EE128 Feedback Control Lecture 14 10 12 2006 Outline The Frequency Response Design Method ch 6 cont Frequency response 6 1 Bode plot techniques 6 1 1 Steady state errors 6 1 2 Neutral stability 6 2 Bode Plot Techniques Advantages of Bode Plots Dynamic compensator design Can be determined experimentally BP of cascade systems simply add Log scale allows wider freq representation Bode form KG s K s z1 s z 2 K j 1 1 j 2 1 K KG j K 0 s p1 s p2 K j a 1 j b 1 K Classes of terms of transfer functions 1 K 0 j n 2 j 1 1 j j 2 3 1 n n 2 1 Bode Plot Techniques Class 1 term Magnitude plot of j n is a straight line with slope n x 20db decade Class 1 terms affect the slope at the lowest frequencies Class 2 3 are constant in that region Procedure 1 Plot K0 at 1 2 Draw line with slope n through 1 log K 0 j n log K 0 n log j Phase horizontal line frequency independent 90 for n 1 180 for n 2 etc Bode Plot Techniques Class 2 term Magnitude of j 1 10 Magnitude 1 j 1 1 1 j 1 j Phase 1 1 0o 1 j 90o 1 j 1 45o break point Phase of j 1 10 Bode Plot Techniques Class 3 term similar to class 2 except for Break point n Magnitude slope factor of 2 40db Phase changes by 180 Transition through break point varies with damping ratio 1 G j at n 2 Summary of Bode Plot Rules MAGNITUDE PHASE 1 Manipulate trans function into Bode form 1 Plot low freq asymptote n x 90 2 Plot low freq asymptote through K0 at 1 with slope n 2 Sketch approx curve changing phase by 90 or 180 at each break point 3 At 1st break point step slope by 1or 2 depending on term order Continue through all break points 3 Locate asymptotes for each individual phase curve so that phase change corresponds to 2 4 Sketch approximate magnitude curve increase 1 4 3db at 1st order numerator break points decrease 0 707 3 db at denominator For 2nd order sketch resonant peak valley using G j 1 2 at denominator or G j 2 at numerator break points 4 Graphically add phase curves from lowest freq asymptote to highest freq Bode Plot examples Real poles zeros KG s KG j 2000 s 0 5 s s 10 s 50 2 j 0 5 1 j j 10 1 j 50 1 Bode Plot examples Complex poles 10 KG s s s 2 0 4 s 4 Complex poles zeros 0 01 s 2 0 01s 1 KG s 2 2 s s 4 0 02 s 2 1 Nonminimum Phase Systems When there are zeros in the RHP At break point phase decreases Example s 1 G1 s 10 s 10 s 1 G2 s 10 s 10 G1 j G2 j Magnitude of G1 and G2 is the same for all frequencies Phase is very different Steady state errors 6 1 2 Recall Steady state error of a feedback system decreases as the open loop gain transfer function increases At very low frequencies the larger the magnitude of the low freq asymptote the lower the steady state error KG j K 0 j n If n 0 type 0 system Kp K0 ess 1 1 Kp If n 1 type 1 system Kv K0 ess 1 Kv A Kv In type 1 systems Kv is the magnitude value of the low freq asymptote at 1rad sec Neutral Stability 6 2 When K is defined such that a closed loop root falls on the imaginary axis Normally the closed loop transfer function of a system is unknown We can evaluate the freq response of the openloop transfer function KG j and test that response For most systems a simply relationship exists between closed loop stability and open loop frequency response Stability condition KG j 1 at G j 180o