On-line controller tuning for unstable systems

Computers chem. EngngVol. 20, No. 3, pp. 301-305, 1996 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0098-1354/96 $15.D0 + 0.00

Pergamon 0098-1354(95)00021-6

ON-LINE

CONTROLLER TUNING SYSTEMS

FOR UNSTABLE

MAHENDRA KAVDIA a n d M. CHIDAMBARAMt

Department of Chemical Engineering, Indian Institute of Technology, Madras 600 036, India (Received 18 April 1994;final revision received 15 February 1995; received for publication 15 February 1995)

Abstraet--A simple method for tuning controllers on-line for unstable processes is proposed. The step response data of the closed loop system with a proportional controller gain is used to identify model parameters of an unstable first order plus time delay transfer function. Reported tuning formulae for PI controllers for unstable systems are applied to obtain the controller parameters. The response of the closed loop system with the controller settings of the identified model is compared with that of the actual system's controller parameters.

INTRODUCTION

PROPOSED METHOD

A variety of on-line tuning of controllers is available. These include loop tuning or the continuous cycling method of Ziegler and Nichols (1942) and the process reaction curve methods of Cohen and Coon (1953) and Ziegler and Nichols (1942). Ekykoff (1974) and Gustavsson et al. (1977) have proposed more elegant methods for closed loop identification to determine a suitable dynamic model. However, these methods are complicated and an accurate process model is not necessary since the objective is controller tuning for a desired closed loop performance rather than for model development. Yuwana and Seborg (1982) have proposed a simple method for identifying a stable first order plus time delay model from the closed loop response for a step change in the set point. By using the standard Ziegler-Nichols tuning formula, PI controllers are designed. In the present work, the method proposed by Yuwana and Seborg (1982) is extended to an open loop unstable process. In this method a step response of the closed loop system with a proportional controller is used to obtain the parameters of the open loop first order time delay unstable transfer function. Using the tuning formulae (Venkatashankar and Chidambaram, 1994; De Paor and O'Malley, 1989) for such unstable transfer function models, the PI controller parameters are calculated. The response of the actual system with the settings is compared with that of the controller settings of the actual process.

For controller design purposes, many of the unstable processes are adequately described by a first order plus delay transfer function: Gp = k exp( - ds)/(rs - 1).

(1)

The closed loop transfer function is given by C(s)/R (s) = g exp( - ds)/[rs - 1 + g exp( - ds)] (2)

where K = kck, R is the set point and C is the process output with a first order Pade approximation, equation (2) becomes g exp( - ds) [1 + 0.5ds] C(s)/R(s)

-

{(rs - 1) (1 + 0.5ds) + K(1 - 0.5ds)}

(3)

g ' exp( - ds) (1 + 0.5ds)

-

[r~s 2+ 2ro~s + 1]

(4)

where K ' = K / ( K - 1)

(5)

rc = [ 0 . S d r / ( g - 1)]0.5

(6)

~= [ r - 0 . 5 d ( K + 1)]/[2dr(K- 1)1°5.

(7)

F o r 0 < ~ < 1, the closed loop system is underdamped. The actual process is put under feedback with a proportional controller and the step response in the set point is obtained. A typical response is shown in Fig. 1. If this response is to be fitted to that of a second order underdamped system, then the following relations are obtained:

t To whom all correspondence should be addressed.

k=c®/[kc(c~ - A ) ]

(8)

r = AtPl P2/:r

(9)

d = 2AtP,/(P2:r) 301

(10)

302

MAHENDRA KAVDIA and M. CHIDAMBARAM

CpI -- ~

Cp2

CMI

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Time Fig. 1. Typical response to a step change in set point and the associated parameters required for identification.

where P, = [(1 - ~2) ( K - 1)] 0.5

with k = l , d = 0 . 5 and r = l . Stability analysis (DePaor and O'Malley, 1989) shows that for stabilization of an unstable first order plus time delay system by the conventional P or PI controllers requires (d/r) < 1. If an approximate knowledge of process gain (k) is known, then the value of kc.mi n c a n be selected (Luyben, 1990) from kc. min> 1 / k . Starting with this value of kc, the closed loop step response similar to that shown in Fig. 1 can be obtained. For the present problem, using a proportional controller with a gain (k¢) of 2 we get the closed loop response for unit step change in the set point. The response is shown in Fig. 1. From Fig. 1 we noted down the values of the parameters cp~, Cp2, cml, At and c= as 3.31, 2.53, 1.17, 1.7 and 2.0 respectively. Using equations (13) and (14) the values of ~ and re are calculated as 0.14375 and 0.5355. Hence using equations (8) to (15) we calculate the values of r, d and k as 1.007, 0.5694 and 1.0 respectively. The actual parameter values of the process are 1, 0.5 and 1. Using the equations given by Venkatashankar and Chidambaram (1994), the controller settings for the fitted transfer function are calculated as

1)0.5+ [ ( K - 1)¢2+ ( K + 1)] 0.5 (11)

kc. min= 1.078,

where c®, ~ and re are calculated from the closed loop response of Fig. 1. These parameters are related to Cp~,Cp2 and Cm~of Fig. 1 ]refer to Yuwana and Seborg (1982) for the method]:

kc.oe~= 1.453,

t'2 = ¢ ( K -

C~ = (Cp2Cpl -- C21)/(Cpl "1- CO2 -- 2Cml)

= - l n ( v ) / [ z c 2 + {ln(v)}z]°5

re = (1 - ¢2)°SAtlzr

U = (C~ -- Cml)/(Cpl -- C~).

k~, rain= 1.055,

(13) (14)

k¢,de~= 1.510,

(15)

Thus from the underdamped response of the closed loop system we can obtain the first order time constant, delay and gain of the open loop unstable transfer function. Recently Venkatashankar and Chidambaram (1994) have proposed tuning formulae for such systems. De Paor and O'Malley (1989) have given a graphical method for the calculation of PI controller settings of an unstable first order plus time delay system. Let us apply the proposed method for two case studies of linear systems and one nonlinear bioreactor control problem. SIMULATION RESULTS

Let us consider a process whose transfer function is given by k exp( - d s ) / ( r s - 1)

(16)

=

1.957,

rl = 10.94,

(17)

whereas the controller settings for the actual process parameters are calculated as

(12)

where

k.....

k..... = 2.15, rl = 12.50.

(18)

The closed loop responses of the actual process with these two sets of controller parameters are evaluated for a unit step change in the set point and the responses are shown in Fig. 2. Figure 2 shows that the controller parameters obtained from the identified model work well on the original process. The effect of measurement noise on the identified model parameters and hence on the controller parameters are evaluated by adding a measurement noise of zero mean and 02 variance to the actual process output. The noise corrupted signal is used for the P control action and for the identification method. Table 1 shows the calculated model parameters and the PI controller parameters for various levels of noise variance. There is no significant change in the identified model parameters and hence in the PI controller parameters. Since the feed-back control action will suppress the effect of measurement noise, the proposed method of identification for controller design purposes is not affected by the measurement noise.

On-line controller tuning for unstable systems The auto tune method (Luyben, 1990) is another closed loop method for identifying the transfer function of the system. Let us briefly discuss the method for identifying a delay plus first order model. In this method the controller is replaced by a relay (of magnitude h) and the closed loop system without any forcing in the set point generates a sustained oscillation in output. The controller ultimate gain (k~,,) and ultimate frequency (~o,) are calculated from the sustained oscillatory response as:

K~,~ = 4h/zra;

~Ou= 2zr/p~

(19)

where a and Pu are the amplitude and period of oscillation respectively. From the initial response, the value of the delay (d) can be found. Using the values of K~.o and too in the following phase angle criterion and magnitude criterion for an unstable delay plus first order system we get the values of system time constant and steady state gain: -

zr + t a n - l(r~oo) - dog~= - ~r

(20)

kk~,J(rzto2 + l ) ° 5 = l .

(21)

By carrying out the auto tune test (with h = 0.1) for the original system with k = 1, d = 0 . 5 and r = 1 we obtain a sustained oscillation in the output. From the initial response, a delay of d = 0.5 is noted. From the oscillatory response we get the following values p , = 3.0 and a = 0,062. Using these values in equation (19) we get k~.u = 2.0536 and to, = 2.09 and from equations (20) and (21) we get r = 0 . 8 2 4 and k = 0.97. Using these identified values in the PI controller design equation (Venkatashankar and

4

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Chidambaram, 1994) we get kc, des= 1.472 and rL = 8.1. The corresponding closed loop servo response is compared with that of the settings based on original system in Fig. 2. The response by the proposed modified Yuwana and Seborg method is closer to that of the controller settings based on the original system. Let us now consider a process whose transfer function is given by

k exp( - ds)/(r,s + 1) (r2s - 1) with k = 1.0, rL = 0.5, rz = 2.0 and d = 0.5. Stability analysis requires r : > r ~ and ( d / r 2 ) < l . With kc = 2.0 the closed loop response for a unit step change in the set point is evaluated and a response similar to Fig. 1 is obtained. From the response the values of CoL, Cp2 , C m l , At and c~ are noted down as 2.98, 2.24, 1.53, 3.7 and 2.0 respectively. Hence the values of ~=0.2236, re= 1.1479 are obtained from equations (13) and (14). Using these values in equations (8) to (15) we get the gain, time constant and the time delay of the fitted transfer function: k exp( - ds)/(rs - 1) where k = l . 0 , r = 2 . 2 6 2 , d = 1.165. The controller parameters are calculated as

kc, rain =

1.06,

kc.des = 1.49,

k ..... = 2.098, rl = 27.425.

Since there is no simple formula available for the controller parameters of the original system, we have to obtain the exact values of controller parameters from the phase angle criterion for the original system with a proportional integral controller: -

tan-L(rlw~) - [er - t a n - t(r2 toe)] - 0.5toe- tan-L(0.2) = - a t .

3

In deriving equation (22) we have used the value of rl as 5/tOc (Venkatashankar and Chidambaram, 1994). Numerical solution of equation (22) gives two values for toc as w d = 0 . 2 2 5 and toc2=0.885. The controller gain can be obtained from

2

k~ = [rho~ + 1]°5/k. l

0

(22)

\

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Time Fig. 2. Closed loop response of example 1 using the controller settings of the original and identified models. - k¢.d~= 1.51, rt= 12,50 designed on original system; - - k~.d~= 1.4528, r~= 10.94 by present method; - . - k~,d¢~= 1.472, r~= 8.1 by auto tune method.

(23)

The values of kc, ~m, k . . . . . and kc, des are calculated as 1.08, 2.18 and 1.531 respectively. The value of rt is obtained (using toot) as 22.32. As can be seen from the values of k¢.aes and r~ of the original process and the fitted first order plus delay transfer function we find a good agreement. The closed loop of the process with these two sets of the controller parameters are simulated for a unit step change in the set point. The responses are shown in Fig. 3. A good response is obtained using the controller settings of

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MAHENDRA KAVDIA and M. CHIDAMBARAM

Table 1. Effect of measurement noise on the identified model parameters for case study l S.No.

STD

cpl

Cml

Cp2

At

k

r

d

kcm

rl

1 2 3 4 5 6

0.00 0.025 0.05 0.10 0.15 0.20

3.31 3.31 3.31 3.32 3.33 3.35

1.17 1.17 1.17 1.18 1.19 1.20

2.53 2.51 2.51 2.51 2.50 2.48

1.7 1.6 1.6 1.6 1.6 1.6

1.00 1.00 1.00 1.00 0.99 0.99

1.01 0.95 0.95 0.95 0.95 0.95

0.569 0.535 0.534 0.533 0.530 0.527

1.453 1.454 1.454 1.456 1.458 1.461

10.94 10.35 10.38 10.45 10.52 10.64

STD--Standard deviation, noise with zero mean and o2 variance.

the identified u n s t a b l e first o r d e r plus time delay model.

T h e steady-state solution of e q u a t i o n s (24) to (26) gives the following t h r e e multiple steady-states:

APPLICATION TO AN UNSTABLE NONLINEAR

[X, S]~ = [0, 4]

(wash-out condition)

[X, S ] 2 = [0.9951, 1.5122]

(unstable)

IX, S ] 3 = [1.5301, 0.1746]

(stable).

BIOREACTOR

T h e p r o p o s e d m e t h o d of identification of an unstable first o r d e r plus time delay system is applied to a n o n l i n e a r c o n t i n u o u s b i o r e a c t o r which exhibits o u t p u t multiplicity. T h e m o d e l e q u a t i o n s are given by (Agrawal a n d Lim, 1986):

dX/dt = (l~- D)X dS/dt = ( S f - S)D - (/zX/),)

T h e dilution rate (D) is c o n s i d e r e d as a m a n i p u l a t e d variable in o r d e r to control the cell mass c o n c e n t r a tion ( X ) at the u n s t a b l e steady-state at X = 0 . 9 9 5 1 . A delay of 1 h is considered in the m e a s u r e m e n t of X. T h e n o n l i n e a r e q u a t i o n s (24) to (26) are solved along with a p r o p o r t i o n a l controller using k~= - 0.704 for a set p o i n t change in X from 0.9951 to 1.294. F r o m the closed loop r e s p o n s e of the deviation variable of X , the values of the p a r a m e t e r s Cp~, co2, Cm~ and At are n o t e d respectively as 0.5231, 0.4278, 0.3035, 2.6. T h e p a r a m e t e r s of the identified

(24) (25)

where

/z=ItmS/(KmWS+gis2).

(26)

T h e m o d e l p a r a m e t e r s are given by (Agrawal a n d Lim, 1986): 3,= 0.4 % g / g ,

Sf=4 %g/g,

Km= 0.12 % g / g ,

(27)

/~m = 0.53 h -t ,

K1 = 0.4545 % g / g .

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Time

Fig. 3. Closed loop response of example 2 using the controller settings of the original and identified model, kc. des= 1.531, rl = 22.32; - - - kc.dcs= 1.49, rl = 27.425.

On-line controller tuning for unstable systems 1.6

305

--

1.5 1.4 ~

.3

M 1.2 1.1

1.0

B

0.9 0

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l0

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Time (h) Fig. 4. Closed loop response of the nonlinear bioreactor using the controller settings of the identified unstable first order plus time delay model, kc = -0.4788, rl = 8.45.

unstable first order plus time delay system are calculated by using equations (8) to (15). The identified parameter values are k = - 6 . 4 4 4 , r = 4.15, d = 1.13. Using the graphical method given by D e Paor and O ' M a l l e y (1989), the controller parameters are calculated as k~.des= - 0 . 4 7 8 8 and rx = 8.45. The closed loop response of the nonlinear model equations (24) to (26) with the designed PI controller are evaluated for a step change in X from 0.9951 to 1.294. A good response is obtained as shown in Fig. 4.

CONCLUSION The proposed method for identifying an unstable first order plus time delay model and using the reported PI controller formula provides a good closed loop response. This is demonstrated with two numerical case studies involving linear systems and one nonlinear bioreactor control problem.

CAC£ 20-3-F

REFERENCES

Agrawal P. and H. C. Lim, Analysis of various control schemes for continuous bioreactors. Adv. Biochem. Biotech. 30, 61-90 (1986). Cohen G. H. and G. A. Coon, Theoretical investigation of retarded control. Trans. A S M E 75, 827-834 (1953). De Paor A. M. and M. O'Malley, Controllers of Ziegler Nichols type for unstable processes. Int. J. Contr. 49, 1273-1284 (1989). Eykhoff P., System Identification. Wiley, New York (1974). Gustavsson I., L. Ljung and T, Soderstrom, Survey paper: Identification of process in closed loop--identifiability and accuracy aspects. Automatica 13, 59-75 (1977). Luyben W. L., Process Modelling, Simulation and Control for Chemical Engineers, pp. 391-397. McGraw Hill, New York (1990). Venkatashankar V. and M. Chidambaram, Design of P and PI controllers for unstable first-order plus time delay systems. Int. J. Contr. 60, 137-144 (1994). Yuwana M. and D. E. Seborg, A new method for on-line controller tuning. AIChE J. 28, 434-440 (1982). Ziegler J. G. and N. B. Nichols, Optimum settings for automatic controllers. Trans. A S M E 64,759-768 (1942).