Multidimensional Behavioural Compensator Design

The aim of this project from the area of mathematical systems theory is to design and parametrise stabilising compensators which solve additional control tasks, for example disturbance rejection, tracking and model matching. The control design problems will be treated in the behavioural framework of multidimensional linear systems. Usually the description of a compensator`s task involves a notion of negligibility, e.g., a given reference signal has to be tracked up to a negligible deviation. To take this into account, we intend to use polynomial matrix methods as well as Serre subcategories and Gabriel localisations. Using these techniques we can treat many different notions of negligibility of signals under a single mathematical framework. Standard examples for negligible signals which can be treated with these methods are the Hurwitz stable ones in the continuous case and the Schur stable ones in the discrete case. Furthermore, the robustness of the compensators under small errors in their implementation as well as under small variations of the data defining the plant will be investigated.

The two main results of this project are on the one hand in the area of mathematical systems theory the design and the parametrisation of stabilising compensators which solve the additional control tasks of disturbance rejection and signal tracking, and on the other hand in the field of computer algebra and symbolic computation a completed theory of Gröbner bases for modules over monoid algebras as well as a new and efficient algorithm for computing these Gröbner bases.The control design problems (first result) have been treated in the behavioural framework for multidimensional linear systems. Usually, the description of a compensator's task involves a notion of negligibility, e.g., a given reference signal has to be tracked up to a negligible deviation. In order to take this into account, we have used the method of Serre categories and Gabriel localisations. This technique has enabled us to treat many different notions of negligibility of signals in a single mathematical framework. Standard examples for negligible signals which can be treated with these methods are the Hurwitz stable ones in the continuous case of linear partial differential equations and the Schur stable ones in the discrete case of partial difference equations.The investigated monoid algebras (second result) consist of polynomials in several variables where only certain exponents? which can also be negative, though? are permitted. It must, however, be ensured that the product of two such generalised polynomials is again an element of the monoid algebra. This requirement leads to the condition that the exponents form a submonoid of the integer grid. For polynomials in several variables, Gröbner bases have a similar significance as the greatest common divisor has for polynomials in one variable, and they are an important standard method in computer algebra. With the results of this project, this method can now be applied in much more general situations.