1        Design via Frequency Response•   How to use frequency response techniques to adjust the    gain to meet a transie...
2Transient Response via Gain Adjustment               Design Procedure:               1.     Draw the Bode magnitude and p...
3                   Lag Compensation•   improves the static error without any resulting instability•   increases the phase...
4Design Procedure1.   Set the gain, K, to the value that satisfies the steady-state error specification     and plot the B...
5The transfer function of the lag compensator is                                     where α > 1.                         ...
6Problem Use Bode diagrams to design a lag compensator to yield a tenfoldimprovement in steady-state error over the gain c...
7Bode plot forK = 5839                                                           2ζ        For 9.5% overshoot, ζ=0.6 and Φ...
8The magnitude at 9.8 rad/s is +24 dBCompensator:•  high break frequency one decade below 9.8 rad/s, i.e., 0.98 rad/s•   l...
9                                  s + 0.98                    GC ( s ) =                                 s + 0.062The com...
10                  Lead Compensation•   increase the phase margin to reduce the percent overshoot•   increase the gain cr...
11Lead Compensator Frequency Response                       where β<1                                      •       frequen...
12Design Procedure1.   Find the closed-loop bandwidth required to meet the settling time, peak     time, or rise time requ...
13Problem Design a lead compensator to yield a 20% overshoot and KV, = 40, witha peak time of 0.1 second.Solution         ...
14•   20% overshoot implies PM=48.10•   original PM=340 at 29.6 rad/s•   Phase contribution from the    compensator = 48.1...
15•     We find compensator break      frequencies from       β =0.42               ωmax= 39 rad/s       1/T = 25.3       ...
16      obtained by      simulation        Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Ni...
17                Lag-Lead Compensation•   Design first the lag compensator to improve the steady-state error and    then ...
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11 elec3114

  1. 1. 1 Design via Frequency Response• How to use frequency response techniques to adjust the gain to meet a transient response specification• How to use frequency response techniques to design cascade compensators to improve the steady-state error• How to use frequency response techniques to design cascade compensators to improve the transient response• How to use frequency response techniques to design cascade compensators to improve both the steady-state error and the transient response Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  2. 2. 2Transient Response via Gain Adjustment Design Procedure: 1. Draw the Bode magnitude and phase plots for a convenient value of gain. 2. Determine the required phase margin from the percent overshoot. − ln(%OS / 100) 2ζ ζ = Φ M = tan −1 π 2 + ln 2 (%OS / 100) − 2ζ 2 + 1 + 4ζ 4 3. Find the frequency, ωΦM, on the Bode phase diagram that yields the desired phase margin, CD 4. Change the gain by an amount AB to force the magnitude curve to go through 0 dB at ωΦM. Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  3. 3. 3 Lag Compensation• improves the static error without any resulting instability• increases the phase margin of the system to yield the desired transient response where α > 1. Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  4. 4. 4Design Procedure1. Set the gain, K, to the value that satisfies the steady-state error specification and plot the Bode plots2. Find the frequency where the phase margin is 50 to 120 greater than the phase margin that yields the desired transient response3. Select a lag compensator whose magnitude response yields a composite Bode magnitude diagram that goes through 0 dB at the frequency found in step 2 as follows: • Draw the compensators high-frequency asymptote to yield 0 dB at the frequency found in step 2; select the upper break frequency to be 1 decade below the frequency found in step 2; select the low- frequency asymptote to be at 0 dB; connect the compensators high- and low-frequency asymptotes with a -20 dB/decade line to locate the lower break frequency.4. Reset the system gain, K, to compensate for any attenuation in the lag network in order to keep the static error constant the same as that found in step 1. Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  5. 5. 5The transfer function of the lag compensator is where α > 1. Gc(s) = (s + 0.1)/(s + 0.01) Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  6. 6. 6Problem Use Bode diagrams to design a lag compensator to yield a tenfoldimprovement in steady-state error over the gain compensated system whilekeeping the percent overshoot at 9.5%. K=583.9 Solution KV = 583.9 / 36 = 16.22, hence for a 10 fold improvement of steady state error KV = 10 x 16.22 = 162.22 Therefore K = 583.9 x 10 = 5839 and Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  7. 7. 7Bode plot forK = 5839 2ζ For 9.5% overshoot, ζ=0.6 and Φ M = tan −1 = 59.20 − 2ζ 2 + 1 + 4ζ 4 We increase phase margin by 100 to 69.20 Φ M = 69.2 0 occurs at a phase angle of - 1800 + 69.20 = −110.80 and the corresponding frequency is 9.8 rad/s Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  8. 8. 8The magnitude at 9.8 rad/s is +24 dBCompensator:• high break frequency one decade below 9.8 rad/s, i.e., 0.98 rad/s• low break frequency is found as -20 dB line intersection with 0 dB and is 0.062 rad/s Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  9. 9. 9 s + 0.98 GC ( s ) = s + 0.062The compensator must have a dc gain of 1, hence, the gain of compensatormust be 0.062/0.98=0.063. Then s + 0.98 GC ( s ) = 0.063 s + 0.062 obtained by simulation Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  10. 10. 10 Lead Compensation• increase the phase margin to reduce the percent overshoot• increase the gain crossover to realize a faster transient response Note: notice that the initial slope, which determines the steady-state error, is not affected by the design Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  11. 11. 11Lead Compensator Frequency Response where β<1 • frequency,ωmax, at which the maximum phase angle, φmax, occurs can be found using ωmax • the maximum phase angle φmax : • compensator’s magnitude at ωmax ωmax is Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  12. 12. 12Design Procedure1. Find the closed-loop bandwidth required to meet the settling time, peak time, or rise time requirement2. Set the gain, K, of the uncompensated system to the value that satisfies the steady-state error requirement.3. Plot the Bode magnitude and phase diagrams for this value of gain and determine the uncompensated systems phase margin.4. Find the phase margin to meet the damping ratio or percent overshoot requirement. Evaluate the additional phase contribution required from the compensator.5. Determine the value of β from the lead compensators required phase contribution.6. Determine the compensators magnitude at the peak of the phase curve7. Determine the new phase-margin frequency by finding where the uncompensated systems magnitude curve is the negative of the lead compensators magnitude at the peak of the compensators phase curve.8. Design the lead compensators break frequencies9. Reset the system gain to compensate for the lead compensators gain. Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  13. 13. 13Problem Design a lead compensator to yield a 20% overshoot and KV, = 40, witha peak time of 0.1 second.Solution Tp = 0.1sec ζ = 0.456 (20% overshoot) ω BW = 46.6rad / s In order to meet the specification Kv=40, K must be set 1440 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  14. 14. 14• 20% overshoot implies PM=48.10• original PM=340 at 29.6 rad/s• Phase contribution from the compensator = 48.1-34+10=24.10• Using for φmax= 24.10, β =0.42• From Gc ( jωmax ) dB = 3.76dB The uncompensated system passes through (-3.76) dB at ωmax= 39 rad/s. Now, we select 39 rad/s as the new phase- margin freq., which will result in a 0dB crossover at 39 rad/s for the compensatedDr Branislav Hredzak system Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  15. 15. 15• We find compensator break frequencies from β =0.42 ωmax= 39 rad/s 1/T = 25.3 1/(βT) = 60.2 - where 2.38 is the gain required to keep the DC gain of the compensator at unity Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  16. 16. 16 obtained by simulation Dr Branislav HredzakControl Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  17. 17. 17 Lag-Lead Compensation• Design first the lag compensator to improve the steady-state error and then design a lead compensator to meet the phase-margin Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.

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