controller gain design

7/21/2019 Controller Gain Design

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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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Principles of Feedback Control

222

- (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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230

-

( )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)

controller gain design

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