A New Method of Parameter Estimation and PID-Controller Tuning Based on the Inflectional Tangent. Prof. Dr.-Ing. Herbert M.

(1)

Prof. Dr. H.M. Schaedel, Cologne University of Applied Sciences

1/16

A New Method of Parameter Estimation and PID-Controller Tuning Based on the

Inflectional Tangent

Prof. Dr.-Ing. Herbert M. Schaedel

Faculty of Information, Media and Electrical Engineering

Innovative Digital Signal Processing and Applications (DiSPA)

Presented at the Workshop Process Automation

20th November 2008 in Köln, Germany

(2)

Step response of a lag process and first order plus dead-time approximation (FOPDT) with identical

T

u

and T

g

K

P

h

1

T

g

t/s

T

u

T

u

: d e ad -tim e

T

g

: b ui ld- u p tim e

( )

1

u s T S g

s

s T

e

G







in flec tio n ta ng en t

h (t)

(3)

3/16

Tuning rules by Ziegler und Nichols (1942)

P- controller

_C



g P u

T

K

K T

PI- controller

_C



0, 9

g P u

T

K

K T

T

r



4, 0

T

u

PID- controller

_C



1, 2

g P u

T

K

K T

T

r



2

T

u

T

d



0, 5

T

u

(4)

Tuning rules by Chien, Hrones and Reswick for optimal set-point control (1952)

aperiodic

20%

overshoot

P- controller

_C



0, 3

g P u

T

K

K T



0, 7

g C P u

T

K

K T

PI- controller

0, 35

1, 2



g C P u r g

T

K

K T

T

0, 6

1, 0



g C P u r g

T

K

K T

T

PID- controller

0, 60

1, 0

0, 5



g C P u r g d u

T

K

K T

T

0, 95

1, 35

0, 47



g C P u r g d u

T

K

K T

T

(5)

5/16

Sum of time constants T

1

as a function of



=T

u

/T

g

for typical plants

0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2 Tu /Tg T1 Tg G P 4 G P 3 G P 1 t r a n s i t i o n GP 1 t o G P 4 G P 5 GP 2

Transfer function of typical plants

 

P sTt P1

K

G

s

e

1 s





 

   



P P2 1 1

K

G

s

1 s



 

 

 







t sT P P3 2 1 1

K e

G

s

1 s





 

 

where



= 0...1

 



P



P 4 n

K

G

s

1 s



 

 

P P5 _n i i 1

K

G

s

1 s

i





















(6)

Tuning rules by Kuh

n

using the sum of time-constants (1995)

normal fast

P-controller

1



C P

K

PD- controller

1

0, 33

_



C P d

K

T

PI- controller

1

0, 5

_



C P r

K

T

1

0, 7

_



C P r

K

T

PID- controller

1

0, 66

0,167

 



C P r d

K

T

2

0,8

0,194

 



R S n v

K

T

(7)

7/16

Parameter estimation through the inflectional tangent

Parameter estimation #1

Parameter estimation #2





1

 

g

 

u g

T



T

₁_

 







T

_g

 

T

_u

T

_g 2 2 2 _{3 3} 1

1

2

0 48









_, 

T

,



2 2 2 2 4 1

1

2

0 37









, 

T

,



> 0,1:









3 3 3 _{2 4} 2 1

-0,1

6

0 36











,

T

,

 



0 1



,



:

T

33



0



> 0,1:









3 3 3 _{2 4} 2 1

-0,1

6

0 37











,

T

,

 







0 1

,

:

T

33



0

;

where



2



1 2 P

h

0,5 1 H

0,4

K





















contribution of build-up time T

g

to sum of time-constants T

1

H

 



1 h K

1 P



e 2, 72 1 h K







1 P



T

u

Lag-time (apparent dead-time)

T

g

build-up time



T

u

/T

g

(8)

Characteristic frequency parameters of a plant for controller tuning

  





t sT P P P 2 2 2 1 2 3 1 2 3

K e

K

G

s

1 s

...

1 sT

s T

sT

...

   





 



Series-expansion of the dead-time term

sT 2 3 2 t 3 t

1

e

T

1 sT

s

...

2

6







1 1  





n _i



_t i

T



T

sum of time constants including dead-time

1 2 2 2 i j 1 1 1

1

2

     



 

n n



_t



n _i



_t i j i i

T

 

T



T

product sum of time constants including dead-time

(9)

9/16

Controller tuning using frequency-domain information according to the principle of the cascaded damping ratio

PI-controller for optimal set-point control (Schaedel 1995)

normal design

Butterworth

sharp design

Tschebyscheff 0.5 db

r C P 1 r 2 2 r 1 2

T

0.5

K

K T

T

2T

  









r C P 1 r 2 2 r 1 1

T

0.375

K

T

   









PID-controller for optimal set-point control (Schaedel 1995)

Broadband design

3 2 3 2 d 2 1 2    



T



T

;

 

C

aT

d 2 2 1 2 r 1 v   







T

;



r



C P 1 r C

3

8

_

 



 

T

K

T



(10)

PI-controller for optimal disturbance rejection (Schaedel 2005)

Butterworth (normal

design)

Tschebyscheff 0.1 db

ITAE

2 1 C 2 P 2 2 2 2 2 r 2 1 1

T

1

K

1

K

2 T

T

4

1 2

T

     

































2 1 C 2 P 2 2 2 2 2 r 2 1 1

T

1

K

0.7

1

K

T

3.11

1 1.43

T

    

































2 1 C 2 P 2 2 2 2 2 r 2 1 1

T

1

K

0.69

1

K

T

3.86

1 1.46

T

     

































PID-controller for optimal disturbance rejection

Butterworth Tschebyscheff

0,1db

ITAE

6 2 C 6 P 3

1

0,1464



1



















T

K

T

6 2 P 6 S 3

1

0, 345

T

1

K

T

 

















6 2 P 6 S 3

1

0, 356

T

1

K

T

 

















8 2 I 9 S 3

0, 0214

T

K

T

 



28 I 9 S 3

0, 0783

T

K

T

 



28 I 9 S 3

0, 069

T

K

T

 



4 2 D 3 1 S 3

1

0,5

T

K

T

K

T

  

















4 2 D 3 1 S 3

1

0,807

T

K

T

K

T

  

















4 2 D 3 1 S 3

1

0,876

T

K

T

K

T

  

















(11)

11/16

Controller-tuning using the extended method of inflectional tangent

Optimal disturbance

rejection

optimal set-point control

P-controller

3 C

1

0,586

0,353



















P

K





3 C

1

0, 586

0, 353



















P

K





PD- controller

C d 2,5 u C d

0, 7

0, 45

0, 26

0, 2







P

K

T











C d 2,5 u C d

0, 7

0, 45

0, 26

0, 2







P

A

K

T









I- controller





I g

0, 5





P

K

 

T

I S





g

0, 5

K



A T





PI-Regler

3 C 2 r u 4

0, 33

2

1

0, 22

3

2,12







 

















P

K

T

















3 C 3 r 3 g

0, 375

0,88

1

1,14

2, 27







 



















P

K

T









 





(12)

PID- Regler

C r u d 2,5 u C d

1, 3

2, 27

0,816

0, 55

0, 25

0, 2















P

K

T



















2 C C r g 2 d u 2 4 C d

0, 52

0, 7

0, 08 :

0, 08

9

0, 08 :

0, 5

0, 333

0, 233

0,123

0, 2





























P P

K

T







 











with

T

u

lag-time and T

g

build-up time



= T

u

/T

g

; h

1

= h (T

u

+T

g

) build-up value

H

 



1 h

1

K

P



e



2, 72 1 h





1

K

P







2 2 1 P

h

0,5 1 H

0,4

K

















(13)

13/16

Simulation results

2

nd

order plant with non-minimum phase term

  





P

1 7, 5s

G

s

1 15s 1 7, 5s







PI-controlled for optimal set-point control

-0,4 -0,2 0 0,2 0,4 0,6 0,8 1 1,2 1,4 0 50 100 150 200

(14)

3

rd

order plant with lead-term

  





P

1 40s

G

s

1 58s 1 6s 1 6s







PI-controlled for optimal set-point control

0

0,2

0,4

0,6

0,8

1

1,2

1,4

0

20

40

60

80

100

120

140

(15)

15/16

Experimental results at a flow process

Online Lab of Jim Henry, University of Tennessee at Chattanooga, http://chem.engr.utc.edu

23 23,5 24 24,5 25 25,5 26 26,5 27 15 15,5 16 16,5 17 17,5 18 t/s F lo w r a te /( lb /m in )

(16)

Characteristic parameters of the step response function K

p

= 0.500 (lb/min)/%

h

1

/K

p

= 0.844 T

u

= 0.32s T

g

= 0.65s



= Tu/Tg = 0.492

Results for the controlled process to a setpoint change

Controller gain (%/lb/min) = 1,1

Integral Time (sec) = 0,42

Initial set-point (lb/min) = 24

Change in set-point (lb/min) = 5

Time for change in set-point (sec) = 15

Valve 1 open (sec) 0

Valve 1 closed (sec) 30

Valve 2 open (sec) 30

Valve 2 closed (sec) 0

2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 t/ s F lo w /( lb /m in ) x (t ) w (t )

(17)

17/16

5 0 5 2 5 4 5 6 5 8 6 0 6 2 1 2 1 4 1 6 1 8 2 0 2 2 2 4 t/s u (t ) / %

(18)

References

Ziegler, J.G. and G.A. Nichols, (1942).

Optimum settings for automatic controllers

, Trans. ASME, 64, pp. 759-768,

New York.

Chien, K.L., Hrones, J.A. and Reswick, J.B.,

„On the automatic control of generalized passive systems“

, Trans. ASME,

74, (1952), pp. 175-185.

Schaedel, H.M. (1997a).

Parameterschätzung über die Wendetangente und direkter Reglerentwurf in den

CAE-Werkzeugen SimTool und SIMID.

2. VDI/VDE Aussprachetag "Rechnergestützter Entwurf von Regelsystemen",

16./17.Sept., Kassel, GMA-Bericht 32, pp. 9-18.

Schaedel, H.M. (1997b).

A new method of direct PID controller design based on the principle of cascaded damping

ratios.

Proc. 4

th

European Control Conference, Brussels, 1.-4. July 1997,, Paper WE-A H4, BELWARE Information

Technology, Waterloo.

Schaedel, H.M. (1998).

Neue Prinzipien des direkten Entwurfs parameteroptimierter Regler für stabile,

schwingungsfähige und instabile Strecken mit dem CAE-Werkzeug SimTool

.

GMA-Kongress ’98 Mess- und

Automatisierungstechnik, Ludwigsburg, VDI-Berichte 1397, pp. 103-110.

Schaedel, H.M. (2003a).

Spreadsheet Control – Prozessidentifikation, Reglerentwurf und Regelkreissimulation mit MS

- Excel.

GMA-Kongress 2003, Automation und Information in Wirtschaft und Gesellschaft, Baden-Baden,

VDI-Berichte 1756, pp. 723-730, (2003).

Schaedel, H.M.(2003b)

.

Processidentification, Controller Tuning and Control Circuit Simulation using MS Excel.

Proc.