Bilinear Techniques and Petrochemical Applications

Copyright

©

IFAC Automatic C0I1Irol in

Petrol eum , Petrochemical ;.lnd Desa linalion

Industries, Kuwait. 1986

BILINEAR TECHNIQUES AND PETROCHEMICAL APPLICATIONS· R. R. Mohler Department uf Electrical and Com puter Engineering, Oregon State U niversity, Condlis , OR 97331 , USA

Abstract. It is shown that bilinear models arise naturally for petrochemical and related process control. Such dynamics occur due to chemical process control. a massaction approximation. catalytic multipliers. convective and conductive heat transfer. distillation columns. mixing. and pump dynamics. In some cases. the system involves cascades of bilinear processes. Still. the subsystems are bilinear and may be so studied by recently developed techniques - some of which are just as useful for coupled bilinear sytems. BLS. These include optimal filtering for state estimation and parameter estimation. Walsh functions for similar applications and Volterra series for structural analysis and system realization. Keywords. Bilinear systems; modeling; control; heat transfer; petrochemistry; system identification; Volterra series; Walsh functions.

INTRODUCTION The purpose of this paper is to show the role of. and to present a base for. the application of bilinear system (BLS) methodologies in petrochemical and related processes. Here. BLS models are described by dx

dt

a

I ji

a

2 ji

Pi

...

L

k=1

Bk

"k

x x

+ Cu

(1)

.,.

aI ij a

~

'<

m

Ax +

xI

2

2 ij

Pj

k 1, •.• ,n+l, k = 1.2. are reaction Here. a ij • i,j constants; Pi'P j characterize two intermediate product s such that PI is the initial and Pn+ 1 the

where x is an n-dimensional state vector; u is an m-control vector. and A"B k • k=I ..... m. and Care matrices of appropriate dimension.

final desired product. This means that the total chain includes n+1 products under the control of the blend u of catalysts xl and x 2 which form the bifunctional catalyst. This mix or blend u controls the process. More exactly. u denotes the active volume fraction of type-x 2 catalytic material with

BLS are summarized by Mohler (1973.1985a.1985b). Numerous methods of analysis have been developed for BLS in recent years. and some of the more appropriate ones are summarized here. with an emphasis on petrochemical applications. But first. consider several basic BLS models which are relevant to the problem at hand.

o <:

u <:

I

CHEMICAL PROCESSES Now. the effective rate constants are BLS are natural models for chemical reactions as a close approximation to the mass-action principle . Furthermore. chemical catalysts become control multipliers for reactions in such systems as catalytic tubular reactors (Gunn and Thomas. 1965). Such reactors. for example. use bifunctional catalysts to reform petroleum napthas. The isomerization of paraffine and the dehydrogenation of cyclohexane to benzene also use a bifunctional catalyst.

I

2

2

• u a j i • (1-u) a ij • and (1-u) a j i . If all reaction types are first order and the reaction steps are carried out in an isothermal tubular reactor. then the concentrations of products Pi.P j • in mole fractions is described by dx at =

Consider one step at a time in the reaction as described by the following:

A x + u B x + c u + d

(2)

where x is an n-state vector; u is a scalar control. and A.B.c.d are of appropriate dimension. Here a I = a 2 = 0 with ii ii n+1

L ~i(t)

i=1

I The research funded by US National Science Grant ECS-8215724.

33

=

0 •

R. R. Mohler

34

The equations governing the flow system are

and thus

X, Constant.

Hence, it is seen that only n concentrations are independent which admits (2) with n state variables. The matrix is related to the rate constants by A = [a ij ], with

(

.(a~i-a~+l'i)' n+l

-l: k=l

2 a

ik

j"l i

n+l,i'

uI,u2 ( 0,

the concentration in pool I

V2

=

the volume of pool 2 the concentration in pool 2

x2 i

> 0;

the volume of pool I

VI xl

2 -a

with the constraints x l ,x2 ) 0; c2 where

c2 = the growth rate in pool 2 the transfer rate between pool I and pool 2

Similarly,

u2 = the transfer rate out of pool I Rewritten in standard fashion, the systems equation become (5)

1,

where

-I)

2 X a n + l ,i

If the desired product Pn + l results from an ir1 2 reversible reaction i.e., a n + ,i = a n + l ,i = 0, l the affine BLS (2) reduces to the homogeneous BLS dx = (A + u B) x • dt

0)

(3)

Note that BI and B2 are dyads.

Hofer (1973) studies the optimal control of this BLS to provide the highest yield at the terminal time, i.e., maximize x n +l(T). As so common for BLS, a singular solution must be considered along with the bang-bang trajectories with 0 ( u ( 1. Espana and Landau (1978) derive similar BLS models for the distillation columns. Details of the model are not presented in the present paper, but chemical concentrations are state variables, and flow rates, heat, and feed flow concentrations are cont rols. FEED FLOW SYSTEMS Conservation of matter results in bilinear processes for feed flow such as appear in petrochemical processing plants. For example, consider Fig. I.

The dyadic struc-

ture of certain BLS make the stabilizing control simpler by such methods as given by Gutman (1981). Even the pumps in the systems may present BLS because of the inertial dependence on the flow rate integration. REACTOR HEAT EXCHANGERS Heat-transfer processes are prevalent in petrochemical plants, and here an introduction is presented to such processes which are BLS. In heat-exchange dynamics, BLS arise if it is assumed that conduction is manipulated or that the convective coefficient is changed such as by altering coolant mass flow rate. For example, consider an energy balance on a perfectly insulated cylinder with generated heat Q (such as a reactor core element) of average temperature Tl and heat capacity cl with negligible axial conduc-

Pool 2

Pool I

tion. Coolant flows through the cylinder, and has average temperature T2 , mass heat capacity c2' specific heat c p ' and mass flow rate w.

First,

assume that the weighted average coolant temperature is given by T 2

=

Ti + 6T o I + 6

where 6 is a constant, and Ti and To are inlet and outlet temperatures, respectively. Then, the energy balance (Mohler, 1973) yields Fig. I.

Feed flow example.

35

Bilillear Techlliques alld Petrochelllical Applicatiolls

~=B"~ dt + CV

Now. the BLS Volterra series has the following sequence of nesting exponentials:

•

WA

where x T = [T l T2 ]. vT

"oh,]

[~Ol"'

B

c /c h 2

-[

C

[Q W].

t

-c /c 2

-1 cl

o c (1 + S)T (Sc ) p i 2

0

-1

j

(8)

L

i=1 (11)

~

Ch and

are nearly constant and design depend-

ent. Although numerous assumptions are made here. more general finite-difference models result in lumped BLS. and spacially-distributed derivations yield bilinear partial differential equations. Here. ch = Ch + c p (I+S)/S • Heating. ventilating. and air conditioning systems. including solar panels. storage tanks. and heat pumps may be similarly modeled by BLS. Such analyses result in significant energy savings over that of the traditional linear optimal control approach.

where C = [cl ••• cm]' x(O) = xo. and Ti ) Ti-l ) 0; i

= 1,2, ••• , ki = 1,2, ••• ,m.

The Volterra series (11) is finite if and only if S = {B.ad~ B}AA is nilpotent where

BLS STRUCTURE AND VOLTERRA SERIES

k

Within each BLS itself there evolves a certain hierarchical structure which is associated with a "canonical" decomposition such as convenient for its Volterra representation (assuming its existence. of course). In this manner. BLS (1). with ouput y = Dx. may be generated by N Y = D

L

" D

x.

1

i=1

L

i=1

(9)

x.

1

ad (.) = [A.[A •••• [A •• ] ••• ]]. and k = 0.1 •••• ; A [A.B] = AB - BA; and {D.E}AA denotes the smallest associative algebra that contains matrices D and E. If [T [T.M]] = O. then [T.M] is nilpotent. Consequently. it is seen that if (1) can be transformed to a "canonical" form with A upper triangular and B strictly upper triangular. then (11) has a finite Volterra series. It is convenient to classify BLS into two subclasses:

where xl

Axl + Cu m

x

2

Ax

2

+

L

k=l

B u x + Cu k k 1

.

m x9.

Ax9. +

L

k=1

Bk uk x9._1 + Cu

1.

weakly BLS with finite Volterra series and solution by decomposition into interconnected linear subsystems. and

2.

strongly BLS with infinite Volterra series or no such representation.

(10)

.

with x(O) = 0 vector for convenience. It is readily seen (Mohler. 1973 and Rao. 1975) that xi' i = 1 ••••• corresponds to the state terms in the Volterra series. and the corresponding kernels are generated as a "nesting" (Rao and Mohler. 1975) of the linear-system impulse responses according to (11) below. The Volterra series for BLS may sometimes be approximated by a finite number of terms (10). and in the case of so-called weakly BLS. may be exactly represented by a finite number. or decomposed into a finite number of linear systems with outputs multipled together to form inputs to successive linear systems according to (10). Consequently. a physical system approximated (or given) by a finite hierarchical structure associated with (10) is conveniently analyzed by linear system theory.

It is seen that factorable Volterra systems may be realized by BLS or by parallel linear systems with outputs multiplied together. Experimental synthesis of the Volterra kernels by correlations of outputs with binary and ternary sequences of a psuedo-random chain is developed by Baheti. Mohler. and Spang (1980). WALSH FUNCTION REALIZATION OF BLS Walsh functions. as a consequence of their convenient computational properties. are very amenable to BLS approximate solutions and for evaluation of parameters. This result is not too surprising since Walsh functions are piece-wise constant. and BLS with such inputs are piece-wise linear time invariant. Convenient group properties and the accurate approximate Walsh function integrals as a sum of a finite number of Walsh functions add to this. The idea is to represent input-output measurements on a finite interval (normalized to the unit interval) by a finite number of Walsh functions.

36

R. R. Mohl e r

Then, replacing the BLS (1) by an equivalent integral equation, products of state and control are approximated by a finite number of Walsh functions, and the integral equation is replaced by an algebraic summation of terms. Parameter identification is thus reduced to an algebraic evaluation by way of the Walsh-function coefficients. This methodology is discussed in detail, along with examples, by Karanam, Frick, and Mohler (1978). Walsh functions, being a complete orthonormal group, conveniently admit a representation of every Lebesgue integrable function f(t) on [0,1] in the form

where the coefficients an are given by

f

1

1/in(t)f(t)dt

o

(n = O,I,Z, ••• )

(13)

Two properties of Walsh functions particularly attractive for our purposes, are the following: 1)

The Walsh functions form an Abelian group (Fine, 1949). Specifically, they are closed under multiplication, with the multiplication rule for Walsh functions expressed as

functions. Convenient algebraic forms and examples for this purpose were developed by Karanam, Frick, and Mohler (1978). A correlation method for such BLS realization is given by Baheti and Mohler (1975). Both methodologies seem convenient for petrochemical application where it is necessary to identify certain relevant parameters. STOCHASTlC BLS BLS filtering and stochastic control were studied by Mohler and Kolodziej (1981,1985). It is shown that conditionally-Gaussian, conditionally linear, and log-normal processes are important in BLS. If it is assumed that certain of the states are measured and fedback through the bilinear control to generate a nonlinear system that is actually conditionally linear (based on the observations), then a finite-dimensional filter may be developed. In certain cases the separation principle follows and a stochastic optimal control may be derived. The method has been applied successfully to tracking and to immunological state estimation by simulation. It has obvious application to coupled BLS in petrochemistry and energy. Again, the petrochemical application seems amenable to such analysis and design. In certain deterministic cases, the corresponding BLS observer may be useful. REFERENCES

The sign $ represents no-carry bitwise moduloZ addition. Z)

Fine (1949) also showed that the integral of a Walsh function can be represented in terms of Walsh functions. Considering the integral t

Jk(t)

=

f

o

1/ik(x)dx

(k

Fine showed that, with k

o .;

(14)

Zn + k',

< Zn,

k'

and

for k

0,1, Z, ••• )

(15)

1.

~

(15) may be manipulated to a more compact form as

f

o

t

'I'(x)dx

P'I'(t)

(16)

Where 'I'(x) is a column vector consisting of Walsh functions and the matrix P is termed 'operational matrix' by Fine (1949). 'I' and P in (16) are of infinite dimension, and finite-dimensional approximations are obtained by restricting'!' to contain only Zn Walsh functions for some integer n ) 1. The resulting square operational matrix p(m) is therefore of order m = Zn and

Consequently, parameter evaluation of (1) can be done algebraicly by an appropriate number of Walsh

Baheti, R. and Mohler, R.R. (1979). "A new crosscorrelation algorithm for Volterra kernel estimation of bilinear systems," IEEE Trans. Auto. Cont. AC24, 661-664. Baheti, R., Mohler, R.R., and Spang, A. (1980). "Second-order correlation method for bilinear system identification," IEEE Trans. Auto. Cont. ACZ5, 1141-1146. Espana, M. and Landau, 1. D. (1978). " Reducedorder models for distillation columns," Automatica 14, 345-357. --Fine, N.J. (1949). "On the Walsh functions," Trans. Amer. Math. Soc. 65, 372-414. Gunn, D.J. and Thomas, W.J. (1965). "Mass transport and chemical reaction in multifunctional catalyst systems," Chem. Engr. ScL 20, 89100. Gutman, P.O. (1981). "Stabilizing controllers for bilinear systems," IEEE Trans. Auto. Cont. AC26, 91Z-922. Hofer,E.p. (1973). "The optimization of bifunctional catalysts in tubular reactors," Proc. Joint Auto. Cont. Conf., Columbus, OH. Karanam, V.R., Frick, P., and Mohler, R.R. (1978). Bilinear system identification by Walsh functions," IEEE Trans. Auto. Cont. AC23, 704-713. Kolodziej, W.J. and Mohler, R.R. (1985). "Optimal estimation and control of conditionally linear systems," SIAM J. Cont. & Optimiz. 23, to appear. Mohler, R.R. (1973). Bilinear Control Processes, Academic Press, New York. Mohler, R.R. (1985a). "Evolution of bilinear systems," IEEE Centennial Book on Control, IEEE Press, New York, to appear. Mohler, R.R. (1985b). "Bilinear systems and control, " Ency. Physical Sci. & Tech., Academic Press, New York, to appear. Mohler, R.R. and Kolodziej, W.J. (1981). " Optimal control of a class of nonlinear stochastic control systems," IEEE Trans. Auto. Cont. AC26, 1048-1053. Rao, K.V. and Mohler, R.R. (1975). "On the synthesis of Volterra kernels of bilinear systems," ACTA 3, 44-50.