CDS 101: Lecture 9.1 PID and Root Locus
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1 CS : Lct 9. P ad Root Loc Richad M. Ma 5 Novmb Goal: fi P cotoll ad dcib how to thm todc th oot loc tchiq ad dcib how to it to choo loop gai Show om of th limitatio of fdback d to RHP pol ad zo Radig: Atom, Sc..,. Optioal: PPH, Sc 3 Advacd: Lwi, Chapt Sc 3. Lct Rviw.: Fqc fom Lat omai Wk ig Loop Shapig fo Stabilit ad Pfomac Stad tat o, badwidth, tackig P ( Mai ida Pfomac pcificatio giv bod o loop taf fctio U cotoll to hap po Gai/pha latiohip cotai dig appoach Stadad compato: popotioal, lad, P P ( 3 5 Nov CS 5 Nov
2 Ovviw: P cotol P P( = K K K p titio Popotioal tm: povid ipt that coct fo ct o tgal tm: i that tad tat o go to zo (if ot, cotol gt bigg ivativ tm: povid aticipatio of pcomig chag A bit of hito o th tm cotol Fit appad i 9 pap b Miok: ictioal tabilit of atomaticall td bodi d th am th tm cotol Alo alizd that mall dviatio (liaizatio cold b d to dtad th (olia tm damic d cotol Utilit of P P cotol i mot commo fdback tct i giig tm Fo ma tm, ol d P o P (pcial ca Ma tool fo tig P loop ad digig gai ( adig 5 Nov CS 3 Fqc domai compatio with P C( Taf fctio fo P cotoll = K K K Roghl qivalt to a P cotoll with lad compatio da: giv high gai at low fqc pl pha lad at high fqc Plac blow did coov fq d p P( H( = Kp K K Pha (dg; Magitd (db = Kp K K = k( T T kt ( / T( / T = T 3 Bod iagam = T 3 3 Fqc (ad/c = T 5 Nov CS 5 Nov
3 Tool fo igig P cotoll d C( P( = K( T T ZiglNichol tig ig P gai bad o tp po Wok OK fo ma plat (a bit ddampd Good wa to gt a fit ct cotoll Fqc domai vio alo it Catio: P amplifi high fqc oi Sol : pol at high fqc Catio: tgato widp Pologd o ca lag itgatd o Effct: gt lag dhoot (to t itgato Sol : mov pol at zo to v mall val Faci ol : atiwidp compatio K =. / a T = * L T = L/ 5 Nov CS 5 a Magitd (db Stp po L Poit of maimm lop Bod iagam Fqc (ad/c Eampl: P ci cotol / m P ( = b/ m a ZiglNichol dig fo ci cotoll Plot tp po, tact L ad a, compt gai L =.9 a =.39 tp lop Pha (dg; Magitd (db P ( P ( Bod iagam 3 Fqc (ad/c 3 Tim (c. Rlt: lggih ica loop gai 5 Nov CS Amplitd Stp Rpo = K( T T K =. / a T = * L T = L/ 5 Nov 3
4 Pol Zo iagam ad Root Loc Plot C( α P( Oigial pol locatio (α = Pol go tabl fo om α Pol zo diagam vifi tabilit Root of PC giv clod loop pol Ca tac th pol a a paamt i chagd: = K( T T α Ral Ai Root loc = loc of oot a paamt val i chagd Ca plot pol locatio fo a igl paamt; jt patdl olv fo oot Commo choic i cotol i to va th loop gai (K mag Ai Pol go to Pol mg ad plit Pol go to tmial val 5 Nov CS 7 Root Loc fo Loop Gai α C( Loop gai a oot loc paamt Commo choic fo cotol dig Spcial popti fo loop gai Root go fom pol of PC to zo of PC Ec pol go to ifiit Ca compt amptot, bak poit, tc V fl tool fo cotol dig MATLAB: loc P( mag Ai Op loop pol locatio (α = Amptot fo c pol at (3 /(PZ Ral ai to th lft of odd # of al pol & zo i o oot loc Clod pol go to op loop zo Ral Ai Additioal commt Althogh loop gai i th mot commo paamt, do t fogt that o ca plot oot v a paamt Nd to lik oot locatio to pfomac 5 Nov CS 5 Nov
5 Scod Od Stm Rpo Scod od tm po Spig ma damic, witt i caoical fom Gidli fo pol placmt ampig atio giv R/m atio Sttig tim dtmid b R(λ H( = = ( j ( j ς ς d ς d = ς d Pfomac pcificatio T./ T 3.9 / ς M p SS = πς / ς Root Loc Edito (C T < ζ.77.5 M p % % Slop.7 mag Ai id gio fo clod loop pol M p <.5 % Nov CS Ral Ai 9 Effct of pol locatio o pfomac da: look at domiat pol Pol at th imagia ai (at to itabilit Aalz ig aalog to cod od tm mag Ai Polzo map Amplitd To: Y( Stp Rpo Fom: U( PZmap complmt ifomatio o Bod/Nqit plot Simila to gai ad pha calclatio Show pfomac i tm of th clod loop pol Paticlal fl fo chooig tm gai Alo fl fo dcidig wh to pt cotoll pol ad zo (with pactic mag Ai mag Ai 5 Nov CS Ral Ai Polzo map Ral Ai Polzo map Ral Ai Amplitd To: Y( Amplitd To: Y( Tim (c. Stp Rpo Fom: U( Tim (c. 3 Stp Rpo Fom: U( Tim (c. 5 Nov 5
6 Eampl: P ci cotol Stat with P cotol dig: / m P ( = b/ m a = K( T T Modif gai to impov pfomac U MATLAB iotool Adjt loop gai (K to dc ovhoot ad dca ttlig tim ζ l tha 5% ovhoot R(p <.5 T l tha c 5 Nov CS Pfomac limit d to RHP pol ad zo Loop gai limitatio RHP pol qi miimm val of loop gai to mov tabl pol to LHP RHP zo limit maimm val of loop gai bfo itabilit occ Sio dig cotait! Watbd ffct M = ma H ( j M = ma H ( j mag Ai Ral Ai Thm: Sppo that P ha a RHP zo at z. Th th it cotat c ad c (dpdig o,, z ch that clog M clog M. M i tpicall mch l tha M mt b lag tha (ic m i poitiv f w ica pfomac i activ ag (mak M ad H mall, w mt lo pfomac (H ica om plac l [watbd ffct] 5 Nov CS 5 Nov
7 Eampl: ctd fa latal poitio cotol θ (, f f ( mgl P ( = ( J d mgl Latal cotol ( Right half pla zo mak dig v tick ig a otpt Magitd (db PC watbd Rdcd itivit btt pfomac p to high fqc mag A Root Loc ig 3 Fqc (ad/c Ral Ai 5 Nov CS 3 Smma: P ad Root Loc P cotol dig V commo (ad claical cotol tchiq Good tool fo chooig gai = K K K p Bod iagam Root loc Show clod loop pol a fctio of a f paamt Pfomac limit RHP pol ad zo plac limit o achivabl pfomac Watbd ffct Pha (dg; Magitd (db 3 = T = T mag Ai 3 3 Fqc (ad/c Ral Ai 5 Nov CS 5 Nov 7
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