Conrol Syem Lecure 9 Frequency eone Frequency eone We now know how o analyze and deign yem via -domain mehod which yield dynamical informaion The reone are decribed by he exonenial mode The mode are deermined by he ole of he reone Lalace Tranform We nex will look a decribing yem erformance via frequency reone mehod Thi guide u in ecifying he yem ole and zero oiion
Sinuoidal Seady-Sae eone co A U A u Conider a able ranfer funcion wih a inuoidal inu: Where he naural yem mode lie Thee are in he oen lef half lane e< A he inu mode and -" Only he reone due o he ole on he imaginary axi remain afer a ufficienly long ime Thi i he inuoidal eady-ae reone The Lalace Tranform of he reone ha ole A z z z K U Y n m 3 Sinuoidal Seady-Sae eone Inu Tranform eone Tranform eone Signal Sinuoidal Seady Sae eone co in in co co A A A u in co A A U N N k k k k k U Y * reone naural reone forced * N N e k e k e k e k ke y SS e k ke y * forced reone naural reone 4
3 Sinuoidal Seady-Sae eone Calculaing he SSS reone o eidue calculaion Signal calculaion [ ] [ ] in co in co lim lim lim e A e A A A U Y k % & ' * % & ' * co A u K k e k e e k e k k L y K K " # $ % & ' co * co A y 5 Sinuoidal Seady-Sae eone eone o i Ouu frequency inu frequency Ouu amliude inu amliude Ouu hae inu hae The Frequency eone of he ranfer funcion i given by i evaluaion a a funcion of a comlex variable a We eak of he amliude reone and of he hae reone They canno indeendenly be varied» Bode relaion of analyic funcion heory co co A y A u 6
Frequency eone Find he eady ae ouu for v Aco _ V L Comue he eady ae ouu A v SS co an L / L V Comue he -domain ranfer funcion T Volage divider T L Comue he frequency reone & L # T, T an $! L % " - [ ] 7 Bode Diagram Log-log lo of magt, log-linear lo of argt veru " Bode Diagram Magniude Phae deg Magniude Phae deg -5 - -5 - -5-45 -9 4 5 6 Frequency rad/ec 8 4
Frequency eone u Aco Aco y Sable Tranfer Funcion Afer a ranien, he ouu ele o a inuoid wih an amliude magnified by and hae hifed by. Since all ignal can be rereened by inuoid Fourier erie and ranform, he quaniie and are exremely imoran. Bode develoed mehod for quickly finding and for a given and for uing hem in conrol deign. 9 Bode Diagram z z zm K n r r r [ ] z z z K z z z m i m n e K r r rn The magniude and hae of when i given by: r r K r K z z r r r z m n z z z, Nonlinear in he magniude m Linear in he hae n 5
Why do we exre Bode Diagram in decibel? log z z z m K r r rn By roerie of he logarihm we can wrie: z z z logr logr logr logr logr logr log log K m m K z z z r r r r r r r r The magniude and hae of when i given by: m z z z K m r Linear in he magniude n? Linear in he hae n n Bode Diagram Why do we ue a logarihmic cale? Le have a look a our examle: T T L Exreing he magniude in : T L L & L # $! % " T Aymoic behavior: log log, * L ' &, L # log$ * '! % " : T % : T log& # log ' / L $ / L log log L LINEA FUNCTION in log!!! We lo a a funcion of log. 6
Bode Diagram Why do we ue a logarihmic cale? Le have a look a our examle: T T L Exreing he hae: L L & T log $ % Aymoic behavior: "" """ 9 L #! an " & $ % L #! " 45 L LINEA FUNCTION in log!!! We lo a a funcion of log. 3 Bode Diagram Log-log lo of magt, log-linear lo of argt veru " Bode Diagram Magniude Phae deg Magniude Phae deg -5 - -5 - -5-45 -/dec -9 4 5 6 Frequency rad/ec 4 7
Bode Diagram Decade: Any frequency range whoe end oin have a : raio A cuoff frequency occur when he gain i reduced from i maximum aband value by a facor / : % log& T # log T log log T 3 MAX MAX MAX ' $ Bandwih: frequency range anned by he gain aband Le have a look a our examle: T * L ' % & # T "! / L T / Thi i a low-a filer!!! Paband gain, Cuoff frequency/l The Bandwih i /L! 5 eneral Tranfer Funcion Bode Diagram & n, # o $ * ' ζ! $ % n n!" m K τ The magniude hae i he um of he magniude hae of each one of he erm. We learn how o lo each erm, we learn how o lo he whole magniude and hae Bode Plo. q Clae of erm: - - 3-4- Ko m τ n &, # $ * ' ζ! $ % n n!" q 6 8
eneral Tranfer Funcion: DC gain Magniude Magniude and Phae: 4 35 3 5 5 5 Ko log K # "! ± π Phaedeg 8 6 4 8 6 4 o if K if K o o > < # Frequencyrad/ec # Frequencyrad/ec 7 eneral Tranfer Funcion: Pole/zero a origin Magniude and Phae: m, m m log π m Magniude/ # # #3 m dec Phaedeg # #4 #6 #8 # #4 # Frequency/rad/ec # # Frequencyrad/ec 8 9
eneral Tranfer Funcion: eal ole/zero Magniude and Phae: τ n n log τ n an τ Aymoic behavior: $$ $ << / τ $$ $ n τ >> / τ n log """ << / τ """ n 9 >> / τ 9 eneral Tranfer Funcion: eal ole/zero Magniude 5 #5 # #5 # #5 #3 #35 n 3 n dec #4 # 3 Frequencyrad/ec n, τ / / τ n 3 gn n
eneral Tranfer Funcion: eal ole/zero Phae4deg # # #3 #4 #5 #6 #7 #8 #9 n 45 # # 3 Frequency4rad/ec n, τ / / τ n 45 n 9 eneral Tranfer Funcion: Comlex ole/zero Magniude and Phae: Aymoic behavior: &, # $ * ' ζ! $ % n n!" / & # q log- $! -. % n " & ζ / # % / n " n q an $! q & #, $ ζ! * % n " * $$ $ << n $$ $ q >> n n q 4log """ << n """ q 8 >> n
Magniude # #4 #6 #8 # eneral Tranfer Funcion: Comlex ole/zero q ζ # # 3 Frequencyrad/ec MAX q 4 dec q,, ζ.5 n n q ζ ζ r n r q ζ. 3 ζ gn q 3 Phaedeg # #4 #6 #8 # # #4 #6 #8 eneral Tranfer Funcion: Comlex ole/zero q 9 # # # Frequencyrad/ec q,, ζ.5 n. / τ q 9 q 8 4
Frequency eone: Pole/Zero in he HP Same. The effec on i ooie han he able cae. An unable ole behave like a able zero An unable zero behave like a able ole Examle: Thi frequency reone canno be found exerimenally bu can be comued and ued for conrol deign. 5 Neural Sabiliy U Y - K oo locu condiion: K, 8 A oin of neural abiliy L condiion hold for K, 8 Sabiliy: A K < K > 8 If K lead o inabiliy If K lead o inabiliy 6 3
Sabiliy Margin The AIN MAIN M i he facor by which he gain can be raied before inabiliy reul. M < M < M i equal o / K K where 8. UNSTABLE SYSTEM a he frequency The PHASE MAIN PM i he value by which he hae can be raied before inabiliy reul. PM < UNSTABLE SYSTEM PM i he amoun by which he hae of -8 when K K exceed 7 Sabiliy Margin M PM 8 4
Frequency eone u Aco Aco y Sable Tranfer Funcion e BODE lo { } Im{ } e NYQUIST lo 9 e Nyqui Diagram { } Im{ } e How are he Bode and Nyqui lo relaed? They are wo way o rereen he ame informaion Im { } e { } 3 5
Nyqui Diagram Find he eady ae ouu for v Aco V _ L Comue he eady ae ouu A v SS co an L / L V Comue he -domain ranfer funcion T Volage divider T L Comue he frequency reone & L # T, T an $! L % " - [ ] 3 Phae deg Magniude Bode Diagram -5 - -5 - -5-45 Bode Diagram Log-log lo of magt, log-linear lo of argt veru " Magniude Phae deg -/dec / L -9 4 5 6 Frequency rad/ec [ ] rad / ec, πf, [ f ] Hz 3 6
7 Nyqui Diagram 33, : T T! " # $ % & L T L T an, 9, : T T L L L L L L T { } { } Im, e L L T L T T!!!!!! L T { } e T { }, Im T - - 3-4- Nyqui Diagram 34 / L { } { } e Im v.
Nyqui Diagram eneral rocedure for keching Nyqui Diagram: Find Find Find * uch ha e{*}; Im{*} i he inerecion wih he imaginary axi. Find * uch ha Im{*}; e{*} i he inerecion wih he real axi. Connec he oin 35 Examle: Nyqui Diagram - - 3-4- : :!!!!!! 3 e Im { } { }, e{ } 36 8
Nyqui Diagram Examle: 37 Nyqui Diagram from Bode Diagram dec 6 dec 9 8 7 9 38 9
Neural Sabiliy U Y - K oo locu condiion: K, 8 A oin of neural abiliy L condiion hold for K, 8 Sabiliy: A K < K > 8 If K lead o inabiliy If K lead o inabiliy 39 Sabiliy Margin The AIN MAIN M i he facor by which he gain can be raied before inabiliy reul. M < M < M i equal o / K K where 8. UNSTABLE SYSTEM a he frequency The PHASE MAIN PM i he value by which he hae can be raied before inabiliy reul. PM < UNSTABLE SYSTEM PM i he amoun by which he hae of -8 when K K exceed 4
Sabiliy Margin M PM 4 Sabiliy Margin /M PM 4
Nyqui Sabiliy Crierion Cae : No ole/zero wihin conour Cae : Pole/zero wihin conour Argumen Princile: A conour ma of a comlex funcion will encircle he origin Z-P ime, where Z i he number of zero and P i he number of ole of he funcion inide he conour. 43 Nyqui Sabiliy Crierion Le u conider hi conour and cloed-loo yem U Y K - The cloed-loo ole are he oluion roo of: The evaluaion of H will encircle he origin only if H ha a HP zero or ole K 44
Nyqui Sabiliy Crierion Le u aly he argumen rincile o he funcion H K. If he lo of K encircle he origin, he lo of K encircle - on he real axi. 45 By wriing Nyqui Sabiliy Crierion b a Kb K K a a we can conclude ha he ole of K are alo he ole of. Auming no ole of in he HP, an encirclemen of he oin - by K indicae a zero of K in he HP, and hu an unable ole of he cloed-loo yem. A clockwie conour of C encloing a zero of K will reul in K encircling he - oin in he clockwie direcion. A clockwie conour of C encloing a ole of K will reul in K encircling he - oin in he counerclockwie direcion. The ne number of clockwie encirclemen of he oin -, N, equal he number of zero cloed-loo ole in he HP, Z, minu he number of ole oen-loo ole in he HP, P: N Z P 46 3
Nyqui Sabiliy Crierion U Y - When i hi ranfer funcion Sable? NYQUIST: The cloed loo i aymoically able if he number of counerclockwie encirclemen N negaive of he oin - by he Nyqui curve of i equal o he number of ole of wih oiive real ar unable ole P. Corollary: If he oen-loo yem i able P, hen he cloed-loo yem i alo able rovided make no encirclemen of he oin - N. 47 4 3 3 Nyqui Sabiliy Crierion 3 4 3 5 3 3 48 4
Secificaion in he Frequency Domain. The croover frequency c, which deermine bandwih BW, rie ime r and eling ime.. The hae margin PM, which deermine he daming coefficien ζ and he overhoo M. 3. The low-frequency gain, which deermine he eady-ae error characeriic. 49 Secificaion in he Frequency Domain The hae and he magniude are NOT indeenden! Bode ain-phae relaionhi: W u o π W u ln M ln dm du u ln / o coh u / du 5 5
Secificaion in he Frequency Domain The croover frequency: c BW c 5 Secificaion in he Frequency Domain The Phae Margin: PM v. M 5 6
Secificaion in he Frequency Domain The Phae Margin: PM v. ζ PM ζ 53 Frequency eone Phae Lead Comenaor T D, α < αt in α in log MAX MAX MAX &, $ log* % T ', # log* ' αt! " I i a high-a filer and aroximae PD conrol. I i ued whenever ubanial imrovemen in daming i needed. I end o increae he eed of reone of a yem for a fixed low-frequency gain. 54 7
Frequency eone Phae Lead Comenaor. Deermine he oen-loo gain K o aify error or bandwidh requiremen: - To mee error requiremen, ick K o aify error conan K, K v, K a o ha e ecificaion i me. - To mee bandwidh requiremen, ick K o ha he oen-loo croover frequency i a facor of wo below he deired cloed-loo bandwidh.. Deermine he needed hae lead α baed on he PM ecificaion. inmax α inmax 3. Pick MAX o be a he croover frequency. 4. Deermine he zero and ole of he comenaor. z/t MAX α / / α T MAX α -/ 5. Draw he comenaed frequency reone and check PM. 6. Ierae on he deign. Add addiional comenaor if needed. 55 Frequency eone Phae Lag Comenaor T D α, α > αt I i a low-a filer and aroximae PI conrol. I i ued o increae he low frequency gain of he yem and imrove eady ae reone for fixed bandwidh. For a fixed low-frequency gain, i will decreae he eed of reone of he yem. 56 8
Frequency eone Phae Lag Comenaor. Deermine he oen-loo gain K ha will mee he PM requiremen wihou comenaion.. Draw he Bode lo of he uncomenaed yem wih croover frequency from e and evaluae he lowfrequency gain. 3. Deermine α o mee he low frequency gain error requiremen. 4. Chooe he corner frequency /T he zero of he comenaor o be one decade below he new croover frequency c. 5. The oher corner frequency he ole of he comenaor i hen / α T. 6. Ierae on he deign 57 9