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EE C128 / ME C134 – Feedback Control Systems
Lecture – Chapter 11 – Design via Frequency Response
Alexandre Bayen
Department of Electrical Engineering & Computer Science
University of California Berkeley
September 10, 2013
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 1 / 38
Lecture abstract
Topics covered in this presentation
I FR compensator design advantages
I Lag, lead, & lag-lead compensators
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Chapter outline
1 11 Design via frequency response11.1 Introduction11.2 Transient response via gain adjustment11.3 Lag compensation11.4 Lead compensation11.5 Lag-lead compensation
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 3 / 38
11 Design via FR 11.1 Intro
1 11 Design via frequency response11.1 Introduction11.2 Transient response via gain adjustment11.3 Lag compensation11.4 Lead compensation11.5 Lag-lead compensation
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11 Design via FR 11.1 Intro
RL vs. FR design, [1, p. 626]
RL
I Stability & TR design via gainadjustment
I Repeated trials
I TR design via cascadecompensation
I Intuitive
I Steady-state error design viacascade compensation
I Repeated trials
FR
I Stability & TR design via gainadjustment
I Read gain from the plots
I TR design via cascadecompensation
I Repeated trials
I Steady-state error design viacascade compensation
I Design derivativecompensation & steady-stateerror jointly
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 5 / 38
11 Design via FR 11.1 Intro
FR design review, [1, p. 627]
Stability
I Nyquist criterion ! stabilityI CL stable if OL stable & OL magnitude FR has a gain less than 0 dB
at the frequency where the phase FR is 180�
TR
I # %OS / " �M
I " speed of response / " bandwidth
Steady-state error
I # steady-state error / " low-frequency magnitude responses
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 6 / 38
11 Design via FR 11.2 Transient response via gain adjustment
1 11 Design via frequency response11.1 Introduction11.2 Transient response via gain adjustment11.3 Lag compensation11.4 Lead compensation11.5 Lag-lead compensation
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 7 / 38
11 Design via FR 11.2 Transient response via gain adjustment
�M -TR-gain relation, [1, p. 627]
Concept
I ⇣ (& %OS) relate to �M
OL TF
G(s) =!2n
s(s+ 2⇣!n)
CL TF
T (s) =!2n
s2 + 2⇣!ns+ !2n
⇣-�M relation
�M = tan�1
✓2⇣q
�2⇣2+p
1+4⇣4
◆Figure: Bode plots showing gainadjustment for a desired �M
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11 Design via FR 11.2 Transient response via gain adjustment
�M -TR-gain relation, [1, p. 627]
Procedure
1. Draw the Bode magnitude &phase plots for a convenientvalue of gain
2. Determine the required �M
from the %OS
Figure: Bode plots showing gainadjustment for a desired �M
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 9 / 38
11 Design via FR 11.2 Transient response via gain adjustment
�M -TR-gain relation, [1, p. 628]
Procedure
3. Find the frequency, !�M , onthe Bode phase diagram thatyields the desired �M
4. Change the gain by an amountto force the magnitude curve togo through 0 dB at !�M . Theamount of gain adjustment isthe additional gain needed toproduce the required �M
Figure: Bode plots showing gainadjustment for a desired �M
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 10 / 38
11 Design via FR 11.2 Transient response via gain adjustment
Example, [1, p. 628]
Example (TR design via gain adjustment)
IProblem: For the position control system, find the value of thepreamplifier gain, K, to yield %OS = 9.5% in the TR for a step input
ISolution: On the board
Figure: System
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 11 / 38
11 Design via FR 11.3 Lag compensation
1 11 Design via frequency response11.1 Introduction11.2 Transient response via gain adjustment11.3 Lag compensation11.4 Lead compensation11.5 Lag-lead compensation
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 12 / 38
11 Design via FR 11.3 Lag compensation
Visualizing lag compensation, [1, p. 630]
Concept
I Improve the staticerror constant by "only thelow-frequency gainwithout anyresulting instability
I " �M of thesystem to yield thedesired TR
Figure: Visualizing lag compensation
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 13 / 38
11 Design via FR 11.3 Lag compensation
Visualizing lag compensation, [1, p. 630]
Procedure
1. Set the gain, K, tothe value thatsatisfies thesteady-state errorspecification andplot the Bodemagnitude andphase diagrams forthis value of gain
Figure: Visualizing lag compensation
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 14 / 38
11 Design via FR 11.3 Lag compensation
Visualizing lag compensation, [1, p. 630]
Procedure
2. Find the frequencywhere �M is 5� to12� greater thanthe �M that yieldsthe desired TR.This compensatesfor the fact thatthe phase of thelag compensatormay still contributeanywhere from�5� to �12� ofphase at !�M
Figure: Visualizing lag compensation
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 15 / 38
11 Design via FR 11.3 Lag compensation
Visualizing lag compensation, [1, p. 630]
Procedure
3. Select a lagcompensatorwhose magnituderesponse yields acomposite Bodemagnitude diagramthat goes through0 dB at thefrequency found inStep 2
Figure: Visualizing lag compensation
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 16 / 38
11 Design via FR 11.3 Lag compensation
Visualizing lag compensation, [1, p. 630]
Procedure
3.1 Draw thecompensator’shigh-frequencyasymptote to yield0 dB for thecompensatedsystem at thefrequency found inStep 2
Figure: Visualizing lag compensation
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11 Design via FR 11.3 Lag compensation
Visualizing lag compensation, [1, p. 630]
Procedure
3.2 Select the upperbreak frequency tobe 1 decade belowthe frequencyfound in Step 2
Figure: Visualizing lag compensation
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 18 / 38
11 Design via FR 11.3 Lag compensation
Visualizing lag compensation, [1, p. 630]
Procedure
3.3 Select thelow-frequencyasymptote to be at0 dB
3.4 Connect thecompensator’shigh- &low-frequencyasymptotes with a�20 dB/decadeline to locate thelower breakfrequency Figure: Visualizing lag compensation
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 19 / 38
11 Design via FR 11.3 Lag compensation
Visualizing lag compensation, [1, p. 630]
Procedure
4. Reset the systemgain, K, tocompensate forany attenuation inthe lag network inorder to keep thestatic errorconstant the sameas that found inStep 1
Figure: Visualizing lag compensation
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11 Design via FR 11.3 Lag compensation
Visualizing lag compensation, [1, p. 630]
Result
I Lag compensatorTF
GC(s) =s+ 1
T
s+ 1↵T
I ↵ > 1
Figure: Visualizing lag compensation
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11 Design via FR 11.3 Lag compensation
Example, [1, p. 632]
Example (Lag compensation design)
IProblem: Use Bode diagrams to design a lag compensator to yield atenfold improvement in steady-state error over the gain-compensatedsystem while keeping %OS = 9.5%
ISolution: On the board
Figure: System
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 22 / 38
11 Design via FR 11.4 Lead compensation
1 11 Design via frequency response11.1 Introduction11.2 Transient response via gain adjustment11.3 Lag compensation11.4 Lead compensation11.5 Lag-lead compensation
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 23 / 38
11 Design via FR 11.4 Lead compensation
Visualizing lead compensation, [1, p. 635]
Concept
I Change the phase diagram
I " gain crossover / " bandwidth
I " �M / # %OS
I " �M / # Tp
I Implement a steady-state errorrequirement ! design a TR
Figure: Visualizing lead compensation
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11 Design via FR 11.4 Lead compensation
Visualizing lead compensation, [1, p. 635]
Concept
I Lead compensator TF
GC(s) =1
�
s+ 1T
s+ 1�T
I � > 1
I Frequency at maximum phaseshift angle
!max
=1
Tp� Figure: FR of a lead compensator for
various values of �
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11 Design via FR 11.4 Lead compensation
Visualizing lead compensation, [1, p. 635]
Concept
I Maximum phase shift angle
�max
= tan�1⇣1��2p�
⌘
= sin�1⇣1��1+�
⌘
I Magnitude at maximum phaseshift angle
|G(j!max
)| = 1p�
Figure: FR of a lead compensator forvarious values of �
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11 Design via FR 11.4 Lead compensation
Visualizing lead compensation, [1, p. 637]
Procedure
1. Find the CL bandwidth required to meet a Ts, Tp, or Tr requirement[1, p. 582]
!BW = !n
q(1� 2⇣2) +
p4⇣4 � 4⇣2 + 2
!n =4
Ts⇣and !n =
⇡
Tp
p1� ⇣2
2. Since the lead compensator has negligible e↵ect at low frequencies,set the gain, K, of the uncompensated system to the value thatsatisfies the steady-state error requirement
3. Plot the Bode diagrams for this value of gain and determine theuncompensated system’s �M
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 27 / 38
11 Design via FR 11.4 Lead compensation
Visualizing lead compensation, [1, p. 637]
Procedure
4. Find the �M to meet ⇣ or %OS requirement. Then evaluate theadditional phase contribution required from the compensator.
5. Determine the value of � from the lead compensator’s required phasecontribution
GC(s) =1
�
s+ 1T
s+ 1�T
�max
= tan�1⇣1��2p�
⌘= sin�1
⇣1��1+�
⌘
6. Determine the compensator’s magnitude at the peak of the phasecurve
|G(j!max
)| = 1p�
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 28 / 38
11 Design via FR 11.4 Lead compensation
Visualizing lead compensation, [1, p. 637]
Procedure
7. Determine the new !�M by finding where the uncompensatedsystem’s magnitude curve is the negative of the lead compensator’smagnitude at the peak of the compensator’s phase curve
8. Design the lead compensator’s break frequencies to find T and thebreak frequencies
GC(s) =1
�
s+ 1T
s+ 1�T
!max
=1
Tp�
9. Reset the system gain to compensate for the lead compensator’s gain
10. Check the bandwidth to be sure the speed requirement has been met
11. Simulate to be sure all requirements are met
12. Redesign if necessary to meet requirements
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11 Design via FR 11.4 Lead compensation
Example, [1, p. 638]
Example (Lead compensation design)
IProblem: Design a lead compensator to yield %OS = 20%,KV = 40, & Tp = 0.1 second
ISolution: On the board
Figure: System
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 30 / 38
11 Design via FR 11.5 Lag-lead compensation
1 11 Design via frequency response11.1 Introduction11.2 Transient response via gain adjustment11.3 Lag compensation11.4 Lead compensation11.5 Lag-lead compensation
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 31 / 38
11 Design via FR 11.5 Lag-lead compensation
Intro, [1, p. 641]
Concept
I What we are not doing: separate lag & lead compensators
1. Design a lag compensator to lower the high-frequency gain, stabilizethe system, & improve the steady-state error
2. Design a lead compensator to meet the phase-margin requirements
I What we are doing: passive lag-lead networkI Eliminates the bu↵er amplifier that separates the lag network from the
lead network
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11 Design via FR 11.5 Lag-lead compensation
Visualizing lag-lead compensation, [1, p. 641]
Concept
I Lag-lead passive compensator TF
GC(s) = GLead
(s)GLag
(s) =
✓s+ 1
TLead
s+ �TLead
◆✓s+ 1
TLag
s+ 1�T
Lag
◆
I 1st term in parentheses: lead compensatorI 2nd term in parentheses: lag compensatorI � replaces � & ↵ of lead & lag networks, respectively
GC(s) =1
�
s+ 1T
s+ 1�T
GC(s) =s+ 1
T
s+ 1↵T
I � & ↵ must be reciprocals of each otherI � > 1 & ↵ > 1 ! � > 1
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 33 / 38
11 Design via FR 11.5 Lag-lead compensation
Visualizing lag-lead compensation, [1, p. 643]
Procedure
1. Using a 2nd-order approximation, find the CL bandwidth required tomeet Ts, Tp, or Tr [1, p. 582]
2. Set the gain, K, to the value required by the steady-state errorspecification
3. Plot the Bode diagrams for this value of gain
4. Using a 2nd-order approximation, calculate the �M to meet the ⇣ or%OS requirement [1, p. 590]
5. Select a new !�M near !BW
6. At the new !�M , determine the additional amount of phase leadrequired to meet the �M requirement. Add a small contribution thatwill be required after the addition of the lag compensator.
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11 Design via FR 11.5 Lag-lead compensation
Visualizing lag-lead compensation, [1, p. 643]
Procedure
7. Design the lag compensator by selecting the higher break frequencyone decade below the new !�M . The design of the lag compensatoris not critical, and any design for the proper �M will be relegated tothe lead compensator. The lag compensator simply providesstabilization of the system with the gain required for the steady-stateerror specification. Find the value of � from the lead compensator’srequirements. Using the phase required from the lead compensator,the phase response curve can be used to find the value of � = ��1.This value, along with the previously found lag’s upper breakfrequency, allows us to find the lag’s lower break frequency.
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 35 / 38
11 Design via FR 11.5 Lag-lead compensation
Visualizing lag-lead compensation, [1, p. 643]
Procedure
8. Design the lead compensator. Using the value of � from the lagcompensator design and the value assumed for the new !�M , find thelower- and upper-break frequency for the lead compensator.
!max
=1
Tp�
9. Check the bandwidth to be sure the speed requirement has been met
10. Redesign if �M or TR specifications are not met, as shown byanalysis or simulation
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11 Design via FR 11.5 Lag-lead compensation
Example, [1, p. 643]
Example (Lag-lead compensation design)
IProblem: Design a passive lag-lead compensator using Bode diagramsto yield %OS = 13.25%, Tp = 2 seconds, & Kv = 12
G(s) =K
s(s+ 1)(s+ 4)
ISolution: On the board
Bayen (EECS, UCB) Feedback Control Systems September 10, 2013 37 / 38
11 Design via FR 11.5 Lag-lead compensation
Bibliography
Norman S. Nise. Control Systems Engineering, 2011.
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