ELEC 37 LECTURE NOTES, WEE Dr Amir Aghdam Cncrdia Univrity Part f th nt ar adaptd frm th matrial in th fllwing rfrnc: Mdrn Cntrl Sytm by Richard C Drf and Rbrt H Bihp, Prntic Hall Fdback Cntrl f Dynamic Sytm by n F Franklin, J David Pwll and Abba Emami-Naini, Prntic Hall Autmatic Cntrl Sytm by Farid lnaraghi and Bnamin C u, Jhn Wily & Sn, Inc, Stability analyi in frquncy dmain cnt d - Fr tability analyi it i vry imprtant t nt that th cld-lp pl ar th zr f and th pn-lp pl ar th pl f r - T u th principl f th argumnt, w nd t dfin a cld path that cntain all pint in th RHP W dfin th fllwing cld path a th Nyquit path r Nyquit cntur: Im{} -plan Radiur r R{} - Nw, upp that Z i th numbr f zr f inid untabl cld-lp pl and P i th numbr f pl f inid inc th pl f and ar qual, thi i in fact numbr f untabl pn-lp pl In tability analyi P i knwn frm th pn-lp tranfr functin Lctur Nt Prpard by Amir Aghdam
and it i dird t find Z Thi can b accmplihd by finding N and uing th principl f th argumnt - W dfin N,, t b th numbr f ncirclmnt f th rigin by th imag f undr - Uing a impl hift in th -plan, it can b aily n that N,, N,, In thr wrd, th numbr f ncirclmnt f th rigin in th -plan i qual t th numbr f ncirclmnt f th pint -, in th -plan Th pint -, i uually rfrrd t a th critical pint - Fr th tability f th cld-lp ytm w mut hav Z r N P - If th pn-lp tranfr functin i tabl, w hav P and thn fr th tability f th cld-lp ytm w mut hav N - It i t b ntd that th imag f th imaginary axi in th Nyquit path undr i th Nyquit diagram f - Th Nyquit tability critrin tat that a cld-lp ytm with ngativ fdback i tabl if and nly if th numbr f cuntrclckwi ncirclmnt f th pint -, in th -plan i qual t th numbr f pl f in th RHP, whr rprnt th pn-lp tranfr functin - Exampl : U th Nyquit critrin t acrtain th tability f th fllwing cld-lp ytm: r t yt - - Slutin: W hav: Lctur Nt Prpard by Amir Aghdam
3 Im{} -plan S R{} Frm th abv figur w knw that P n pn-lp pl in th RHP Nw, w hav t find th map f th Nyquit path undr Fr th prtin f th Nyquit path which i n th imaginary axi frm t, th map i in fact th Nyquit diagram f W hav: 5 9 R{ } <, 4 Im{ } <, Th prtin f th Nyquit path that i a circl > > φ r r, 9 φ 9, will b mappd int th rigin f th -plan thi i alway th ca fr trictly prpr tranfr functin Thrfr, th Nyquit map will b a fllw: Lctur Nt Prpard by Amir Aghdam
4 A it can b n frm th diagram, th numbr f ncirclmnt f th critical pint -, by th imag f undr i qual t zr r N,, S, frm th principl f th argumnt w will hav: Z N P Thi impli that th cld-lp ytm ha n pl in th RHP and, it i untabl Fr thi impl xampl, withut uing th Nyquit critrin, w knw that th charactritic quatin f th cld-lp ytm i S, th charactritic quatin ha a rt at and th cld-lp ytm i untabl Nyquit critrin can b vry uful fr mr cmplx ytm in gnral - If thr i a cntant gain in th pn-lp tranfr functin, th charactritic quatin will b In thi ca, fr Nyquit tability analyi, n can find th map f th Nyquit path undr imilar t th prviu ca but th critical pint will b intad f Thi can b n frm th fllwing quatin: Lctur Nt Prpard by Amir Aghdam
5 Thi impli that th numbr f ncirclmnt f th rigin by th imag f undr i qual t th numbr f ncirclmnt f th pint, by th imag f undr In thr wrd: N,, N,, N,, S, in gnral w will u, a th critical pint - It i t b ntd that th tability analyi uing th Nyquit critrin can b gnralizd t th nn-unity fdback ytm If th frward path tranfr functin i dntd by and th tranfr functin in th fdback path i dntd by H, th charactritic quatin will b H Thrfr, th nly diffrnc i that fr th tability analyi w mut find th imag f undr H intad f - Exampl : U th Nyquit critrin t acrtain th tability f th fllwing cld-lp ytm: 6 r t y t 3 - - Slutin: W hav: Im{} -plan -3 - - R{} Lctur Nt Prpard by Amir Aghdam
6 Sinc thr i n pn-lp pl inid th Nyquit path, w hav P Frm th rult f Exampl 95, w hav th fllwing Nyquit diagram map f th imaginary axi undr : Th numbr f ncirclmnt f th critical pint, in thi xampl dpnd n th valu f W will hav th fllwing ca nt that P : Fr < < r quivalntly < <, w will hav:,, Z N P Th cld - lp ytm i tabl N Fr < < r quivalntly >, w will hav: N,, Z N P Th cld - lp ytm ha untabl pl 3 Fr < < r quivalntly <, w will hav: N,, Z N P Th cld - lp ytm ha untabl pl Lctur Nt Prpard by Amir Aghdam
7 4 Fr < < r quivalntly < <, w will hav:,, Z N P Th cld - lp ytm i tabl N - Opn-lp pl n th imaginary axi: Sinc th Nyquit path mut nt pa thrugh any pl and zr f th charactritic quatin, if th pn-lp tranfr functin ha pl n th imaginary axi, w will hav t u mall micircl a hwn in th fllwing figur t g arund th pl Im{} -plan R{} - Exampl 3: U th Nyquit critrin t acrtain th tability f th fllwing cld-lp ytm: r t y t - - Slutin: W will u th fllwing Nyquit path: Lctur Nt Prpard by Amir Aghdam
Lctur Nt Prpard by Amir Aghdam 8 Th Nyquit diagram can b btaind a fllw: 9 << 8 >> > < > <, } Im{, } R{ 4 Sinc th micircl i vry cl t th pl, it will b mappd int infinity but w nd t find th hap f th map Fr thi purp, w can ch thr pint A, B and C a hwn in th figur and find th angl f th imag f th pint Aum that th radiu f th mall micircl i qual t Dfin A, B and C a th imag f A, B and C, rpctivly W will hav: 9 : : 9 : M C C M B B M A A Im{} R{} -plan - A B C
9 W knw that th imag f a curv in th cmplx plan undr a ratinal functin i a cnfrmal map Thi man that th angl f th -plan cntur will b rtaind in th -plan On can u thi rult t implify th prc f finding th imag f th crnr f th Nyquit path Fr xampl, aum that uing th tchniqu givn fr drawing th Nyquit diagram, aum that th imag f th imaginary axi undr th functin i btaind If w mv frm tward n th imaginary axi, w will hav t turn 9 dgr t th lft at th C pint W hav xactly th am thing in th imag f th Nyquit path undr at th C pint W will nw u th Nyquit critrin fr th tability analyi f Exampl 3 Sinc thr ar n pl f th pn-lp tranfr functin inid th Nyquit path, w hav P W will hav th fllwing tw ca: Fr < < r quivalntly >, w will hav:,, Z N P Th cld - lp ytm i tabl N Fr < < r quivalntly <, w will hav: Lctur Nt Prpard by Amir Aghdam
N,, Z N P Th cld - lp ytm ha untabl pl Cnditinal tability - Cnidr th fllwing cld-lp ytm: φ C r t y t - whr C rprnt th nminal pn-lp tranfr functin fr and φ - Th trm φ can rprnt th mdling rrr f th prc Aum that th nminal cld-lp ytm crrpnding t and φ i tabl - Fr φ, th largt and mallt valu f which rult in a tabl cldlp ytm ar calld upward gain margin and dwnward gain margin, rpctivly - Fr, th largt valu f φ which rult in a tabl cld-lp ytm i calld pha margin - Th pha margin and gain margin ar indicatd in th fllwing Nyquit diagram: R C plan φ - d d Im Lctur Nt Prpard by Amir Aghdam
Aum that th cld-lp ytm i tabl Frm thi figur w that rtating th Nyquit plt by any angl l than φ will nt chang th numbr f ncirclmnt f th critical pint Thi impli that th pha margin i qual t φ On th thr hand, if th pn-lp tranfr functin C i multiplid by any valu gratr than d and l than, th numbr f d ncirclmnt f th critical pint will nt chang Thi man that th upward gain margin i qual t gain margin i qual t d < < d and, d r lg d in th db cal and th dwnward d < < d d r lg d in th db cal Nt that - A a dignr, w want th pha margin and upward gain margin t b a larg a pibl and dwnward gain margin t b a mall a pibl t mak ur that th cld-lp ytm will rmain tabl in th prnc f uncrtainty in th plant mdl - A ytm with a nnzr dwnward gain margin and finit upward gain margin i calld a cnditinally tabl ytm Uually fr many practical ytm th dwnward gain margin i qual t and th trm gain margin i rfrrd t th upward gain margin - Pha margin and upward gain margin ar dntd by PM and M, rpctivly - PM and M rprnt th rlativ pitin f th Nyquit plt with rpct t th critical pint and ar ud a quantitativ maur fr rlativ tability In thr wrd, PM rprnt th angl by which th Nyquit diagram huld rtat, uch that th Nyquit plt pa thrugh th critical pint M rprnt th gain by which th Nyquit diagram huld b multiplid uch that th Nyquit plt pa thrugh th critical pint Lctur Nt Prpard by Amir Aghdam