LINEAR CONTROL SYSTEMS Ali Karimpour Assistant Professor Ferdowsi University of Mashhad

Lecture 21 Ali Karimpour Dec Lecture 21 Frequency Domain Analysis Topics to be covered include: v Frequency domain analysis of control systems. v Frequency domain specification. u Peak of resonance and resonance frequency. u Bandwidth. u Effect of adding poles and zeros to open loop transfer function. u Effect of adding poles and zeros to closed loop transfer function.

Lecture 21 Ali Karimpour Dec One degree-of-freedom configuration T(s) S(s) S(s) is : Sensitivity Function T(s) is : Complementary Sensitivity Function Why? Since S+T=1

Lecture 21 Ali Karimpour Dec One degree-of-freedom configuration Where T(s) and S(s) are How to derive command tracking How to derive disturbance rejection How to derive noise attenuation

Lecture 21 Ali Karimpour Dec One degree-of-freedom configuration Derive command tracking and disturbance rejection

Lecture 21 Ali Karimpour Dec One degree-of-freedom configuration Derive command tracking and disturbance rejection How to derive noise attenuation

Lecture 21 Ali Karimpour Dec One degree-of-freedom configuration Ideal |T| for command tracking, disturbance rejection and noise attenuation Ideal |L| for command tracking, disturbance rejection and noise attenuation Actual |T| for command tracking, disturbance rejection and noise attenuation Actual |L| for command tracking, disturbance rejection and noise attenuation

Lecture 21 Ali Karimpour Dec A prototype second order system. یک سیستم نمونه درجه c e r c r Frequency (rad/s)

Lecture 21 Ali Karimpour Dec Drawing procedure

Lecture 21 Ali Karimpour Dec Drawing procedure + - c e r

Lecture 21 Ali Karimpour Dec Frequency domain specification Peak of resonance ( ) Resonance frequency ( )

Lecture 21 Ali Karimpour Dec Frequency domain specification Closed-loop bandwidth ( ) -3 Open - loop bandwidth ( ) Gain crossover frequency ( )

Lecture 21 Ali Karimpour Dec Introducing a prototype second order system. + - c e r c r Resonance frequency ( ) Peak of resonance ( )

Lecture 21 Ali Karimpour Dec Introducing a prototype second order system. + - c e r c r Closed-loop bandwidth ( ) -3

Lecture 21 Ali Karimpour Dec Introducing a prototype second order system. + - c e r c r

Lecture 21 Ali Karimpour Dec Adding poles to closed loop transfer functions اضافه کردن قطب به تابع انتقال حلقه بسته c2c2 r τ=0τ=0 τ=0.5 τ=1.0 τ= 5.0 τ= 2.0 P.O. t r System speed BW

Lecture 21 Ali Karimpour Dec Adding zeros to closed loop transfer functions اضافه کردن صفر به تابع انتقال حلقه بسته c1c1 r c2c2 r P.O. t r System speed BW

Lecture 21 Ali Karimpour Dec Adding poles to open loop transfer functions اضافه کردن قطب به تابع انتقال حلقه باز - c2c2 e r τ=0τ=0 τ= 1.0 τ= 2.0 τ= 5.0 P.O. t r System speed More problem as poles go to ?? BW

Lecture 21 Ali Karimpour Dec Adding zeros to open loop transfer functions اضافه کردن صفر به تابع انتقال حلقه باز - c2c2 e r P.O. t r System speed BW τ=0 τ=0.5 τ=2.0 τ=0.2 τ=5.0 τ=10