controller gain design
TRANSCRIPT
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KD 4.5.5.14
4.6 (Selecting controller gains)
(PID-control) , (dominant time constant) (steadystateerror) (rise time),(settling time),(maximum overshoot) (bandwidth), (resonantfrequency) (peak amplitude)
1) (Equilibrium specifications)- (stability)- (steady state error)
2) (transient specifications)- (speed of response)- (degree
of damping)
3) (Sensitivity specification)
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- (sensitivity to parameter variations)
- (sensitvity to model inaccuracies)
- (Noise rejection)
(bandwidth)
4) (nonlinear effects)- (stability)- (final control element capabilities)
(standard industrial controller) (tunning)
(Performance Index criteria)
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(Optimal Control) (performance index) (performance index)
(costing function)
(optimal control system)
0
J e(t)dt
= 4.6.1(Integral Absolute Error), IAE
0
J t e(t)dt
= 4.6.2(Integral Time multiplied Absolute Error), ITAE
2
0
J e (t)dt
= 4.6.3(Integral Square Error), ISE2
0
J te (t)dt
= 4.6.4(Integral Time multiplied Square Error), ITSE
e(t) (e(t)) t
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e(t) ft
(IAE)
(ITAE) t t (highlyunderdamped) (highly overdamped)
ITAE (overshoot)(osillating) (well-damped oscillations)
(ISE)( ITSE) (IAE) ( ITAE) 1)
2) 3)
4.6.1
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(Advanced Control System)
K
p
x
x
x( )0
x x u+ = 4.6.5 u Kx= () 4.6.6
(performance index)
2 21 2
0
J w x (t) w x (t) dt
= + 4.6.7
w1w2(weighting factors).
u Kx=
x x Kx+ = 4.6.8
(1 K)tx(t) x(0)e += 4.6.9
x(t) (performance index)
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2 2
1 2
1J x (0) w w (1 K)
2(1 K) = + + +
4.6.10
K J
J
0K
=
2
2
J
0x
4.6.11
1+K > 0
1
2
wK 1
w= + 4.6.12
K w1w2
w10 x(t)
K 1 x(t)=x(0) ( )x t w2= 0 K
x(t)0 K 0 < K < x x w1w2
K
K
K w1 w2(performance
index) (simple system)
4.6.1
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(optimization) (performance index) ITAE
4.6.1 ITAE
C s
R s
a
s a s a s a s an
n n n
n n
n
( )
( ) = + + + + + = 11
2
2
1
2
, an
s n+
s sn n2 21 4+ +.
s s sn n n
3 2 2 31 75 215+ + +. .
s s s sn n n n
4 3 2 2 3 421 3 4 2 7+ + + +. . .
s s s s sn n n n n5 4 2 3 3 2 4 52 8 5 0 5 5 3 4+ + + + +. . . .
s s s s s sn n n n n n6 5 2 4 3 3 4 2 5 63 25 6 60 8 60 7 45 3 95+ + + + + +. . . . .
4.6.2 (normalized form)
1G(s)
s(s 1)=
+
(PD-control) 4.6.1 KpKDITAE
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1G(s)
s(s 1)=
+
4.6.1
2pDs (1 K )s K 0+ + + =
4.6.1 ITAE
2 2
n ns 1.4 s 0 + + =
2rad/sec n= 2rad/sec
D D
2P
1 K 1.4 2 2.8 K 1.8
K 2 4
+ = = =
= =
4.6.1 n (Closed-loopnatural frequency) P-control 4.6.1
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4.7 -(Ziegler-Nichols) PID (Ziegler-Nichols Tuning of PID regulators)
-(Ziegler-Nichols) PID(transient response) (steady state) - (Model-based control) --
(dynamic equation) (transfer function)
(empirically based rules) -(Ziegler-Nichols) 40(Classical Control)
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( )P PG s K=
( )PI p r1G s K 1 Ts
= +
( ) ( )pPD dG s K 1 T s= +
( ) dpPDfilterD
T sG s K 1
s 1
= + +
( ) pPID dr
1G s K 1 T s
T s
= + +
( ) dpPIDfilterr D
T s1G s K 1
Ts s 1
= + + +
( ) s ssPIDseriess s
I D sG s K 1 1
s D s 1
= + + +
( ) p ppPIDparallelp p
I D sG s K
s D s 1= + +
+
(low pass filter) fast mode
4.5.4.1 (setpoint change) (Saturated Control) (intergral wind-up) 7
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- IAE (systemresponse) (unit-step input) -(Ziegler-Nichols)
(quarter-decay)
(overshoot)
25%4.7.2 - (Processreaction method) (Ultimate cycle method)
(Process reaction method)(Process reaction curve)
(Signal Recorder)Storage Oscilloscope, Analog recoder, Digital recorder
( )sGOutputUnit Step
- (input signal)
s (Process reaction
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curve)(FirstorderSystem)
dt sY(s) Ke
U(s) s 1
=
+
4.7.1
( first ordersystem) (transportation lag, td)
P,PI PID() RL P, PIPID4.7.1
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(Untimate cycle method) 25%
(quarter-decay) 4.7.2 =0.21 (Proportionalcontrol) 4.7.1 K(oscillate) (unstable) K (Closed-loop Characteristic equation) (Imaginary Axis) K Ku
(period of oscillation)Pu
(ultimateperiod) (amplitude) Ku Pu
P, PI PID 4.7.1
4.7.1
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100%
25%
Period
4.7.2
4.7.1 -(The Ziegler-Nichols Rules)
Ip pD DI
K1G(s) K 1 T s K K s
T s s
= + + = + +
(P-Control)
p
1K
RL= p pu
K 0.5K=
(PI-Control)
p 0.9K RL= p puK 0.45K=
IT 3.3L= I uT 0.83P=
(PID-Control)
p
1.2K
RL= p pu
K 0.6K=
IT 2L= I uT 0.5P=
DT 0.5L= uDT 0.125P=
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4.7.1 (Third order plant) 0 1 2 3a , a ,a , a
+ + +3 2
3 2 1 0
1
a s a s a s a
3 23 2 1 0 pa s a s a s a K 0+ + + + = 4.7.2
KpPu(ultimate period)
Kpu
Kp(sustainedoscillation) s (ultimate period) us j= u 2 2us = 3 3us j= s
( ) ( )2 30 pu 2 u 1 u 3 ua K a j a a 0 + + = 4.7.3
4.7.3 (real part)(imaginary part)
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2
0 pu 2 ua K a 0+ = 4.7.4
( )2u 1 3 ua a 0 = 4.7.5
4.7.5
1u
3
aa
= 4.7.6
4.7.64.7.4 Kpu
(ultimate gain)
2 2 1pu 2 u 0 0
3
a aK a a a
a= = 4.7.7
Pu(ultimate period)
uu
2P
= 4.7.8
(sustained oscillation) (Real Axis) s(negativerealpart) 1 us j= 2 us j= s
( )( )( )u u 3s j s j s s 0 + =
3 2 2 23 u 3 us s s s s 0 + = 4.7.9
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233
as
a=
3 2
1G(s)
s 2s s 1=
+ + +
3 2 1a 1, a 2, a 1= = = 0a 1=
-(The Ziegler-Nichols Rules)(PID-control)
p
I
D
K 0.6 1 0.6
T 0.5 2 3.142
T 0.125 2 0.785
= =
= =
= =
I
p D
K1G(s) 0.6 1 0.785s K K s
3.142s s
= + + = + +
4.7.3 4.7.4 (PID-
control) -(Ziegler-Nichols Rules)
3 2
1
s 2s s 1+ + +
10.6 1 0.785s
3.142s
+ +
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4.7.3 4.7.1 3 2
1G(s)
s 2s s 1=
+ + +
4.7.4 4.7.1 3 2
1G(s)
s 2s s 1=
+ + +
(PID-Control) = 10.6 1 0.785s3.142s
+ +
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(Untimate cycle method) Matlab/Simulink Slider Gain Scope 4.7.5 Simulink Slider Gain Scope Scope
(Autoscale)
4.7.6 1 1.16 Scope Ku 1 4.7.1-(The Ziegler-Nichols Rules)-
5 6 7 frequency response
4.7.5 Simulink 4.7.1
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Slider Gain= 1
Scope
Slider Gain= 1.16
Scope4.7.6 Slider Gain Scope
4.7.2-
( )( )
3
1G s
s 1=
+
(Untimate cycle method) -(Imaginary axis)
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( ) ( )
3
Pu Pu u3
11 K 0 K j 1
s 1+ = = +
+
( ) ( )3 3 2
Pu u u u uK j 1 j 3 3 1 = + = + +
puK 8= u 3 = rad/sec Pu(Ultimate period) 3.63 4.7.1
p
I
D
K 0.6 8 4.8
T 0.5 3.63 1.81
T 0.125 3.63 0.45
=
=
=
I
p DK1G(s) 4.8 1 0.45s K K s
1.81s s = + + = + +
( )3
1
s 1+
14.8 1 0.45s
1.81s
+ +
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Matlab/Simulink
( ) dpPIDfilterI D
T s1G s K 1
T s s 1
= + + +
D d0.1 *T 0.045 = =
( ) ( )( )
( )( ) ( )( )
2 DDd 2I I
pPIDfilter 3 3
D
1T s 1 sT T 52.8s 109.23s 58.93G s G s K
s s 1 s 1 s s 22.2 s 1
+ + + + + +
= =+ + + +
Matlab >> sys1=tf([52.8 109.23 58.93],poly([0 -22.2 -1 -1 -1]));
>> syscl1=feedback(sys1,1);
>> step(syscl1) ()
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dT 1=
( ) ( )
( )
( )( ) ( )( )
2 DDd 2
I IpPIDfilter 3 3
D
1T s 1 s
T T 52.8s 50.65s 26.52G s G s K
s s 1 s 1 s s 10 s 1
+ + + + + + = =
+ + + +
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(fine tune)