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 2016
EE 3CL4, §9
2 / 30
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
4 / 30
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(jω
, where ωx is
x )|
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
EE 3CL4, §9
5 / 30
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
6 / 30
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
8 / 30
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
Lead Compensators
(s+z)
• Gc (s) = Kcs+p
, with |z| < |p|, alternatively,
• Gc (s) = Kαc 1+sατ
1+sτ , where p = 1/τ and α = p/z > 1
• Bode diagram (in the figure, K1 = Kc /α):
EE 3CL4, §9
9 / 30
Lead Compensation
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
• What will lead compensation, do?
• Phase is positive: might be able to increase phase
margin φpm
• Slope is positive: might be able to increase the
cross-over frequency, ωc , (and the bandwidth)
EE 3CL4, §9
10 / 30
Lead Compensation
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
• Gc (s) = Kαc 1+sατ
1+sτ
• By making the denom.
real,
can show that
∠Gc (jω) = atan
ωτ (α−1)
1+α(ωτ )2
√
1
• Max. occurs when ω = ωm = τ √
=
α
• Max. phase angle satisfies tan(φm ) =
zp
α−1
√
2 α
α−1
• Equivalently, sin(φm ) = α+1
√
• At ω = ωm , we have |Gc (jωm )| = Kc / α
EE 3CL4, §9
11 / 30
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
Bode Design Principles (lead)
• Set the loop gain so that desired steady-state error
constants are obtained
• Insert the compensator to modify the transient
properties:
• Damping: through phase margin
• Response time: through bandwidth
• Compensate for the attenuation of the lead network, if
appropriate
To maximize impact of phase lead, want peak of phase near
ωc of the compensated open loop
EE 3CL4, §9
12 / 30
Design Guidelines
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
1
2
3
4
5
6
7
8
9
10
For uncompensated (i.e., proportionally controlled)
closed loop, set gain Kp so that steady-state error
constants of the closed loop meet specifications
Evaluate the phase margin, and the amount of phase
lead required.
Add a little “safety margin” to the amount of phase lead
From this, determine α using sin(φm ) = α−1
α+1
To maintain steady-state error const’s, set Kc = Kp α
Determine (or approximate) the frequency at which
Kp G(jω) has magnitude −10 log10 (α).
If we set ωm of the compensator to be this frequency,
then Gc (jωm )G(jωm ) = 1 (or ≈ 1). Hence, the
compensator will provide its maximum phase
contribution at the appropriate frequency
√
√
Choose τ = 1/(ωm α). Hence, p = ωm α.
Set z = p/α.
(s+z)
Compensator: Gc (s) = Kcs+p
.
EE 3CL4, §9
13 / 30
Example
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
5
• Type 1 plant of order 2: G(s) = s(s+2)
• Design goals:
• Steady-state error due to a ramp input less than 5% of
velocity of ramp
• Phase margin at least 45◦ (implies a damping ratio)
• Steady state error requirement implies Kv = 20.
• For prop. controlled Type 1 plant: Kv = lims→0 sKp G(s).
Hence Kp = 8.
• To find phase margin of prop. controlled
loop
we need
40
to find ωc , where |Kp G(jωc )| = jωc (jωc +2) = 1
• ωc ≈ 6.2rad/s
• Evaluate ∠Kp G(jω) = −90◦ − atan(ω/2) at ω = ωc
• Hence φpm, prop = 18◦
EE 3CL4, §9
14 / 30
Example
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
• Let’s go for a little more, say 30◦
Lead
Compensators
• So, want peak phase of lead comp. to be 30◦
Lag
Compensators
• φpm, prop = 18◦ . Hence, need 27◦ of phase lead
α−1
• Solving α+1
= sin(30◦ ) yields α = 3. Set Kc = 3 × 8
• Since 10 log10 (3)
= 4.8 dB we should choose ωm to be
40
= −4.8 dB
where 20 log10 jωm (jω
m +2)
• Solving this equation yields ωm = 8.4rad/s
√
• Therefore z = ωm / α = 4.8, p = αz = 14.4
• Gc (s) = 24(s+4.8)
s+14.4
120(s+4.8)
• Gc (s)G(s) = s(s+2)(s+14.4)
, actual φpm = 43.6◦
• Goal can be achieved by using a larger target for
additional phase, e.g., α = 3.5
EE 3CL4, §9
15 / 30
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
Bode Diagram
EE 3CL4, §9
16 / 30
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
Step Response
EE 3CL4, §9
17 / 30
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
Ramp Response
EE 3CL4, §9
18 / 30
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
Ramp Response, detail
EE 3CL4, §9
20 / 30
Lag Compensators
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
• Gc (s) =
Kc (s+z)
s+p , with |p| < |z|, alternatively,
Kc α(1+sτ )
1+sατ , where z = 1/τ and α = z/p
• Gc (s) =
>1
• Bode diagrams of lag compensators for two different
αs, in the case where Kc = 1/α
EE 3CL4, §9
21 / 30
Tim Davidson
What will lag compensation do?
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
• Since zero and pole are typically close to the origin,
phase lag aspect is not really used.
• What is useful is the attenuation above ω = 1/τ :
gain is −20 log10 (α), with little phase lag
• Can reduce cross-over frequency, ωc , without adding
much phase lag
• Tends to reduce bandwidth
EE 3CL4, §9
22 / 30
Tim Davidson
Qualitative example
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
• Uncompensated system has small phase margin
• Phase lag of compensator does not play a large role
• Attenuation of compensator does:
ωc reduced by about a factor of a bit more than 3
• Increased phase margin is due to the natural phase
characteristic of the plant
EE 3CL4, §9
23 / 30
Tim Davidson
Bode Design Principles (lag)
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
For lag compensators:
• Set the loop gain so that desired steady-state error
constants are obtained
• Insert the compensator to modify the phase margin:
• Do this by reducing the cross-over frequency
• Observe the impact on response time
Basic principle: Set attenuation to reduce ωc far enough so
that uncompensated open loop has desired phase margin
EE 3CL4, §9
24 / 30
Design Guidelines
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
1
2
3
4
For uncompensated (i.e., proportionally controlled)
closed loop, set gain Kp so that steady-state error
constants of the closed loop meet specifications
Evaluate the phase margin, analytically, or using a
Bode diagram. If that is insufficient. . .
Determine ωc0 , the frequency at which the
uncompensated open loop, Kp G(jω), has a phase
margin equal to the desired phase margin plus 5◦ .
Design a lag comp. so that the gain of the compensated
open loop, Gc (jω)G(jω), at ω = ωc0 is 0 dB
• Choose Kc = Kp /α so that steady-state error const’s
are maintained
• Place zero of the comp. around ωc0 /10 so that at ωc0 we
get almost all the attenuation available from the comp.
• Choose α so that 20 log10 (α) = 20 log10 (|Kp G(jωc0 )|).
With that choice and Kc = Kp /α, |Gc (jωc0 )G(jωc0 )| ≈ 1
• Place the pole at p = z/α
(s+z)
• Compensator: Gc (s) = Kcs+p
EE 3CL4, §9
25 / 30
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
Example, same set up as lead
design
5
• Type 1 plant of order 2: G(s) = s(s+2)
• Design goals:
• Steady-state error due to a ramp input less than 5% of
velocity of ramp
• Phase margin at least 45◦ (implies a damping ratio)
• Steady state error requirement implies Kv = 20.
• For prop. controlled Type 1 plant: Kv = lims→0 sKp G(s).
Hence Kp = 8.
• To find phase margin of prop. controlled
loop
we need
40
to find ωc , where |Kp G(jωc )| = jωc (jωc +2) = 1
• ωc ≈ 6.2rad/s
• Evaluate ∠Kp G(jω) = −90◦ − atan(ω/2) at ω = ωc
• Hence φpm, prop = 18◦
EE 3CL4, §9
26 / 30
Example
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
• Since want phase margin to be 45◦ , we set ωc0 such that
∠G(jωc0 ) = −180◦ + 45◦ + 5◦ = −130◦ . =⇒ ωc0 ≈ 1.5
• To make the open loop gain at this frequency equal to 0 dB,
the required attenuation is 20 dB. Actual curves are around
2 dB lower than the straight line approximation shown
• Hence α = 10. Set Kc = Kp /α = 0.8
• Zero set to be one decade below ωc0 ;
• Pole is z/α = 0.015.
• Hence Gc (s) =
0.8(s+0.15)
s+0.015
z = 0.15
EE 3CL4, §9
27 / 30
Tim Davidson
Example: Comp’d open loop
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
4(s+0.15)
• Compensated open loop: Gc (s)G(s) = s(s+2)(s+0.015)
• Numerical evaluation:
• new ωc = 1.58
• new phase margin = 46.8◦
• By design, Kv remains 20
EE 3CL4, §9
28 / 30
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
Step Response
EE 3CL4, §9
29 / 30
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
Ramp Response
EE 3CL4, §9
30 / 30
Tim Davidson
Frequency
Domain
Approach to
Compensator
Design
Lead
Compensators
Lag
Compensators
Ramp Response, detail