Chapter 6 Control Systems Design by Root-Locus Method. Lag-Lead Compensation. Lag lead Compensation Techniques Based on the Root-Locus Approach.

Transcription

1 hapter 6 ontrol Sytem Deign by Root-Lou Method Lag-Lead ompenation Lag lead ompenation ehnique Baed on the Root-Lou Approah. γ β K, ( γ >, β > ) In deigning lag lead ompenator, we onider two ae where γ βand γ β ae I γ β. In thi ae, the deign proe i a ombination of the deign of the lead ompenator and that of the lag ompenator. he deign proedure for the lag lead ompenator follow: Lag lead ompenation ehnique Baed on the Root-Lou Approah.. From the given performane peifiation, determine the deired loation for the dominant loed-loop pole.. Uing the unompenated open-loop tranfer funtion (), determine the angle defiieny φ if the dominant loed-loop pole are to be at the deired loation. he phae-lead portion of the lag lead ompenator mut ontribute thi angle φ. 3. Auming that we later hooe uffiiently large o that the magnitude of the lag portion β 3 Lag lead ompenation ehnique Baed on the Root-Lou Approah.

2 Lag lead ompenation ehnique Baed on the Root-Lou Approah. Lag lead ompenation ehnique Baed on the Root-Lou Approah. hooe the value of uh that where K and γ are already determined in tep 3. Hene, given the value of K v the value of β an be determined from thi lat equation. hen, uing the value of β thu determined, 6 Lag lead ompenation ehnique Baed on the Root-Lou Approah. Lag lead ompenation ehnique Baed on the Root-Lou Approah. 7 8

3 Lag lead ompenation ehnique Baed on the Root-Lou Approah. Example: Lag-Lead ompenator ae: γ β 9 R ( ) + ( ) he loed-loop tranfer funtion 0. ( ) ; ζ 0.; ω 0. + n rad e 0 Example: Lag-Lead ompenator ae: γ β Example: Lag-Lead ompenator ae: γ β he dominant loed-loop pole are It i deired ytem of the dominant loed-loop pole, 0. ± j.983 ζ 0. ω n rad e he damping ratio ζ 0. he undamped natural frequeny ω n rad e he tati veloity error ontant i 8 e - K v 80 - e We ue a lag-lead ompenator having the tranfer funtion γ β K, ( γ >, β > ) 3

4 z lead Example: Lag-Lead ompenator ae: γ β p lead 0. A B + K 0 ( 0.),. ± j.33 Example: Lag-Lead ompenator ae: γ β hu, the phae lead portion of the lag-lead ompenator beome K K γ 0. θ AB. 33 tan 60 B B B. 3 where, γ 0. 0 Example: Lag-Lead ompenator ae: γ β Example: Lag-Lead ompenator ae: γ β Next we determine the value of K K K ( 0.) ( ) ( 0.) A A A 3 B.+ j A3 A A B + K 0 ( 0.) hu K γ K A A A 3 B 6. 6

5 Example: Lag-Lead ompenator ae: γ β Example: Lag-Lead ompenator ae: γ β Firt the value of β i determined to atify the requirement on the tati veloity error ontant: K v lim 0 Hene, β i determined a lim K () 0 ( 0.) β γ β Kv lim ( 6.).008β β.97 7 Finally, we hooe the value of large enough o that and We may hooe.97.+ j.33 <.97 <.+ j Example: Lag-Lead ompenator Now the tranfer funtion of the deign lag-lead ompenator i given by 0. () Example: Lag-Lead ompenator he new damping ratio i ζ0.9 he ompenated ytem will have the open-loop tranfer funtion () ( 0. ) 9 New loed-loop pole are loated at -.±j.7 0

6 Example: Lag-Lead ompenator Example : Lag-Lead ompenator ae: γ β 0. ( ) he loed-loop tranfer funtion R ( ) + ( ) ; ζ 0.; ω 0. + n rad e Example : Lag-Lead ompenator ae: γ β It i deired ytem of the dominant loed-loop pole ζ 0. ω n rad e Example : Lag-Lead ompenator ae: γ β z lead. + K ( 0.) p 8.6 lead A 0,. ± j.33 K v 80 - e We ue a lag-lead ompenator having the tranfer funtion β β K, ( β > ) 3 θ B AB. 33 tan 3. B B B 6. 6

7 Example : Lag-Lead ompenator ae: γ β Example : Lag-Lead ompenator ae: γ β he phae lead portion of the lag-lead network thu beome z lead p lead. 0. β 8.6 β 3.6 For the phae lag portion, we may hooe 0 zlag 0. p lag β Sine the requirement on the tati veloity error ontant i 80 e -, We have K lim v 0 0. lim K 8K 80 K 0 0 hu, the lag-lead ompenator beome Example : Lag-Lead ompenator ae: γ β Example : Lag-Lead ompenator ae: γ β he ompenated ytem will have the open-loop tranfer funtion R ( ) ( ) ( ) he loed-loop tranfer funtion ( + 0.) ( ) New loed-loop pole and zero are loated at p 3.968; p 0.00; p3,.307 ± j.9 ( ) + ( ) ( + )( + )( + ) lim E lim lim z.; z 0. e he teady-tate error of the ytem for a unit-ramp input i he tati veloity error ontant, K v i define by Kv 77.6 e e

8 Example : Lag-Lead ompenator ae: γ β Example : Lag-Lead ompenator ae: γ β