Upcoming SlideShare
×

# 11 elec3114

994 views
940 views

Published on

Published in: Education
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total views
994
On SlideShare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
146
0
Likes
0
Embeds 0
No embeds

No notes for slide

### 11 elec3114

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.