controllability and observability of heat exchanger networks in the time-varying parameter case

Pergamon

0967-0661(95)00144-1

ControlEng. Practice, Vol. 3, No. 10, pp. 1409-1419, 1995 Copyright © 1995 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0967-0661/95 $9.50 + 0.00

CONTROLLABILITY AND OBSERVABILITY OF HEAT EXCHANGER NETWORKS IN THE TIME-VARYING

PARAMETER CASE

E.I. Varga, K.M. Hangos and F. Szigeti

Systems and Control Laboratory, Computer and Automation Research Institute, H-1518 Budapest, P.O. Box 63, Kende u. 13-17, Hungary

(Received July 1994; in final form April 1995)

Abstract. In this paper the structural controllability and observability of heat exchanger networks axe determined from qualitative information about the heat exchangers and their network topology. The heat exchanger network is modelled as a time-varying linear system based on first engineeriiI, g principles where the effect of the flowrate variations is described by unknown time--varying parameters, inlet temperature variations are regarded as disturbances and external heaters/coolers are considered as input variables. Necessary and sufficient conditions for structural controllability and observability of heat exchanger networks are derived, based on an extended Kalman-type rank criterion for linear systems with time-varying parameters. Determining the structural controllability and observability of heat exchanger networks needs only checks for the input and output connectability of the network in both constant and time-varying parameter cases. The results are extended to the more-practical case where bypass ratios are also used as control variables and where more than one time-varying parameter enters into the state-space matrices.

Keywords. Dynamic models, Chemical processes, Structural properties, Digraphs, Process control.

1. I N T R O D U C T I O N

The advanced control of complex linear or linearized dynamic systems often has to cope with special problems caused by the size of the system in question. Based on the preliminary available technological knowledge, however, it is possible to predict the presence or absence of important control properties such as stability, observability or controllability.

Heat exchangers are widely used in process indus- tries, mainly arranged in units containing several (5 - 200) of them. The design of these heat exchanger networks is far from being trivial. The main goal of the design is to find energy- or cost- opt imal systems, but these turn out to be diffi- cult to operate and /or control. Therefore, there is an urgent need to analyze heat exchanger networks from the viewpoint of their possible control properties and problems. There are several papers about the static structural operabili ty analysis of heat exchanger networks (see, e.g., (Calandranis and Stephanopoulos, 1986)) giving results on the

analysis of feasible regions and static disturbance- rejection properties.

The dynamic control properties of heat exchanger networks, however, have been investigated by only a few authors. Wolff et al. (1991) presented various dynamic models of single heat exchangers and used a multiple mixing tank model for single- and double-pass heat exchangers. They generated a dynamic model of heat exchanger networks from a network structure including a linearized model of the units. The control of heat exchanger networks has been investigated using bypasses, while dynamic considerations using open-loop transfer function and interaction and pairing considerations by relative gain array have also been taken into account.

Mathisen et al. (1991, 1992b) defined controllability as the dynamic resilience of heat exchanger networks. They reviewed different controllability measures and used them to select bypasses and appropriate pairings for the control of heat exchanger networks. The flow rate dependence of

1409

1410 E.I. Varga et al.

the heat transfer coefficients is included in their model, and this is found to have a significant effect on control.

Mathisen et al. (1992a) suggested several opti- mization problems to deal with a four-way trade- off between flexibility (static performance specifications), controllability (dynamic performance specifications), investment cost (installation of bypasses and control system) and operating cost (utility requirements and control system mainte- nance) affected by bypasses (number of bypasses, type of bypasses and bypass placement) in a heat exchanger network.

The structural stability of heat exchanger networks has been investigated in (Hangos, 1991, 1992) based on the influence graph (linearized) model. It has been shown that, irrespective of the network topology, a heat exchanger network must not be structurally unstable.

The dynamics of the so-called "heat exchanger superstructure" has been studied by numerical simu- lation and by qualitative prediction using a signed directed graph (SDG) model in (Varga and Han- gos, 1992). In the same paper the effect of network topology on the structural control properties of heat exchanger networks has been investigated in the constant-parameter linearized case.

The main purpose of this paper is to describe some attempts to analyze the structural control properties of heat exchanger networks with time-varying parameters, i.e. such control properties (stability, observability and controllability) as can be determined from qualitative information about the heat exchangers and their network topology. For this purpose heat exchanger networks are modelled as time-varying linear systems where the effect of the flow rate variations is described by unknown time-varying parameters. Their controllability and observability in the Kalman sense (i.e. state controllability and observability) are investigated, in both constant and time--varying parameter cases. From a practical point of view the output controllability is more important; however, the state controllability and observability (at least for one state) is a necessary precondition. The main result of this paper is that in the time- varying parameter case (which is usually the case in reality) one can obtain equivalent conditions for the state controllability and observability of the heat exchanger networks to those in the constant- parameter case.

The paper is organized as follows: First, state- space models of heat exchanger networks are derived and discussed. The effect of the network topology on the state-space model is investigated in the next section. Thereafter, the structural

control properties of the heat exchanger network are analyzed in both the constant-parameter and time-varying parameter cases. Finally, some conclusions are drawn.

2. CONSTRUCTION OF THE OVERALL SYSTEM MATRICES FROM THE UNIT MODELS AND FROM THE NETWORK

TOPOLOGY

Heat exchanger networks can be regarded as networks of heat exchanger units, where the physical connections (pipes, pumps etc.) between the units serve as mass and/or energy transfer channels causing no delay or other dynamic effect. The effect of delays caused by transport through pipes has been investigated elsewhere (Mathisen et al., 1994). Assuming this simplified structure, units possess their own dynamic system models and various input and output ports (or channels) for the aforementioned connections (Stephanopoulos et al., 1990).

2.1. Simple model of a heat exchanger cell

2.1.1. ODE model from first engineering principles. The simplest model of a heat exchanger cell consists of two perfectly stirred tanks with in- and outflows, connected by a heat transfer area between them. The liquid volume in each tank is assumed to be constant (i.e., the in- and outflow rates are equal). The heat exchanger cell will be denoted on the figures by the symbol shown on Fig. 1.

Vh , Th i

T ¢ O 4

• ~ -- Tho

Tci, vc

Fig. 1. The symbol of the heat exchanger cell

The model of the heat exchanger is in the form of two energy balances:

dT~o _ vc (rc~ - T c o ) + ~U" (Tho - T~o) (1)

dt Vc cp~p~ V~

dTho Vh Tho) + UA (Too - Tho) dt = ~hh (Thi -- cphPh~

Note that the above model of the heat exchanger can be regarded as oversimplified because it is

(2)

Heat Exchanger Networks 1411

not, able to describe such a fundamental dynamic property as a delay caused by the distributed-

parameter natlure of a heat exchanger and the

effects caused by concurrent or countercurrent flows. etc.

2.1.2. State-space model. The above model (Eqs (1) and (2)) fixes the state vector of the heat ex-

changer cell as z(t) = [Thor TcOIT, and leaves the possibly time-varying variables (w,, w,,, T’i, T,i) open for being disturbances (z(t)), input variables (u(t)) or time-varying constants depending on the modelling goal and assumptions.

Assuming that the flow rates ve(t) and u*(t) are modelled as time-varying parameters while the input temperatures are regarded as disturbances,

i.e. t(t) = [Thi, T,ilT, then the state space model (l-2) of the heat exchanger cell is linear. With the notation for the unit dependent constants r,

and Th

J__ UA -- TV CPY PY”, ’

p=c,h (3)

the following state A and disturbance L matrices result:

AZ at+*) $ ( - 7,

L=($ %, ( ,) g+*

c

(4)

The state matrix of the heat exchanger cell with time-varying parameters can be decomposed as A(t) = A1 + w(t)Az + w(t)Aa with cell-dependent constant matrices

The output variables of the heat exchanger cell are the outflow stream temperatures y(t) =

[Th,, LIT, and the output matrix is the identity

matrix: CT = Iz. Finally, the state-space model of the heat exchanger cell is as follows with the above-mentioned matrices:

i=Ax+Lt y=cx

2.2. Cascade model of heat exchangers

It is well known that any distributed-parameter heat exchanger can be described within a pre- scribed accuracy by a so-called “cascade model” consisting of a sequence of the above heat exchanger cells. In the case of heat exchangers with plug Ilow in each of the sides, the dynamic be- haviour can be approximated by the structure of Fig. 2.

Fig. 2. Cascade model of a countercurrent heat exchanger

The plug flow heat exchanger substructure can

be regarded as a fixed and structured element of

a heat exchanger network. As an example, the state-space matrices of a countercurrent heat exchanger approximated by 3 cells (Fig. 2) are given in the following form:

x(t) = [Tit), Tc2’ Ti;) T,‘,?, T,(i) Td:‘lT h$” ’ 4i) = [T/c, Teal (7)

y(t) = [Tizo, T,,lT = [T,$), T:,l’lT

APf =

LPf =

with

Al Aa 0 A12 A2 A32

0 A23 A3

A12

0

-421 I

010000

000010

(8)

a=3++ b=3*, c=-l-, $=_L c ‘A 7, ’

(9)


2.3. C o n n e c t i o n s

Before the detailed modelling of the connections between heat exchangers, the original problem statement of the heat exchanger network synthesis problem should be recalled: given vh hot and v¢ cold streams with their flow rates and input temperatures to the network i i ( vh ,T~ i , i=l,..,Vh; v~,T~i, i=l,. . ,ve) and target temperatures for the output temperatures of the streams (T~o , i=l,..,Vh; T~o, i=l,. . ,vc ), design an optimal (with respect to some cost function) heat exchanger network which performs the heat exchanging task. External heaters or coolers can also be used when they are needed.

A connection of a new heat exchanger to the network involves specifying the position of its hot and cold sides on suitable streams and giving its neigh- bouring heat exchangers on these streams. The series connection is shown in Fig. 3, where the hot side of the j t h heat exchanger cell has been connected after the hot side of the kth cell.

Fig. 3. Series connection of two heat exchangers

This simple connection implies the following cor- respondence of the network variables:

T (k) _ T( i ) _ T(J)

The case of mixing is similar but there is no new input variable here. Fig. 4b shows the mixing of the ith and j th hot streams into the kth stream. The relations between the input and output stream variables are as follows:

4 k ) = ¢ : ) , (12)

Using the series, splitting and mixing connections one can develop all the possible heat exchanger configurations for a given number of heat exchangers and streams. The automatic and energy- optimal generation of heat exchanger network configurations uses the design superstructure as a building element (Floudas and Grossmann, 1987). The generated network is optimal for multiperiod operation (i.e., in a specified parameter range) and the splitting ratios are used to change the network between different periods of operation. As an example the design superstructure consisting of two heat exchangers is shown in Fig. 5, @l~ich involves" all the possibilities of connecting the two heat exchangers through their hot stream.

) = vl k ), h, = r i ko ). (lO)

According to the preliminary assumption that the physical connections between units does not cause any dynamic effect, splitting and mixing can be regarded as special connections with multiple outputs and multiple inputs, respectively. Fig. 4a shows the splitting of the ith hot stream into two streams, j t h and kth.

(a) (1 - . ) . 4 vl (b)

. , ¢h . . . .

Fig. 4. The mixing (a) and splitting (b) connection

The splitting ratio denoted by c~ will be a new input variable and the relations between the input and output stream variables are the following:

v 1 | T I c ei (1 - Z)

(lv L (1 Vh Thin

Fig. 5. The design superstructure of two heat exchangers

By selecting appropiate values for the splitting ratios a, fl or 7, the following simple connections can be described:

• by-pass (fl = O, 7 = 0), • series connection (~ = O,/3 = 1, 7 = 1), • parallel connection (/3 = O, 7 = 1).

For the case of 7 = 1 the state--space model of the design superstructure is given as

x ( t ) - rT(U T, (1) T (~) T, (2)IT - - L ho ' co , ha ' co j ,

Z/''~l,) m r , rr(1) rr(2)lT - " [ l h i ' l c i , ± e i J '

y(t) = [Ibm,leo , co J

H e a t E x c h a n g e r N e t w o r k s 1413

these simplifications the state-space model of the half heat exchanger cell is as follows (in the case of a cooler):

(13)

x( t) = Tho, u(t) = Too, z( t ) = (15)

A1 0 ) A = A21 A2 '

0 0 V ~ " v (~)

o o L = ~ 0 0

v(2) o o (coco) C = 0 1 0 0 ,

0 0 0 1

(14)

with

( _ , 1 ) A1 vF' ~

v (x) i ' I _~_~

---- i v (2) i | '

0 0 c l = ( 1 - a ) ( 1 - f l ) , c 2 = a + ~ - a f t .

2.4. Input variables, disturbances

Heat exchanger networks are usually controlled by two means, by

• utility heat exchangers (coolers and heaters), • bypasses, where the manipulated variable is

the ratio of the bypass flow to the flow through the heat exchanger.

Considering only the first possibility, i.e. using only coolers and heaters as inputs, the resulting overall model of the heat exchanger network will be linear. However, this is only for the sake of simplicity, because the resulting bilinear model can be linearized around the given steady state in the general case. Therefore, a similar state-space model is obtained in both cases.

The dynamic effect of a utility heat exchanger is described by (possibly a cascade of) so-called "half heat exchanger cell(s)", where the side of the utility heat exchanger connected to a network stream is modelled in the usual way (using Eq. (1) or (2)) and the outer side is simply characterized by its (manipulated) output temperature. With

AhC Vh 1 B h c 1 L h c Yh - - , = - - , = - - . ( 1 6 )

vh rh rh Vh

2.5. The state-space model of the overall heat exchanger network

The overall heat exchanger network consists of two principal structural components: single heat exchangers (units) and the network topology. A unit (heat exchanger, utility heater/cooler) is built up from elementary heat exchanger cells (or half heat exchanger cells) through cascade-type models (see plug flow substructure). From this point of view the cascade model of heat exchangers (countercurrent, concurrent, etc.) can be regarded as a network of cells with a special topology. The overall network topology gives the connections between the input and output ports of the units. A simple connection starts at an output port and ends in an input port of the same type. The splitting and mixing connections connect several output and input ports, respectively.

The system matrices of the overall heat exchanger network can be constructed algorithmically from the given system matrices of units, their input/output ports and the network topology (Han- gos, 1991). An input port may be any input variable and an output port may be any output variable.

The connection algorithm consists of the following steps:

• Create the unconnected overall system matrices. The number of state, input and output variables is equal to the sum of that of each unit involved in the network. The initial system matrices A °, B °, L ° and C o will be block diagonal with the individual system matrices in the blocks, and the order of blocks is the same in each matrix.

• Check thai the physical types of the input and output ports to be connected are the same.

• Unify the input and output variables corresponding to the input and output ports to be connected.

The resulting overall state matrix is block diagonal with blocks of state matrices of the individual units and there are nonzero entries outside the blocks corresponding to the connections. A sim-


pie connection means one nonzero entry which is located in the column of that state variable from which the output port could be reached, and in the row of that state variable which could be reached from the input port. A splitting or mixing connection causes two or more entries in the corresponding row and column, respectively.

• s-rank(lAB]) = n.

Definition 3 A class of systems is said to be output-connectable if in the digraph G([Q]) =

G ( 00 [A][C]) the re i s ' f°r each state vertex' a

path from this state vertex to at least one of the output vertices.

3. STRUCTURAL CONTROLLABILITY AND OBSERVABILITY IN THE

CONSTANT-PARAMETER CASE

Assuming that the flow rates are constant, i.e. vth')(t)" = Vth s),' V~J)(t) = V! j), then the system matrices of the units and the overall heat exchanger network are time-invariant:

Theorem 2 (Reinschke, 1988) A class of systems characterized by the (r + n) * n structure matrix

pair [ CA 1 ! is s-observable iff

• it is output-connectable, and

. s A = n.

&(t) = Ax(t) + Lz(t) + Bu(t) u(t) = C (O.

(17)

Applying the definitions and theorems above to the special case of heat exchanger networks with constant flow rates, the following properties can be proven.

The structural controllability and observability can be determined using the graph-theoretic approach (Reinschke, 1988). The structural properties hold for a class of systems which axe described by numerical realizations of a given structural state-space model. The elements of a structure matrix [Q] are fixed either at zero or at indeterminate values which are assumed to be independent of one another.

The structural state-space model can be repre- sented by a directed graph (or digraph) where the nodes correspond to the states, inputs and outputs: there is a directed edge between two nodes if the corresponding structure matrix element is not zero (i.e., indeterminate). The digraph G rep- resenting the structure matrix [Q] is denoted by G[Q] where [Q] is the so-called "occurrence matrix" of the graph.

Definition 1 A set of independent entries of [Q] is defined as a set of indeterminate entries, no two of which lie on the same line (row or column). The structural rank (s-rank) of [Q] is defined as the maximal number of elements contained in at least one set of independent entries.

Definition 2 A class of systems is said to be input-connectable if in the digraph G([Q]) =

G ( [Z] [B] ~ there is, for each state vertex, a \ 0 0 ]

path from at least one of the input vertices to the chosen state vertex.

Theorem 1 (Reinschke, 1988) A class of systems characterized by the n * (n + m) structure matrix pair [A, B] is s-controllable if and only if

Proposition 1 The n* n state space structure matrix [A] of the overall heat exchanger network has full s-rank, i.e. s-rank([A]) = n.

Proof The state space matrices Ai of the overall heat exchanger networks with parameters of different numerical values are all numerical realizations of the following structure matrix [A] (i.e. Ai E [A]): [A] is block diagonal with blocks of indeterminate entries, outside the blocks there are indeterminate entries corresponding to the connections and all other entries are fixed at zero. The entries of Ai are independent because the units are independent of each other; they depend only through connections. According to Defini- tion 3. the indeterminate entries in the main diagonal can be chosen as a set of indeterminate entries, no two of which lie on the same line, so the s-rank of [A] is n. *

Corollary 1 A class of overall heat exchanger networks characterized by the structure matrix pair [A B] is s-controllable iff it is input-connectable.

Corollary 2 A class of overall heat exchanger networks characterized by the structure matrix pair

[ C ] is s-°bservable iff it is °utput-c°nnectable" A

Therefore, there results a very simple criterion for structural controllability and observability as well. The condition of input-connectability and output-connectability is easy to check: it is necessary to find paths which connect each state vertex with one of the input /output vertices in the digraph G([Q]). This is a standard task of algo- rithmic graph theory (see, e.g., (Roberts, 1976)).

. it is input-connectable, and


For a large system it is advantageous to use the notion of a strong component for deciding input or output connectability. The strong component (or strongly connected component) is a subgraph in a digraph where there is at least one directed path from each node to every other node. This means that every node is connected with every other node, so from the viewpoint of input and output connectability a strong component can be regarded as a simple node. Therefore instead of the original digraph its condensed graph will be used to check input and output connectability, where the strong components are replaced with single nodes, keeping all edges starting and finish- ing at this component. For example, the graph of the countercurrent heat exchanger is a strong component in the graph of the heat exchanger network.

where ai(t) are scalar functions of time, and Ai are time-independent and independent of each other.

The Lie algebra £ (see, e.g., (Pontryagin, 1946), or (Jacobson, 1979)) generated by the matrices Ai (i = 1, ..,m) under the commutator product [Ai,Aj] = AiAj - AjAi is of finite dimension k (k > m). £ C_ R~xn; hence, obviously k < n.

Proposition 2 (Szigeti, 1992) Let A1, . . . ,Ak be the basis of the Lie algebra generated by A(t). Suppose that there exist coefficients cq, -. . , c~k, oTt, • •., oTk satisfying that

the pairs ( ~/k= 1 aiAi, B ),

( ~/k=l ~iAi , C ) are controllable and ob-

servable, respectively;

there is no non-trivial polynomial P satisfying the algebraic differential equation P(g,g,-- .) = 0, where g is the solution of a differential equation over an interval [0, T];

4. STRUCTURAL CONTROLLABILITY AND OBSERVABILITY IN THE TIME-VARYING

PARAMETER CASE

If the flow rates are time-dependent then the model of the overall heat exchanger network is in the form of the following time-dependent linear system:

x(t) = A(t)x(t) + L(t)z(t) + Bu(t) = ( 1 8 )

where the matrices A(t) and L(t) are time- dependent, while the matrices C and B are not.

In the case of time-varying linear systems an ex- tension of the Kalman-type rank condition with an additional differential-algebraic condition for controllability and observability has been derived (Szigeti, 1992), which is briefly summarized here.

Consider a time-dependent linear system

then the time-varying system (19) is controllable and observable over [0, T], respectively.

The above proposition is applied to the overall heat exchanger network with time-varying parameters. Let [A] be a structure matrix corresponding to the state matrix of the overall heat exchanger network with time-varying parameters (Eq.(18)). Suppose that [A] = [A1] + ' - " + [Am] is a pairwise disjoint decomposition of [A], that is the sets of elements of [Ai] different from zero are disjoint for each different pair of i , j E {1,2, . . - ,m}. Then, consider system (18) such that Ai E [Ai] are numerical realizations of the corresponding structure matrix, satisfying that [A] = )-~i~1 [Ai] is of full rank. Then, using Corollaries 1, 2 and Proposi- tion 2, the following proposition can be obtained immediately.

Proposition 3 Suppose that a time-varying heat, exchanger network (18) satisfies that

• the decomposition [A] = [A1] + . . . + [A,~] is pairwise disjoint,

• the network is input (output) connectable,

= A(t)x(t) + B(t)u(t) u(t) = C (t) (19)

and write the time-dependent state matrix A(t) in the form of

m

A(t) = ai(t)A (20) i=l

• there is no non-trivial polynomial P satisfying the algebraic differential equation P(g,g, . - . ) = 0, where g is the solution of a differential equation, associated to the Lie algebra £ generated by A(t) = ~i':~=1 ai(t)Ai over [0, 7"];

then, the time varying heat exchanger network (18) is s-controllable and s-observable, respectively.


4.1. Difference between constant-parameter and time-varying parameter cases

5. HEAT EXCHANGER NETWORK EXAMPLE FOR CHECKING INPUT AND

OUTPUT CONNECTABILITY

It has been shown already that for heat exchanger networks the generating structure matrices of the state space matrix [A] has full s- rank. If in addition the system is input-, output-

'~ A connectable, then the pairs ( ~'~i=i[ i], [B] ) and ( ~im=i[Ai], [C] ) are always s-controllable and s-observable, respectively. However, such time-varying systems can easily be shown where constant linear combinations of the generating structure matrices do not give controllable, observable pairs, but the time-varying system is controllable and observable.

Consider a simple case study presented in (Sagli et al., 1990). The best 6-unit design resulting from the mathematical programming method consists of three heat exchangers, two heaters and one cooler (Fig. 6).

H2

Example Let

0 1 1 ) A = 1 0 0 ,

1 0 0 C = ( 1 0 0 ) ; (010) A1 = 1 0 0 ,

0 0 0 A = At + As

B = (1)

0 , 0

0 0 1 ) As = 0 0 0 ,

1 0 0

The pairs (aiA1 + a2A2, B) and (aiAi + a2A2, C) are uncontrollable and unobservable Val, a2 E 7~. However, the time-varying system

~(t) = (al(t)A1 + a2(t)A2) x(t) + Bu(t) (21) ~(t) = C~(t)

Fig. 6. Heat exchanger network example

It is shown on this example how to construct the system matrices of the overall network from that of units and the network topology. The digraph model of the network is generated from structure matrices. The system structure matrices of countercurrent heat exchangers modelled as a cascade of three cells are as follows (cf Eq. (8) and (9)) where the indeterminate entries are denoted by the symbol *:

[A vf ] =

• 0001) 0 • * 0 * 0 0 • 0 * * 0 0 0 * * 0 0 0 * 0 * 0 0 0 0 *

(23)

may be controllable and observable, because the Lie algebra l: generated by A(t) is 3-dimensional with three basis elements At, As and

0 0 0 ) A3 = 0 0 1 (22)

0 - 1 0

[L p/] =

/i ° 0 0 0

[CV/]= ( 0 * 0 0 0 0 0 0 0 • 0)0 , i = 1 , 2 , 3

\

For this, the the pairs [ ~ ' = 1 c~iAi, B ) ,

( ~ = i 6 q A i , C ) are controllableandobserv-

able, respectively ; therefore, the controllability and observability of the time-varying system depend only on the differential algebraic condition of Proposition 2 which can be fulfilled by the terms at(t), a2(t). Note that the above example system is neither structurally stable, nor does it sat- isfy any conservation laws (Hangos, 1991,1992); therefore it is almost never found in chemical engineering practice.

(,oo) [A he]= * * 0 , [LhC]=

0 * * (24)

[B he]= , [C he]=(0 0 , )

Applying the connection algorithm described in Section 2.5 to the matrices above the resulting system structure matrices are in the following block

matrix form:

[A] = ([A0~I] 0 [A13] 0 0

[API] 0 0 0 [A32] [A pI] 0 0

/ [A41] 0 0 [Ah¢] 0 / 0 [A41 ] 0 0 [A hc ] \ o 0 [A6a] 0 0

/ [L,] [[L2]

ILl= /[L~] ,

[c]=([c,] o o

with the block matrices

0 0 0

[B]= [B41 [B~] [Bd

[C~] [C~]

ILl] =

[B4] =

(;00000) ~ 0 0 0 0 0

[A13]= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 * 0 0

(; [A32] = O0

[A41] = ( !

0 0 0 0 0

0 0 0 0 0

0 0 0

:,1¢ 0 0

0 0 0 0 0 0 0 0 0 0 0 0 ' 0 0 0 0 0 0

(; [L3]=

* 0 O) * 0 0 , * 0 0

(

000;)0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

°°.i) 0 0 0 0 0 0 ooo ) 0 0 0 0 0 0

Heat Exchanger Networks

[Ah¢] ]

(25)

[C6])

(i* 0i} [L21= o o 0 0 0 0 0 •

1417

oo;) 0 0 0 0 0 0 0 0

(i* o) [Bs] = * 0 • 0

[C4] =

[el] =

(! (i* 0

0 0

0 0 0

0 0 0 0 0 0 0 0

[C5] =

[cd = o 0 0

0 0 0

(!°i)oO°

0 0

, Hi, Ci

tr 7- ~ a ,

HEi

Fig. 7. The digraph of the heat exchanger and the utility heat exchangers

Fig. 7. shows the elements of the digraph of the heat exchanger network example: the digraph of the heat exchangers and the utility heat exchangers as half heat exchangers. The overall digraph describes the system given by matrices (25). The nodes of the digraph correspond to states and the directed edges correspond to nonzero elements in system structure matrices. The inputs (ui , i = 1 , . . . , 3 ) , disturbances ( z j , j = 1 , . . - , 4 ) and outputs (yk, k = 1 , . . . , 4) are the manipula- tors of heaters/coolers, the temperatures of input streams and the temperatures of output streams of the network, respectively.

The subgraph of each heat exchanger is a strong component, so it can be condensed into one node and denoted by a filled box. The subgraph of the utility heat exchangers containing input variables (one for each) are denoted by a filled cir- cle, while output variables are depicted as tri- angles and disturbances as boxes. The resulting condensed graph is shown on Fig. 8. In order to check input and output connectability directed paths are searched from inputs to state nodes and from state nodes to outputs.

Structural observability There is a path from any of the state nodes to at least one of the measured


HE1 .[>

HE3 yl

y3 ul C1 z3

~ HE2 z4 u3 y2 H2

Y4 U2

6.1. Apphcation /or integrated process and control system synthesis

Process synthesis is the first step of process design when only the operating units and their connections, i.e. the flowsheet has been determined. The process flowsheet can be described by a directed bipartite graph, the P-graph (Friedler, et al., 1992, 1993) which contains the materials and the operating units involved as its vertices.

Fig. 8. The condensed graph of the heat exchanger network example

outputs, so the network is output connectable and therefore structurally observable.

Structural controllability There are only paths from inputs to the state nodes of the heaters and the cooler, so the states of the full heat exchangers are not input connectable and therefore the network is structurally not controllable. In practice, however, only the input-output controllability is required. It is fulfilled for the three outputs (Yi, i = 2, 3, 4) which are input connectable. How- ever, to control the output Yl one has to place a bypass where the input variable is the bypass fraction (see Wolff, et al., 1991; Mathisen, et al., 1991, 1992a, 1992b).

6. CONCLUSIONS AND DISCUSSION

A simple method for analyzing the structural controllability and observability of heat exchanger networks is proposed in this paper, using only qualitative information about heat exchangers and network topology. Based on the structural properties of the overall state-space model of the network and on the necessary and sufficient conditions for structural controllability and observability it is shown that in both the constant and time-varying parameter cases the structural controllability and observability can be determined only from the input and output connectability of the network. The results are derived for the case when no bypass is used as input. The results can be easily extended to the general case by lin- earizing the bilinear model of the heat exchanger around a given steady state.

Output controllability is more important for practical purposes than state controllability and observability, but these are necessary preconditions (for at least one state) for the output controllability. The simplicity of the method and its need for only qualitative information makes it a good candidate for evaluating the control properties of a heat exchanger network, even in the early stages of the design, and possibly in interaction with the design method or tool.

The conventional approach is to design a control system for a process with its structure already fixed or to evaluate the influence of this structure on the design of its control system; it appears that no serious attempt has been made to fully and systematically integrate process design with design of its control system. A graph-theoretical approach to integrated process and control system design has been proposed in (Hangos, et al., 1994). The foundation of this integration is the above well-established, graph-theoretic approach to process synthesis in conjunction with the analysis of structural controllability based on digraph- type process models.

A directed bipartite graph, the CP-graph, has been introduced for unambiguous representation of an integrated process and control system (IPCS) structure. The notion of the CP-graph has given rise to a set of axioms for describ- ing the combinatorially feasible and controllable structures. The maximal controllable structure of an IPCS synthesis problem has been defined as the union of combinatorially feasible and controllable IPCS structures; obviously, the optimal IPCS structure must be a substructure of this maximal controllable structure. Thus, the mathematical programming model, e.g., the MINLP model, of an IPCS synthesis problem should be derived from the maximal controllable structure. The fundamental combinatorial algorithm of IPCS synthesis, i.e., algorithm CMSG, for identifying this maximal controllable structure has been formu- lated.

7. ACKNOWLEDGEMENT

This work has been supported by the Hungarian National Science Foundation (OTKA grant num- bers 2577 and T017108 ), which is gratefully ac- knowledged).

8. REFERENCES

Calandranis, J. and G. Stephanopoulos (1986). Structural Operability Analysis of Heat Ex- changer Networks. Chem. Eng. Res. Des, 64, 347-364.


Floudas C.A. and I.E. Grossmann (1978). Au- tomatic Generation of Multiperiod Heat Exchanger Network Configurations. Comp. Chem. Eng., 11, 123-142.

Friedler F., K. Tarjan, Y.W. Huang and L.T. Fan (1992). Graph-Theoretic Approach to Pro- cess Synthesis: Axioms and Theorems. Chem. Engng. Sci., 47, 1973-1988.

Friedler F., K. Tarjan, Y.W. Huang and L.T. Fan (1993). Graph-Theoretic Approach to Process Synthesis: Polynomial Algorithm for Maximal Structure Generation. Comput. chem. Engng., 17(9), 929-942.

Hangos, K.M. (1991). Structural Properties of Complex Linear Dynamic Systems with Ap- plication to Heat Exchanger Networks. Re- search Report No. 41/1991. Computer and Automation Institute of Hungarian Academy of Sciences.

Hangos, K.M. (1992). Structural Control Proper- ties of Technological Networks. Mdrds ds Au- .tomatika, 40, 12-15.

Hangos, K.M., F. Friedler, J.B. Varga and L.T. Fan (1994). A Graph-Theoretic Approach to Integrated Process and Control System Syn- thesis. Prep. of 1FAC Workshop on Integra- tion of Process Design and Control, 58-63.

Jacobson, N. (1979). Lie Algebras. Dover Publi- cations, Inc., New York.

Mathisen, K.W., S. Skogestad and E.A. Wolff (1991). Controllability of Heat Exchanger Networks. AIChE Annual Mtg., Los Angeles, USA.

Mathisen, K.W., S. Skogestad and T. Gunder- sen (1992a). Optimal Bypass Placement in Heat Exchanger Networks. AIChE Spring Nat. Mtg., New Orleans, USA.

Mathisen, K.W., S. Skogestad and E.A. Wolff (1992b). Bypass Selection for Control of Heat Exchanger Networks. European Symposium on Computer Aided Process Engineering-1 (ESCAPE-l), (ed. R. Gani), Supplement to Comp. Chem. Eng., 263-272.

Mathisen, K.W., M. Morari and S. Skogestad (1994). Dynamic Models for Heat Exchangers and Heat Exchanger Networks. Comp. chem. Engng., 18, $459-$463.

Pontrjagin, L. (1946). Topological Groups. Princeton, University Press.

Reinschke, K.J. (1988). Multivariable Control. A Graph-theoretic Approach. In: Lecture Notes in Control and Information Sciences (ed. M.Thoma and A.Wyner), Springer-Verlag.

Roberts , S.F. (1976). Discrete Mathematical Models with Applications to Social, Biological and Environmental Problems. Prentince Hall, Englewood Cliffs (New Yersey).

Sagli B., T. Gundersen and T.F. Yee (1990). Topology Traps in Evolutionary Strate- gies for Heat Exchanger Network Synth~ sis. EFChE Conference on Computer Appli- cations in Chemical Engineering, (ed. H.Th. Bussemaker and P.D. Iedema), Elsevier, Am- sterdam, 51-58.

Stephanopoulos, G., G. Henning and H. Leone (1990). MODEL.LA. A Modeling Language for Process Engineering - II. Multifaceted Modeling of Processing Systems. Comp. Chem. Eng., 14, 847-869.

Szigeti, F. (1992). A Differential-Algebraic Con- dition for Controllability and Observability of Time Varying Linear Systems. Submitted paper to 31th IEEE Conf. on Decision and Control.

Varga E.I. and K.M. Hangos (1992). The Effect of Heat Exchanger Network Topology on the Network Control Properties. 2nd IFAC Workshop on System Structure and Control, (Prague, Czechoslovakia), 220-223.

Wolff E.A., K.W. Mathisen and S. Skogestad (1991). Dynamics and Controllability of Heat Exchanger Networks. Proc. of Computer Ori- ented Process Engineering (COPE-91), (ed. L. Puigjaner and A. Espuna), 11%122.

controllability and observability of heat exchanger networks in the time-varying parameter case

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