Design of an extended Smith controller with gain adaptation

ControlEng. Practice,Vol. 3, No. 10, pp. 1467-1470, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0967-0661/95 $9.50 + 0.00

Pergamon 0967-0661 (95)00151-4

DESIGN OF AN EXTENDED SMITH CONTROLLER WITH GAIN ADAPTATION U. Ebach* and A. Griiser** *Fachhochschule Koblenz, Alte Poststrasse 37, D-57581 Katzwinkel, Germany **Universitdt Bremen, lnstirut fiir Automatisierungstechnik, Lohmannstr. 61, D-56567 Neuwied, Germany

(Received March 1995; in final form June 1995)

Abstract: For the design of the benchmark test controller a combination of well-known techniques and an extension of the standard Smith. predictor controller is used. The selection of the design methods was governed by a practical point of view. All methods used as well as the optimisation procedure, should be usable with the tuning tools available to the service and commissioning staff in a "normal" DCS system. Black-box approaches and the design of a self-adjusting controller have been avoided. Keywords: non-linear decoupling, extended Smith predictor, gain estimation

1.

INTRODUCTION

with n equal time constants and may be expanded by an additional dead time (']7T model ).

The design of the controller is divided into five steps: • estimation of the dynamic system by analysing the step response, • eliminating the nonlinearity due to the measurement by an inverse nonlinearity, • decoupling of the remaining linear 2 x 2 system, • design of a Smith predictor controller for the decoupled system, • extensions of the controller to manage the retention variation.

2.

With the known machine parameters, length and speed, the transport delay from the valves to the gauge can be computed. In the model the delay time used is TT = 50.9 s. Because of the recirculation flows PTnTT models are not sufficiently accurate. It is necessary to use a more compheated model structure, also based on the time-percent characteristic. Because of the saturation of the ash content in the white water system the model parameters, especially the amplification, vary. It is at least necessary to work with different controller amplifications for alternating steps of the aim values to take the saturation effect into consideration.

SYSTEM IDENTIFICATION

To get a mathematical description of the paper machine, linear models with a transport delay are approximated (Schwarze, 1962). The method of the time-percent-characteristic is a quick and easy usable tool for the approximation. The method leads to 171",- models of the type

G(s) =

K

3.

NONLINEARITY AND DECOUPLING

Due to the definition of the control variables, dry weight (dw) and ash content (ac), the system has to be described by a nonlinear 2 x 2 system with strong couplings. A mass balance for fibre and filler leads

(1)

(l+s.T) n 1467

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~hiok-

~.:..

1100

12OO

1300

MOO

--r::']._~-~

_r~

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k~O0

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Fig, 3 • P-canonical structure and series decoupling of the linearized model The dynamics of the dry weight path is described by Ifi00

ram/me

Gll=

Fig, 1 • Comparison between model and plant dynamics

0.9913 + 0.0146 ./e_50.9 s 1+ 24.84s (1+ 17.57s)10 J

(3)

and the ash content path by 0.751 0.3872 "~ 509. G22=~l+3--~s)3 + (l+1-~.Ss)2j ~(4) dw

k ~W,,Im

NL"

All time constants are in seconds.

NL

To reduce the influence of G21, the thick stock to filler coupling~ a lead lag element R 21 is introduced. From G12 = 0 follows R12 = 0. R 11 and R22 may be set to 1. Fig, 2" Schematic diagram of the control loops 4. to a linear 2 x 2 system. The nonlinearity is caused only by the measurement. Introducing an inverse nonlinearity leads to a linear 2 x 2 model. The inverse nonlinearity may be calculated from the definition of the measurement fibre,,, = dw. 10 -~ M S " W . O _ ac . I O-2 )

(2) f i l l e r ~ . dw.lO . . ~ .M S . W

Q

with the machine parameters W - width of the paper Q - flow of the pulp.

ac. 10 -2

MS - machine speed

Dividing the original model into a linear model of the machine and a non-linear model for the measuring gauge, the inverse nonlinearity NL* computes the fibre and filler flow for given aim values of dry weight and ash content. Fig, 2 shows the control loops with the split model of the machine. The introduction of NL* leads to a linear behaviour of the system. Due to the ash in the thick stock flow there is still a coupling from thick stock to filler, represented by G21 in the linear 2x2 system. G12 may be ignored. The linearized machine is described by a P-canouical structure shown in Fig. 3.

TIlE STANDARD SMITH PREDICTOR CONTROLLER

For systems with dominating dead times a Smith predictor controller achieves better results than a standard controller. The predictor consists of the controller C and two additional feed back loops containing a dead-time-free model G* of the plant and the dead time TT of the system. The controller has to be designed for G*. It is necessary to identify G*(s) and T T with high accuracy. A large deviation between the real plant and the plant model in the controller may lead to an unstable closed loop. An additional dead time of 15 see (haft the scan time) has to be introduced to describe the scanning gauge. This dead time is a44ed to the transport delay. The Smith controller uses a corrected dead time of 65.9 see.

4.1 Combination o f open~closed loop control

To get the system to the operating point, open-loop control is used (SIMULINK causes problems if a start from a predefined operating point is demanded). The controller is also expanded by switching elements. Depending on the operating situation the controller is bumplessly switched between open/closed-loop control and the tracking mode, a situation quite similar to the real operation of a DCS system.

Design of an Extended Smith Controller 4.2 Controller design

1469

time, shown in Fig, 5. The estimation of the disturbance

The parameters of the controller C are determined from the Bode plots for a 60 ° phase margin. T N = o3-1 ,

z=G2-1.[y.(G~G2 - G ) + z.G2]

(6)

is exact if G 1 and G 2 are identical to the original plant G, and ffthe inversion of G 2 is possible.

Lt. But due to the dead time the inversion is impossible. However, the dead time in G 1 is small compared with the wavelength of the retention variation and can be ignored. As a simplification, and to avoid a transfer function G s with several zeros, a lead-lag feed-forward controller is used. Fig. 4 : Standard Smith predictor controller

I

K

tu

For C the same parameters are used as for the ordinary Smith predictor.

e

NL I ITa¢ I

appre~~_

wlgtul medd

'I

I

'

'i

eK I[

Fig` 6: Extended Smith predictor for disturbance feed-forward control

Fig. 5: Estimation of an additive disturbance

where [G(Jo3) I is -3 dB. For C in the dw- and acloops the controller parameters are calculated to 1+ 303s Cdw(s) = 0.656 - 30.3s

(5) 1 + 52.7s Cac( s ) = 1.og - 52.7s 4.3 Retention variation as an additive disturbance Because of the large transport delay standard controller designs do not produce satisfactory results. A controller with dead time compensation may be able to handle the ordinary control subjects, but gives poor results for disturbances. ~ r (1994) discusses an extension of the Smith predictor for disturbance feed-forward control. The paper deals with a measurable disturbance. But in case of the retention the disturbance cannot be measured, and must be estimated. To extract the disturbance from the measured signal, an observer is built, based on the models described above (Ebach, 1994). It is known that the retention affects only the wire section. With the machine parameters speed, width, length and flow, it is possible to describe the behaviour of the dryer sections by a simple dead

Fig. 7 shows that the extended Smith predictor leads to better results. It is important to take into account, that the retention varies. At t = 2000 see the retention is close to the nominal point. In this case the extended and standard Smith predictor create the same output.

5.

ADAPTATION OF THE CONTROLLER GAIN

The retention affects primarily the wire section and influences mainly the ash path. The time constants of the ash path may be assumed to be unaffected, but the amplification varies in the range 1:3. A standard Smith controller is not able to compensate for the variation in amplification. It is necessary to estimate the amplification of the ash path, and to tune the controller parameters according to the estimated value. Fig. 8 shows the block diagram. Because of additional disturbances and a deviation between plant and model, the observer has to be expanded by some heuristic error-detection circuits. To avoid division by zero the signal of the ac-loop is observed and the detector locked in case of failure. As above, a highly accurate model of the unaffected ac-path is needed. Setpoint steps in ash content cause peaks in the calculated signal. A peak filter limits the slope of the signal, and a median filter smoothes it.

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Fig. 7: Response to a dry weight step by standard and extended Smith-Predictors

Fig. 9:

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2000 3000 t / sec ->

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Computedamplification variation K=I/K a 6.

CONCLUSION

lt'~elwe

Fig, 8: Detection of ac-path amplification To get a simple controller structure, an additional tuning parameter, K, behind the Smith controller is introduced and K = 1/Ka is estimated directly, where K a is the amplification of the ash path. This leads to a constant amplification of the ash path, and the Smith controller becomes unaffected Fig, 9 shows the estimated amplification variation, K. The spikes occur mainly during the setpoint changes. They are caused by differences between plant and model behaviour. Filtering eliminates the spikes and disturbances of the original signal but lead to a phase displacement. Therefore it is useful not to enlarge the integration time of the median filter longer than the dead time of the system. The adaptation results in a almost constant amplification of the ash content loop in spite of the retention variation. In the case of controller design the open-loop amplification is constant. A new model approximation of the plant is necessary. Based on this model the controller parameters are determined for a phase margin of max. 85 °. The optimised controller is calculated to l+50s

Ca,= 0.54- ~

50s

(7)

In all cases, the sinusoidal retention variation in the range 0.3 - 0.8 causes design problems and the results exceed the time-domain tolerances demanded. It may be possible to improve the result by using a better adapted model structure, and perhaps with a better adapted observer structure. Taking the situation in a factory into account, the optimisation of more complicated models is very limited. Even the necessary tuning expenditure for the controller suggested here may exceed the possibilities. The most promising way seems to be the measurement and online control of the retention. Even if the measurement takes approximately 1 min, it is possible to reduce the retention variation and to ease the controller tuning.

REFERENCES E l ~ h , U. (1994). Entwurf eines nichtlinearen Mehrgrdflenreglers flit eine Papiermaschine. Diplomarbeit, FH-Rheinland-Pfalz, Abt. Koblenz. Grfiser, Axel (1994). Erweiterung des SmithPrddiktors bei Stdrgrdflenaufichaltung. atp 1/94 P.46-51 Schwarze, G. (1962), Bestimmung der regelungstechnischen Kennwerte aus der Obertragungsfunkt~on ohne Wendetangenten-Konstruktion. msr 5 P.447-449