Feedback Linearisability of a Nonlinear Heat Exchanger

Copyright © IFAC Adaptive Systems in Control and Signal Processing, Budapest, Hungary, 1995

FEEDBACK LINEARISABILITY OF A NONLINEAR HEAT EXCHANGER M.H.R. Fazlur Rahman and R. Devanathan School of Electrical and Electronic Engineering Nanyang Technological Univer"ity Nanyang Avenue, Singapore 2263.

Abstract: Necessary and sufficient conditions are derived for a nonlinear heat exchanger model to be feedback linearisable. The conditions are provided in terms of parameters directly related to heat transfer parameters. As part of the result, certain useful properties on the independence of distributions with respect to input are also derived. An example is provided to illustrate the theory. Key Words- Process Control, Feedback Linearisation, Nonlinear, Distributions, Modelling.

1. INTRODUCTION

the paper can be extended to other process plants, for example, those involving mass transfer operations, in a straightforward way through the generality of the methodology used for the model development.

Feedback linearisation, as means to a control design methodology, has attracted a great deal of research interest in recent years. Linearisation has been studied by many authors (Jakubczyk, 1987; Lee et al., 1987; Levin and Narendra, 1993; Sluis, 1993) and approximate linearisation has been investigated in (Guzzella and Isidori, 1993; Lee and Marcus, 1986) using state feedback and coordinate change. Nonlinear decoupling via feedback by differential geometric approach has been accomplished in (Isidori et al., 1981). Feedback linearisation of DC motors is described in (Oliver, 1991). In this paper, we consider the feed back linearisability of a heat exchanger model. Heat exchanger is considered since it is one of the inherently nonlinear systems in the process industries.

The paper is organised as follows. The next section illustrates the modelling of a nonlinear heat exchanger using lumped approximation. Section 3 deals with the general aspects of feedback linearisation. Feedback linearisation of a heat exchanger model is discussed in Section 4. Necessary and sufficient conditions for feedback linearisation of the heat exchanger are discussed in that section. Finally in Section 5, a simple example illustrates the use of the theory. 2. SYSTEM MODELLING In a heat exchanger (Shinskey, 1988; Silebi and Schiesser, 1992) liquid flows through the inner tube and it is heated by another liquid that flows cocurrently around the tube as shown in Fig. 1.

Though our interest in the paper is not modelling a heat exchanger per se, we give a brief account of modelling techniques applied to a heat exchanger. A heat exchanger is identified by an orthogonal parameter estimation method in (Liu et al., 1987; Billings and Fadzil, 1985). It is also described by a nonlinear distributed parameter system using linear nonparametric and parametric models in (Franck and Rake, 1985). A dynamic model of air-to-air heat exchanger in a linear difference equation form with pure delay has been proposed in (Radchenko, 1990). A state space model of a counter flow and parallel flow heat exchanger has been developed in (Jonsson et al., 1992).

Shell Tube

~

~--------;;----,-+-"'--t--------'-T\

-

FIG. 1. Concurrent Shell and Tube Heat Exchanger The following main assumptions are made: (i) the physical and chemical properties of the fluids under consideration should be constant, (ii) the variation in fluid velocity and temperature radially is negligible, (ill) the heat transfer to the surroundings is not significant and (iv) the overall heat transfer co-efficient is constant. Considering energy balance on the tube side (either shell or inner tube) fluid, on an element ~x,

The main contribution of the paper is to provide explicit necessary and sufficient conditions for feedback linearisation of a nonlinear model of a heat exchanger in terms of parameters directly related to heat transfer parameters. A different method for feedback linearisation can also be found in (Fazlur Rahman and Devanathan, 1994). It is expected that the results of 413

at a distance x-~x at time t and assuming that the velocity of fluid averaged across the tube is constant, one can write that a(a~xpGpT)

at

= avpGp[TX _ AX -

model, forward difference equation form of obtained by equating

t = T(k + 1) -

Tx ] + U1rd~x~T (2.1)

T 1j(k + 1)

= PITIJ(k) + qlT2J(k) + rlTIJ-I(k)

T2j(k + 1)

= P2T2j(k) + q2T 1j(k) + r2T2J -

Pi

= 1-

hI

(bi + hVi) '

x(k

(2.3)

=

aiPiGpi

hlbi,

hlv.

ri

= h'

i=I.2

(3.11)

(2.4)

- al(x,u), f (x,U ) ax ' J=I.2 .....N

=

+ 1) = I[x(k),u(k)]

%

i=I.2;

qi

If the feedback law can be achieved, linear theory can be used to control and stabilise the system around an equilibrium point, for example at the origin. For a system to be locally feedback linearisable, the necessary and sufficient conditions (Jakubczyk, 1987; Levin and Narendra, 1993) are given below, where the argument k is dropped for clarity. Let

and

aTij _ Tij - TiJ-l ax h

(k) (2.10)

around an equilibrium state is discussed. x(k) represents the state variable at time k. The question arises as to whether the system (3.11) is feedback linearisable through the following two transformations: (i) transformation of coordinates in the state space z = 4>( x) with 4>(.) invertible and continuously differentiable, (ii) a feedback law u(k) = ~[x(k), w(k)]

The initial and boundary conditions are T; (x, to) li(X); Ti(O, t) = gi(t), i = 1,2. The heat exchanger is divided into small incremental elements at x= 0, h, 2h, ... , N h uniformly. The system state T represents the temperature at each node. T = [Tll ,T12 , ••• ,T1N T21,T22, ... ,T2Nf, where TiJ' (i = 1,2; j = 0,1,2, ... , N) is the temperature at the jth point of the ith tube. T 10 (= gl(t)) and T 20 (= g2(t)) are considered inputs. The system order is 2N. Let

bi=~, '=1,2

1

(2.9)

3. FEEDBACK LINEARISATION In this section, linearisation (Jakubczyk, 1987; Hunt et al., 1986) of the nonlinear system

(2.2)

U1rd (T1 _ T2 ) a2P2Gp2

(2.8)

where

°

ax

T(k)

where hI is the sampling interval along the time axis. Writing (2.6) and (2.7), in the difference equation form, we have

Using subscripts 1 and 2 to correspond to the shell and inner tubes respectively, and considering the limit as ~x -+ and assuming no acceleration of the fluid, (2.1) can be used to derive the following equations for the shell tube and inner tube side fluids respectively.

= -V2 aT2 +

can be

hI

where a - cross sectional area of tube (cm 2 ); p - fluid density (gm/cm 3 ); d - internal diameter of the tube (cm); Gp - heat capacity (cal/gm K); T - temperature of the tube under consideration (K); t - time (sec); U - overall heat transfer coefficient(cal/sec cm 2 K); v fluid velocity (cm/sec); ~T - change in temperature (K); ~x - incremental distance (cm)

aT2 at

t

(2.5)

- al(x, u) I .. ( x,U ) au

(3.12)

Define the following distributions (Isidori, 1989) on 'Rn depending on u:

(2.2) and (2.3) can then be written as ~o(x,u) = ~.+I(x, u)

°

=

1;-1 (x, U)[~i(f(X, u), u)+ lm/.. (x, u)],

i=0.1.2 ..... n

(3.13) where lm I .. (x, u) denotes the image of I .. and 1;-1 V denotes the counter-image of the subspace V under the linear map 1%. ~i (x, u) are distributions depending on u and it can be shown that ~o(x, u) C ~I (x, u) C ~2(X,U)"', where ~, attains maximum rank after at most n steps, where n is the dimension of the state vector x. Denoting the Jacobian matrix of I by dl = (f%, I .. ), the following theorem is stated without proof from (Jakubczyk, 1987).

where dot above a variable denotes differentiation with respect to time and j 1,2, ... ,N throughout the paper unless otherwise specified.

=

The continuous model of (2.6) and (2.7) will now be discretised with respect to time, before feedback linearisation is attempted. The reason for this is that the linearisation analysis based on the discrete model of the heat exchanger will eventually be applied in the context of linearisation of an Artificial Neural Network (ANN) model of the heat exchanger (which is discrete in nature being based on the available discrete input-output data of an industrial heater) which is our future work. To discretise the heat exchanger

Theorem 1: Let (x = 0, u = 0) be an equilibrium point of Equation (3.11). A".mme that rank dl(O, 0) = n. Then "ydem (3.11) i" locally feedback lineari"able at (0,0) if] the di"tribution" ~1(X,U), ~2(X,U), . ", ~n-I (x, u), are of con"tant dimen"ion and are independent of u in a neighbourhood of (0,0) and dim ~n(O,O)=n. 0

414

A

4. FEEDBACK LINEARISATION OF HEAT EXCHANGER MODEL Identifying the state vector T with x, for the system shown in (2.9) and (2.10) we have from (3.12),

a E R"', u E S, PE Rn}

(4.14)

o

o]

r'

g

(4.24) (4.25)

where a' (u) is a p tuple vector given by a'(u)

=

r

= Im{BM(u)}

{P I P = BM(u)a,

where PI = pdN + L 1, p. = P2 IN + L 2, P2 = q1IN, P3 q2 IN and Li is a N x N matrix with elements ri, i = 1, 2 just below the main diagonal and rest of the elements being zero. IN is an identity matrix of order N. fu(T, u) = G is a 2N x 4 matrix such that

o

Im{A(u)}

P = A(u)a = BM(u)a = Ba'(u)

F1 h(T, u) = P = [ F 3

G- [ 0e

=

= M(u)a

(4.26)

While a freely ranges over all of RP, a'(u) also ranges over all of RP at least for some u E S, since M (u) has a rank p for some u E S. Then P will range over A = RP S; Rn. The basis vectors for the subspace A will correspond to the columns of B and since B is independent of u, A is also independent of u and of constant dimension p.

(4.15)

where To prove the necessity, given that A = RP S; Rn and is independent of u, one can find the basis vectors bJ , j = 1,2,"', p which are independent of u such that A = span {bj(U)}~=l' Since A = Im{A(u)}, A( u) ClUl be put as

and

A(u)

= BM

(4.27)

where B = [b 1 , b2 ," " bp] and M is a p x m matrix. Premultiplying (4.27) by B T , we have

It can be shown that

B T A(u)

where

= B T BM

(4.28)

T

A1,A2=

(PI

+ P2) ± J(P1

+ 4q1 q2

- P2)2

2

The p x p matrix (B B)-l exists since B is of rlUlk p. Therefore,

(4.17)

(4.29)

Since the two factors on the RHS of (4.16) are coprime (Gantmacher, 1964), one can construct invariant subspaces R;', i=1.2 which are spanned by the sets of vec(i) (i) (i) . 1 tors zN ,zN_1,"',Zl , i=1.2 respecbvey such t hat ,-I ( Ai -

( A-1 i

-

p-1)J

p-1)N ZN (i)

(i) _

(i)

ZN - ZN-J'

-

0

. .=1.2,

.

, .=1,2

J=1.2 .. ··,N-1

and M is thus a function of u. Let M = M(u). Hence from (4.27), we have A(u) BM(u). Given that A RP, A(u) has a maximum rlUlk p for some u E S. Then B is of rank p implies that M (u) has a rank p for some u E S. Hence the proof. 0

=

(4.18) (

4.19

=

Lemma 2: : Given a Jordan block matrix In(u) such that,

)

Z~), i= 1,2 are the generator vectors with minimal annihilating polynomials of degree N each, and

In(u)

= -y(u)In + D

(4.30)

where D is a n X n matrix whose elements immediately above the main diagonal are each -1 and the rest of the elements are zero and -y is a scalar function of u, such that -y(u) '" 0, 'V u E S. Let Q(u) be a n X m matrix, m $ n. If At{u) = In(u)Q(u), then Al = Im{A1(u)} is of constant dimension p and independent of u ifJ

(4.20) Certain lemmas are proved first before the final result. Lemma 1: : Let a n x m matrix A(u), m $ n, whose elements, in general, are functions of u, u E S, S being a neighbourhood of the equilibrium point Ua, be such that it has a maximum rank p $ m for some u E S. Then

Q(u)

= BM(u)

(4.31 )

and

A

= Im

{A(u), u E S}

(4.21) DQ(u)

is of constant dimension p and independent of u ifJ A(u)

= BM(u)

(4.22)

where B is an n X p matrix, of rank p, all of whose elements are independent of u and M(u) is a p x m matrix whose elements can be functions of u and which has a rank of p for some u E S.

= BM(u)

(4.32)

Proof: Necessity: By Lemma 1, (4.33) where B 1 is a n x p matrix of rank p and independent of u and M 1 (u) is a p x m matrix of rlUlk p for some u E S. Let Cl be a n x (n - p) matrix such that

Proof: In order to prove sufficiency, assume A(u)

= BM'(u)

where B is a n x p matrix of rank p and independent of u and M(u) and M'(u) are p x m matrices with M(u) having a rank p for some u E S.

(4.23) 415

cT B l = 0 (4.33) by cT, we

Premultiplying

= Bi+lY;(U), i = 1,2,,,, ,n-1

(4.46)

RI = Bi+lZi+I(U), i= 1,2,"',n-1

(4.47)

DB. (4.34) have

cT AI(u) = cT BIMI(u) = 0

where Xi, Y; and Zi+l are of appropriate dimensions and

(4.35)

Using (4.30), we can write (4.35) as

rank

cT Al(u) = cT In(u)Q(u) =

C'[[(-y(u)In

for some

(4.36)

=

(4.38)

= min{rankBl,rankM(u)}

=

[In(U)Ai(U): In(u)Q(u)]

=

[In(U)Ai(U): AI(u)]

=

[-Y(U)Ai(U)

+ DAi(U)

: Al (u))(4.49)

Substituting for Ai(U) and AI(u) in (4.49) we have

=

((-Y(U)BiMi(U)

+ DBiMi(U))

BIMI(u)}

(4.39)

: (4.50)

To prove necessity, assume

For some u E S, p = min{p, rankM(u)}

(4.40)

(4.51 )

Knowing that rank M(u) :5 p from (4.40), we have rank M(u) = p for some u E S.

where Bi+l is a n x mi+l matrix of rank mi+l and independent of U and Mi+l(U) is a mi+l x m(i + 1) matrix of rank mi+l :5 m(i + 1) for some U E S. Let Ci+l be a n x (n - mi+d matrix 3

To prove sufficiency,

AI(u)

(4.48)

u E S.

Ai+1

This implies that the columns of Q(u) and DQ(u) lie in Im{Bd. That is, (4.31) and (4.32) are satisfied for B = B l for p x m matrices M(u) and M'(u) respectively. Since 6 1 = RP, rank Al (u) = p for some u E S. Since In(u) is nonsingular VuE S, it follows that rank Q(u) = p for some u E S. From (4.31), rank Q(u)

:

Proof: (4.44) can be written as

Since (4.36) is to be true for all -y(u), u E S it follows that (4.37) cTQ(u) 0

cT DQ(u) = 0

+ Y;(U)Mi(U))

Zi+I(u)M(u) ) = m'+l

+ D)Q(u)]

o

and

(-y(u)Xi(u)Mi(U)

= In(u)Q(u) = (-y(u)I n + D)Q(u)

(4.41)

(4.52)

Substituting (4.31) and (4.32) into (4.41),

AI(u) = [-Y(u)BM(u)

+ BM'(u)] =

Premultiplying (4.50) by C41 and using (4.51) and (4.52), we have

BM"(u) (4.42)

where

M"(u) = -y(u)M(u)

+ M'(u)

C41Ai+I(U) = 0

(4.43)

~ ((-Y(U)C41BiMi(U) + C41 DBiMi(U)) : C41BIMl(U)} = 0 (4.53) Since (4.53) is to be true for all -y(u),u E S, it follows that

Hence from Lemma 1, 6 1 = Im {Al(u)} is independent of u. B is of rank p and M(u) is of rank p for some u E S. This means that the n x m matrix Q(u)( BM(u)) for some u E S contains p columns which correspond to p linearly independent combinations of the basis vectors {bj};"=l where B = [b l ,b2 , .. ·,b m ]. Hence Q(u) has rank p for some u E S. Since In(u) is nonsingular VuE S, Al(u) = In(u)Q(u) has a rank p for some u E S. Hence 6 1 also has a constant dimension p. Hence the result. 0

=

C4IBiMi(u) = 0

(4.54)

C41DB.M.(u) = 0

(4.55)

C41BIMI(U) = 0

(4.56)

and

Lemma 3: : Let 6i+l

=

Im{A.+ I } i

= 1,2,'"

= I n(U){6i + Im Q(u)}, ,n -1

Since the m. x mi matrix, Mi(U) has a rank m, for some u E S, it follows that

(4.44)

where 6 1 = Im{AI(u)} i! a! defined in Lemma 2 and = Im{A.(u)} is such that Ai(U) = BiMi(U), where B , is an n x mi matrix of rank m. :5 n and independent of u and M.(u) is a mi x mi matrix, mi :5 mi, of rank mi for some u E S. Then 6i+l is of constant dimension mi+1 and independent of u ifJ there exists a n x mi+l matrix B i+ l of rank mi+l :5 nand independent of u, such that,

6i

Bi=B,+IXi(U), i=2,3,"',n-1

:5 mi, (4.57)

where M/(u) is a mi x mi nonsingular submatrix of M i(u) for some u E S. It then follows that Cl'+l B, o. Similarly it follows that C41 DBi 0 and C'~lBl o. Hence, (4.45), (4.46) and (4.47) follow for some Xi, Y; and Zi+l matrices of appropriate dimensions.

=

(4.45 ) 416

=

=

for the distributions A~i), i = 1,2; j = 1,2" .. ,N to be of constant dimension and independent of u. One can form distributions

Substituting for Bi, DBi and B l into (4.50), it follows that

=

Ai+I{U)

Bi+d(-r(u)Xi(U)Mi(U) " . -

,,(I)

~J -

L.J,.J

+ L.J,.J' ,,(2)

J. - " 1 2 ... , N

+Y;(U)Mi(U)) : Zi+I{u)Ml(u)}

=

Bi+IMi+I{U)

~ (B4IBi+I)M'(u)

The distributions A J C R 2N , j = 1,2,·,·, N are by construction of constant dimension and are independent of u and correspond to the system (2.9) and (2.10). It is clear that, AN = R 2N iff A~) = RN, i = 1,2. Hence the result of Theorem 2 follows from Theorem 1 in respect of the system (2.9) and (2.10). 0

(4.58)

= (B4IBi+I)Mi+I(U)

(4.59)

where,

=

M'(u)

[1'(u)Xi(u)Mi(U) + Y;(U)Mi(U) :

Zi+I(U)MI{u)]

(4.60)

5. EXAMPLE Assuming the numerical values as below:

(B41Bi+I)-1 exists since the n x mi+l matrix Bi+l is of rank mi+l, mi+l ::; n. Hence, M'(u)

al = 50 cm 2; a2 = 2 cm 2; Pi = 1.0 gm/cm 3 , i=1,2; Cpi = 1.0 cal/gm K, .=1,2; d l = 2.0 cm; d2 = 0.4 cm; U = 0.05 cal/sec cm 2 K; hI = 100; h = 100.

= Mi+I{U)

Thus M'(u) is such that, rank M'(u) = mi+l for some u E S.

Let N = 1 and considering only input VI, one can consider other inputs to be physically constant. Equations (2.9) and (2.10) reduce to

For sufficiency substitute expressions Bi' DBi and B l into (4.50)

=

Ai+I{U)

T u (k+l)

[(1'(U)Bi+lXi(U)Mi(U) T 21 (k

+Bi+l Y;(U)Mi(U)): Bi+l Zi+l (u)M(u)]

= [ 0.3:'1~ VI ~2~;4];

(4.62)

>.-1

= .! (-(1.76 + 2

where

Mi+I{U)

=

(4.63)

Bi+l is a n x mi+l matrix of rank mi+l. Mi+l(U) as per (4.48) has rank m'+l for some u E S. Using Lemma 1, it follows that Ai+l is independent of u and of constant dimension mi+l. Hence the result. 0

J (l) ~

o J;:)

]

;

G

dl) ] = [ (2)

] (5.67)

V01)

3.45

1 ](5.68) '1>.;1 - 3.45

where '1 = (4.45 - 3.45vI). In terms of zP) and z~2) as the basis vectors,

(5.69) For i

= 1, to satisfy

Lemma 2,

(4.64)

where J~), i = 1,2 corre6ponds to the Jordan form defined in Lemma 2 and partitioning of p-l and G are done in terms ofi 1,2 as shown in (4.64). Then the discrete time 6ystem is locally feedback linearisable ifJ the conditions of Lemma 2 and Lemma 3 in terms ofQ = d i), In(u) = J~), and l' = >'i l i = 1,2. are satisfied and that A~) = RN for i = 1,2.

and

Proof: Nonsingularity of P ensures that the rank condition of Theorem 1 is satisfied. As shown in Lemmas 2 and 3, conditions in terms ofQ = di), In(u) = J~), and l' >'i l , i 1,2 are both necessary and sufficient

Since D

Q

=

=

vI) ± (14.1376 - 5.04vI + -2.76 + 2.14vl

[.>,' ~

(1): (2)]_ [ Zl • Zl -

Theorem 2: Con6ider the di6crete time heat exchanger model (2.9) and (2.10). Auuming that p-l exid6. Let the matrice6 p-l and G be expre66ed in terms of the basis vectors z;i), i = 1,2; j = 1,2,···, N such that they are of the form

=[

= [ -~u

G

The rank condition of Theorem 1 is satisfied at any point where det P = 2.14vl - 2.76", o. On evaluating (4.18), one can obtain the generator vectors as

[(-r(u)Xi(u)Mi(U) + Y;(U)Mi(U)) :

Zi+I(u)M(u)]

p-l

(5.66)

3.14Tu (k) - 2. 14T21 (k)

The eigenvalues of matrix p-l are

+Y;(U)Mi(U)): Zi+l(U)M(u)] Bi+lMi+l(U)

+ 1) =

It follows from (4.14) and (4.15) that,

Bi+l[(-r(U)Xi(U)Mi(U)

=

(5.65)

(4.61)

p

=

Ai+I(U)

= 0.38Tu(k)+0.62T21(k)-VITu(k)

= =

T

{-2.14 - >.;1 (2.14VI - 2.76)} 1 1 (2.1 4Vl - 2.76)(>'2 - >'1 ) BM(u) (5.70) Q

_

"'1 -

u

where

B

=

417

= 0, DQ = o.

= Tu

To satisfy lemma 3

binary multifrequency signals. In IFA C Proceedings on Identification and System Pammeter Estimation, pages 1859-1864, York, UK.

=

Gantmacher, F. R. (1964). The Theory of Matrices, volume 1. Chelsea Publishing Company.

=

Guzzella, L. and Isidori, A. (1993). On approximate linearization of nonlinear control systems. International Journal Robust and Nonlinear Control, 3:251-276.

where

(1)

El

= Tu

and MI1)(u)

= {..\)1[-2.14 -

Hunt, L. R., Luksic, M., and Su, R. (1986). Exact Iinearization of input-output systems. International Journal of Control, 43:247-255.

..\21(2.1~V1 - 2~76)l}

(2.14vl-2.76)(..\2

-..\1 )

Isidori, A. (1989). Nonlinear Control Systems An Introduction. Springer-Verlag.

Similarly for i = 2, it can be seen that Lemmas 2 and 3 are satisfied. Hence 6~i) = RI, i = 1,2 are independent of u. Therefore

Isidori, A., Krener, A. J., Gori-Giorigi, C., and Monaco, S. (1981). Nonlinear decoupling via feedback: A differential geometric approach. IEEE Tmnsactions on Automatic Control, AC26, No.2:331-345.

Hence the condition for Theorem 2 is satisfied and the system (5.65) and (5.66) is locally linearisable. For the input transformation, w(k)

v1(k)

= T ll (k)

Jakubczyk, B. (1987). Feedback linearization of discrete-time systems. Systems and Control letters, 9:411-416.

(5.72)

Jonsson, G., Palsson, O. P., and Sejling, K. (1992). Modeling and parameter estimation of heat exchangers-a statistical approach. Journal of Dynamic Systems, Measurement, and Control,

(5.65) and (5.66) can be written as T(k

+ 1) = H 1T(k) + K 1w(k)

(5.73)

114:673~79.

where

Lee, H. G., Arapostathis, A., and Marcus, S. I. (1987). Linearization of discrete-time systems. International Journal of Control, 45, no.5:1803-1822.

and

Lee, H.-G. and Marcus, S. I. (1986). Approximate and local linearizability of non-linear discretetime systems. International Journal of Control, 44:1103-1124.

0.38

H1 = [ 3.14

0.62 ]. -2.14 '

(5.73) is in the linearised form under the linearising transformation given by (5.72).

Levin, A. U. and Narendra, K. S. (1993). Control of nonlinear dynamical systems using neural networks: Controllability and stabilization. IEEE Tmnsactions of Neuml Networks, 4, no.2:192206.

6. CONCLUSION A nonlinear model for a heat exchanger is derived in a form which is applicable to a class of process plants. Distribution theory is used to derive the necessary and sufficient conditions for the linearisability of the model explicitly in terms of the heat transfer parameters of the model. In the process, we have derived some useful properties on the independence of distributions with respect to input. The results of the paper will be useful in the control of the heat exchanger through the linearisation technique.

Liu, Y. P., Korenberg, M. J., Billings, S. A., and Fadzil, M. B. (1987). The nonlinear identification of a heat exchanger. In Proceedings of the 26th Conference on Decision and Control, pages 1883-1888, Los Angeles. Oliver, P. D. (1991). Feedback linearization of de motors. IEEE Tmnsaetions on Industrial Electronics, 38, no.6:498-501. Radchenko, I. F. (1990). A dynamic model of a heatexchanger apparatus. Soviet Journal of Automation and Information Sciences (English tmnslation of Avtomatika), 23(4):93-96.

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