Chapter 9. Design via Root Locus

Transcription

1 Chapter 9 Deign via Rt Lcu

2 Intrductin Sytem perfrmance pecificatin requirement imped n the cntrl ytem Stability Tranient repne requirement: maximum verht, ettling time Steady-tate requirement :.. errr actual ytem perfrmance: determined by dminant cled-lp ple (A) deired ytem perfrmance: repreented by a cled-lp ple (B) - If A B, perfrmance pecificatin are met. - If nt, need t change dminant C.L. ple lcatin. Hw? Ex) Wuldliketpeeduptherepnefthe ytem withut affecting the percent verht maintain i current damping but fater ettling time Increae the gain lcate the CL ple at A but nt atifactry yyet, need a CL ple at B but a ple at B i nt n the rt lcu imple gain adjutment change the lcatin f CL ple nly n the rt-lcu

3 Hw t change dminant C.L. ple lcatin? A. imple gain adjutment (i.e. increae the gain ) : change the lcatin f CL ple nly n the rt-lcu B. Cmpenatin : alter the verall ytem behavir by adding ple and zer that the cmpenated ytem ha a new rt lcu that ge thrugh the deired ple lcatin fr me value f gain Cmpenatr - PID cmpenatr (r cntrller) - Lead cmpenatr - Lag cmpenatr - Lead-Lag cmpenatr Phyical implementatin f cmpenatr - Op-amp amp, RC-netwrk: analg - PC, micrprcer: digital

4 RC netwrk fr Lead/Lag cmpenatr If R C > R C : lead R C > R C : lag Lag-Lead cmpenatr

5 Effect f adding Ple & Zer - Cled-lp lp ple that have dminant effect n the tranient-repne repne behavir - CL ple nearet the jω when the rati f real pat f the CL ple > 5 effect f adding ple - pull the rt lcu t the right, thereby tend t lwer the relative tability f the ytem - lw dwn the ettling time Adding ple & zer Mdify the rt-lcu hape! effect f adding zer - pull the rt lcu t the left, thereby tend t make the ytem mre table - peed up the ettling time (phyically equivalent t adding a derivative (anticipatry) cntrl actin

6 Tw cnfiguratin f cmpenatin Gain (a) () Cacade cmpenatin Prduce a cntrl ignal that will reduce the errr t zer r t a mall value baed n the errr ignal (b) Parallel r Feedback cmpenatin Bth cmpenatr add the pen-lp ple and/r zer Creating a new rt lcu paing thrugh the deired cled-lp ple lcatin

7 Imprving Steady-State Errr via Cacade Cmpenatin Objective T imprve the teady-tate errr withut appreciably affecting the tranient repne ) Ideal integral cmpenatin r Ideal prprtinal-plu-integral (PI) cntrller Ue pure integratr (i.e. add an pen-lp, frward-path ple at the rigin) Increae ytem type reduce the.. errr t zer mut be implemented with active netwrk uch a amplifier ) Lag cmpenatr Place the ple near the rigin De nt drive the.. errr t zer but yield a meaurable.. errr reductin - Nt change the lcatin f the dminant CL ple - Increae the pen-lp gain a much a needed can be implemented with a le expenive paive netwrk nt requiring the pwer urce Name f the cmpenatr a. Prprtinal cntrl ytem feed the errr frward t the plant b. Integral cntrl ytem feed the integral f the errr t the plant c. Derivative cntrl ytem feed the derivative f the errr t the plant

8 A. Ideal Integral Cmpenatin (r PI cntrller) Imprving teady-tate errr increae ytem type by placing an pen-lp ple at the rigin Aume A i a cled-lp ple that generate the deirable tranient repne i.e. need t have a rt-lcu pa thrugh pint A Add an integratr (i.e. ple at the rigin) t increae ytem type Rt lcu withut cmpenatr t increae the ytem type, add a ple at the rigin angle cnditin at A i nt 80 Rt lcu n lnger pa thrugh pint A

9 Hw t meet angle cnditin, then? T cancel ut the angular cntributin f added ple Add a zer cle t the added ple at the rigin angular cntributin f added zer and ple cancel ut A i till n the rt lcu gain cntributin (length) f added zer and ple cancel ut ame gain at A mean, maintain deired tranient repne with the ame gain Finally, imprved the.. errr withut appreciably affecting the tranient repne Ideal integral cmpenatr Have a ple at the rigin and a zer cle t the ple

10 Ex 9.) hw the additin f the ideal integral cmpenatr reduce the.. errr fr a tep input withut appreciably affecting the tranient repne f a ytem perating with a damping rati 0.74 Uncmpenated cled-lp ytem v. cled-lp ytem with ideal integral cmpenatin

11 a. Rt lcu fr uncmpenated ytem Dminant ecnd rder ple Can be neglected t apprximate a a nd rder ytem Third ple fr the ame gain 64.4 Steady-tate errr e( ) p + lim ( + )( + )( + 0) 0

12 b. Rt lcu fr cmpenated ytem Added ple at the rigin Added d zer at -0. 0cle t the rigin i Gain cntributin 0 Apprximately the ame with the uncmpenated ytem gain cmpenated 58. p uncmpenated 64.4 G( ) H ( ) ple length zer length neglect Cled-lp pple at zer at -0. Ple-zer cancellatin fr the ame gain 58. due t &, cled-lp ple and gain are apprximately the ame a the uncmpenated ytem CL ple and gain mean cmpenated tdytem ha imilar il tranient t repne with uncmpenated tdytem Hwever, cmpenated ytem will have a zer teady-tate errr t a tep input unlike the uncmpenated ytem thank t the ple at the rigin

13 Step repne Zer.. errr Similar tranient repne up t 3ecnd After that, integratr lwly cmpenate the errr -Q.E.D.-

14 Implementing the ideal integral cmpenatr Tranfer functin f an ideal integral cmpenatr adding a ple at the rigin and a zer G c ( z) z + + ( ) + + adjut the value f zer by varying / (Errr + integral f errr) are fed frward t the plant PI cntrller

15 B. Lag Cmpenatin Ideal integral cmpenatr place the ple a ple at the rigin require an active integratr Lag cmpenatr place the ple near the rigin nt at the rigin need nly paive netwrk eay and inexpenive t implement Static errr cntant fr the uncmpenated type ytem z z L v p p L Lag cmpenatr Effect f lag cmpenatr n teady-tate errr Static ti errr cntant t fr the ytem with the lag cmpenatr z z zc zc ( L) vn v ( p p ) pc p > L c Cmpenated ytem ha increaed tatic errr cntant Imprved teady-tate errr.. errr v v

16 Rt lcu: a. befre lag cmpenatin b. after lag cmpenatin at at Aume P t be the dminant ple Effect f lag cmpenatr n tranient repne if z c p c Angular cntributin f cmpenatr 0 i.e. lcatin f p lcatin f p ple length zer length i.e. Gain fr p Gain fr P virtually n effect n the lcatin f the dminant cled lp ple i.e. n effect n the tranient repne

17 Requirement fr Lag cmpenatr Lag cmpenatr huld atify the bth fllwing cnditin T minimize the effect n the tranient repne, z c p c T maximize the teady-tate errr imprvement, mut be large Bth z c and p c huld be cle t the rigin z c p c Ex) if z c -0.0, p c bth cle virtually n effect n tranient repne but rati i 0 achieve 0 time better.. errr

18 ex 9.) Deign a cmpenatr t imprve the teady-tate errr by a factr f 0 if the ytem i perating with a damping rati f 0.74 fr the fllwing ytem Uncmpenated ytem Frm ex 9.) e( ) p + lim ( + )( + )( + 0) 0 0 time imprvement 0.08 e ( ) t arg et + p, t arget e t arget ( ) p, t arget e ( ) t arget z 9.59 c c pn p, t arg et pn p. 3 pc 8.3 z p c p Arbitrarily elect p c 0. 0 z c.3p 0. c

19 Nw, the cmpenated ytem Search the dminant ple n ζ 0.74 line atifying angle cnditin ±j3.836 with a gain 58. ( + 0.0)( + )( + )( + 0) ( + 0.) ( ± j rd & 4 th CL ple -.55, -0.0 fund by earching the real axi fr a gain th CL ple at -0.0 cancel the cmpenatr zer at -0. Leaving 3 CL ple ame with the uncmpenated ytem ple apprximately the ame tranient repne But.. errr f the cmpenated ytem e cmpenated ( ) p, cmpenated Almt /0 time le.. errr

20 Nte. Finding the dminant CL ple pair n a certain damping line Ex) Find the dminant CL ple pair n a ζ damping line O.L.T.F. G( ) ( + )( + 4)( + 6) C.L.T.F. C( ) R( ) G( ) + G( ) ( + α)( ζω + ω ) n n ( + α)( ω + ω ) ( ω n + α ) + (0.76 ωn α + ωn ) + αωn n n Slve 3 equatin fr 3 unknwn( ω, α ) n, 0.76ω n + α ωnα + ωn 4 αωn + ω. 833 α n dminant CL ple pair n ζ damping line ζωn ± jωn ζ.007 ± j.67

21 Time repne f uncmpenated and cmpenated ytem Similar tranient repne Better.. errr What if cmpenatr ple and zer are 0 time cler than the prped deign 4th CL ple at th CL ple at time cler t the rigin prduce a lnger tranient repne take lnger t reach the final value - Bth will reach the ame final value eventually

22 Ogata Ex. 7-) Deign a cmpenatr uch that the cled-lp ytem ha v 5 and keeping the damping ratin f the riginal ytem Open-lp TF G( ).06 ( + )( + ) ( + zc ) Gc ( ) ( + p ) c Cled-lp TF C( ) G( ).06 R( ) + G( ) ( + )( + ) ( j0.5864)( j0.5864)( ) Dminant CL ple f the riginal ytem ± j ζωn ± jωn ζ ω n ζ 0.49 Let the frm f the lag cmpenatr v z c p c Arbitrarily elect z G ( ) ( c v, deired v, deired pc v c ( + p z c p z c 0 p c c c ) ) 0 ( Q v lim G ( ) 0.53 ) 0 G c ( ) 05) ( ) ( )

23 New Open-lp TF G c ()G() mut atify the magnitude cnditin G ( ) G ( ) G c.03( ) ( ) ( + )( + ) ζ 0.49 line Search the dminant ple n ζ 0.49 line atifying angle cnditin -0.3±j ±j0 with a gain ( + 0.0)( + )( + )( + 0).06( + 0.) ± j Lag cmpenatr G c ( ) ( ) ( ) 005) G c ( ) G( ).03( ) ( )( + )( + )( + 0) v f the cmpenated ytem v lim G( ) lim G 0 0 c.03( ) ( ) G( ) ( ) ( + )( + ) 5.5

24 Imprving Tranient Repne via Cacade Cmpenatin Tranient repne pecificatin deirable percent verht + hrter ettling time Hw t achieve? Tw technique ) Ideal derivative cmpenatin r prprtinal-plu-derivative (PD) cntrller add pure differentiatr t the frward path equivalent t adding a zer require an active netwrk fr realizatin differentiatin make nie amplificatin ) Lead cmpenatr apprximate differentiatin with a paive netwrk

25 Ideal Derivative Cmpenatin (PD cntrller) Tranient repne determined by cled-ple lcatin Deired pecificatin crrepnd t a pecific cled-ple lcatin If thi CL ple i n the rt lcu, imple gain adjutment If nt rehape the rt lcu t pa thrugh the deired lcatin by adding ple and zer Tranfer functin f a cmpenatr adding a ingle zer G ( ) + z c ( c Sum f pure differentiatr and a pure gain Ideal derivative r PD cntrller

26 Ex) PD cntrller with different cmpenating zer at -, -3, -4 fr a ytem perating with ζ 0.4 All fur cae have dminant ple n the ame damping rati line percent verht will be the ame Dminant CL ple f cmpenated ytem ha Skip Increaed real part hrter ettling time increaed imaginary part hrter peak time T p π ω ω d Imprved.. errr But nt alway with PD cntrller

27 Ex) 9.3 deign a PD cntrller t yield a 6% verht, with a threefld reductin in etting time 6% verht ζ Uncmpenated ytem Find dminant CL ple pair n damping line -.05±j.064 Cmpenated ytem T ζω.05 Lcatin f dminant CL ple t atify ettling time pec.? n T, cmpenated σ cmpenated.07 σ cmpenated Real part f the dminant ple f cmpenated ytem

28 imaginary part f the dminant ple f cmpenated ytem ω 3.63tan( ) ω d, cmpenated Deired CL ple p deired -3.63±j6.93 p deired Uncmpenated rt lcu de NOT pa thrugh p deired Need cmpenatin where t lcate z c t make the rt lcu pa thrugh p deired? Angle cnditin fr p deired angle t zer angle t ple 80 θ 3 θ θ z θ θ z Let angle f a vectr cnnecting p deired d and z c (mewhere n real axi) θ z θ z θ θ θ Larger than 90 mewhere right t tan( ) z c Z c

29 - Rt lcu fr the cmpenated ytem Fater!

30 Implementing the ideal derivative cmpenatr (PD cntrller) ( ) ( G c + + ) z c Add zer + lp gain tuning c.f.) G ( ) + jut add zer c z c : Che t meet the required lp-gain Same rle a rt lcu gain PD cntrller Drawback f ideal derivative cmpenatr (PD cntrller) - Require an active circuit t perfrm the differentiatin - Amplifie the nie

31 Lead Cmpenatin Active ideal integral cmpenatr apprximated with a paive lag netwrk Active ideal derivative cmpenatr apprximated with a paive lead netwrk Paive netwrk Cannt prduce a ingle zer Intead, prduce a cmpenatr zer and ple If the ple i far left frm the zer, angular cntributin i till pitive a PD cntrller (a ingle zer) cntribute pitive angle Advantage ver PD ) N additinal pwer upplie needed ) Nie due t differentiatin i reduced Diadvantage ) Additinal ple de nt reduce the number f branche that may cr the imaginary axi while a zer reduce the number f branch that may cr the imaginary axi

32 Prcedure fr lead cmpenatr deign Select the deired dminant cled-lp ple Arbitrarily elect either z c r p c 3 Find angular cntributin f thi z c r p c plu ytem pen-lp ple and zer 4 Find remaining cmpenatr ple r zer uch that it cmpenate the difference between 80 and the angle btained in the tep 3. ( θ θ ) + θ θ θ ( k + )80 80 ( Net angular cntributin f the cmpenatr θ θ θ θ c - Different chice f z c and p c are pible a lng a the cmpenatr cntribute θ c

33 Ex 9.4) Deign a lead cmpenatr fr the fllwing ytem t reduce the T half while maintaining 30% verht 30% verht ζ Find dminant CL ple pair n damping line -.007±j.064 T 4 ζω n ) Find the deired CL ple t atify ettling time pec. Real part σ cmpenated σ.986 T, cmpenated cmpenated Imaginary part.04 tan( ) 5. 5 ω d, cmpenated.04 ) Select z c -5 arbitrarily 3) Angular cntributin f z c and pen lp ple(0, -4, -6) ) Select p c uch that it cntribute tan 7.3 p.04 c 4.96 p c

34 Nte. Finding the dminant CL ple pair n ζ damping line O.L.T.F. G( ) ( + )( + 4)( + 6) C.L.T.F. C( ) R( ) G( ) + G( ) ( + α)( ζω + ω ) n n ( + α)( ω + ω ) ( ω n + α ) + (0.76 ωn α + ωn ) + αωn n n Slve 3 equatin fr 3 unknwn( ω, α ) n, 0.76ω n + α ωnα + ωn 4 αωn + ω. 833 α n dminant CL ple pair n ζ damping line ζωn ± jωn ζ.007 ± j.67

35 Ogata Ex. 7-) Deign a cmpenatr uch that the cled-lp ytem ha 4 & ω ζ 0. 5 n Open-lp TF Cled-lp TF Dminant CL ple 4 G() ( ) ( + ) C( ) G( ) R( ) + G( ) ( + + j 3)( + j ± j 3 σ ± jζω ζω ± jω ζ n n n 3) Uncmpenated (riginal) ytem ω n ζ 0.5 Deired dminant CL ple d ζ dωn, d ± jωn, d ζ d ± j4 0.5 ± j 3 ω n, d 4 ζ d 0.5 ) Select z c -.9 arbitrarily 3) Angular cntributin f z c and pen lp ple(0, -) d ± j ) Select p c uch that it cntribute ( ) 3 tan p 5.4 c p c p c z c 0

36 Nw, new pen lp TF f the cmpenated ytem mut atify the magnitude cnditin G c ( +.9) 4 ( ) G( ) ( + 5.4) ( + ) ( + 5.4) ( + ) 4( +.9) ± j Lead cmpenatr G c 4.68( +.9) ( ) ( + 5.4) Lead cmpenated ytem G c ( ) G( ) 4.68( +.9) ( + 5.4) 4 ( + )

37 Imprving Steady-State Errr and Tranient Repne Want t imprve bth teady-tate errr and tranient repne? - cmbine cntrller fr each tak - which ne firt? Den t matter But, generally Prprtinal-plu-integral-plu-derivative plu integral plu (PID) cntrller deign an active PD cntrller firt, then deign an active PI cntrller L l d t Lag-lead cmpenatr deign paive lead cmpenatr firt, then deign a paive lag cmpenatr

38 PID Cntrller Deign PID cntrller Tranfer functin G c ( ) Tw zer + ne ple at the rigin (ne zer + ne ple at the rigin frm ideal integral cmpenatr; ne zer frm ideal derivative cmpenatr) Cmpenatr deign prcedure ) Evaluate the perfrmance f the uncmpenated ytem ) Deign the PD cntrller t meet the tranient repne pecificatin. 3) Check if all requirement have been met by imulatin 4) Redeign if nt atifactry 5) Evaluate the teady-tate errr 6) Deign the PI cntrller t yield the required teady-tate errr 7) Determine the gain,,, and 3. 8) Check if all requirement have been met by imulatin 9) Redeign if nt atifactry

39 Ex) Deign a PID cntrller t achieve /3 f the uncmpenated ytem peak time at 0% verht and with zer teady-tate tate errr fr a tep input Step ) Uncmpenated ytem perfrmance 0% verht ζ ln(% OS /00) ζ π + ln (% OS /00) Find dminant CL ple pair n damping line -5.45±j0.57 with a gain.5 T p π π ω d A third fr the ame gain.5 Step ) find the deired CL ple lcatin π (0.97) Tp deired ωd, ω 3, deired d, deired Real part σ ω deired 8.3 tan( ) d, deired 5.87

40 Deign PD cntrller determine the lcatin f z c Angle cntributin f pen lp ple drawn frm the deired ple z c need t cmpenate tan8.37 z 8.3 c 55.9 z c PD cntrller deigned d G PD ()(+55.9) ( Step 3 & 4) imulate and cnfirm all the pecificatin are met Peak time reductin &.. errr imprvement

41 Step 5) deign PI cntrller t have zer teady-tate errr fr a tep input Che z c t be b cle l t t the rigin i i Rt lcu fr the PID cmpenated ytem G PI ( ) Step 6) determine the gain,, and 3 f the PID cntrller G PID fund graphically r by magnitude cnditin ( )( + 0.5) ( ) 4.6( )( + 0.5) 4.6( ) 59.5, 5 8.6, G PID ( )( + 0.5) ( + 8) ( ) G ( ) ( + 3)( + 6)( + 0) Step 7) PID cntrller imprved.. errr withut changing the tranient repne f PD cntrlled ytem

42 Lag-Lead Cmpenatr Deign Lag-lead cmpenatr deign paive lead cmpenatr firt, then deign a paive lag cmpenatr Cmpenatr deign prcedure ) Evaluate the perfrmance f the uncmpenated ytem ) Deign the lead cmpenatr t meet the tranient repne pecificatin. 3) Check if all requirement have been met by imulatin 4) Redeign if nt atifactry 5) Evaluate the teady-tatetate errr 6) Deign the lag cmpenatr t yield the required teady-tate errr 7) Check if all requirement have been met by imulatin 8) Redeign if nt atifactry

43 Ex) Deign a lag-lead cmpenatr fr ) the ettling time reductin by half at 0% verht ) the tenfld imprvement in teady-tate tate errr fr a ramp input Step ) Uncmpenated ytem perfrmance find the dminant ple n ζ line -.794±j3.50 with a gain 9. T 4 ζω n Step ) find the deired CL ple lcatin 4.30 T, deired ( ζω ) deired ( ζω ) n n deired ω d ( ζω ) tan( ) d, deired d n deired d Deign the lead cmpenatr t achieve the deired CL ple Arbitrarily che z c -6 cancel the O.L. ple tan( ) p c p c Step 3 & 4) check the deign with imulatin Rd Reduced dt

44 ( ζω n ) deired ω d, deired ( ζωn ) deired tan( ) d ± j ( + 6)( + 0) d z c d d ± j z c

45 Step 5) evaluate teady-tate errr Tranfer functin f uncmpenated p & lead cmpenated p ytemy 9. G( ) ( + 6)( + 0) G LC 977 ( ) ( + 0)( + 9.) Obtained by magnitude cnditin G LC () type I fr bth Steady-tate errr (/ v ) v, G 3.0 v, G LC 9 Lead cmpenatr imprved.. errr 6.794/3.0. Requirement tenfld f uncmpenated v,target 3.0 Lag cmpenatr till need t imprve 4.73(3.0/6.794 ) time f lead cmpenated ytem Step 6) deign lag cmpenatr t imprve teady-tate errr Arbitrarily elect p 0. 0 z c G lag z c ( ) ( ) ( + 0.0) Q z c v, t arget p c v Open-lp tranfer functin f the lag-lead cmpenated ytem G LLC ( ) ( ) ( + 0)( + 9.)( + 0.0)

46 ( ) Frm the magnitude cnditin, G LLC ( ) ( + 0)( + 9.)( + 0.0) 97 Finally, pen-lp tranfer functin f the lag-lead cmpenated ytem G LLC ( ) 97( ) ( + 0)( + 9.)( + 0.0)

47 Rt lcu fr lag-lead cmpenated ytem Fater peak and ettling time Imprvement in teady-tate errr fr a ramp input

48 4 Ogata Ex 7-3)Deign an apprpriate cmpenatr fr the ytem G( ) t achieve ( + 0.5) ζ deired 0.5, ωn, deired 5rad /, v, deired 80 Step ) Uncmpenated ytem perfrmance v 4 lim 8 0 ( + 0.5) 4 C. L. T. F. dminant ple: -0.5±j.983 ω, ζ n Step ) find the deired CL ple lcatin d.5 ± j4.33 Deired dminant ple: d -.5±j4.33 Deign the lead cmpenatr t achieve the deired CL ple 5 0 Arbitrarily che z c -0.5 cancel the O.L. ple 4 p c zc d ( + 0.5) 0 0 d z c tan(80 0 ) p c p c 5.05

49 Step 3 & 4) check the deign with imulatin Nw pen-lp pt.f. f the lead cmpenated ytem i ( + 0.5) 4 GLC ( ) Gc ( ) G( ) ( ) ( + 0.5) G LC () mut atify the magnitude cnditin G LC ( + 0.5) 4 ( ) ( ) ( + 0.5) d Step 5) evaluate teady-tate tt errr ( ) 4 d 6.5 Steady-tate errr (/ v ) v 4 lim 8 0 ( + 0.5) v, G LC 4.95 Requirement v,target 80 Lag cmpenatr till need t imprve 6.6(80/4.95) 6(80/4 time f lead cmpenated ytem

50 Step 6) deign lag cmpenatr t imprve teady-tate errr Arbitrarily elect z c G lag p c ( + 0.) ( ) ( ) Q z c v, t arget 6. 6 pc v Finally, pen-lp tranfer functin f the lag-lead cmpenated ytem G LLC ( ) ( + 0.5) ( ) ( + 0.) ( ) New dminant cled-lp ple f the cmpenated ytem.43± j new d Meet the required perfrmance pecificatin The effect f the third CL ple f the cmpenated ytem very mall becaue

51

52 Ntch Filter High-frequency vibratin mde mdeled a pair f cmplex ple near the imaginary axi (i.e. lw damping, high-frequency) May mve cler t imaginary axi r even cr int RHP Make k difficult t btain deired drepne Ntch filter cancel Niy undeirable e repne e Clear deired repne

53 Summary

54

55 Feedback Cmpenatin S far, rehaping the rt lcu t pa thrugh hthe deired d CL ple wa dne by cacading a cmpenatr in the frward path nly. What if having a cmpenatr in the feedback lp? parallel (feedback) cmpenatin May mre cmplex than cacade due t minr lp But may yield fater repne fr ilatin Generally it i a rate enr (differentiatr) t e.g. tachmeter, t rategyr Ilatin: effect f the dynamic repne f a certain ytem n ther dynamic i reduced r minimized when it dynamic are much fater than ther Ex) entire aircraft pitch angle cntrlled by the entire cled-lp ytem aer-urface pitin cntrlled by minr lp huld h ldbe ilated frm the entire aircraft dynamic huld be much fater than the entire aircraft dynamic

56 Apprach G G() / /G equivalent H() Open-lp TF with feedback H() Withut feedback H() G( ) H ( ) H G f c + GG G G [ f H c + G ] G ( ) H ( ) GG Effect f adding feedback H() Replace the ple and zer f G with the ple and zer f [ H ] f c + G Similar t cacade cmpenatin rehaping the rt lcu t pa thrugh the deired CL ple by adding ple and zer via H()

57 Ex 9-7) Deign rate feedback cmpenatr t reduce T by a factr f 4 maintaining 0% verht 0% verht 7.33 Step ) Uncmpenated ytem perfrmance dminant ple alng the 0% verht: -.809±j3.53 T 4 ζω n Step ) find the deired CL ple lcatin tan Angle cnditin T ( ) Cmpenatr zer need t z c tan( Magnitude cnditin f d 4. ) 7.36 z c. 4 T, deired 0.55 ω n Deired dminant ple: d -7.36±j4., ζ 0.5 T ( ) f [ (75 + )] f d f

58 Steady-tate errr (/ v ) v lim 0 [ (75 + )] f 75 + f 4.8 Figure 9.54 Imprved but nt atify the required pecificatin but mre practical than cacade PD cmpenatr which i niy and nt alway practical Cmpare with the reult f Prb. 8 (cacade PD cmpenatr)

59 equivalent ( + 5)( + 5) equivalent ( + 5)( + 5) f T ( ) f + ( + 5)( + 5) f d

60 Apprach (ame a Ogata Velcity feedback) Deign a minr lp tranient repne eparately frm the cled-lp ytem repne G minr () Minr lp TF frward path TF f the entire feedback ytem G minr G ( ) ( ) + G ( ) H Cled-lp l ple f minr lp pen-lp ple f the entire feedback ytem Adjutable with minr lp gain Change plant ple thrugh a gain adjutment nly nt by adding ple and zer a in cacade cmpenatin Finally change the cled-lp ple by the lp gain, a in cacade cmpenatin Feature Cacaded cmpenatin rehape the rt lcu with additinal ple and zer Feedback cmpenatin apprach Change plant ple thrugh a gain adjutment nly nt by adding ple and zer a in cacade cmpenatin f c ( )

61 Ex) deign minr-lp feedback cmpenatin t yield ) a damping ratin f 0.8 fr the minr lp and ) a damping ratin f 0.6 fr the cled-lplp ytem Fig Adjut f t lcate the minr-lp ple. Adjut t yield the deired cled-lp repne TF f the minr lp G minr () ( + 5)( + 5) G minr ( ) [ ( ( + 5)( + 5) f f )] Ple f G minr() fund by rt-lcu f pen lp TF Interectin f ζ 0.8 line with the rt lcu give Fig the cmplex ple at -0±j7.5 with a gain f 8.5 Thi ple wuld be the pen-lp ple f the entire feedback cntrl ytem (-0±j7.5, 0) Final rt lcu drawn with pen lp ple at -0±j7.5, 0 Fig Interectin f ζ 0.6 line with the final rt lcu give the cmplex ple at ±j6.046 with a gain 64.3 Reult much fater repne & maller teady-tate errr Fig. 9.58