Transcript
Page 1: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Lecture 7:

PID Tuning

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Page 2: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Objectives

• Describe and use the two methods of Ziegler-Nichols to tune PID controllers.

• Use the process reaction curve (step response) to fit a FOPDT model to the system.

• List some guidelines to design and implement a good step experiment.

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Page 3: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

PID TUNING

• How do we apply the same equation to many processes?• How to achieve the dynamic performance that we desire?

TUNING!!!

t

dI

c dt

CVddtteteKtu

0

')'(

1)()(

The adjustable parameters are called tuning constants. We can match the values to the process to affect the dynamic performance 3

Page 4: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

PID TUNING

Trial 1: unstable, lost $25,000

0 20 40 60 80 100 120-40

-20

0

20

40S-LOOP plots deviation variables (IAE = 608.1005)

Time

Con

tro

lled

Var

iabl

e

0 20 40 60 80 100 120-100

-50

0

50

100

Time

Man

ipul

ated

Va

riabl

e

Trial 2: too slow, lost $3,000

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1S-LOOP plots deviation variables (IAE = 23.0904)

Time

Con

tro

lled

Var

iabl

e

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Time

Man

ipul

ated

Va

riabl

e

0 20 40 60 80 100 1200

0.5

1

1.5S-LOOP plots deviation variables (IAE = 9.7189)

Time

Con

tro

lled

Var

iabl

e

0 20 40 60 80 100 1200

0.5

1

1.5

Time

Man

ipul

ated

Va

riabl

e

Trial n: OK, finally, but took way too long!!

Is there an easier way than

trial & error?

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Page 5: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Ziegler Nichols’ First method

When to use the first method?

The first method is applicable for processes whose “process reaction curve” (open-loop step response) is “S-shaped”.

DYNAMIC SIMULATION

Time

0 5 10 15 20 25 30 35 40 45 50-0.5

0

0.5

1

1.5

Time

Con

trol

led

Var

iabl

e

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

Man

ipul

ated

Var

iabl

e

S-shaped

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Page 6: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

EMPIRICAL MODEL BUILDING PROCEDURE

Process reaction curve - The simplest and most often used method. Gives nice visual interpretation as well.

1. Start at steady state

2. Single step to input

3. Collect data until steady state

4. Perform calculations

T

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Page 7: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Ziegler Nichols’ First method

How to use the first method?• Apply a step input to the process (open-loop).• Record the process reaction curve.• Fit a FOPDT model to the “process reaction curve”.

7

1

s

Ke

U

YG

Ls

PRC

Page 8: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Ziegler Nichols tuning rules

• With the aid of the following table find the controller parameter corresponding to the FOPDT model obtained.

8

1

s

Ke

U

YG

Ls

PRC

sT

sTKsG d

ipc

11)(

Page 9: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

How to fit a FOPDT model to the process reaction curve?

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Page 10: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

EMPIRICAL MODEL BUILDING PROCEDURE

-5

5

15

25

35

45 in

pu

t v

ari

ab

le i

n d

ev

iati

on

(%

op

en

)

-5

-1

3

7

11

15

ou

tpu

t v

ari

ab

le i

n d

ev

iati

on

(K

)

0 10 20 30 40 time (min)

Process reaction curve - Method I

S = maximum slope

L

igureshown in fL

S

K

/

/

Data is plotted in deviation variables

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Page 11: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

EMPIRICAL MODEL BUILDING PROCEDURE

-5

5

15

25

35

45 in

pu

t v

ari

ab

le i

n d

ev

iati

on

(%

op

en

)

-5

-1

3

7

11

15

ou

tpu

t v

ari

ab

le i

n d

ev

iati

on

(K

)

0 10 20 30 40 time (min)

Process reaction curve - Method II

%63

%28%63 )( 5.1

/

tL

tt

K

0.63

0.28

t63%t28%

Data is plotted in deviation variables

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Page 12: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Recommended

EMPIRICAL MODEL BUILDING PROCEDURE

Process reaction curve - Methods I and II

The same experiment in either method!

Method I

• Prone to errors because of evaluation of maximum slope

Method II

• Simple calculations

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Page 13: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Notes on experiment design

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Page 14: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

EMPIRICAL MODEL BUILDING PROCEDURE

Process reaction curve

-5

5

15

25

35

45

inp

ut

va

ria

ble

in

de

via

tio

n (

% o

pe

n)

-5

-1

3

7

11

15

ou

tpu

t v

ari

ab

le i

n d

ev

iati

on

(K

)

0 10 20 30 40 time (min)

Is this a well designed experiment?

Input should be close to a perfect step; this was basis of equations. If not, cannot use data for process reaction curve.

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Page 15: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

EMPIRICAL MODEL BUILDING PROCEDURE

-5

5

15

25

35

45

inp

ut

va

ria

ble

, %

op

en

-5

-1

3

7

11

15

ou

tpu

t v

ari

ab

le,

de

gre

es

C

0 10 20 30 40 time

Process reaction curve

Should we use this data?

The output must be “moved” enough. Rule of thumb:

Signal/noise > 5

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Page 16: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

EMPIRICAL MODEL BUILDING PROCEDURE

Process reaction curve

-5

5

15

25

35

45

inp

ut

va

ria

ble

, %

op

en

-5

-1

3

7

11

15

ou

tpu

t v

ari

ab

le,

de

gre

es

C

0 10 20 30 40 time

Plot measured vs predicted

measured

predicted

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Page 17: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Example

Let us apply ZN first method to the following process

1) Approximate the process with a FOPDT model using the two-points method.

2) Find the PID controller parameters recommended by ZN’s first method.

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)2)(1(

1)(

sssG

Page 18: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Answer • The step response of the given process is given by

• Using partial fractions

• Hence the time domain step response is given by

• Which has a steady state value of 0.5. • Therefore, we need to find the time at which the response becomes

approximately 0.14 and 0.31 (28% and 63%, respectively) 18

ssssY

1.

)2)(1(

1)(

2

5.0

1

15.0)(

ssssY

tt eety 25.05.0)(

Page 19: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Answer, continued • We can write the following equations:

• Which can be rewritten as (where we defined A = e-t1 and B = e-t2 )

• These are simple quadratic equations which can be solved to give

19

22

11

22

21

5.05.0315.0)(

5.05.014.0)(tt

tt

eety

eety

037.02037.02

072.02072.0222

22

22

11

BBee

AAeett

tt

56.121.0

75.047.0

%632

%281

ttB

ttA

Page 20: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Answer, continued

Applying a step input and recording the process reaction curve gives:

t28% = 0.75 sec,

t63% = 1.58 sec.

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Page 21: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Answer, continued

• The FOPDT parameters are then:

• Then, the controller parameters are obtained as

34.0

24.1)( 5.1

5.0/

%63

%28%63

tL

tt

K

17.0

68.0

75.8

d

i

p

T

T

K

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Page 22: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Ziegler Nichols’ 2nd method (Ultimate-Cycle Method)

While the first Ziegler-Nichols method is used in open-loop configuration, the second method is used in closed-loop.

When to use the 2nd method?

• If the process is open loop unstable, or,• If it is stable but does not give S-shaped step response.

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Page 23: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Procedure of ZN 2nd method

1. Put the process under closed-loop control (Use only a proportional controller).

2. Create a small disturbance in the loop by changing the set point.

3. Adjust the proportional gain, increasing and/or decreasing, until the oscillations have constant amplitude.

4. Record the gain value (Kcu) and period of oscillation (Tu).

5. Use the table to find the controller parameters.

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Page 24: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

The sustained oscillation

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Page 25: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Example

Let us apply ZN’s 2nd method to the following process

1) Find the ultimate gain and period.

2) Find the PID controller parameters recommended by ZN’s second method.

Then use MATLAB to plot the step set-point and disturbance responses of the closed loop system using the designed PID controller.

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2)12(

1)(

sssG

Page 26: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

• Using proportional controller Kc, the characteristic equation of the closed-loop system is

Writing the Routh array:

The system is stable if Kc < 1. So, the ultimate gain Kcu =1.

c

c

c

Ks

Ks

Ks

s

0

1

2

3

1

4

14

Answer

26

044

0)12(

0)12(

11

23

2

2

c

c

c

Ksss

Kss

ssK

Page 27: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Answer, continued

• When Kc = 1, Routh array becomes

The third row is zero. So, the auxiliary equation obtained from the second row is

27.sec56.12

2rad/sec5.0

5.0

equationauxiliary thesolvingby

014 2

uu

TT

js

s

1

0

14

14

0

1

2

3

s

s

s

s

Page 28: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

The sustained oscillation 28

Page 29: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

• Using the ZN 2nd method, the PID controller parameters are calculated as:

57.1

28.6

6.0

D

I

cK

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Page 30: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Another method to find the ultimate gain, Kcu

Using the root locus method

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syms ss=tf('s');G=1/(s*(2*s+1)^2);rlocus(G)

Page 31: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

The open loop response:

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Page 32: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

The closed-loop set-point step response

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Page 33: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

The closed-loop disturbance step response

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Page 34: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

close all

% Simulate

t=0:0.01:70;

s=tf('s');

G = 1/(s*(2*s+1)^2);

figure(1)

step(G,t)

% The FOPDT parameters

Ku = 1; Pu = 12.54;

% The PID parameters using ZN first method

Kc = 0.6*Ku; tauI = 0.5*Pu; tauD = 0.125*Pu;

KI=Kc/tauI; KD=Kc*tauD;

Gc = pid(Kc,KI,KD,0.01);

% Set point step response

cloop = Gc*G/(1+Gc*G);

figure(2)

step(cloop,t)

% Disturbance step response

cloop_dist = G/(1+Gc*G);

figure(3)

step(cloop_dist,t)34

MALAB code for this example

Page 35: Lecture 7: PID Tuning 1. Objectives Describe and use the two methods of Ziegler-Nichols to tune PID controllers. Use the process reaction curve (step

Comments on ZN tuning rules

• It is realized that the responses are oscillatory.

• Generally, Ziegler-Nichols tuning is not the best tuning method.

• However, these two guys were real pioneers in the field! It has taken 50 years to surpass their guidelines.

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