Sampling. Controller Implementation of Control. Algorithm. Aliasing. Aliasing Example. Example Prefiltering. Prefilters

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1 Conrollr Implmnaion of Conrol Algorims Sampling Compr A/D Algorim D/A Procss y Aliasing Compr Implmnaion of Conrol Algorims Sampld conrol ory Discrizaion of coninos im dsign Exampl: PID algorim discrizaion conrol mods ning AD-convrr acs as samplr A/D DA-convrr acs as a old dvic Normally, zro-ordr-old is sd picwis consan conrol signals Aliasing 5 Tim ω N = ω s / = Nyqis frqncy, (ω s = sampling frq.) Frqncis abov Nyqis frqncy ar foldd and appar as low-frqncy signals. T fndamnal alias frqncy for a frqncy f > f N is givn by f = (f + f N ) mod ( f s ) f N Aliasing Exampl Fd war aing in a sip boilr Prssr Sam Fd Valv war Tmprar Pmp Condnsd war 38 min min To boilr Tmprar Abov: f =.9, f s =, f N =.5, f =. Prssr 3. min 4 Tim Prfilrs Ani-aliasing filr Analog low-pass filr a liminas all frqncis abov Nyqis frqncy (a) Exampl Prfilring (b) Analog filr -6 ordr Bssl or Brwor Difficlis wi canging (sampling inrval) Digial Filr Fixd, fas sampling wi fixd analog filr Conrol algorim a a slowr ra ogr wi digial LP-filr Easy o cang sampling inrval 3 3 (c) (d) 3 3 Tim Tim ω d =.9, ω N =.5, ω alias =. 6 ordr Bssl wi ω B =.5 T filr may av o b incldd in dsign. 5 6

2 Dsign of Digial Conrollrs Sampld Conrol Tory Digial conrollrs can b dsignd in wo diffrn ways: Discr im dsign sampld (digial) conrol ory sif opraors (z-ransforms) (k) =k y(k)+k (k ) a dsign paramr Coninos im dsign + discrizaion Laplac ransform U(s) =G c (s)e(s) approxima coninos dsign fas, fixd sampling 7 Compr Clock A-D { } { ( k )} y( () y( ) k ) Algorim D-A Procss T basic ida: Look a sampling insancs only! 8 Sampld Conrol Tory Disk Driv Exampl k ( ) ( ) Hold k D-A Procss Compr y( ) A-D y( ) Samplr y k y k Conrol of arm of a disk driv Coninos im conrollr G(s) = k Js U(s) = b a U c(s) s + b s + a Y(s) Discr im conrollr (coninos im dsign + discrizaion) Sysm ory analogos o coninos im linar sysms ( k )= ( b a c( k ) y( k )+x( k )) x( k + ) =x( k )+ ((a b)y( k ) ax( k )) Br prformanc can b acivd Problms wi inrsampl bavior 9 Disk Driv Exampl Incrasd sampling priod y: = adin(in) :=*(b/a*c-y+x) do() x:=x+*((a-b)*y-a*x) Clock Algorim a) =.5/ω b) =.8/ω (a) Op (b) Op Sampling priod =./ω Op.5 5 Inp Tim (ω ) Inp Tim (ω ) Inp.5 5 Tim (ω )

3 Dad-ba conrol =.4/ω Br prformanc? ( k )= c ( k )+ c ( k ) s y( k ) s y( k ) r ( k ) Sampling of Sysms Look a sysm from poin of viw of compr Posiion Clock Vlociy Inp Tim (ω ) Howvr, long sampling priods also av problms opn loop bwn sampls disrbanc and rfrnc cangs a occr bwn sampls 3 will rmain ndcd nil nx sampl {( k )} () y() { y( k )} D-A Sysm A-D Zro-ordr-old sampling of a sysm L inps b picwis consan Look a sampling poins only Us linariy and calcla sp rsponss wn solving sysm qaion 4 Sampling a coninos-im sysm Sysm dscripion T Gnral Cas dx d = Ax()+B() y() = Cx()+D() Solv sysm qaion x() = A( k) x( k )+ = A( k) x( k )+ = A( k) x( k )+ k k k A( s ) B(s ) ds = Φ(, k )x( k )+Γ(, k )( k ) A( s ) ds B( k ) ( cons.) As ds B( k ) (variabl cang) wr x( k+ ) = Φ( k+, k )x( k )+Γ( k+, k )( k ) y( k ) = Cx( k )+D( k ) Φ( k+, k ) = A(k+ k) k+ k Γ( k+, k ) = As ds B 5 6 Priodic sampling Assm priodic sampling, i.. k = k, n x(k + ) = Φx(k)+ Γ(k) y(k) = Cx(k)+D(k) wr Φ = A Γ = As ds B NOTE: Tim-invarian linar sysm! I is also possibl o sampl a sysm wi im dlay Hnc Exampl: Sampling of dobl ingraor dx = d y = x + x Φ = A = Γ = s ds = Svral ways o calcla Φ and Γ. Malab 7 8

4 Sabiliy rgion In coninos im sabiliy rgion is complx lf alf plan, i. sysm is sabl if all pols ar in lf alf plan. In discr im sabiliy rgion is ni circl. Conrol Dsign A larg variy of conrol dsign mods ar availabl in digial conrol ory,.g.: sa-fdback conrol pol-placmn LQ conrol obsrvr-basd sa fdback conrol LQG conrol op fdback conrol... Cors in Compr-Conrolld Sysms. 9 Sampling Inrval Discrizaion of Coninos Tim Dsign Nmbr of sampls pr ris im, T r, of closd loop sysm N r = T r 4 Basic idas: Rs dsign () A-D H(z) G(s) { ( k) } { y( k) } Algorim D-A y() Wi long sampling inrvals i may ak long bfor disrbancs ar dcd Clock G(s) is dsignd basd on analog cniqs Wan o g: A/D + Algorim + D/A G(s) Mods: Approxima s, i.., G(s) H(z) Or mods Approximaion Mods Sabiliy of Approximaions Forward Diffrnc (Elr s mod) dx() d Backward Diffrnc dx() d Tsin: x( + ) x() s = z x() x( ) s = z z s = z z + 3 How is coninos-im sabiliy rgion (lf alf plan) mappd? Forward diffrncs Backward diffrncs Tsin 4

5 Sampling Inrval An Exampl: PID Conrol T fasr br. Rl-of-mb: ω c.5.5 wr ω c is cross-ovr frqncy of coninos-im sysm ( frqncy wr gain is ) Sbsanially largr n wi discr-im dsign. Mor robs agains variaions in sampling inrval. T olds conrollr yp T mos widly sd Plp & Papr 86% Sl 93% Oil rfinris 93% Mc o larn!! 5 6 T Txbook Algorim Proporional Trm () = (() + T I (τ )dτ + d() TD ) d max U(s) = (E(s) + st I E(s) + T D se(s)) min Proporionalband = P + I + D max > = + < < min < 8 7 Propris of P-Conrol Errors wi P-conrol.5.5 S poin and masrd variabl Conrol variabl 4 c=5 c=5 c= c= c= c= Conrol signal: Error: Error rmovd if:. qals infiniy. = = + = 5 5 Solion: Aomaic way o obain saionary rror incrasd mans fasr spd, incrasd nois snsiiviy, wors 9 sabiliy 3

6 Ingral Trm Aomaic Rs b +st i = + = ( + Ti ) ()d (PI) c + Saionary rror prsn d incrass incrass y incrass rror is no saionary U = E+ U + st i ( )U = + st i U = st i U + st i + st i + s i U = ( + )E st i 3 3 Propris of PI-Conrol S poin and masrd variabl.5 Ti= Ti= Ti=5.5 Ti= 5 5 Conrol variabl Ti= Ti= Prdicion A PI-conrollr conains no prdicion T sam conrol signal is obaind for bo s cass: Ti=5 Ti= 5 5 I P I P id id rmovs saionary rror smallr T I implis wors sabiliy, fasr sady-sa rror rmoval Drivaiv Par Propris of PD-Conrol Rglrfl () + T d d() d S poin and masrd variabl Td=. Td=.5 () ( + T d).5 Td= 5 5 id 6 4 Conrol variabl Td=. Td=.5 Td= P: () =() PD: ( ) d() () = ()+T d ( + T d ) d T d = Prdicion orizon T D oo small, no inflnc T D oo larg, dcrasd prformanc In indsrial pracic D-rm is ofn rnd off. 36

7 Alrnaiv forms So far w av dscribd dirc (posiion) vrsion of PID conrollr on paralll form Or forms: sris form U = ( + st )( + st D )E I = ( + T D T + st + st D )E I I Diffrn paramr vals incrmnal (vlociy) form U = s ΔU ΔU = (s + T I + s T D + st D /N )E Ingraion xrnal o algorim (.g. sp moor) or inrnal Algorim Modificaions Limiaions of drivaiv gain Modificaions ar ndd o mak conrollr pracically sfl Limiaions of drivaiv gain Drivaiv wiging Spoin wiging Handl conrol signal limiaions W do no wan o apply drivaion o ig frqncy masrmn nois, rfor following modificaion is sd: st D st D + st D /N N = maximm drivaiv gain, ofn 39 4 Drivaiv wiging Spoin wiging An advanag o also s wiging on spoin. T spoin is ofn consan for long priods of im Spoin ofn cangd in sps D-par bcoms vry larg. Drivaiv par applid on par of spoin or only on masrmn signal. rplacd by β = (y sp y) = (β y sp y) st D D(s) = + st D /N (γ Y sp(s) Y(s)) Ofn, γ = in procss conrol, γ = in srvo conrol A way of inrodcing fdforward from rfrnc signal (posiion a closd loop zro) Improvd s-poin rsponss. 4 4

8 .5 Spoin wiging.5 S poin and masrd variabl ba= ba=.5 ba= Conrol variabl Conrol Signal Limiaions All acaors sara. Problms for conrollrs wi ingraion. Wn conrol signal saras ingral par will conin o grow ingraor (rs) windp. Wn conrol signal saras ingral par will ingra p o a vry larg val. Tis may cas larg ovrsoos..5.5 Op y and yrf ba= ba=.5 ba= Conrol variabl Ani-Rs Windp Tracking Svral solions xis: conrollrs on vlociy form (Δ is s o if saras) limi spoin variaions (saraion nvr racd) condiional ingraion (ingraion is swicd off wn conrol is far from sady-sa) racking (back-calclaion) wn conrol signal saras, ingral is rcompd so a is nw val givs a conrol signal a saraion limi o avoid rsing ingral d o,.g., masrmn nois, rcompaion is don dynamically, i.., rog a LP-filr wi a im consan T (T r ) Tracking Tracking y Tds = r y T i s v Acaor s T.5 y = r y T ds Acaor modl Acaor.5 3 T i s T + s

9 Discrizaion Discrizaion I-par: I() = T I (τ )dτ P-par: P (k) =(β y sp (k) y(k)) Forward diffrnc di d = T I I( k+ ) I( k ) = T I ( k ) I(k+) := I(k) + (*/Ti)*(k) T I-par can b prcalclad in UpdaSas 49 Backward diffrnc T I-par canno b prcalclad, i(k) = f((k)) Ors 5 Discrizaion Discrizaion D-par (assm γ = ): st D = D + st D /N ( Y(s)) T D dd N d + D = T dy D d Forward diffrnc (nsabl for small T D ) Backward diffrnc T D D( k ) D( k ) + D( k ) N y( k ) y( k ) = T D T D D( k )= T D + N D( k ) T D N T D + N (y( k) y( k )) 5 Tracking: v := P + I + D; := sa(v,max,min); I := I + (*/Ti)* + (/Tr)*( - v); 5 Bmplss Transfrs Bmplss Mod Cangs Avoid bmps in conrol signal wn canging opraing mod (manal - ao - manal) canging paramrs canging bwn diffrn conrollrs Canging opraing mod y Iss: Mak sr a conrollr sas av corrc vals, i.., sam vals bfor and afr cang Incrmnal Form: + y sp y MCU Inc PID M A s 53 54

10 Bmplss Mod Cangs Bmplss paramr cangs Dirc Posiion form: + T m y sp y T s PD T i s T + M A + A cang in a paramr wn in saionariy sold no rsl in a bmp in conrol signal. For xampl: v := P + I + D; I := I +(*/Ti)*; or v := P + (/Ti)*I + D; I := I + *; T lar rsls in a bmp in if or Ti ar cangd Bmplss paramr cangs Canging Conrollrs Mor involvd siaion wn spoin wiging is sd. T qaniy P + I sold b invarian o paramr cangs. Conrollr Conrollr Swic Procss I nw = I old + old (β old y sp y) nw (β nw y sp y) 57 Similar o canging bwn manal and ao L conrollrs rn in paralll L conrollr a is no aciv rack on a is aciv. Alrnaivly, xc only aciv conrollr and iniializ nw conrollr o is corrc val wn swicing (savs CPUim) 58 PID cod PID-conrollr wi ani-rs windp and manal and ao mods (γ = ). y = yin.g(); = yrf - y; D = ad * D - bd * (y - yold); v = *(ba*yrf - y) + I + D; if (mod == ao) { = sa(v,max,min)} ls = sa(man,max,min); O.p(); I = I + (*/Ti)* + (/Tr)*( - v); if (incrmn) { inc = } ls {if (dcrmn){ inc = -} ls inc = ;} man = man + (/Tm) * inc + (/Tr) * ( - man) yold = y ad and bd ar prcalclad paramrs givn by backward diffrnc approximaion of D-rm. 59