Elementary nonlinear decoupling control of composition in binary distillation columns

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UTTERWQRTH El N E M A

J. Proc. Cont. Vol. 5, No. 4, pp. 241-247, 1995 N

Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0959-1524/95 $10.00 + 0.00

N

0959-1524(95)00016-X

Elementary nonlinear decoupling control of composition in binary distillation columns Jens G. Balchen and Bjarne Sandrib University of Trondheim, NTH, Department of Engineering and Cybernetics, N-7034, Trondheim, Norway Elementary nonlinear decoupling (END) is a model based control algorithm intended to decouple and linearize a nonlinear multivariable process in order to achieve better control than can be obtained by conventional decentralized linear feedback control. The application of END to the composition control of a theoretical binary distillation column illustrates that the quality achievable is very high.

Keywords: Nonlinear control; decoupling; distillation

The control of composition in binary distillation columns has been the subject of research for many years and a large number of contributions with different control strategies have been presented. A comprehensive review of such contributions is given in Skogestad'. One control strategy which has so far not been tested for the composition control of binary distillation columns is nonlinear decoupling. This control strategy attempts to linearize and decouple (diagonalize) the process in order that high performance linear control can be applied 2~. Since a distillation column controlled by the reflux of liquid top product and boil-up vapour flow of bottom product is highly nonlinear and has a large number of state variables the theory of elementary nonlinear decoupling (END) introduced by Balchen seems very appropriate.

where x,u and v are state, control and disturbance vectors respectively and f is a set of nonlinear functions. In a distillation column the dimension (n) of the state vector will usually be very much larger than the dimension (r) of the control vector. This has severe consequences in the development of most model based control strategies. As will be shown in the following, the consequences in E N D are less pronounced. A common phenomenon in nonlinear modelling of dynamic processes is that the system is nonlinear in the states and disturbances but linear in the control variable such that: = f(x,v) + B(x)u

(2)

In E N D one defines a property transformation z = d(x) which is commonly replaced by a simple linear transformation:

Elementary nonlinear decoupling (END)

z : Dx

End is a version of the original nonlinear decoupling algorithm that solves the invertibility problem by designing the property space which is the object of the linearization and decoupling. It has some similarity with input-output linearization z in which the invertibility problem is solved by differentiating the output variables a number of times. In Balchen 5 it is claimed that E N D is the most realistic solution in practical systems. A nonlinear dynamic process is described by:

(3)

in which the property vector z may be interepreted as the properties one wants to control. E N D intends to determine the control action (u) which will drive the dynamic system in such a way that i follows a desired trajectory zd. From Equations (2) and (3) it follows that: = Dx = Df(x,v) + DB(x)u

(4)

Equation (4) can be solved with respect to u yielding:

= f (x,u,v)

(1)

241

Elementary nonlinear decoupling controk J.G. Balchen and B. Sandrib

242

u = (DB('))'(id- Df(x,v))

(5)

in which i has been replaced by zd. Equation (5) will only have a solution as long as the matrix DB(.) has an inverse. This puts restrictions on both D and B. The first restriction is that DB(.) must be a square matrix, implying that dim z = dim u. If dim u > dim z one has an optimization problem which will not be dealt with here. Since in general B(.) is given for a certain process, one has to design the D matrix such that DB(.) becomes non-singular and the resulting system has acceptable dynamic properties. Applying Equations (5) and (2) yields:

= ( I - B(DB)-'D)f(x,v) + B(DB)-'z a

(6)

which when linearized becomes

= ( I - B(DB)4D)(Ax + Cv) + B(DB)-'zd

(7)

assuming for simplicity that B(x) = B = constant. Multiplying Equation (6) by D gives: i =Dx. =ia

(8)

which shows that the system has been decoupled and linearized and is replaced by r integrators between zdand z. ia can be regarded as the new control input and z as the new property output. The dynamics of the system is determined by the differential equation (6) and approximately by Equation (7) with eigenvalues of the matrix:

(I -B(DB)-'D)A

(9)

characterizing the system stability. From Equation (7) the main system transfer matrix is given by: x(s) = H(s)ia(s)

(10)

where:

l

H(s) = ( s I - ( I - B(DB) 'D)A)-'B(DB) '

(11)

This transfer matrix converted into the frequency response matrix HOw) can be characterized by means of its maximal and minimal singular values &nOto) and ~(joJ). This gives a method for designing dynamic behaviour by choosing different D matrices. The contents of Equations (5) and (3) can be expressed in a block diagram as shown in Figure 1. The blocks below the dashed line reflect the mathematical model of the process and, as can be seen, the full state (x) and the disturbance (v) are assumed to be available. In order to achieve this, a state estimator is employed which may be, for example, an augmented Kalman filter based on the measurements (y). Note that the quantity ~ acting upon the nonlinear function represents the measurable part of the process disturbance (v). In this way feed forward is realized in a very direct and correct manner. The design of the property transformation D has the goal of making the system invertible with acceptable dynamic properties. One could think of another transformation z° = D°x which expresses the desirable but nonrealizable output vector. The reason for the nonrealizability is that the matrix D°B most often is singular. When introducing the difference Az = z - z° and the frequency response matrix AH(joJ) defined by: (12) one can find the transformation matrix D which minimizes some scalar measure of the matrix AH(jo~). Such a measure is the /-/=-norm given by - max~&~(jw). In the following, design of D is carried out in a rather simple manner by choosing a matrix which has only a few elements different from those of D O such that the dynamic behaviour of the system is acceptable. Figure 2 shows the proposed feedback control around the system where the END algorithm has been applied. There is an inner loop (I) containing a control matrix G and with a reference vector z0 and output z. This loop will have a nearly perfect response with very high bandwidth. The outer feedback loop (II) containing a control matrix G°, a reference vector z~ and an output z°will be slightly degraded relative to loop (I) particularly with respect to bandwidth, but with a performance which is in many cases greatly improved compared to a system containing only loop (II). Zo

. ~ ~ I - - q ~ I END ) _ _ ~ A l g o r i t h m

Figure 1 Nonlinear decoupling structure

Figure 2 Feedback control loops

-"

Elementary nonlinear decoupling control: J.G. Balchen and B. Sandrib

A dynamic model of a binary distillation column

2.7

&

2.6

A number of state space models for distillation columns are available in the literature with varying degrees of details in the models. Skogestad' presents arguments for different kinds of simplifications which may be made without losing too much of the important dynamics of the column. A common simplification is to neglect the vapour hold up and the energy hold up in the column. The important dynamic effects left in the model are liquid dynamics and composition dynamics. The result is a dynamic model with a reasonable complexity with 2N + 4 state variables, where N is the number of trays in the column. This type of model presented in the Appendix will be assumed in the following paragraphs even though other models could just as well have been used.

243

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Illustration of E N D control of a distillation column

-4

A simulation study has been performed using a distillation column as described in the Appendix. The total system employing an augmented Kalman filter as

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shown in Figure 1 and with feedbacks as shown in Figure 2 is studied. Figure 3 shows 6"n(jto) and _o-H0w) for two choices of D as given in the Appendix of which the latter is implemented in the system. The performance of the Kalman filter is illustrated first without any nonlinear decoupling and feedback. Figure 4 shows estimator responses to step changes in relative volatility (a) in the process. In Figure 5 nonlinear decoupling is employed and responses in z, z" and u are shown following step changes in t 0 (Figure 5a) and v (Figure 5b). In Figure 6 diagonal, proportional control is employed in loop I. z and the responses of the 'realizable' non-linear decoupling are illustrated following steps in z °. In Figure 7 diagonal PI-control is employed in loop II and responses following step changes in z°o (Figure 7a) and disturbances v (Figure 7b) are shown. Figure 8 shows responses of an ordinary well-tuned PI-two point controller (top concentration - reflux, bottom concentration - boil up) with weak nondynamic feed-forward following a step change in feed

flow (vl).

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Maximaland minimal singularvalues from ~ to x_. (Lower curve implemented) Figure 3

This investigation shows that the END principle yields superior control quality compared to conventional approaches both with regard to the suppression of disturbances and following set point changes. Further studies of the influence of different kinds of model errors are suggested.

Elementary nonlinear decoupling control: J.G. Balchen and B. Sandrib

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Elementary nonlinear decoupling control: J.G. Balchen and B. Sandrib

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Acknowledgement

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The simulation study reported is in part (BS) supported by STATOIL and (JGB) by SINTEF. (a)

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E l e m e n t a r y n o n l i n e a r d e c o u p l i n g c o n t r o l : J.G. B a l c h e n a n d B. S a n d r i b

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= KB(L ~ - F B - B)

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(15)

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(16)

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(17)

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(18)

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(19)

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(20)

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(22)

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(23) (24)

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(25)

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(26)

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Figure 8 Conventional control system responses

References l Skogestad, S in 'Proc. IFAC symp. DYCORD '92' (Ed J.G. Balchen), IFAC Symposia Series no. 2, Pergamon Press, Oxford, 1992 2 Isidori, A. 'Nonlinear control systems,' Springer-Verlag, Berlin, 1989 3 Balchen, J.G., Lie, B. and I. Solberg in '10th IFAC World Congress on Automatic Control,' Munich, 1987 also Model. Identif Contr. 1987, 9(3) 137 4 Balchen, J.G. Model. Identif Contr. 1991, 12(2) 81 5 Balchen, J.G. Model. Identif Contr. 1993, 14(4) 219 6 Di Ruscio, D. 'Dynamic mathematical models for a binary distillation process,' Report 87-64-U, Dept. of Engineering Cybernetics, The Norwegian Institute of Technology, Trondheim. (in Norwegian)

Appendix The model to be used here is developed in Di R u s c i o 6 and uses s t a n d a r d self-explanatory n o t a t i o n as follows:

+ RxD-MNXN

(27)

Accumulator: D = KD( V N - (R + D)

(28)

M ~ = MOD + D/KD

(29)

MoxD = Vsy N - (R + D ) x D - M D X D

(30)

A n equilibrium relationship between v a p o u r a n d liquid concentration is assumed by:

Yi--

tXX i

1 + (or - 1 ) x i

(31)

where a is relative volatility. The model consists o f 2 N + 4 differential equations and 2 N + 3 algebraic equations. Parameters which have to be specified in the model are:

Elementary nonlinear decoupling control: J.G. Balchen and B. Sandrib N u m b e r of trays, N = 10 C o l u m n feed tray, f = 5 Relative volatility, o~ = 2.5 T r a y constant, Ki = 1 (s ') Liquid mass in reboiler, MoB = 7.2 (mol) Liquid mass on tray, M0~ = 1.8 (mol) Liquid mass in accumulator, Moo = 1.1 (mol) T h e m a n i p u l a t e d variables are: Reflux R = 1.025 (mol s -') (Ul) V a p o u r flow from reboiler, VB = 1,2 (mol s ') (u2) Disturbances: Feed flow, F = 0.3 (moi s -~) (vt) Feed concentration, xv = 0.5 (mol mol ') (vz) Z 1 ---~ XB

247

Z 2 -~ X D + d2.23 D

T h e B(x) and D matrices are as follows:

2 b2.2 0 . . . 0

0

where the nominal values are: b22.1 = 0 . 0 5 ,

D =

b23.1 = -

0

1.0,

hi. 2 = -

o o o 0...0 d2.23

where

dz.23 = - 0 . 0 1 : i m p l e m e n t e d dzz3 = + 0.01 : tested in Figure 3.

1,

b22

= - 0.045