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INC 341 PT & BPINC 341 PT & BP
INC341Design Using Graphical Tool
(continue)
Lecture 10
INC 341 PT & BP
Improving Both Steady-State Error and Transient Response
• PI, Lag improve steady-state error
• PD, Lead improve transient response
• PID, Lead-lag improve both
(PID = Proportional plus Intergal plus Derivative controller)
INC 341 PT & BP
PID Controller
s
K
Ks
K
KsK
s
sKKsKsK
s
KKsGc
)(
)( 3
2
3
1232
3213
21
INC 341 PT & BP
PID controller design
1. Evaluate the performance of the uncompensated system
2. Design PD controller to meet transient response specifications
3. Simulate and Test, redesign if necessary
4. Design PI controller to get required steady-state error
5. Find K constant of PID
6. Simulate and Test, redesign if necessary
INC 341 PT & BP
Example
Design PID controller so that the system can operatewith a peak time that is 2/3 of uncompensated system,at 20% OS, and steady-state error of 0 for a step input
INC 341 PT & BP
sec297.057.10
dpT
Step 1
• %OS = 20% damping ratio = 0.456
Ѳ = 62.87
• Search along ther line to find a point of 180 degree (-5.415±j10.57)
• Find a correspoding K=121.51
• Then find the peak time
INC 341 PT & BP
INC 341 PT & BP
• Decrease peak time by a factor of 2/3 get imaginary point of a compensator pole:
• To keep a damping ratio constant, real part of the pole will be at
• The compensator poles will be at -8.13±j15.867
867.15)297.0)(3/2(
p
d T
13.8)87.62tan(
d
Step 2
INC 341 PT & BP
92.55
)37.18tan(13.8
87.15
c
c
z
z
ดPD controller is (s+55.92)
• Sum of the angles from uncompensated poles and zeros to the test point (-8.13±j15.867) is
-198.37• The contribution angle for the compensator zero
is then 180-198.371 = 18.37
INC 341 PT & BP
INC 341 PT & BP
Step 3• Simulate the PD compensated system to
see if it reduces peak time and improves ss error
INC 341 PT & BP
s
ssGPI
5.0)(
Step 4• design PI compensator (one pole at origin
and a zero near origin; at -0.5 in this example)
• Find a new point along the 0.456 damping ratio line (-7.516±j14.67), with an associate gain of 4.6
INC 341 PT & BP
INC 341 PT & BP
s
ss
s
sss
ssKsGPID
)96.2742.56(6.4
)5.0)(92.55(6.4
)5.0)(92.55()(
2
K1 = 259.5, K2 = 128.6, K3 = 4.6
Step 5
• Evaluate K1, K2, K3 of PID controller
• Compare to s
sKKsKsGc
2321)(
INC 341 PT & BP
Step 6
INC 341 PT & BP
Lead-Lag Compensator Design
Same procedures as in designing PID:– Begin with designing lead compensator to get
the desired transient response– design lag compensator to improve steady-
state error
INC 341 PT & BP
Example
Design lead-lag compensator so that the system can operate with 20% OS, twofold reduction in settling time, and tenfold improvement in steady-state error for a ramp input
INC 341 PT & BP
Step 1
• %OS = 20% damping ratio = 0.456
Ѳ = 62.87
• Search along ther line to find a point of 180 degree (-1.794±j3.501)
• Find a correspoding K=192.1
• Then find the settling time
sec230.2794.1
44
sT
INC 341 PT & BP
INC 341 PT & BP
Step 2• Decrease settling time by a factor of 2 get a
real part of a compensator pole:
• To keep a damping ratio constant, imaginary part of the pole will be at
• The compensator poles will be at -3.588±j7.003
588.3)230.2)(2/1(
44
sT
003.7)87.62tan(588.3 d
INC 341 PT & BP
1.29
)35.15tan(588.3
003.7
c
c
p
p
Lead compensator is
• Select the compensator zero at -6 to coincide with the open-loop pole
• Sum of the angles from uncompensated poles and zeros to the test point (-3.588±j7.003) is
-164.65• The contribution angle for the compensator zero
is then 180-164.65 = 15.35
)1.29(
)6(
s
s
INC 341 PT & BP
Then find a new K at the design point (K=1977)
INC 341 PT & BP
Step 3Simulate the lead compensated system
INC 341 PT & BP
Step 4
• Originally the uncompensated system has the transfer function:
1.192,)10)(6(
)(
Ksss
KsG 201.3
106
1.192
vK
INC 341 PT & BP
• After adding the lead compensator, the system has changed to
• Static error constant, Kv, is then 6.794 (lead compensator has improved ss error by a factor of 6.794/3.201=2.122)
• So the lag compensator must be designed to improve ss error by a factor of 10/2.122=4.713
)1.29)(10(
1977)(
ssssGLC
713.4,)(
)()(
c
c
c
clag p
z
ps
zssG
INC 341 PT & BP
Step 5• Pick a pole at 0.01, then the associated zero will
be at 0.04713
• Lag-lead compensator
• Lag-lead compensatated open loop system
)01.0)(101.29(
)04713.0)(6()(
ss
sssG laglead
)01.0(
)04713.0()(
s
ssGlag
)01.0)(101.29)(10(
)04713.0()(
ssss
sKsGLLC
INC 341 PT & BP
INC 341 PT & BP
INC 341 PT & BP
Step 6
INC 341 PT & BP
Conclusions
INC 341 PT & BP
INC 341 PT & BP
Feedback Compensation
Put a compensator in the feedback path
INC 341 PT & BP
Tachometer
Tachometer generates a voltage output proportionalto input rational speed
Popular feedback compensator, rate sensor
INC 341 PT & BP
rate feedback
INC 341 PT & BP
ExampleDesign a feedback compensatorto decrease settling timeby a factor of 4 andkeep a constant %OS of 20
INC 341 PT & BP
%OS = 20% damping ratio = 0.456 Ѳ = 62.87
Search along the daping ratio line to get a summationof angle of 180 degrees at -1.809±j3.531
sec21.2809.1
44
sT
Find the corresponding K from the magnitude rule
settling time
841.257
531.3191.13531.3191.3531.3809.1 222222
K
Step 1
INC 341 PT & BP
Step 2
Reduce the Settling time by a factor of 4
236.7)21.2)(4/1(
44
sT
123.14)87.62tan(236.7 d
A new location of poles is at -7.236±j14.123
INC 341 PT & BP
At dominant pole -7.236±j14.123, KG(s)H(s) has a net angle of = -277.33 needs an additional angle from zero of 277.33-180 = 97.33
42.5
)33.97180tan(236.7
123.14
c
c
z
z
Find the corresponding K to the pole at -7.236+j14.123using the magnitude rule:
)15)(5(
)42.5(
sss
sK
K = 256.819
INC 341 PT & BP
211.1388
185.0
42.51
819.256
1
1
K
K
K
KK
f
f
f
Feedback block is 0.185(s+5.42)
INC 341 PT & BP
Physical System Realization
R1
R2C
Vi(s) Vo(s)
22
1
1
( )
sR CR
C sR s
PI Compensator
INC 341 PT & BP
R1
R2
C
Vi(s) Vo(s)
2 2
1 2
1 2
1
( )1
( )
c
sR R C
G sR R s
R R C
Lag Compensator
INC 341 PT & BP
R1
R2
C
Vi(s) Vo(s) 21
1( )cG s R C s
RC
PD Compensator
INC 341 PT & BP
R1
R2C
Vi(s) Vo(s)
1
1 2
1
( )1 1
,
c
cc c
c
sRC
G ssRC R C
s zz p
s p
Lead Compensator
INC 341 PT & BP
R1
R2C2
Vi(s) Vo(s)
C1
2 1 1 22 1
1 2
1
( )c
R C RCG s R C s
R C s
PID Compensator