2009 Spring ME451 - GGZ Page 1Week 14-15: Frequency Design
Control Design based upon
• RL (Root Locus)
• Frequency Response Design– Bode Diagram (BD)
– Nyquist Approach
Control Design (CD) Control Design (CD) –– Control Design ProcessControl Design Process
1. Modeling
Mathematical modelMathematical model
2. Analysis
ControllerController
3. Design/Synthesis
4. Implementation
PlantPlantOutputOutput
ActuatorActuatorControllerController
SensorSensor
InputInput
Control Method to be discussed:
• PID control
• Gain compensation
• Phase lead/lag compensation
• Lead/lag compensation
2009 Spring ME451 - GGZ Page 2Week 14-15: Frequency Design
-
tt--domain:domain:
ss--domain:domain:
CD CD –– PID ControllerPID Controller
)(ty)(sP
)(tr
sKi/
pK
sKd
4342143421321
DerivativeIntegal
0alProportion
)()()()(
dt
tdeKdeKteKtu
d
t
ip++= ∫ ττ
++=++= sK
sKKsK
s
KKsC
D
I
pd
i
p
11)(
)(te)(tu
2009 Spring ME451 - GGZ Page 3Week 14-15: Frequency Design
• Most popular in process and robotics industries
– Good performance
– Functional simplicity (Operators can easily tune.)
• To avoid high frequency noise amplification, derivative term
is implemented as
with τd much smaller than plant time constant.
• PI controller
• PD controller
CD CD –– PID Controller RemarksPID Controller Remarks
1+≈
s
sKsK
d
d
dτ
s
KKsC i
p+=)(
sKKsCdp
+=)(
2009 Spring ME451 - GGZ Page 4Week 14-15: Frequency Design
• We plot y(t) for step reference r(t) with
– P controller
– PI controller
– PID controller
CD CD –– A Simple Example A Simple Example (1)(1)
-
)(ty)(sP
)(tr
sKi/
pK
sKd
)(te)(tu
3)1(
1)(
+=
ssP
2009 Spring ME451 - GGZ Page 5Week 14-15: Frequency Design
• Simple
• Steady state error
– Higher gain gives
smaller error
• Stability
– Higher gain gives
faster and more oscillatory response
0 5 10 15 200
0.2
0.4
0.6
0.8
1
1.2
1.4
CD CD –– A Simple Example A Simple Example (P Controller (2))(P Controller (2))
pKsC =)(
1=p
K
2=p
K
5=p
K
2009 Spring ME451 - GGZ Page 6Week 14-15: Frequency Design
0 5 10 15 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
• Zero steady state error(provided that CL is
stable.)
• Stability
– Higher gain gives faster and more
oscillatory response
CD CD –– A Simple Example A Simple Example (PI Controller (3))(PI Controller (3))
s
KKsC i
p+=)(1=
pK
1=iK
5.0=iK
2.0=iK
0=iK
2009 Spring ME451 - GGZ Page 7Week 14-15: Frequency Design
0 5 10 15 200
0.5
1
1.5
2
• Zero steady state error (due to integral control)
• Stability
– Higher gain gives
more dampedresponse
• Too high gain worsen
performance.
CD CD –– A Simple Example A Simple Example (PID Controller (4))(PID Controller (4))
sKs
KKsC
d
i
p++=)(5.1,5.2 ==
ipKK
0
1
2
4
=
=
=
=
d
d
d
d
K
K
K
K
2009 Spring ME451 - GGZ Page 8Week 14-15: Frequency Design
• Model-based
– Root locus
– Frequency response approach
– Useful only when a model is available
– Necessary if a system has to work at the first trial
• Empirical (without model)
– Ziegler-Nichols tuning rule (1942)
– Simple
– Useful even if a system is too complex to model
– Useful only when trial-and-error tuning is allowed
CD CD –– How to Turn PID ParametersHow to Turn PID Parameters
2009 Spring ME451 - GGZ Page 9Week 14-15: Frequency Design
• Step response method (for only stable systems)
tt
OpenOpen--loop step responseloop step response
Steepest tangentSteepest tangent
PID parametersPID parameters
CD CD –– ZieglerZiegler--Nichols PID Tuning Rules Nichols PID Tuning Rules (1)(1)
)1
1()( sTsT
KsCd
I
p++=
)(ty
τ
α− ττα
τα
α
5.02/2.1
3/9.0
/1
Type
PID
PI
P
TTKDIP
2009 Spring ME451 - GGZ Page 10Week 14-15: Frequency Design
• Ultimate sensitivity method
ClosedClosed--loop step responseloop step response
with a gain controller with a gain controller
Increase gain and find Increase gain and find KKcc
generating oscillation generating oscillation
(marginally stable case).(marginally stable case).
PID parametersPID parameters
CD CD –– ZieglerZiegler--Nichols PID Tuning Rules Nichols PID Tuning Rules (2)(2)
)1
1()( sTsT
KsCd
I
p++=
CCC
CC
C
DIP
TTKPID
TKPI
KP
TTK
125.05.06.0
8.04.0
5.0
Type
)(ty
CT
t
2009 Spring ME451 - GGZ Page 11Week 14-15: Frequency Design
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
• Step response method • Ultimate sensitivity
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
PP
PIPI
PIDPID
PP
PIPI
PIDPID
CD CD –– A Simple Example A Simple Example (Revisited (5))(Revisited (5))
2009 Spring ME451 - GGZ Page 12Week 14-15: Frequency Design
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
0.790.79--0.280.28
CD CD –– OL Step Response for OL Step Response for ““Step Response methodStep Response method””
2009 Spring ME451 - GGZ Page 13Week 14-15: Frequency Design
0 5 100
0.5
1
0 5 100
0.5
1
0 5 100
1
2
0 5 100
0.5
1
1.5
CD CD –– CL Step Responses forCL Step Responses for ““Ultimate Sensitivity methodUltimate Sensitivity method””
1=P
K
4=P
K
2=P
K
8=P
K 8=C
K
7.3=C
T
2009 Spring ME451 - GGZ Page 14Week 14-15: Frequency Design
• One example: Using OP amp
--
++--
++
Exercise: Derive this!Exercise: Derive this!
CD CD –– PID Controller Realization PID Controller Realization
++
+−
sCRsCR
C
C
R
R
21
12
2
1
1
2 1
)(tvi )(tv
o
1C 2
C
2R
1R
3R
4R
2009 Spring ME451 - GGZ Page 15Week 14-15: Frequency Design
• PID control
– Most popular controller in industry
– Model-free methods for design are available.
– Simple controller structure
– Simple controller tuning
– Widely applicable
• Ziegler-Nichols tuning rules provide a starting point for
fine tuning, rather than final settings of controller
parameters in a single shot.
CD CD –– PID Control Summary and Exercise PID Control Summary and Exercise
2009 Spring ME451 - GGZ Page 16Week 14-15: Frequency Design
• Z: # of CL poles in open RHP
• P: # of OL poles in open RHP (given)
• N: # of clockwise encirclement around -1
by Nyquist plot of OL transfer function L(s)
(counted by using Nyquist plot of L(s))
Remark: N = -1: a counter-clockwise encirclement
CD CD –– NyquistNyquist Stability Criterion (Review)Stability Criterion (Review)
0: stable is system CL =+=⇔ NPZ
2009 Spring ME451 - GGZ Page 17Week 14-15: Frequency Design
• IF P=0 (i.e., if L(s) has no pole in open RHP or stable)
This fact is very important since openThis fact is very important since open--loop systems loop systems
in many practical problems have no pole in open RHP!in many practical problems have no pole in open RHP!
CD CD –– NyquistNyquist Stability Criterion: A Special CaseStability Criterion: A Special Case
0stable is system CL =+=⇔ NPZ:
0stable is system CL =⇔ N
2009 Spring ME451 - GGZ Page 18Week 14-15: Frequency Design
ReRe
ImIm
ReRe
ImIm
CL stableCL stable CL unstableCL unstable
CD CD –– Examples with P = 0 (stable OL system) Examples with P = 0 (stable OL system)
)( ωjL )( ωjL
2009 Spring ME451 - GGZ Page 19Week 14-15: Frequency Design
• Nyquist stability criterion gives not only absolute but also
relative stability.
– Absolute stability: Is the closed-loop system stable or
not? (Answer is yes or no.)
– Relative stability: How “much” is the closed-loop
system stable? (Margin of safety)
• Relative stability is important because a math model is never accurate.
• How to measure relative stability?
– Use a “distance” from the critical point -1.
– Gain margin (GM) & Phase margin (PM)
CD CD –– NyquistNyquist Stability Remarks Stability Remarks
2009 Spring ME451 - GGZ Page 20Week 14-15: Frequency Design
• Phase crossover
frequency ωp:
• Gain margin (in dB)
• Indicates how much OL
gain can be multiplied without violating CL
stability.
NyquistNyquist plot of plot of L(sL(s))
CD CD –– NyquistNyquist Gain Margin (GM) Gain Margin (GM)
180)( −=∠p
jL ω
)(
1log20 20
pjL
GMω
=
2009 Spring ME451 - GGZ Page 21Week 14-15: Frequency Design
ReRe
ImIm
ReRe
ImIm
CD CD –– GM Examples GM Examples
dB6)(
1log20
2
10≈=
43421p
jLGM
ω
2
1)( −=
pjL ω
)( ωjL )( ωjL
3
1)( −=
pjL ω
dB5.9)(
1log20
3
10≈=
43421p
jLGM
ω
2009 Spring ME451 - GGZ Page 22Week 14-15: Frequency Design
Same gain margin,Same gain margin,
but different relative stabilitybut different relative stability
Gain margin is often inadequateGain margin is often inadequate
to indicate relative stabilityto indicate relative stability
Phase margin!Phase margin!
CD CD –– Why GM Alone is Inadequate Why GM Alone is Inadequate
2009 Spring ME451 - GGZ Page 23Week 14-15: Frequency Design
• Gain crossover
frequency ωg:
• Phase margin
• Indicates how much OL
phase can be added
without violating CL
stability.
NyquistNyquist plot of plot of L(sL(s))
CD CD –– NyquistNyquist Phase Margin (PM) Phase Margin (PM)
1)( =∠g
jL ω
o
gjLPM 180)( −∠= ω
2009 Spring ME451 - GGZ Page 24Week 14-15: Frequency Design
CD CD –– PM Example PM Example
)50)(5(
2500)(
++=
ssssL
dB80.14182.0
1log20 10 ==GM
2009 Spring ME451 - GGZ Page 25Week 14-15: Frequency Design
• Advantages
– Nyquist plot can be used for study of closed-loop stability, for open loop systems which is unstable and
includes time-delay.
• Disadvantage
– Controller design on Nyquist plot is difficult.
(Controller design on Bode plot is much simpler.)
We translate GM and PM on We translate GM and PM on NyquistNyquist plot plot
into those in Bode plot!into those in Bode plot!
CD CD –– NyquistNyquist Plot Remarks Plot Remarks
2009 Spring ME451 - GGZ Page 26Week 14-15: Frequency Design
ωωgg
ωωpp
GMGM
PMPM
CD CD –– Bode Diagram Relative Stability Bode Diagram Relative Stability
)( ωjL∠
)(log2010
ωjL
ω
ω
2009 Spring ME451 - GGZ Page 27Week 14-15: Frequency Design
• Advantages
– Without computer, Bode plot can be sketched easily.
– GM, PM, crossover frequencies are easily determined
on Bode plot.
– Controller design on Bode plot is simple.
• Disadvantage
– If OL system is unstable, we cannot use Bode plot for
stability analysis.
CD CD –– Bode Diagram Remarks Bode Diagram Remarks
2009 Spring ME451 - GGZ Page 28Week 14-15: Frequency Design
100
101
102
103
-100
-50
0
100
101
102
103
-180
-270
ωωgg ωωpp
GMGM
PMPM
CD CD –– Bode Diagram Example Bode Diagram Example
)50)(5(
2500)(
++=
ssssL
2009 Spring ME451 - GGZ Page 29Week 14-15: Frequency Design
10-1
100
-20
0
20
10-1
100
-180
-90
PMPM
GMGM
Time delay reducesTime delay reduces
relative stability!relative stability!
Delay timeDelay time
CD CD –– Bode Diagram Relative Stability Bode Diagram Relative Stability (time Delay)(time Delay)
)2)(1(
1)(
++=
ssssL
)2)(1()(
++=
−
sss
esL
s
lag phase deg 3.57 dTω
2009 Spring ME451 - GGZ Page 30Week 14-15: Frequency Design
1 0-2
1 0-1
1 00
1 01
1 02
-1
-0 .5
0
0 .5
1
1 0-2
1 0-1
1 00
1 01
1 02
-6 0 0 0
-4 0 0 0
-2 0 0 0
0
• TF
Huge phase lag!Huge phase lag!
As can be explained with As can be explained with NyquistNyquist stability criterion, stability criterion,
this phase lag causes instability of the closedthis phase lag causes instability of the closed--loop system,loop system,
and hence, the difficulty in control.and hence, the difficulty in control.
(rad) )( , ,1)()( TjGjGesG Ts ωωωω −=∠∀=⇒= −
CD CD –– Body Diagram of A Time DelayBody Diagram of A Time Delay
2009 Spring ME451 - GGZ Page 31Week 14-15: Frequency Design
ωωggωωpp
GMGM
PMPM
CD CD –– Body Diagram Unstable CL Case Body Diagram Unstable CL Case
ω
ω
)( ωjL∠
)(log2010
ωjL
2009 Spring ME451 - GGZ Page 32Week 14-15: Frequency Design
• Relative stability: Closeness of Nyquist plot to the critical
point -1
– Gain margin, phase crossover frequency
– Phase margin, gain crossover frequency
• Relative stability on Bode plot
• We normally emphasize PM in controller design.
CD CD –– Body Diagram Summary and Exercises Body Diagram Summary and Exercises
2009 Spring ME451 - GGZ Page 33Week 14-15: Frequency Design
Design specifications in time domainDesign specifications in time domain
(Rise time, settling time, overshoot, steady state error, etc.)(Rise time, settling time, overshoot, steady state error, etc.)
Desired closedDesired closed--loop loop
pole location pole location
in sin s--domaindomain
Constraints on openConstraints on open--loop loop
frequency response frequency response
in sin s--domaindomain
Root locus shapingRoot locus shaping Frequency response shapingFrequency response shaping
(Loop shaping)(Loop shaping)
Approximate translationApproximate translation
CD CD –– Control Design Comparison Control Design Comparison
2009 Spring ME451 - GGZ Page 34Week 14-15: Frequency Design
• Given G(s), design C(s) that satisfies time domain specs,
such as stability, transient, and steady-state responses.
• We learn typical qualitative relationships between open-
loop Bode plot and time-domain specifications.
PlantPlantControllerController
OL:OL:
CL:CL:
CD CD –– Feedback Control System Design Feedback Control System Design
)(ty)(tr)(sG)(sC
)()(:)( sCsGsL =
)(1
)(:)(
sL
sLsT
+=
2009 Spring ME451 - GGZ Page 35Week 14-15: Frequency Design
SteadySteady--state accuracystate accuracy
SensitivitySensitivity
Disturbance rejectionDisturbance rejection
Noise Noise
reductionreduction
TransientTransient
Response speedResponse speed
Transient Transient
OvershootOvershootRelative stabilityRelative stability
Relative stabilityRelative stabilityNoise Noise
reductionreduction
CD CD –– Typical Desired OL Body DiagramTypical Desired OL Body Diagram
)( ωjL∠
)(log2010
ωjL
2009 Spring ME451 - GGZ Page 36Week 14-15: Frequency Design
For steadyFor steady--state accuracy, state accuracy,
L should have high gain at low frequencies.L should have high gain at low frequencies.
yy((tt)) tracks tracks rr((tt)) composed of composed of
low frequencies very well.low frequencies very well.
CD CD –– Steady State Accuracy Steady State Accuracy (1)(1)
PlantPlantControllerController
OL:OL:
CL:CL:
)(ty)(tr)(sG)(sC
)()(:)( sCsGsL =
)(1
)(:)(
sL
sLsT
+=
)( Large ωjL)(log2010
ωjL
1)(1
)()( ≈
+=
ω
ωω
jL
jLjY
2009 Spring ME451 - GGZ Page 37Week 14-15: Frequency Design
• Step r(t)
Increase
• Ramp r(t)
Increase • Parabolic r(t)
Increase
For For KvKv to be nonzero,to be nonzero,
L must contain L must contain
at least one integrator.at least one integrator.
For Ka to be nonzero,For Ka to be nonzero,
L must contain L must contain
at least two integrators.at least two integrators.
<<--2020 <<--4040
CD CD –– Steady State Accuracy (2)Steady State Accuracy (2)
)0(: LKp
=
)(log2010
ωjL )(log2010
ωjL )(log2010
ωjL
ω ω ω
)(lim:0
ssLKs
v→
= )(lim: 2
0sLsK
sa
→=
2009 Spring ME451 - GGZ Page 38Week 14-15: Frequency Design
Noise Noise
reductionreduction
TransientTransient
Response speedResponse speed
Transient Transient
OvershootOvershootRelative stabilityRelative stability
Relative stabilityRelative stabilityNoise Noise
reductionreduction
SteadySteady--state accuracystate accuracy
SensitivitySensitivity
Disturbance rejectionDisturbance rejection
CD CD –– Typical Desired OL Body Diagram Typical Desired OL Body Diagram (Revisited)(Revisited)
)( ωjL∠
)(log2010
ωjL
2009 Spring ME451 - GGZ Page 39Week 14-15: Frequency Design
• For illustration, we use the feedback system:
CD CD –– A 2A 2ndnd Order System ExampleOrder System Example
PlantPlantControllerController
)(ty)(tr)(sG)(sC
)2()()(:)(
2
n
n
sssCsGsL
ςω
ω
+==
22
2
2)(1
)(:)(
nn
n
sssL
sLsT
ωςω
ω
++=
+=
2009 Spring ME451 - GGZ Page 40Week 14-15: Frequency Design
0 5 10 150
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
For small percent overshoot, For small percent overshoot,
L should have larger phase margin.L should have larger phase margin.
10-1
100
101
-20
0
20
10-1
100
101
-180
-160
-140
-120
-100
CL step responseCL step responseOL Bode plotOL Bode plot
PMPM
CD CD –– Percent OvershootPercent Overshoot
1=n
ω5.0
3.0
1.0
=
=
=
ς
ς
ς
2009 Spring ME451 - GGZ Page 41Week 14-15: Frequency Design
Noise Noise
reductionreduction
TransientTransient
Response speedResponse speed
Transient Transient
OvershootOvershootRelative stabilityRelative stability
Relative stabilityRelative stabilityNoise Noise
reductionreduction
SteadySteady--state accuracystate accuracy
SensitivitySensitivity
Disturbance rejectionDisturbance rejection
CD CD –– Typical Desired OL Body Diagram Typical Desired OL Body Diagram (Revisited)(Revisited)
)( ωjL∠
)(log2010
ωjL
2009 Spring ME451 - GGZ Page 42Week 14-15: Frequency Design
0 5 10 150
0.2
0.4
0.6
0.8
1
1.2
1.4
100
-20
0
20
10-1
100
101
-180
-160
-140
-120
-100
For fast response, For fast response,
L should have larger gain crossover frequency.L should have larger gain crossover frequency.
CL step responseCL step responseOL Bode plotOL Bode plot
CD CD –– Response SpeedResponse Speed
3.0=n
ω3
2
1
=
=
=
ς
ς
ς
2009 Spring ME451 - GGZ Page 43Week 14-15: Frequency Design
Noise Noise
reductionreduction
TransientTransient
Response speedResponse speed
Transient Transient
OvershootOvershootRelative stabilityRelative stability
Relative stabilityRelative stabilityNoise Noise
reductionreduction
SteadySteady--state accuracystate accuracy
SensitivitySensitivity
Disturbance rejectionDisturbance rejection
CD CD –– Typical Desired OL Body Diagram Typical Desired OL Body Diagram (Revisited)(Revisited)
)( ωjL∠
)(log2010
ωjL
ω
ω
ω
2009 Spring ME451 - GGZ Page 44Week 14-15: Frequency Design
• We require adequate GM and PM for:
– safety against inaccuracies in modeling
– reasonable transient response
• It is difficult to give reasonable numbers of GM and PM for
general cases, but usually,
– GM should be at least 6dB
– PM should be at least 45deg
(These values are not absolute but approximate!)
• In controller design, we are especially interested in PM
(which typically gives good GM).
CD CD –– Relative StabilityRelative Stability
2009 Spring ME451 - GGZ Page 45Week 14-15: Frequency Design
Noise Noise
reductionreduction
TransientTransient
Response speedResponse speed
Transient Transient
OvershootOvershootRelative stabilityRelative stability
Relative stabilityRelative stabilityNoise Noise
reductionreduction
SteadySteady--state accuracystate accuracy
SensitivitySensitivity
Disturbance rejectionDisturbance rejection
CD CD –– Typical Desired OL Body Diagram Typical Desired OL Body Diagram (Revisited)(Revisited)
)( ωjL∠
)(log2010
ωjL
ω
ω
2009 Spring ME451 - GGZ Page 46Week 14-15: Frequency Design
yy((tt)) is not affected by is not affected by nn((tt))
composed of high frequencies.composed of high frequencies.
noisenoise
For noise rejection, For noise rejection,
L should have small gain at high frequencies.L should have small gain at high frequencies.
CD CD –– Noise RejectionNoise Rejection
PlantPlantControllerController
)(ty)(tr)(sG)(sC
)(tn
)( Small ωjL)(log2010
ωjL
0)(1
)()( ≈
+−=
ω
ωω
jL
jLj
N
Y
ω
2009 Spring ME451 - GGZ Page 47Week 14-15: Frequency Design
• Reshape Bode plot of G(jω) into a “desired” shape of
by a series connection of appropriate C(s).
CD CD –– Frequency Shaping (Loop Shaping)Frequency Shaping (Loop Shaping)
noisenoisePlantPlantControllerController
)(ty)(tr)(sG)(sC
)(tn
)(tddisturbancedisturbance
)()(:)( ωωω jCjGjL =
ω
ω
2009 Spring ME451 - GGZ Page 48Week 14-15: Frequency Design
• Bode plot of a series connection G1(s)G2(s) is the addition of
each Bode plot of G1 and G2.
– Gain
– Phase
• We use this property to design C(s) so that G(s)C(s) has a
“desired” shape of Bode plot.
CD CD –– Advantages of Body DiagramAdvantages of Body Diagram
)(log20)(log20)()(log202101102110
ωωωω jGjGjGjG +=
)()()()(1121
ωωωω jGjGjGjG ∠+∠=∠
2009 Spring ME451 - GGZ Page 49Week 14-15: Frequency Design
SteadySteady--state accuracystate accuracy
SensitivitySensitivity
Disturbance rejectionDisturbance rejection
Noise Noise
reductionreduction
TransientTransient
Response speedResponse speed
Transient Transient
OvershootOvershootRelative stabilityRelative stability
Relative stabilityRelative stabilityNoise Noise
reductionreduction
CD CD –– Typical Shaping Goal Typical Shaping Goal (Review)(Review)
)( ωjL∠
)(log2010
ωjL
ω
ω
2009 Spring ME451 - GGZ Page 50Week 14-15: Frequency Design
• We use simple controllers for shaping.
– Gain
– Lead and lag compensators
CD CD –– Simple ControllersSimple Controllers
noisenoiseStable PlantStable PlantControllerController
)(ty)(tr)(sG)(sC
)(tn
)(tddisturbancedisturbance
KsC =)(
1
1
poly.)order -(1st
poly.)order -(1st)(
+
+==
ps
zs
sC
2009 Spring ME451 - GGZ Page 51Week 14-15: Frequency Design
dBdB
degdeg
CD CD –– Bode Plot of a GainBode Plot of a Gain
KsC =)(
K10
log20
2009 Spring ME451 - GGZ Page 52Week 14-15: Frequency Design
10-2
10-1
100
101
102
103
-100
-50
0
50
100
10-2
10-1
100
101
102
103
-180
-160
-140
-120
-100
In case of In case of K K > 1,> 1,
�� Gain increases Gain increases
uniformly, but phase uniformly, but phase
does not change.does not change.
�� Typically, Typically,
�� (Steady state) (Steady state) LL(0)(0)
�� (Speed) (Speed) ωωgg
�� (Stability & (Stability &
overshoot) PMovershoot) PM PMPM
CD CD –– Effect of a Gain Effect of a Gain C(sC(s) of ) of L(sL(s) )
)0()( >= KsC
)(10)( sGsL =
)25(
2500)(
+=
sssG
2009 Spring ME451 - GGZ Page 53Week 14-15: Frequency Design
10-2
10-1
100
101
102
103
-20
-15
-10
-5
0
10-2
10-1
100
101
102
103
-60
-40
-20
0
10-2
10-1
100
101
102
103
0
5
10
15
20
10-2
10-1
100
101
102
103
0
20
40
60
LeadLead compensatorcompensator LagLag compensatorcompensator
PHASE LEADPHASE LEAD PHASE LAGPHASE LAG
MEMORIZE THESE MEMORIZE THESE
SHAPES!!!SHAPES!!!
CD CD –– Bode Diagrams of a Lead and Lag Bode Diagrams of a Lead and Lag C(sC(s))
pz <
1
1)(
+
+=
ps
zs
sC
zp <
2009 Spring ME451 - GGZ Page 54Week 14-15: Frequency Design
Noise Noise
reductionreduction
Relative stabilityRelative stability
Gain+lagGain+lag
LeadLead
LagLag
LeadLead
CD CD –– Guideline of a Lead and Lag DesignGuideline of a Lead and Lag Design
)( ωjL∠
)(log2010
ωjL
ω
ω
2009 Spring ME451 - GGZ Page 55Week 14-15: Frequency Design
Destabilizing effectDestabilizing effect
•• Decreasing Decreasing ωωωωωωωωgg
Select z much (at least 1 Select z much (at least 1
decade) less than decade) less than ωωgg
10-2
10-1
100
101
102
103
-20
-15
-10
-5
0
10-2
10-1
100
101
102
103
-60
-40
-20
0
CD CD –– Effect of a Lag Effect of a Lag C(sC(s) on ) on L(sL(s) )
p
z10
log20
2009 Spring ME451 - GGZ Page 56Week 14-15: Frequency Design
30/1
3/1
1
110)(
s
s
LagsC
+
+=
10-2
10-1
100
101
102
103
-100
-50
0
50
100
10-2
10-1
100
101
102
103
-180
-160
-140
-120
-100
PM: 28 deg at PM: 28 deg at
ωωgg=47 =47 rad/srad/s
PMPMPM: 27 deg at PM: 27 deg at
ωωgg=47 =47 rad/srad/s
CD CD –– Lag + Gain Lag + Gain C(sC(s) Design) Design
)(sG)25(
2500)(
+=
sssG
)()( sCsGLag
)(sG
)()( sCsGLag
2009 Spring ME451 - GGZ Page 57Week 14-15: Frequency Design
Noise Noise
reductionreduction
Relative stabilityRelative stability
Gain+lagGain+lag
LeadLead
LagLag
LeadLead
CD CD –– Guideline of a Lead and Lag Design Guideline of a Lead and Lag Design (revisited)(revisited)
)( ωjL∠
)(log2010
ωjL
ω
ω
2009 Spring ME451 - GGZ Page 58Week 14-15: Frequency Design
10-2
10-1
100
101
102
103
0
5
10
15
20
10-2
10-1
100
101
102
103
0
20
40
60
Stabilizing effectStabilizing effect
Increasing Increasing ωωωωωωωωgg
Select Select z&pz&p around around ωωgg
CD CD –– Effect of a Lead Effect of a Lead C(sC(s) on ) on L(sL(s) )
2009 Spring ME451 - GGZ Page 59Week 14-15: Frequency Design
101
102
103
-60
-40
-20
0
20
101
102
103
-180
-160
-140
-120
-100
PM: 28 deg at PM: 28 deg at
ωωgg=47 =47 rad/srad/s
PMPMPM: 47 deg at PM: 47 deg at
ωωgg=60 =60 rad/srad/s
CD CD –– Example of a Lead DesignExample of a Lead Design
)(sG
)()( sCsGLead
)()( sCsGLead
)(sG
1.94
21.38
1
1)(
s
s
LeadsC
+
+=
)25(
2500)(
+=
sssG
2009 Spring ME451 - GGZ Page 60Week 14-15: Frequency Design
10-2
10-1
100
101
102
103
0
5
10
15
20
10-2
10-1
100
101
102
103
-60
-40
-20
0
20
40
CD CD –– LeadLead--Lag Compensator Lag Compensator
321321Lead
s
s
LagGain
s
s
sC1.94
21.38
30/1
3/1
1
1
1
110)(
+
+⋅
+
+⋅=
+
2009 Spring ME451 - GGZ Page 61Week 14-15: Frequency Design
10-2
10-1
100
101
102
103
-100
-50
0
50
100
10-2
10-1
100
101
102
103
-180
-160
-140
-120
-100
PM: 28 deg at PM: 28 deg at
ωωgg = 47 = 47 rad/srad/s
PMPMPM: 47 deg at PM: 47 deg at
ωωgg = 60 = 60 rad/srad/s
CD CD –– Example of a LeadExample of a Lead--Lag DesignLag Design
)()( sCsG
)(sG
)(sG
)()( sCsG
)()()( sCsCsCLagLead
=
)25(
2500)(
+=
sssG
2009 Spring ME451 - GGZ Page 62Week 14-15: Frequency Design
0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
Uncompensated (Uncompensated (C(sC(s) =) =1)1)
LeadLead--lag compensatedlag compensated
Less overshoot is due to larger PM.Less overshoot is due to larger PM.
Faster response is due to larger Faster response is due to larger wwgg. .
CD CD –– Step ResponsesStep Responses
2009 Spring ME451 - GGZ Page 63Week 14-15: Frequency Design
0.48 0.485 0.49 0.495 0.50.48
0.485
0.49
0.495
0.5
Uncompensated (Uncompensated (C(sC(s)=1) )=1)
KKvv
= 100= 100
LeadLead--lag compensated lag compensated
KKvv= 1000= 1000
Ramp referenceRamp reference
Smaller steadySmaller steady--state error is due to larger state error is due to larger KKvv. .
CD CD –– Ramp ResponsesRamp Responses
2009 Spring ME451 - GGZ Page 64Week 14-15: Frequency Design
• Frequency shaping (Loop shaping) on Bode plot
• Effect of lead, lag, and lead-lag compensators
• Qualitative explanation
• In actual design, one needs to use Matlab.
• Next, more detail about
– Lag design
– Lead design
CD CD –– Loop Shaping SummaryLoop Shaping Summary
2009 Spring ME451 - GGZ Page 65Week 14-15: Frequency Design
Noise Noise
reductionreduction
TransientTransient
Response speedResponse speed
Transient Transient
OvershootOvershootRelative stabilityRelative stability
Relative stabilityRelative stabilityNoise Noise
reductionreduction
SteadySteady--state accuracystate accuracy
SensitivitySensitivity
Disturbance rejectionDisturbance rejection
CD CD –– Typical Desired OL Body DiagramTypical Desired OL Body Diagram
)( ωjL∠
)(log2010
ωjL
ω
ω
2009 Spring ME451 - GGZ Page 66Week 14-15: Frequency Design
• Sensitivity indicates the influence of plant variations (due to temperature, humidity, age.) on closed-loop performance.
• Sensitivity function
For sensitivity reduction, For sensitivity reduction,
L should have large gain L should have large gain
at low frequencies.at low frequencies.
CD CD –– Sensitivity ReductionSensitivity Reduction
)(log2010
ωjL
)(1
1
)()(1
1
)(/)(
)(/)(:)(
sLsCsGsGsG
sTsTsS
+=
+=
∂
∂=
0)(1
1)( ≈
+=
ωω
jLjS)( Large ωjL
ω
2009 Spring ME451 - GGZ Page 67Week 14-15: Frequency Design
yy((tt)) is not affected by is not affected by dd((tt))
composed of low frequencies.composed of low frequencies.
For disturbance rejection, For disturbance rejection,
L should have large gain at low frequencies.L should have large gain at low frequencies.
CD CD –– Disturbance RejectionDisturbance Rejection
Stable PlantStable PlantControllerController
)(ty)(tr)(sG)(sC
ω
)(tddisturbancedisturbance
)(log2010
ωjL )( Large ωjL
0)(1
1)( ≈
+=
ωω
jLj
D
Y
2009 Spring ME451 - GGZ Page 68Week 14-15: Frequency Design
• Unwanted signals
• Examples
– Wind turbulence in airplane altitude control
– Wave in ship direction control
– Sudden temperature change outside the temperature-
controlled room
– Air pressure brake to DC motor
– Bumpy road in cruise control
• Often, disturbance is neither measurable nor predictable.
(Use feedback to compensate it!)
CD CD –– DisturbanceDisturbance
2009 Spring ME451 - GGZ Page 69Week 14-15: Frequency Design
SteadySteady--state accuracystate accuracy
SensitivitySensitivity
Disturbance rejectionDisturbance rejection
Noise Noise
reductionreduction
TransientTransient
Response speedResponse speed
Transient Transient
OvershootOvershootRelative stabilityRelative stability
Relative stabilityRelative stabilityNoise reductionNoise reduction
• Next, frequency shaping (loop shaping) design
CD CD –– SummarySummary
)( ωjL∠
)(log2010
ωjL
ω
ω
ω
2009 Spring ME451 - GGZ Page 70Week 14-15: Frequency Design
CD CD –– Body Diagram of a Lead/Lag Body Diagram of a Lead/Lag C(sC(s) (Review)) (Review)
10-2
10-1
100
101
102
103
-20
-15
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-5
0
10-2
10-1
100
101
102
103
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-40
-20
0
10-2
10-1
100
101
102
103
0
5
10
15
20
10-2
10-1
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101
102
103
0
20
40
60
LeadLead compensatorcompensator LagLag compensatorcompensator
PHASE LEADPHASE LEAD PHASE LAGPHASE LAG
MEMORIZE THESE MEMORIZE THESE
SHAPES!!!SHAPES!!!
pz <
1
1)(
+
+=
ps
zs
sC
zp <
2009 Spring ME451 - GGZ Page 71Week 14-15: Frequency Design
dBdB +20+20
degdeg+45+45
dBdB
--2020
degdeg
--4545
Lead (z < p)Lead (z < p)
Lag (p < z)Lag (p < z)
dBdB+20+20
degdeg
+45+45
dBdB
--2020
degdeg
--4545
--4545 +45+45
CD CD –– StraightStraight--Line ApproximationsLine Approximations
1
1)(
+
+=
ps
zs
sG
1)( +=z
ssG
1
1)(
+=
ps
sG
2009 Spring ME451 - GGZ Page 72Week 14-15: Frequency Design
• Design C(s) so that L(jω):=G(jω)C(jω) has a desired shape.
• We study the design of simple compensators:
– Gain compensator (Today)
– Lag compensator (Today)
– Lead compensator (Next lecture)
CD CD –– Frequency Shaping (Loop Shaping)Frequency Shaping (Loop Shaping)
Stable PlantStable PlantControllerController
)(tr)(sG)(sC
2009 Spring ME451 - GGZ Page 73Week 14-15: Frequency Design
Noise Noise
reductionreduction
Relative stabilityRelative stability
Gain+lagGain+lag
LeadLead
LagLag
LeadLead
CD CD –– Guideline of LeadGuideline of Lead--Lag Design (Review)Lag Design (Review)
)( ωjL∠
)(log2010
ωjL
ω
ω
2009 Spring ME451 - GGZ Page 74Week 14-15: Frequency Design
• Consider a system
• Analysis for C(s) = 1
– Stable
– PM at least 12 deg
– GM at least 3.5 dB
These values are too These values are too
small for good small for good
transient response!transient response!
10-2
10-1
100
101
102
-100
-50
0
50
10-2
10-1
100
101
102
-250
-200
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-100
CD CD –– An Example (LeadAn Example (Lead--Lag Design)Lag Design)
Stable PlantStable PlantControllerController
)(tr)(sG)(sC
)(ty
)2)(1(
4)(
++=
ssssG
2009 Spring ME451 - GGZ Page 75Week 14-15: Frequency Design
• PM is specified to be 50 deg.
• In this example, to increase PM by gain compensation,
we need to lower the gain curve.
CD CD –– Gain Margin Compensation Gain Margin Compensation (Example (2))(Example (2))
2009 Spring ME451 - GGZ Page 76Week 14-15: Frequency Design
10-2
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100
101
102
-100
-50
0
50
10-2
10-1
100
101
102
-250
-200
-150
-100Uncompensated (Uncompensated (C(sC(s)=1))=1)
Gain compensatedGain compensated
Low freq. gain Low freq. gain
decreases.decreases.
CD CD –– Bode Diagram for Bode Diagram for C(sC(s) = 0.286 ) = 0.286 (Example (3))(Example (3))
2009 Spring ME451 - GGZ Page 77Week 14-15: Frequency Design
0 5 10 150
0.2
0.4
0.6
0.8
1
1.2
1.4
K=0.455 (PM=35deg)K=0.455 (PM=35deg)
K=0.286 (PM=50deg)K=0.286 (PM=50deg)
K=0.158 (PM=65deg)K=0.158 (PM=65deg)
CD CD –– Step Responses Step Responses (Example (4))(Example (4))
2009 Spring ME451 - GGZ Page 78Week 14-15: Frequency Design
dBdB
degdeg
--4545
--2020
10-2
10-1
100
101
102
103
-20
-15
-10
-5
0
10-2
10-1
100
101
102
103
-60
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-20
0
CD CD –– PhasePhase--Lag Compensator Lag Compensator (Review)(Review)
zpsGps
zs
<<+
+= 0,
1
1)(
z
p10log20
2009 Spring ME451 - GGZ Page 79Week 14-15: Frequency Design
Step 1: To satisfy low frequency requirement, adjust DC gain of OL system by a constant gain K.
• Analysis for C(s) = 1
– Stable
– PM at least 12 deg
– GM at least 3.5 dB
We try to design phaseWe try to design phase--lag lag C(sC(s) which gives) which gives
•• PM 50degPM 50deg
•• Low frequency gain same as the original plant.Low frequency gain same as the original plant.
CD CD –– PhasePhase--Lag Compensator Lag Compensator C(sC(s) Design) Design
Stable PlantStable PlantControllerController
)(tr)(sG)(sC
)2)(1(
4)(
++=
ssssG
2009 Spring ME451 - GGZ Page 80Week 14-15: Frequency Design
10-2
10-1
100
101
102
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-50
0
50
10-2
10-1
100
101
102
-250
-200
-150
-100
OKOK
CD CD –– PhasePhase--Lag Design Step 1 Lag Design Step 1 ((C(sC(s) = 1)) = 1)
1 with )( =KsKG
2009 Spring ME451 - GGZ Page 81Week 14-15: Frequency Design
Step 2: Find the frequency ωg (which will become gain crossover frequency after compensation) where
In this example,
Note: The reason of +5 deg is explained later.
CD CD –– PhasePhase--Lag Design Step 2 Lag Design Step 2 ((C(sC(s) = 1)) = 1)
4.0=g
ω
PM required : ,5180)(m
o
m
o
gjG φφω ++−=∠
{oooo
g
m
jG 125550180)( −=++−=∠φ
ω
2009 Spring ME451 - GGZ Page 82Week 14-15: Frequency Design
10-2
10-1
100
101
102
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-50
0
50
10-2
10-1
100
101
102
-250
-200
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-100
PM=55PM=55
CD CD –– PhasePhase--Lag Design Step 2 Lag Design Step 2 ((C(sC(s) = 1)) = 1)
)(sKG
gω
2009 Spring ME451 - GGZ Page 83Week 14-15: Frequency Design
Step 3:
dBdB
--2020
degdeg
--4545
For small phase lag at For small phase lag at ωωgg
For setting new gain crossover at For setting new gain crossover at ωωgg
CD CD –– PhasePhase--Lag Design Step 3 Lag Design Step 3 ((C(sC(s) = 1)) = 1)
)55.4
04.0(
)(
1.0==
g
g
jKGp
ω
ω
)04.0(1.0 ==g
z ω
)(
1log20log20
1010
gjKGz
p
ω=
2009 Spring ME451 - GGZ Page 84Week 14-15: Frequency Design
10-2
10-1
100
101
102
-100
-50
0
50
10-2
10-1
100
101
102
-250
-200
-150
-100
PM=50PM=50
CD CD –– PhasePhase--Lag Design Step 3 Lag Design Step 3 ((C(sC(s) = ) = CClaglag(s(s))))
)(sKG
1
1)(
0088.0
04.0
+
+=
s
s
LagsC
)()( sCsKGLag
2009 Spring ME451 - GGZ Page 85Week 14-15: Frequency Design
0 5 10 150
0.5
1
1.5
2
C(sC(s)=0.286 (PM=50deg, )=0.286 (PM=50deg, ωωωωωωωωgg=0.5)=0.5)
C(sC(s)=)=CCLagLag(s(s) (PM=52.3deg, ) (PM=52.3deg, ωωωωωωωωgg=0.4)=0.4)
Small overshoot is due to larger PM.Small overshoot is due to larger PM.
Slower response is due to smaller Slower response is due to smaller wwgg. .
C(sC(s)=1)=1 (PM=12deg, (PM=12deg, ωωωωωωωωgg=1.1)=1.1)
CD CD –– PhasePhase--Lag Design Step ResponsesLag Design Step Responses
2009 Spring ME451 - GGZ Page 86Week 14-15: Frequency Design
67 67.5 68 68.5 69 69.5 7064
65
66
67
68
69
70
Smaller steadySmaller steady--state error is due to larger state error is due to larger KvKv. .
C(sC(s)=0.286 ()=0.286 (KvKv=0.572)=0.572)
Ramp referenceRamp reference
C(sC(s)=)=CCLagLag(s(s) () (KvKv=2)=2)
C(sC(s)=1)=1 ((KvKv=2)=2)
CD CD –– PhasePhase--Lag Design Ramp ResponsesLag Design Ramp Responses
2009 Spring ME451 - GGZ Page 87Week 14-15: Frequency Design
• Gain controller design in Bode plot
– Gain changes uniformly over frequencies.
– Phase does not change.
• Lag compensator design in Bode plot
– Lag compensator can be used for
• Improving PM by maintaining low freq. gain, or
• Improving low freq. gain by maintaining PM
• Low freq. gain determines steady state error, disturbance
rejection, while PM does overshoot.
• Next, lead compensator design
CD CD –– PhasePhase--Lag Design SummaryLag Design Summary
2009 Spring ME451 - GGZ Page 88Week 14-15: Frequency Design
• What is problematic?
– Nyquist stability criterion says that, for closed-loop
stability, Nyquist plot of open-loop system must
encircle -1 point.
– It is hard to translate this condition into Bode plot.
• To use FR technique…
UnstableUnstableControllerController
Stabilize first!Stabilize first!
CD CD –– If If G(sG(s) has OL RHP Poles) has OL RHP Poles
)(sG)(sC
)(sH
2009 Spring ME451 - GGZ Page 89Week 14-15: Frequency Design
1. Select z near uncompensated ωg.
In the example, ωg = 1.14. So, select, for example, z = 1.
2. Select p > z by trial-and-error.
3. Check PM and settling time. If not satisfactory, move the pole p. If moving pole does not give the desired results,
try to move the zero z.
CD CD –– PhasePhase--Lead Design ProcedureLead Design Procedure
2009 Spring ME451 - GGZ Page 90Week 14-15: Frequency Design
10-2
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100
101
102
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-50
0
50
10-2
10-1
100
101
102
-250
-200
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PM=50PM=50
CD CD –– PhasePhase--Lead Design ExampleLead Design Example
)(sG
1
1)(
50+
+=
sLead
ssC
)()( sCsGLead
57.1=g
ω
2009 Spring ME451 - GGZ Page 91Week 14-15: Frequency Design
0 5 10 150
0.5
1
1.5
2C(sC(s)=0.286 (PM=50deg, )=0.286 (PM=50deg, ωωωωωωωωgg=0.5)=0.5)
C(sC(s)=)=CCLagLag(s(s) (PM=52.3deg, ) (PM=52.3deg, ωωωωωωωωgg=0.4)=0.4)
C(sC(s)=1)=1 (PM=12deg, (PM=12deg, ωωωωωωωωgg=1.1)=1.1)
C(sC(s)=)=CCLeadLead(s(s) (PM=50deg, ) (PM=50deg, ωωωωωωωωgg=1.6)=1.6)
Faster response is due to larger Faster response is due to larger wwgg..
(Compare crossover frequencies.) (Compare crossover frequencies.)
CD CD –– PhasePhase--Lead Design Step ResponsesLead Design Step Responses
2009 Spring ME451 - GGZ Page 92Week 14-15: Frequency Design
69 69.2 69.4 69.6 69.8 7067
67.5
68
68.5
69
69.5
70
C(sC(s)=0.286 ()=0.286 (KvKv=0.572)=0.572)
Ramp referenceRamp reference
C(sC(s)=)=CCLagLag(s(s) () (KvKv=2)=2)
C(sC(s)=1)=1 ((KvKv=2)=2)
C(sC(s)=)=CCLeadLead(s(s) () (KvKv=2)=2)
CD CD –– PhasePhase--Lead Design Ramp ResponsesLead Design Ramp Responses
2009 Spring ME451 - GGZ Page 93Week 14-15: Frequency Design
10-2
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102
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-50
0
50
10-2
10-1
100
101
102
-250
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-100
We recover the gain loss We recover the gain loss
at gain crossover freq.at gain crossover freq.
by adding a gain.by adding a gain.
CD CD –– Phase LeadPhase Lead--Lag Design Example (1)Lag Design Example (1)
)(sG
)()( sCsGLead
)()()( sCsCsGLagLead
2009 Spring ME451 - GGZ Page 94Week 14-15: Frequency Design
10-2
10-1
100
101
102
-100
-50
0
50
10-2
10-1
100
101
102
-250
-200
-150
-100
PM=51PM=51
IncreasedIncreased
CD CD –– Phase LeadPhase Lead--Lag Design Example (2)Lag Design Example (2)
)(sG
)()(4)( sCsCsCLagLeadLL
=
)()( sCsGLL
43.1=g
ω
2009 Spring ME451 - GGZ Page 95Week 14-15: Frequency Design
0 5 10 150
0.5
1
1.5
2
C(sC(s)=0.286)=0.286
C(sC(s)=)=CCLagLag(s(s))
C(sC(s)=1)=1
C(sC(s)=4C)=4CLeadLead(s)C(s)CLagLag(s)(s)
CD CD –– Phase LeadPhase Lead--Lag Design Step ResponsesLag Design Step Responses
2009 Spring ME451 - GGZ Page 96Week 14-15: Frequency Design
69 69.2 69.4 69.6 69.8 7067
67.5
68
68.5
69
69.5
70
C(sC(s)=0.286 ()=0.286 (KvKv=0.572)=0.572)
Ramp referenceRamp reference
C(sC(s)=)=CCLagLag(s(s) () (KvKv=2)=2)
C(sC(s)) = 1 (= 1 (KKvv=2)=2)
C(sC(s)=4C)=4CLeadLead(s)C(s)CLagLag(s) (s)
((KvKv=8)=8)
CD CD –– Phase LeadPhase Lead--Lag Ramp ResponsesLag Ramp Responses
2009 Spring ME451 - GGZ Page 97Week 14-15: Frequency Design
• Lead compensator can be used for improving
• Gain crossover frequency
• Phase margin
by maintaining low frequency gain,
• Lead-lag compensator can improve
– Transient (ωg for speed, PM for overshoot)
– Steady state (low frequency gain for error constant)
• Next, case studies
– Antenna azimuth position control
– Hard disk drive control
CD CD –– LeadLead--Lag Compensator SummaryLag Compensator Summary