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Control SystemsChibum Lee -Seoultech
Outline
Lead compensator design in frequency domain
Lead compensator design steps.
Example on lead compensator design.
Control SystemsChibum Lee -Seoultech
Frequency Domain Design
Frequency response approach can impose the transient
response performance indirectly
Bode diagrams: useful for the frequency domain
specification
Nyquist plot: useful for system (relative) stability analysis.
Control SystemsChibum Lee -Seoultech
Lead Compensator
Appropriate improvement in transient response and
a small change in steady-state accuracy
Polar plot (KC = 1)
• 𝛼 determines the maximum phase
angle 𝜙𝑚 which occurs at 𝜔𝑚
)10(1
1
1
1)(
Ts
Ts
KTs
TsKsG ccc Zero:-
1
𝑇Pole: -
1
𝑇
1
1)(
Tj
TjKjG cc
112sin
1 1
2
m
Control SystemsChibum Lee -Seoultech
Lead Compensator
Bode diagram (KC = 1, 𝛼=0.1)
• Lead compensator: High-pass filter
• Goal of the lead compensator: to provide sufficient phase-lead
angle to offset the excessive phase lag in the uncompensated
system
11.0
11.0)(
Tj
TjjGc
Tm
1
1
1)(
Tj
TjKjGc
Control SystemsChibum Lee -Seoultech
Lead Compensator
The maximum phase lead angle 𝜙𝑚 occurs at 𝜔𝑚
• To increase phase margin, magnitude of the compensated system to
cross 0 dB at this frequency 𝜔𝑚
• but need to care for magnitude distortion added by lead compensator
-100
-50
0
50
Magnitude (
dB
)
0.01 0.1 1 10 100-270
-225
-180
-135
-90
-45
Phase (
deg)
Bode Diagram
Gm = 9.93 dB (at 4.01 rad/s) , Pm = 104 deg (at 0.443 rad/s)
Frequency (rad/s)𝜔𝑐: Gain crossover frequency
Phase Margin )()(180 cc jHjGPM
Control SystemsChibum Lee -Seoultech
Lead Compensator
The magnitude of the lead controller at:
• This is the amount that the lead compensator will shift the
magnitude plot. To have the gain crossover point (0 dB) at the
right point, we have to make sure that the geometric frequency
mean falls at the point where the open-loop uncompensated
system is 20 log10( 𝛼) below 0 dB.
• If is absorbed into the control gain, then the previous maximum
magnitude would be
1
1
1
1)(
1 j
j
Tj
TjjG
T
c
m
Tm
1
1
Control SystemsChibum Lee -Seoultech
Lead Compensator Design
Step 0: Assume following lead compensator
• Open-loop Plant:
• Compensator:
• Loop Gain of Compensated System:
with
( )G s
1
1 1( ) ( ) ( ) ( )
1 1C
Ts TsG s G s K G s G s
Ts Ts
1( ) ( )G s KG s
)(1
1
1
1)( KK
Ts
TsK
Ts
TsKsG ccc
Control SystemsChibum Lee -Seoultech
Lead Compensator Design Steps
Step 1: Determine K to satisfy static error constants
(Kp or Kv)
Step 2: Using this K, draw a Bode diagram of G1(s), and
evaluate the phase margin
Step 3: Determine the necessary phase angle needed to
meet design specs.
Control SystemsChibum Lee -Seoultech
Lead Compensator Design Steps
Step 4: Determine by using
( 𝜙𝑚=required phase lead angle + 5~12deg)
Find the frequency where:
Select this frequency as the new gain crossover frequency, where the maximum phase shift occurs
Step 5: Determine the lead compensator
1sin
1m
1 10
1( ) 20logG s
Tcm
1
Zero @ -1
𝑇Pole @ -
1
𝑇
Control SystemsChibum Lee -Seoultech
Lead Compensator Design Steps
Step 6: Solve for
Step 7: Check the gain margin. If it is not satisfactory,
one may have to iterate.
KKc
Control SystemsChibum Lee -Seoultech
Example
Ex.
Design a lead compensator to give Kv = 20 s-1,
PM 50 deg, and GM 10 dB
• Step 0: Find low-frequency gain:
Using
4
( )2
G ss s
1( )
1C C
TsG s K
Ts
1
4( ) ( )
2C
KG s KG s K K
s s
Control SystemsChibum Lee -Seoultech
Example
• Step 1:
Gain requirements:
• Step 2: draw a Bode diagram of G1(s),
and evaluate the phase margin
0 0
1 4lim ( ) lim 2
1 2C
s s
Ts KKv sG s G s s K
Ts s s
2 20 10K K
)2(
40)(1
jjjG
Control SystemsChibum Lee -Seoultech
Example
• Step 3: Determine the necessary phase angle needed to meet design specs.
𝜙𝑚= 33 (necessary)+5 (additional) deg
(extra 5 deg for compensation of the shift in gain crossover frequency.)
• Step 4: Determine
1 sin 381 sin1sin 0.24
1 1 sin 1 sin 38
mm
m
Control SystemsChibum Lee -Seoultech
Example
• Step 5: Determine the corner frequencies
First we find 𝜔𝑐 by noting that
The graph gives
at
The zero at 1/T
The pole at 1/T
12.04 6.2 dB
1 10
1( ) 20logG s
rad/sec9 cm
dBjG 2.6)(1
1 1 9 .24 4.41 /C Cw w rad secTT
1 9 / .24 18.4 /Cwrad sec
T
Control SystemsChibum Lee -Seoultech
Example
• Step 6: Solve for KC = K/
• Step 7: Verify the design
PM 50 deg and GM infinite…
10/.24 41.7CK K
4.41( ) 41.7
18.4C
sG s
s
Control SystemsChibum Lee -Seoultech
Example
• Nyquist
Control SystemsChibum Lee -Seoultech
Lecture Outline
Lag compensator design in frequency domain
Lag compensator design steps.
Example on lag compensator design.
Control SystemsChibum Lee -Seoultech
Lag Compensator
An appreciable improvement in steady-state accuracy
at the expense of increasing the transient-response time
Polar plot (KC = 1)
)1(1
1
1
1)(
Ts
Ts
KTs
TsKsG ccc
Zero:-1
𝑇Pole: -
1
𝑇
1
1)(
Tj
TjKjG cc
Control SystemsChibum Lee -Seoultech
Lag Compensator
Bode diagram (KC = 1, 𝛽=10)
• Lead compensator: Low-pass filter
• Goal of the lead compensator: to provide attenuation in high
frequency range to give a system sufficient phase margin
110
110)(
Tj
TjjGc
1
1)(
Tj
TjKjG cc
Control SystemsChibum Lee -Seoultech
Lag Compensator Design Steps
Step 0: Assume the following lag compensator
• Compensator:
• Loop Gain of Compensated System:
with
Step 1: Determine K to satisfy static error constants.
1( ) ( )G s KG s
)(1
1
1
1)( KK
Ts
TsK
Ts
TsKsG ccc
)(1
1)(
1
1)()( 1 sG
Ts
TssG
Ts
TsKsGsGc
Control SystemsChibum Lee -Seoultech
Lag Compensator Design Steps
Step 2: Find C for G1(s)
• Check PM and GM to see if they meet specs
• If not, find the frequency
where = -180 deg + required PM
• Required PM = specified PM + 5~12 deg. for safety margin
• This frequency is the new gain crossover frequency C
Step 3: Choose the corner frequency =1/T of the zero
• We want to change the magnitude plot without changing the
phase plot at the new crossover frequency
• Therefore, choose the zero at 1/T to be around 1 decade below
the new corner frequency C
1( )G jw
Control SystemsChibum Lee -Seoultech
Lag Compensator Design Steps
Step 4: Determine and the pole location...
• We now examine to find out how much it is greater
than 0 dB.
• Choose and then the pole is at 1/T
Step 5: Form the lag compensator...
• The actual compensator gain
)(1 cjG
101 log20)(dB0 cjG
KKc
Control SystemsChibum Lee -Seoultech
Example
Ex.
Design a lag compensator Kv = 5 sec-1,
PM > 40 degrees and GM > 10 dB
• Step 0:
)1(,1
1
)(
Ts
Ts
KsG cc
)15.0)(1(
1)(
ssssG
)15.0)(1()()(1
sss
KsKGsG
Control SystemsChibum Lee -Seoultech
Example
• Step 1:
Gain
• Step 2: draw a Bode diagram
of G1(s)
5
)15.0)(1(lim)(lim
)(1
1lim)()(lim
01
0
100
K
Ksss
sKssG
sGTs
TsssGssGK
ss
sc
sv
)15.0)(1(
5)(1
ssssG
Control SystemsChibum Lee -Seoultech
Example
• Step 3:
The phase margin is –20 degrees which means the unity feedback of the gain adjusted uncompensated system G1(s) is unstable.
If we want to have a phase margin of 40 deg, we should add another 5~12 deg to be safe.
Choose corner frequency =1/T=0.1 rad/sec (zero of the lag compensator)
The phase of G1(s) is –128 degaround 0.5 rad/sTherefore, the new crossover frequency c should be about 0.5 rad/s.
Control SystemsChibum Lee -Seoultech
Example
• Step 4:
Based on the new cross over
frequency, the zero should be placed
at = 1/T = 0.05 rad/s about a
decade below c.
The gain of G1(s) at c = 0.5 rad/s is
about 20 dB so that’s how much the
lag compensator must attenuate.
The pole of the lag compensator
10201
log20 10
rad/sec 01.01
T
Control SystemsChibum Lee -Seoultech
Example
• Step 5:
100
110
1
)(
s
s
KsG cc
5.010
5
KKc
)15.0)(1)(1100(
)110(5)()(
ssss
ssGsGc
Control SystemsChibum Lee -Seoultech
Example
Lag controller causes slower transients
Control SystemsChibum Lee -Seoultech
Outline
Lead-Lag compensator
Qualitative system responses.
Control SystemsChibum Lee -Seoultech
Lead-Lag Compensators
Whether to use the Lead or Lag controller depends on
the nature of your plant
• Lead for improved transient performance
• Lag for improved steady-state performance
when you need improved transient performance and
steady-state tracking lead-lag compensator
)1,1(1
11
)(
2
2
1
1
Ts
Ts
Ts
Ts
KsG cc
Control SystemsChibum Lee -Seoultech
Lead Lag Compensators
Phase lead adds phase at the uncompensated gain
crossover frequency thereby increasing the phase margin
Phase lag provides attenuation allowing an increase in
gain at low frequency to improve steady-state
performanceKc = 1, γ = β = 10, T2 = 10T1.
Kc = 1, γ = β.
Control SystemsChibum Lee -Seoultech
Lead Lag Compensators
Qualitative system responses
uncompensated lead lag lead-lag
)(ty )(ty )(ty )(ty
)(ty )(ty )(ty )(ty