Polynomial Algebraic Methods for Modeling, Analysis and Control of Distributed Physical Systems

The research described in this proposal aims at providing theoretical tools and algorithms based on multivariable polynomial algebra to deal with some specific problems regarding the stability, modeling, and control of distributed systems.Linear distributed (also called ``N-D'') systems arise naturally when considering the modeling of physical phenomena whose evolution depends not only on time but also on space, for example in image processing, seismology, circuit theory, flow-control, and iterative learning control. In the research proposed here we aim at studying open distributed systems described by systems of linear constant-coefficient partial difference/differential equations. A system of this sort is described by a set of linear, constant-coefficient partial differential/difference equations and it is naturally associated with a polynomial matrix in N indeterminates, in the sense that properties of the dynamical system are reflected in the algebraic properties of the polynomial matrix. The interplay of systems dynamics and functionals typical of many applications (for example in stability theory, in optimal control, etc.) also can be described effectively by polynomial matrices (in 2N indeterminates). Using the framework outlined above, we aim at using polynomial algebraic methods for the computation of first-order representations; and for the study of functionals arising in the analysis of physical systems. In both areas the expertise of the Principal Investigator and of the Host Investigator is considerable and complementary. The project is articulated in two work packages. 1) COMPUTATION OF FIRST-ORDER REPRESENTATIONS OF 2-D SYSTEMS First-order representations, corresponding to state-space equations in the 1-D case, are of primary importance for simulation, filtering, and so forth because of the ease of update. However they cannot be considered a starting point for the description of a system, but they need to be constructed from a higher-order model consisting of the interconnection of simpler subsystems, for example obtained from a library of standard models for certain components. The problem thus arises of how to compute a first-order representation of a system described by a set of higher-order partial differential or difference equations. In this work package we aim at investigating the computation of first-order representations for distributed systems. Crucial in this investigation is the notion of state map, developed in the 1-D case by the principal investigator, and the notion of Markovianity for 2-D systems, studied by the Host Investigator. The possibility of deriving automatically a first order representation from a set of higher-order partial differential/difference equations is particularly relevant for applications in simulation of 2-D systems. 2) FUNCTIONALS IN THE DESCRIPTION AND ANALYSIS OF PHYSICAL SYSTEMSThe interplay of dynamics and functionals is often considered in systems and control, for example in stability analysis, in the modeling of physical quantities, and in optimal control. In this research stream we investigate some themes related to this area, in particular the development of polynomial algebraic algorithms to compute conserved- and zero-mean quantities for 2-D systems; the development of algorithms for the solution of important polynomial matrix equations arising in the analysis and control of 2-D systems, for example the Lyapunov equation; and a representation-free approach to Hamiltonian systems. Automating the computation of these quantities opens up the possibility of using computer algebra for energy-based modeling and control methods such as those used in engineering practice, to be integrated in computer-aided simulation and analysis tools.

The work outlined in this proposal has the potential to make a major impact in a scientific area, that of 2-D systems and signals, which is of interest to a number of scientific and engineering disciplines, from physics to circuit theory, from flow-control to radar signal propagation. Despite the importance of these applications, this is an an area where much basic work remains to be done before its potential can be fully realized. Computer algebra provides the opportunity to use computational tools of remarkable power which fit as a hand fits a glove many of the theoretical methods already available, or to be developed within this project. In the following we first describe the scope for impact in general terms, before moving to specific plans for dissemination of the results of this research. The proposed research aims at providing theoretical and algorithmic tools for the solution of specific problems in the modeling, analysis and control of 2-D systems using polynomial algebra. The first research stream considered in this proposal deals with the setting up of first-order models from higher-order ones. This important problem arises whenever a complex model, coming for example from the linearization of a system of nonlinear partial differential equations such as those arising in flow-control, needs to be simulated. Consequently, any advancement in this research stream is bound to have important effects in all those areas where an automatic derivation of easy-to-update dynamical models is important, most prominently the simulation and filtering of 2-D systems and signals as they arise in seismology, in aerospace- and naval engineering, etc. The second research stream is equally relevant for practical applications. The use of conservation laws, equipartition of energy principles, and so forth has a long history of application in engineering and in physics. To name but one recent application, energy-flow methods have been recently used in order to do model reduction, i.e. the derivation of a simplified model from a complex one, so as to obtain a mathematical description more amenable to be used efficiently for control and simulation. The potential of the techniques studied in this research stream for application in such different areas as civil and aerospace engineering, for example for the analysis of the vibrations of complex structures, is still to be exploited. Finally, the last research stream to be investigated comprises issues which are of paramount importance whenever a system is to be controlled in order to obtain a desired behaviour. This is a problem of great importance for example in the stabilization of vibrating structures. The results of the research proposed here will be submitted for publication in the leading international control journals. Moreover, partial results will be presented during the leading international conferences in the field, thus ensuring on the one hand the widest possible dissemination of results, and on the other hand guaranteeing contacts with the most prominent researchers. The results will also be presented on a more local scale in seminars and national conferences. Moreover, the School of Electronics and Computer Science of the University of Southampton stipulates that all publications and preprints be made publicly available via the e-prints archive eprints.ecs.soton.ac.uk. There will be beneficiaries of the research also outside of academia. Given the importance within the project of the development of algorithms, its relevance for the engineering community is very high. We also plan to develop software so as to put the algorithms developed here to the test of actual engineering models, and to provide the scientific and engineering community with computer-algebra-based software tools that can be used in practice. This software will be made publicly available through the Web pages of the University of Southampton.