the desired loctios The closedloop system pole loctios c be rbitrrily plced if d oly if the system is cotrollble Whe the full stte is ot vilble for feedbck, we utilize observer The observer desig process is described d the pplicbility of Ackerm s formul is estblished The stte vrible compestor is obtied by coectig the fullstte feedbck lw to the observer We cosider optiml cotrol system desig d the describe the use of iterl model desig to chieve prescribed stedystte respose to selected iput commds
Pole Plcemet Usig Stte Feedbck The sttespce desig method is bsed o the poleplcemet method d the qudrtic optiml regultor method The pole plcemet method is similr to the rootlocus method I tht we plce closedloop poles t desired loctios The bsic differece is tht i the rootlocus desig we plce oly the domit closed loop poles t the desired loctios, while i the poleplcemet method we plce ll closedloop poles t desired loctios The stte vrible feedbck my be used to chieve the desired pole loctios of the closedloop trsfer fuctio T(s) The pproch is bsed o the feedbck of ll the stte vribles, d therefore u = K Whe usig this stte vrible feedbck, the roots of the chrcteristic equtio re plced where the trsiet performce meets the desired respose
Stte Vrible Compestor Employig FullStte Feedbck i Series with Full Stte Observer u A System Model Bu C y Cotrol Lw K ˆ Observer + ˆ Aˆ Bu Lyˆ Compestor C 3
Cotrollbility d Observbility The cocept of cotrollbility d observbility were itroduced by Klm i 96 They ply importt role i the desig of cotrol systems i stte spce The coditios of cotrollbility d observbility my gover the eistece of complete solutio to the cotrol system desig problem The solutio of the problem my ot eist if the system is ot cotrollble 4
Cotrollbility A system described by the mtrices (A, B) c be sid to be cotrollble if there eists ucostried cotrol u tht c trsfer y iitil stte () to y other desired loctio ( Tht mes tht over time, some or ll of the sclr time fuctios i u c be rbitrrily lrge i mgitude P If c P A c Bu B AB A is ozero, BA B Aother method of determiig whether system is cotrollble is to drw the stte vrible flow digrm d determie whether the cotrol sigl, u, hs pth to ech stte vrible If pth to ech stte eists, the system my be cotrollble (Cotrollbility mtri P) thesystem is cotrollble 5
6 Emple of Cotrollble system! is ozero P Determit of P B A, AB, B ) ( ) ( ) ( c c 3 u s s s s T s U s Y o
Cotiue Ucotrollble system hs subsystem tht is physiclly discoected from the iput For prtilly cotrollble system, if the ucotrollble modes re stble d the ustble modes re cotrollble, the system is sid to be stbilized For emple such system u Is ot cotrollble The stble mode tht correspods to the eigevlue of is ot cotrollble The ustble mode tht correspods to the eigevlue of is cotrollble Such system c be mde stble by the use of suitble feedbck Therefore the system is stbilizble 7
Observbility All the roots of the chrcteristic equtio c be plced where desired i the sple if, d oly if, system is observble d cotrollble Observbility refers to the bility to estimte stte vrible Thus we sy system my be observble if the output hs compoet due to ech stte vrible A system is observble if, d oly if, there eists fiite time T such tht the iitil stte ( c be determied from the observtio history y( give the cotrol u( Cosider the sigleiput, sigleoutput system A Bu d y C Thissystem is observble is ozero, C CA Q CA where whe thedetermit of Q 8
9 Emple of Observble System! thesystem is observble! d, Det Q, Q d CA CA d C A
Is this System Cotrollble d Observble? y u Sice the rk of the mtri B AB is, the system is fully stte cotrollble To test theobservbility coditio, emie the rk of [C AC] is, the system is observble
FullStte Feedbck Cotrol System to chieve the desired pole loctios of the closed loop system First we should ssume tht ll the sttes re vilble for feedbck The system iput u( is give by u = K Determiig the gi mtri K is the objective of the fullstte feedbck desig procedure A System Model Bu y Cotrol Lw K
Compestor Desig Itegrted FullStte Feedbck d Observer u Cotrol Gi K ˆ + A BK LC Observer Gi L y + ˆ BK LC A BK LC
u( Kˆ( Feedbck Lw Employ stteestimte ˆ Aˆ Bu L( y Cˆ) (From the observer, see the previos lecture) ˆ ( A BK LC )ˆ Ly e ˆ A Bu Aˆ Bu Ly e ( A LC ) e A Bu; y C A Bu (A BK) BKe A BK e A BKˆ; ˆ( i the feedbck cotroli plce of Det( si ( A BK)) From full sttefeedbck We eed to verify tht whe usig the feedbck cotrol lw we reti the stbility u Kˆ BK A LC e ˆ e ( LC ˆ (time derivtive of estimtio error) 3
The closedloop system hs o iput The objective is to miti zero output Becuse of disturbces the output will devite from zero The ozero output will be retured to zero referece iput becuse of the stte feedbck scheme A Bu; u Ku A Bu ( e ( ABK) t ( t Give the pir ( A, B), Whe r( for ll t t A BK ( A BK) det( λi ( A BK)) If ll the roots of the chrcteri stic equtio liei theleft hlf ple the theclosed loop system is stble ) s t we c determie K to plce ll thesystem closed loop poles i theleft hlf ple if thesystem is completely cotrollble The dditio of referece iput c be cosidered s u( K( Nr( where r( is the referece iput the cotrol desig problem is regulr 4
The gol is to verify tht, with the vlue of u( we reti the stbility of the closedloop system d the observer The chrcteristic equtio ssocited with the previous equtio is So if full sttefeedbck lw d secod to the desig of i the left hlf i the left hlf Δ(s) Det( si (A BK)) Det( si ( A LC)) the roots of Determie K both prtsof the theoverll systemis stble ple d plce the roots properly to meet the desig criteri ple d plce the roots to chieve the bove equtio (first relted DesigProcedure such tht Det( si (A BK)) Determie L such tht Det( si ( A LC)) the observer, hs hs roots roots observer performce 3 Coect the observer to the full sttefeedbck lw usig u( to Kˆ( 5
Desig of Third Order System d y d y 5 3 3 dt dt Select the stte vribles s : A BK A BK ( k) 3 Chose 8for 4 8 s 96s 36 s 48 The we require k y u The chrcteri stic equtio is Det [A BK] s Chose the desired chrcteri stic equtio : T 4 Thestep respose hs o overshoot A BK ; Thestte feedbck miiml overshoot If 94; k (3 k ) 44s 79; k dy / dt; (5 k s s s ; If we chose 6, the we hv s 3 y; u A Bu 3 5 If the stte vrible feedbck mtri K is : K k s dy dt 8s 78 3 3 78 mtri is 3 ) 3 d k (5 k y / dt k 3 3 ) s d u K (3 k ) s ( k we wt settlig timeequl tos, the ; ) 6
Ackerm s Formul For sigleiput, sigleoutput system, Ackerm s formul is useful for determiig the stte vrible feedbck mtri K u [ k q( ) q( A) K k ] P A k Give the desired Thestte feedbck K [ c A ] chrcteri q( A) gi mtri is stic equtio A I 7
Observer Desig If the system is completely observble with give set of outputs, the it is possible to determie or estimte the sttes tht re ot directly mesured u y Estimte of the mtri ˆ ˆ Observer Aˆ Bu L ~ y ~ y y Cˆ + C L is the observer gi mtri d to be determied 8
The gol of the observer is to provide estimte ˆ so tht ˆ Wedo ot kow ( t estimte ˆ( t e( t Tke the timederivtive of the estimtio error of the previous equtio ˆ s t ( ˆ( The observer desig should produce e( s e ˆ A Bu L( y Cˆ) e( ( A LC) e( ) precisely; we should provide itil ) to the observer The observer estimtio error is e A Bu Aˆ Bu L( y Cˆ) e( s t s Det( I ( A LC)) hs its ll roots i theleft hlf ple 9
E3: A system is described by the mtri equtio Determie whether the system is cotrollble d observble P c B Sice P c C Q CA 3 AB Sice Det Q is equl isot equl The observbility mtri is ; u y 6 3 to zero, to zero, thesystem is cotrollble therefore thesystem is uobservble!
E4: A system is described by the mtri equtio Determie whether the system is cotrollble d observble First fid P c 4 A Sice Det P C Q CA thecotrollbility mtri AB c, The observbility mtri is 4 ; d u thesystem is ucotrollble Sice Det Q, thesystem is uobservble y