Graph Modelling Approach Application To A Distillation Column  

Vincent Hovelaque,Christian Commault, Jean-Michel Dion, Mehrdad Bahar,  Jan Jantzen  

Laboratoire d'Automatique de Grenoble 
INPG/UJF/CNRS UMR 5528 
ENSIEG 
38402 St Martin d'Heres 
FRANCE 
commault@lag.ensieg.fr 
Technical University of Denmark 
Department of Automation 
Bldg. 326, DK-2800 Lyngby 
DENMARK 
jantzen@iau.dtu.dk 


Abstract: In this paper, structured systems described by state space models are considered. For these systems, the entries of the state space model matrices are supposed to be either fixed zeros or free independent parameters. With such systems, one can associate a directed graph which is a useful tool to study some control properties of systems. In this context, we present here an illustrative application of disturbance rejection and input-output decoupling problems on a distillation column model. 

Keywords: Structured systems, graph theory, input-output decoupling, disturbance rejection 

Vincent Hovelaque graduated in Applied Mathematics and Informatics in 1992. He received the DEA degree of Operations Research from the Institut National Polytechnique de Grenoble, France, in 1994. He is now a Ph.D student at the Institut National Polytechnique de Grenoble attached to the Laboratoire d'Automatique de Grenoble from l'Ecole Nationale Superieure d'Agronomie de Rennes. His research interests include the analysis of structured systems via graph theoretical and geometrical points of view. 

Christian Commault received the Electrical Engineer degree, the Docteur-Ingenieur degree and the Docteur d'Etat degree from the Institut National Polytechnique de Grenoble in 1973, 1978 and 1983 respectively. From 1974 to 1976, he taught at the Dakar Institute of Technology , Senegal. Since 1979, he has taught automatic control and manufacturing systems at the Ecole Nationale Superieure d'Ingenieurs Electriciens de Grenoble. In 1978, he spent one year as a visiting researcher at the Mathematics Institute of Groningen , The Netherlands. From 1986 to 1988, he worked at the Renault Research Center on design methods for manufacturing systems. His research interests are in linear multivariable control and performance evaluation of production systems 

Jean-Michel Dion was born at La Tronche, France, in 1950. He received the BSc. degree in Mathematics in 1972. He received the These de 3eme cycle and These d'Etat degrees both from the Institut National Polytechnique de Grenoble in 1977 and 1983 respectively. Since 1979, he has been a researcher at the Centre National de la Recherche Scientifique where currently he is Directeur de Recherche and Head of the Laboratoire d'Automatique de Grenoble. He is also vice president of the Institut National Polytechnique de Grenoble. His current research interests include linear systems, robustness and time-delay systems. 

Mehrdad Bahar was born in 1965. He received the MSc. degree in Electrical Engineering from the Technical
 

Jan Jantzen, born in 1953, is an Associate Professor at the Technical University of Denmark (DTU). He received the MSc. degree in Electric Power Engineering from DTU (1979) and Ph.D in Systems Science from DTU (1982). From 1979 to 1982 he was systems designer at LK-NES, Inc. For two years (1982-1983) he was Queen's Quest Visiting Scholar at Queen's University, Kingston, Canada. As computer consultant, he spent the next two years (1984-1985) with SimCorp, Inc. In 1986 he was visiting scientist at IBM T.J.Watson Research Center, and Assistant Professor at DTU. Later (1990) he became Associate Professor and (1993) part time consultant. 

1.Introduction
In this paper  we consider linear systems represented by a quadruplet (A,B,C,E) where the entries of (A,B,C,E) are either fixed zeros or free parameters. With such systems, called structured systems, one can associate a directed graph in a natural way [11,12,13]. One can study structural properties, i.e. properties which are true for almost all values of the parameters. Most of these properties can be obtained from properties of the associated graph. Structural properties have been extensively studied during the last twenty years following [11]. 

For such systems, the generic infinite structure can be deduced from the associated graph [5,13,21]and corresponds to sets of vertex disjoint input-output paths. As an application, the structural solvability conditions of classical control problems can easily be checked on the associated graph. For instance, the disturbance rejection problem has been considered in [4,5,9,21,22]. 

The decoupling problem has been studied in [6,12]. Efficient algorithms to determine this infinite structure and to solve control problems have been proposed in Differently, \cite{RE:88,SCEV:81} proposed some graphical structural studies for finding the feedback configuration of control problems like input-output decoupling and disturbance rejection. Structural numerical techniques have been developed for finding such feedbacks as those which have led to a MATLAB toolbox . 

The purpose of this paper is to combine the above two structural techniques in order to improve the existing procedures for input-output decoupling and disturbance rejection. The goal is to combine generic infinite structure conditions for such control problems, with graphical techniques for finding the feedback configuration.\\ These approaches are illustrated on a 13 tray binary distillation column model represented by classical state space equations.\\ The outline of this paper is as follows. We recall first some basic properties of applied graph theory and some results in structured system analysis. In Section 3, the distillation column model is presented and its associated graph is depicted. In Section 4, we discuss the input-output decoupling problem, show the existence of a feedback control law and compute it. Section 5 deals with the state feedback disturbance rejection problem. 


We show that this problem is generically solvable only if one of the disturbances is available for measurement, and calculate the feedback control law by both graphical and geometrical approaches. We end this paper with some concluding remarks in Section 6. 

























6.Concluding Remarks.

The above results present a graphical approach to the generic input-output decoupling problem and the generic disturbance rejection problem for a distillation column plant. 
Even if the parameters of a state space model were undecided, such that numerical computer routines cannot be applied, the structure of the model would still provide valuable information. The "digraph approach" focuses on the pattern of non-zero entries in the model, and it provides sufficient structural conditions for decouplability and disturbance rejection. 
For the considered distillation column, the analysis shows that after input-output decoupling, the system can be decomposed into a number of sub-systems, making it easier to study such problems as disturbance rejection and pole placement. The structural form of the feedbacks for input-output decoupling or disturbance rejection is given. A parameterization of such feedbacks in terms of the entries of the original system is also provided. 
The graph approach may also provide a way to cope with uncertainty and varying parameters. In practice the structure of a model may be known from professional insight into the physics of the system, but the parameters may vary \mbox{according} to the operating point or temperature variation. The digraph approach can nevertheless test necessary conditions for decoupling purely on structural grounds. The approach is typically applied in the following situations: the parameters of a model are undecided or vary; there is large sensitivity towards small variations; the parameters have been decided, but, the necessary matrix operations are numerically intensive. 

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