In this talk, we give a very general overview of the theory of multidimensional systems using a behavioral framework. This framework relies on the idea that control systems are described by equations, but their properties of interest are most naturally expressed in terms of the set of all solutions to the equations. This is formalized by the relatively new notion of the system behaviour (denoted by B), due to Willems. We will focus our attention on systems described by partial differential equations with constant coefficients and in the relation between finitely generated modules and these behaviours. This enables the application of the huge and powerful machine of commutative algebra to problems in multidimensional linear systems theory. The main difficulty in using this relation is that it is highly abstract. However we will see how it is possible to give interpretation of some algebraic theory. We will explain the mathematical background needed and show how structural properties such as controllability or autonomy can be describe in terms of some algebraic properties.

In particular I will investigate in some detail the decomposition of a given behavior into the sum of finer components. It is inmediatly apparent that decomposition is a powerful tool for the analysis of the system properties. It is, indeed, of particular interest, the case of multidimensional systems where a description of the nD systems trajectories is quite complicated and decomposing the original behavior into smaller components seems to be an effective way for simplifying the systems analysis.

Another motivation for considering this problem is that the problem is also an interesting question from a purely mathematical point of view and allows different approaches.

The autonomous-controllable decomposition has played a significant role in the theory of linear time-invariant systems. Such decomposition expresses the idea that every trajectory of the behavior can be thought of as the sum of two components: a free evolution, only depending on the set of initial conditions, and a force evolution, due to the presence of a soliciting input. In the case of 1D systems, this sum is direct, i.e.

where B_cont and B_aut represent the controllable and autonomous part of B respectively.

However, this decomposition is, in general, not longer direct for n \geq 2, and we may have that the controllable part of B, (which is uniquely defined for a given B) intersects all possible autonomous parts involved in the controllable-autonomous decomposition [5, 6, 7].

Some of the results I will give are already known, and my main contribution will be a complitely new approach to the problem using algebraic geometry, which hopefully opens new ways to tackle open problems in multidimensional systems theory.

Finally we provide an algorithm to effectively solve our problem.

References

[1] J. Wood, E. Rogers, and D. H. Owens, Modules and Behaviours in nD Systems Theory, Mult. Systems and Signal Processing, 2000.
[2] J. C. Willems, On Interconnections, Control, and Feedback, IEEE Trans. on Auto. Contr.
[3] U. Oberst, Multidimensional Constant Linear Sys- tems, Acta Applicandae Mathematicae, vol. 20, 1990.
[4] H. Pillai and S. Shankar, A Behavioural Approach to Control of Distributed Systems, SIAM Journal on Control and Optimization, vol. 37, 1999.
[5] Wood, Jeffrey and Rogers, Eric and Owens, David H., Controllable and autonomous nD linear systems, Multidimens. Systems Signal Process., vol. 10, 1999.
[6] Valcher, Maria Elena, On the decomposition of two- dimensional behaviors, Multidimens. Systems Signal Process., vol. 11, 2000.
[7] Bisiacco, Mauro and Valcher, Maria Elena, Two-dimensional behavior decompositions with finite-dimensional intersection: a complete characterization, Multidimens. Systems Signal Process., vol. 16, 2005.