Example on PID controller selection

Transcription

1 Examl on P controllr lction Problm tatmnt: conidr th following two unity fdbac ytm: Sytm #: firt ordr ytm Sytm #: cond ordr ytm () + - C() () + - C() quirmnt:. Find th tady tat rror ( ) for th following ca: a. r(t) = b. r(t) = t c. r(t) =t /. add a P or P controllr to in atify th following condition: For ytm #: Condition St am Parabolic Stady tat rror <. <. <. Th maximum ovrhot (%) < 5 < 5 < 5 Th ttling tim ().5 i tim () For ytm #: Condition St am Parabolic Stady tat rror <. <. <. Th maximum ovrhot (%) < 5 < 5 < 5 Th ttling tim () i tim () Pag of 3

2 Solution:- For ytm #: G For th t inut r(t) = or () = /, th tady tat rror can b calculatd uing Eq.: ; lim G( ) lim () For th ram inut r(t) = t or () = /, th tady tat rror can b calculatd uing Eq.: ; v lim G( ) lim () v For th arabolic inut r(t) = t / or () = / 3, th tady tat rror can b calculatd uing Eq3: ; a lim G( ) lim (3) a A n from quation, and 3, th tady tat rror or thi ytm do not atify condition numbr. So, dfiantly w nd a controllr for ytm For ytm #: G For th t inut r(t) = or () = /, th tady tat rror can b calculatd uing Eq.4: ; lim G( ) lim (4) For th ram inut r(t) = t or () = /, th tady tat rror can b calculatd uing Eq.5: ; v lim G( ) lim (5) v Pag of 3

3 For th arabolic inut r(t) = t / or () = / 3, th tady tat rror can b calculatd uing Eq.6: ; a lim G( ) lim (6) a A n from Eq.4, th tady tat rror i zro but thi do not man that w do not nd a controllr bcau w nd to atify th othr condition: max. Ovrhot, ri tim and ttling tim. On th othr hand, th ram and arabolic inut nd a controllr. A MATLAB cod wa ud to draw both ytm for th thr inut and th rult ar hown in Fig.. Fig.. tim ron for ytm (th ur raw) and ytm (th lowr raw). Th rd lin i ytm ron and th blu lin i th rfrnc inut. A n from Fig., whn unity t inut i alid to th firt ytm, th outut rach a valu qual ½ which i half of th original inut. Thi rult i rovd by Eq. whr th rror i ½. On th othr hand, th cond ytm how that thi ytm will rach th rfrnc inut vntually which i rovd by Eq.4 whr th rror i zro. Alo it obviou that whn th inut of th firt ytm (firt ordr ytm) i ram or arabolic, th outut divrg from th inut which i rovd by Eq and 3. Pag 3 of 3

4 For th cond ytm (cond ordr ytm), th ram inut how a convrgnc to th inut but with rror qual. For xaml, th tady tat ron i hown in Fig.. for ram inut how that at tim qual c which can b conidrd nough to tady tat condition th outut qual 9 whil th inut qual and o th rror i which i rovd by Eq.5. From rviou analyi, w how that for both firt and cond ordr ytm a controllr i ndd to atify th condition rquird rviouly. Add P controllr for both ytm Th tranfr function of P controllr (Gc) i givn a: Gc()= +. whn thi controllr i addd to th ytm thy bcom: Sytm #: firt ordr ytm Sytm #: cond ordr ytm () + - C() () + - C() Now, th rror calculation mut b rformd again but thi tim th P controllr i addd For ytm #: Error condition:- For th t inut r(t) = or () = /, th tady tat rror for th modifid ytm can b calculatd uing Eq.7: ; lim G( ) lim. (7) From Eq.7. w can find our firt condition which i >999. Thi condition can b atifid by chooing =. Stability condition:- Aftr adding Gc() to th ytm, th tranfr function M() = C() / () i calculatd a: Pag 4 of 3

5 C G M (8) G Th charactritic quation of th ytm i: (9) Eq.9 giv th cond condition (tability condition). Bcau and ar ral numbr, mut b ngativ ral numbr to inur tabl ytm and thi can b atifid whn > or bcau th valu of i gratr than +999 (i.. th nominator i alway ngativ thrfor th dominator mut b oitiv). To draw th tim ron, a MATLAB imulin i ud with valu of = and = -.5. Fig how th bloc diagram and Fig.3 how th tim ron for thi ytm. Fig.. imulin bloc diagram for th firt ytm whn th inut i unit t, = and = -.5 Fig.3. tim ron for th firt ytm with P controllr whn th inut i unit t, = and = -.5 Pag 5 of 3

6 Howvr, if th tim i incrad to c intad of.c and th tady tat ron i magnifid a hown in Fig.4, th ron will ocillat in bhavior how intability. Thi bhavior can b ignord if th tady tat rror i l than. and w can aum that th ytm rach tady tat aftr.5 c a hown in Fig.3. Fig.4. magnifid tim ron for th firt ytm with P controllr whn th inut i unit t, = and = -.5 n arat xrimnt, highr valu of wr ttd and th rult how that a incra, th ocillation frquncy incra and both ttling and ri tim dcra. For th ram inut r(t) = t or () = /, th tady tat rror can b calculatd uing Eq.: ; v lim G( ) lim () v t i obviou hr that adding a P controllr do not olv th roblm. Thi can b concludd for th arabolic inut too a hown in Eq. ; a lim G( ) lim () a A a concluion, P controllr i not uitabl for firt ordr ytm whn th inut i ithr ram or arabolic. Pag 6 of 3

7 For ytm #: G For th t inut r(t) = or () = /, th tady tat rror can b calculatd uing Eq.3: ; lim G( ) lim (3) A n w hav to giv no xtra car about th tady tat rror. Howvr, w nd to tt ytm tability. Th tranfr function M() = C() / () i calculatd a: G C G M Th charactritic quation of th ytm i: (4) (5) Thi i a quar quation and o two root for may b found or th tability can b xamind uing outh mthod a illutratd in th following tabl: + To inur tabl ytm, th valu of and ( +) mut b oitiv and o: > and > -. Th valu of i chon a.5 and th valu of wr itratd and th rult ar hown in tabl blow: Max. ovrhot (%) tr () t() Pag 7 of 3

8 A n from tabl, all th hadd row atify all th condition: max. Ovrhot <5%, t r <5c and t < c. Fig.5 how th tim ron whn =.5 and =.5. Fig.5. tim ron for th cond ytm with P controllr whn th inut i unit t, =.5 and =.5 For th ram inut r(t) = t or () = /, th tady tat rror can b calculatd uing Eq.6: ; v lim G( ) lim. (6) v So > to atify th tady tat condition. For tability condition, th charactritic quation do not bn affctd by th ty of inut and o th rviou tability condition: > and > - or > and > - to atify both rror and tability condition ar valid. Chooing = and =.5 will atify all condition a hown in Fig.6 whr th rd lin in th magnifid ictur how th rfrnc inut. Pag 8 of 3

9 Fig.6. tim ron for th cond ytm with P controllr whn th inut i unit ram, = and =.5. For th arabolic inut, th tady tat rror can b calculatd uing Eq.7: ; a lim G( ) lim (7) a From Eq.7 thr i no bnift from adding P controllr whn th inut i arabolic bcau thr will b in th nomntor ma a = and th tady tat rror go to infinity. Adding P controllr Th tranfr function of P controllr (Gc) i givn a: Gc()= + /. whn thi controllr i addd to th ytm thy bcom: Sytm #: firt ordr ytm Sytm #: cond ordr ytm () + - / C() () + - / C() Now, th rror calculation mut b rformd again but thi tim th P controllr i addd in imilar way a w hav don with P controllr. Th nxt tabl ummriz th nw chang aftr adding th P controllr Pag 9 of 3

10 Sytm G() M() = C()/() Charactritic Eq. Stability cond > and > > and > Error analyi: Nxt tabl how th rror condition for th nw rood ytm Sytm St am Parabolic According to rviou tabl w can conclud th following:. P controllr can not hl u to control th firt ytm whn th inut i arabolic. May b addition to cond intgral controllr can hl.. For t inut, both ytm how xact tady tat olution ( =) 3. For ram inut, th firt ytm nd to b controlld to atify th rror condition whil tady tat rror in th cond ytm i cancld by th rnc of th controllr whatvr th valu of and P. 4. Error of th cond ytm whn th inut i arabolic can b controlld uing P controllr Now to atify th othr rquirmnt, w nd to ittrat for th valu of and for ach ca individually. Pag of 3

11 Ca : ytm + t inut + P controllr An ittration valu of and P wr ttd and th rult ar hown in th nxt tabl P Max. ovrhot (%) tr () t() A n, th valu of = 75 and = will atifay all th condition. Fig.7 how th tim ron for th valu Fig.7. tim ron for th firt ytm with P controllr whn th inut i unit t, = and =. Pag of 3

12 Ca : ytm + ram inut + P controllr To atify th tability condition, > and > - or > and > - to atify both tability and rror condition. Arbittary valu of = and P = will atify allcondition a hown in Fig.8. Fig.8. tim ron for th firt ytm with P controllr whn th inut i unit ram, = and =. Ca 3: ytm + t inut + P controllr Aftr many ittration, th valu of P =.5 and =. atifay all condition and Fig.9 how th tim ron undr th valu Fig.9. tim ron for th cond ytm whn th inut i unit t, =.5 and =.. Pag of 3

13 Ca 4: ytm + ram inut + P controllr Aftr many ittration, th valu of P = and =. atifay all condition and Fig. how th tim ron undr th valu Fig.. tim ron for th cond ytm whn th inut i unit t, = and =.. Ca 5: ytm + arb inut + P controllr Aftr many ittration, th valu of P = and = atifay all condition and Fig. how th tim ron undr th valu Fig.. tim ron for th cond ytm with P controllr whn th inut i ram t, = and =. Pag 3 of 3