D Compenation ehnique Performane peifiation for the loed-loop ytem Stability ranient repone Æ, M (ettling time, overhoot) or phae and gain margin Steady-tate repone Æ e (teady tate error) rial and error approah to deign Performane peifiation Root-lou or Frequeny repone tehnique Synthei Analyi of loed-loop ytem Are peifiation met? No Ye
D2. Proportional Control Bai Control E() M() () - M E ( ) ( ) m ( t) e( t) 2. Integral Control E() M() () - M ( ) E( ) m ( t) e( t) dt Integral ontrol add a pole at the origin for the open-loop: ype of ytem inreaed, better teady-tate performane. Root-lou i pulled to the left tending to lower the ytem relative tability.
D3 3. Proportional Integral Control - E() p i M() () M( ) p i i p m( t) pe() t () E( ) ie t dt A pole at the origin and a zero at i are added. p 4. Proportional Derivative Control E() p d M() () - M ) E( ) ( m( t) e () t p d p d de() t dt Root-lou i pulled to the left, ytem beome more table and repone i ped up. Differentiation make the ytem enitive to noie.
D4 5. Proportional Derivative Integral (PID) Control M() - E() i d () p M( ) i p dd E( ) de () () t m ( t ) p e t d i e() t dt dt More than 50% of indutrial ontrol are PID. More than 80% in proe ontrol indutry. When () of the ytem i not known, then initial value for p, d, i an be obtained experimentally and than fine-tuned to give the deired repone (Ziegler-Nihol). 6. Feed-forward ompenator
D5 E() () M() () - Deign () uing Root-Lou or Frequeny Repone tehnique.
D6 Frequeny repone approah to ompenator deign Information about the performane of the loed-loop ytem, obtained from the open-loop frequeny repone: Low frequeny region indiate the teady-tate behavior. Medium frequeny (around - in polar plot, around gain and phae roover frequenie in Bode plot) indiate relative tability. High frequeny region indiate omplexity. Requirement on open-loop frequeny repone he gain at low frequeny hould be large enough to give a high value for error ontant. At medium frequenie the phae and gain margin hould be large enough. At high frequenie, the gain hould be attenuated a rapidly a poible to minimize noie effet. Compenator lead:improve the tranient repone. lag: improve the teady-tate performane at the expene of lower ettling time. lead-lag: ombine both
D7 Lead ompenator () a a a > 0 and 0 < α < Pole and zero of the lead ompenator: a Frequeny repone of (jω): he maximum phae-lead angle φ m our at ω m, where: α inφ m α and logω m log log ω 2 a Æ m a
D8 Sine jω jωa ω ω m a the magnitude of (jω) at ω m i given by: ( jω ) a m Polar plot of a lead network a( jω ) ( jωa ) where 0 < a < i given by
D9 Proedure: Lead ompenation baed on the frequeny repone. Determine the ompenator gain α atifying the given error ontant. 2. Determined the additional phae lead φ m required ( 0%~5%) for the gain adjuted ( α()) open-loop ytem. 3. Obtain α from in φ m 4. Find the new gain ro over frequeny from a ( jω ) 0loga 5. Find from and tranfer funtion of () a a a ω and () a a eneral effet of lead ompenator: Addition of phae lead near gain roover frequeny. Inreae of gain at higher frequenie. Inreae of ytem bandwidth.
D0 Example: Conider - () () where ( ) 4 ( 2) Performane requirement for the ytem: Steady-tate: v 20 ranient repone: phae margin >50 gain margin >0 db Analyi of the ytem with () For v 20 Æ 0 hi lead to: phae margin 7 gain margin % Deign of a lead ompenator: () a a 4 a 2. v () () 2a 20 Æ α0 lim0
D 2. From the Bode plot of α(jω), we obtain that the additional phae-lead required i: 50-7 33. We hooe 38 (~33 5%) 3., a in φ m in 38 Æ α 0.24 a 4. Sine for m, the frequeny with the maximum phae-lead angle, we have: jω jω a m m a We hooe, the new gain roover frequeny o that ω m ω and () () hi give that: a ( jω ) ha to be equal: a 6.2dB jω 40 ( jω 2) jω From the Bode plot of α(jω) we obtain that jω 40 ( jω 2) 6.2dB at 9 rad/e
D2 5. hi implie for ω 9rad/e a 0.24 Æ 4.4 and 20 2 a () 4.7 4.7 4.4 8.4 he ompenated ytem i given by: 4.7( 4.4) - 8.4 4 ( 2) he effet of the lead ompenator i: Phae margin: from 7 to 50 Æ better tranient repone with le overhoot. : from 6.3rad/e to 9 rad/e Æ the ytem repone i fater. ain margin remain \. v i 20, a required Æ aeptable teady-tate repone.
D3 Bode diagram for a ( jω) 40 jω(jω 2) Bode diagram for the ompenated ytem ( jω ) ( jω ) 4.7 jω jω 4.4 8.4 4 jω (jω 2)
D4 Lag ompenator β () β > 0, > Pole and zero: β β Frequeny repone: Polar plot of a lag ompenator. (j)/(j)
D5 Bode diagram of a lag ompenator with β -20log Magnitude of (j)/(j)
D6 Proedure: Lag ompenation baed on the frequeny repone. Determine the ompenator gain β to atify the requirement for the given error ontant. 2. Find the frequeny point where the phae of the gain adjuted open-loop ytem ( β()) i equal to -80 the required phae margin 5 ~ 2. hi will be the new gain roover frequeny. 3. Chooe the zero of the.425038,947 %,9,-4:9 otave to deade below. 4. Determine the attenuation neeary to bring the magnitude urve down to 0dB at the new gain roover frequeny ( jω ) 20 log β β Æ 5. Find the tranfer funtion (). eneral effet of lag ompenation: Dereae gain at high frequenie. Move the gain roover frequeny lower to obtain the deired phae margin.
D7 Example: Conider - () () where () ( )( 0.5 ) Performane requirement for the ytem: Steady tate: v 5 ranient repone: Phae margin > 40 ain margin > 0 db Analyi of the ytem with () v () 5 lim 0 for 5, the loed-loop ytem i untable Deign of a lag ompenator: () β β β
D8. v () () 5 lim0 β 2. Phae margin of the ytem 5() i -3 Æ the loed-loop ytem i untable. From the Bode diagram of 5(jω) we obtain that the additional required phae margin of 40 2 52 i REWDLQH DW rad/e. he new gain roover frequeny will be: 0.5 rad/e 3. 3ODFH WH ]HUR RI WH ODJ FRPSHQVDWRU DW % rad/e( at about /5 of ). 4. he magnitude of 5(jω) at the new gain roover frequeny 0.5 rad/e i 20 db. In order to have a the new gain roover frequeny, the lag ompenator mut give an attenuation of -20db at ω. herefore - 20log - 20 db Æ 0 5 5. 0. 5 β, pole : 0. 0 and () 0.5 0. 0.0 β
D9 Bode diagram for: (jω) 5(jω) (gain-adjuted β(jω) open-loop tranfer funtion), (jω)/ (jω)/5 (ompenator divided by gain β 5), (jω)(jω) (ompenated open-loop tranfer funtion) he effet of the lag ompenator i: he original untable loed-loop ytem i now table. he phae margin ƒ Æ aeptable tranient repone. he gain margin % Æ aeptable tranient repone. v i 5 a required Æ aeptable teady-tate repone. he gain at high frequenie ha been dereaed.
D20 Lead-lag ompenator () 2 2 2 2 a β γ γ β β γ, 2 > 0, > and γ > Frequeny repone: Bode diagram of a lag-lead ompenator given by () 2 2 β γ with, 0 and 2 0
D2 Polar plot of a lag-lead ompenator given by () 2 γ β with and 2 Comparion between lead and lag ompenator Lead ompenator Lag ompenator o High pa o Low pa o Approximate derivative plu o Approximate integral plu proportional ontrol proportional ontrol o Contribute phae lead o Attenuation at high frequenie o Inreae the gain roover frequeny o Move the gain-roover frequeny lower o Inreae bandwidth o Redue bandwidth