Automatica, Vol. 17, No. 2, pp. 371 377. 1981 Printed in Great Britain 0005-1098/81/020371-07 $02.00/0 Pergamon Press Lid q 1981 International Federation of Automatic Control Brief Paper A Graph Theoretic Approach to Muitivariable Control System Design* C. SCHIZAS and F. J. EVANSt Key Words---Multivariable control systems; graph theory; controllability; large scale systems; distur- bance rejection. Abstract--A graph theoretic approach is described for the design of multivariable control for large systems as an alternative to geometric methods. An example is given for a distillation column to demonstrate the technique, with a particular reference to aspects of disturbance rejection and the possibilities for pole assignment. 1. Introduction A RECENT PAPER (Takamatsu et al., !979) described the applications of the geometric approach of Wonham (Wonham, 1979) to the design of a multi-variable controller for a distillation column. Such application studies are not common, and a contributing factor to this may well be the abstract nature of the underlying theory. It is suggested in this paper that the mapping of such problems into a more tangible graph-theoretic form can produce a more lucid and flexible design procedure. However, the ease of design which the procedure offers is dependent on the adoption of a process model (i.e. the choice of a particular state vector) that has the closest relationship with the physical aspects of its operation. If such a model is mapped into a digraph then the properties of this physical structure will also appear in the digraph itself. Therefore any visual inspection of this digraph will also reveal a realistic relationship between states, inputs, distrubances and output on the real process. It is accepted that any linear transformation of states, for whatever reason, will almost certainly increase the complexity of the representative digraph, and thus destroy the ability to relate back easily to the physical process any observations. This is not to say that a reduction of the digraph com- plexity is in itself a desirable objective, as is clearly de- monstrated if the eigenvector similarity transformation is adopted to give a diagonal form. The same weakness of loss of physical significance occurs in this case as in one where the digraph had been made unnecessarily complex. In order that a useful comparative study be made it is proposed to adopt the distillation column model referred to above, and to perform a parallel design study to that already deacribed by Takamatsu et al. In this study the modelling of the plant is discussed in detail, and the model so derived supports the validity of the digraph approach, since the states are the conventionally accepted variables for this type of plant. *Received February 11 1980, revised July 31 1980, revised September 23 1980. The original version of this paper was not presented at any IFAC Meeting. This paper was recom- mended for publication in revised form by associate editor A. van Cauwenberghe. ?Department of Electrical and Electronic Engineering, Queen Mary College, University of London, Mile End Road, London E1 4NS, England. 371 2. Theoretical preliminaries 2.1 Equations into digraphs. Kevorkian has previously sug- gested how both linear and non-linear state space equations can be mapped into a directed graph (digraph) (Kevorkian, 1975). Furthermore, the use of such a graph has already been exploited in studies of controllability and observability in large scale systems (Franksen et al., 1979a, b). A short summary of this earlier work will be given for the sake of completeness. The digraph formulation to which we refer is a diagram- matic representation of the information flow process which occurs during the numerical solution (i.e. the evolution in time) of the system. To illustrate this mapping we will restrict ourselves to the conventional linear state space form ~=Ax+Bu y=Cx + Du (1) although, as Kevorkian has shown there is no such restriction in principle. The extensions of this approach to non-linear problems remains to be done. if we consider first the auto- nomous case ~k=Ax. We can rewrite this in the form Fi(xi,x i ..... xn)=O i=1 ...n as in the following example Ft = --~j +x2 +x3-0 F2=-x2+xt+Xz+X3=0 F3= - .'(3 -~- x3 = O. We will now form the following "occurrence array' XI X2 X3 "~'1 '(2 -'(3 F l 0 1 1 @ 0 0 F 2 1 1 1 0 C) 0 F3 0 0 1 0 0 @ DI C) 0 0 1 0 0 D 2 0 C) 0 0 1 0 D 3 0 0 @ 0 0 1 in which the D~ rows represent the implicit relationship existing between ~ and xv For each functional row an 'output' is shown ringed, and each function and its output is now mapped into a digraph shown in Fig. 1. We shall now assume that as the arc ~-~x~ must always exist that a 'contraction' of Fig. 1 is now possible into a reduced digraph shown in Fig. 2. It is now immediately apparent that Fig. 2 could have been