EE Control Systems LECTURE 11

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1 Up: Moy, Ocor 5, 7 EE Corol Sy LECTUE Copyrigh FL Lwi 999 All righ rrv POLE PLACEMET A STEA-STATE EO Uig fc, o c ov h clo-loop pol o h h y prforc iprov O c lo lc uil copor o oi goo y- rcig y uyig h y- rror Movig h Pol Uig Fc A ipor propry of fc i h i ov h y pol Th x xpl how how o u hi for fc ig Expl - Movig h Pol Uig Fc A op-loop pl h rfr fucio Thi y h ul coplx pol pir 3 ± j r K u y i ir o fc copor K how i h figur, whr w w o u h proporiol-riviv P fc

2 K P wih h riviv gi P h proporiol gi W w o lc h gi o oi h prforc pcificio: clo-loop POV of 4% lig i of c A POV of 4% corrpo o pig rio of ζ / Sic 5τ, lig i of c τ / 5 4 c Thi giv rl pr of h pol of α 5 τ Th rgio of h -pl pcifi y hi prforc pc i how i h figur Th clo-loop pol u o h 45 o li how o h lf of α 5 jω 45 o σ -5 W c ify h pc y lcig α 5 ζ 77, α 5, ω 7678 Th, h ir clo-loop ζ chrcriic polyoil i Δ α ω 5 35 Th clo-loop rfr fucio i T P P O h h pol y plc rirrily ir y lcig h wo P gi To oi h ir clo-loop chrcriic polyoil, 5 35 P

3 olv o g h P gi 6, 5 P Sy-S Error Fc c iprov h rcig cpilii of pl y ig h y rror llr u y A op-loop y i how fi h rcig rror u y For goo rcig, h rror houl ll, for h h cul oupu y follow h ir co u Th rcig rror i fou E U U U U To fi h y- rror i rpo o ui p co o u h Fil Vlu Thor o oi E U Thrfor, h y- rror i qul o zro oly if h C gi i xcly qul o Thi i ifficul rquir o i prcic ic, for ic h C gi y p o o rior prr which y vry wih prur, c figur To oi or rou rcig y ig, fc how i h r K u y Th clo-loop rfr fucio i giv y 3

4 T Howvr, o fi h y- rror, o ri h rfr fucio fro r o h rror Uig Mo' forul, hi i giv y E To fi h y- rror i rpo o ui p co, lc u h Fil Vlu Thor o oi E K K Thrfor, h y- rror i ll if h C loop gi i iply lrg ough O c gur hi y iply lcig h copor K, lrg vlu of co proporiol gi Sy-S Error Sy Typ r K u y A clo-loop rcig y i how i h figur A rcr i o ur h h cul pl oupu y follow or rc h ir rfrc co r Trcig y r ig y uyig h y- rror propri of h clo-loop y Th rcig rror i fi r y L h pl giv i h for Th copor i giv K, 4

5 5 which i -igror wih gi of o h h gi h iffr ui pig o h vlu for i i iiol, whil h ui c -, o o Th loop gi i K o for rliv gr rquir w > W u h pl o o hv y pol h origi A y wih pol h origi i cll yp y Th rfr fucio fro rfrc co o rror i giv uig Mo' forul K E Accorig o h Fil Vlu Thor, h y- rror i giv y K E Thi p o h co r wll h y yp vluig hi, o h u o h pcil for w lc for, o h, po o Ui Sp Poiio Co L h rfrc co h ui p, which corrpo o poiio co Th, f, corrpoig o yp zro y, h h y- rror o poiio co i qul o O c hi ll y lcig lrg gi o h i uch grr h o h o u ow h ig of, ic h corol gi u hv h ig

6 y r ozro y- rror o p co f, corrpoig o yp o y, h h y- rror o poiio co i qul o fc, for y o h zro y- rror Thrfor, for yp y, h y- rror i rpo o ui p c ll ir y icrig h gi For yp or lrgr y, h y- rror i rpo o ui p i qul o zro Thi h rcig y c follow poiio co prfcly log hr i l o igror i h fforwr ph po o Ui p Vlociy Co L h rfrc co h ui rp, which corrpo o vlociy co Th, f, corrpoig o yp zro y, h h y- rror i rpo o ui rp co i qul o f, corrpoig o yp o y, h h y- rror i qul o O c hi ll y lcig lrg gi o h i uch grr h o h o u ow h ig of, ic h corol gi u hv h ig 6

7 y r ozro y- rror o vlociy co f, corrpoig o yp wo y, h h y- rror i qul o fc, h y- rror i zro for y Thrfor, for yp y, h y- rror i rpo o ui rp i ifiiy Such y co follow vlociy co For yp y, h y- rror i rpo o ui rp c ll ir y icrig h gi For yp or lrgr y, h y- rror i rpo o ui p i qul o zro Thi h rcig y c follow poiio co prfcly log hr r l wo igror i h fforwr ph rl Mol Pricipl Th irl ol pricipl i vry ipor i fc ig h rcig y c follow prfcly co r log h fforwr ph coi ol of h co Thu, poiio followr o hv l o igror i h fforwr ph, vlociy followr l wo igror, c 7

8 Expl - Sy-S Error Alyi r K y hi clo-loop y, h pl i h corollr i h igrl copor K wih h igrl gi Th rcig rror i giv y r y Thi xpl phiz h o u of fi ulipl rfr fucio i h igr o lyz i fc, i hi xpl w hll hv o fi four rfr fucio Fi h Sy-S Error i po o Ui Sp iurc To fi h rqu rror, / Th rfr fucio fro o i giv y E E Uig ow h Fil Vlu Thor yil E 8

9 9 Th y i of yp, o h rror wr ui p iurc i qul o zro o h hi h oupu y go o zro, ic h co i r Fi h Sy-S Error i po o Ui Sp Co r To fi h rqu rror, / Th rfr fucio fro r o i giv y E E Uig ow h Fil Vlu Thor yil E Th y i of yp, o h rror wr ui p co i qul o zro o h hi h oupu y go o o, ic h co i ru - c Fi h POV For Ui Sp Co Th clo-loop rfr fucio fro h co r o h oupu y i giv y K K Th clo-loop chrcriic polyoil i Δ, coprig hi o h r for ω ζω Δ o h ω ζ ω O c hrfor lc h igrl gi o oi y ir vlu of pig rio, hc of POV which i giv y

10 % / ζ πζ POV Fi h Oupu y if, u u r Th rfr rlio w h wo ipu h oupu i giv y K K K or Sig ow, o oi For h giv vlu of o y ow prfor PFE ri y Expl 3- or Trcig Sy h xpl w hv o fr, hr i uiy gi our rcig loop o h h rror ppr xplicily i h loc igr Thi i vry y o u Mo' Forul o ri h rfr fucio for y- rror lyi Th figur how or rcig y wih o uiy gi our loop Th rcig rror i fi y r, K y r

11 u i co rw i h figur Suppo h h pl i h corollr i h igrl copor K wih h igrl gi To fi h y- rror i rpo o ui p co, / Th rfr fucio fro r o y i giv y ow, copu E Thi i h rror o which o prfor h Fil Vlu Thor fc, E Thu, hi i vry ig i o yp y, ic h igror o o ppr i h fforwr ph

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