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EE 3CL4, §9
1 / 30
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators

EE3CL4:
Introduction to Linear Control Systems
Section 9: Design of Lead and Lag Compensators using
Frequency Domain Techniques

Tim Davidson
McMaster University

Winter 2013

EE 3CL4, §9
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Outline

Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators

1 Frequency Domain Approach to Compensator Design

Lag
Compensators

2 Lead

Compensators

3 Lag

Compensators

EE 3CL4, §9
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Frequency domain analysis

Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators

• Analyze closed loop using open loop transfer function
•
•
•
•
•

L(s) = Gc (s)G(s)H(s).
Nyquist’s stability criterion
1
Gain margin: |L(jωx )| , where ωx is the frequency at which ∠L(jω) reaches −180◦
Phase margin, φpm : 180◦ + ∠L(jωc ), where ωc is the frequency at which |L(jω)| equals 1
Damping ratio: φpm = f (ζ)
Roughly speaking, settling time decreases with increasing bandwidth of the closed loop

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Bode diagram

Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators

L(jω) =

1 jω(1 + jω)(1 + jω/5)

• Gain margin ≈ 15 dB
• Phase margin ≈ 43◦

EE 3CL4, §9
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Compensators and Bode diagram Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators

• We have seen the importance of phase margin
• If G(s) does not have the desired margin,

how should we choose Gc (s) so that
L(s) = Gc (s)G(s) does?
• To begin, how does Gc (s) affect the Bode diagram
• Magnitude:

20 log10 |Gc (jω)G(jω)|
= 20 log10 (|Gc (jω)| + 20 log10 |G(jω)|
• Phase:

∠Gc (jω)G(jω) = ∠Gc (jω) + ∠G(jω)

EE 3CL4, §9
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Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators