Algebraic issues in linear multi-dimensional system theory

1-D Multivariable system theory has been developed richly over the past fifty years using various approaches. The classical approach includes the matrix fraction description (MFD), the state-space approach etc., while the behavioural approach is relatively new. Nowadays, however there is an enormous need to develop this theory for systems where information depends on more than one independent variable i.e. the n-D system theory (n ≥ 2), due to the vast number of applications for these kind of systems. By contrast to the 1-D system theory, the n-D system theory is less developed and its main aspects are not yet complete, where generalising the results from 1-D to n-D has proved to be not straight forward nor smooth. This could be attributed to the n-D polynomial matrices which are the basic elements used in the analysis of n-D systems. n-D polynomial matrices are more difficult to manipulate when compared to the 1-D polynomial matrices used in the analysis of 1-D systems, because the ring of n-D polynomials to which their elements belong does not possess many of the favourable properties which the ring of 1-D polynomials possesses. The work proposed in this thesis considers the Rosenbrock system matrix and the matrix fraction description approaches to the study of n-D systems. [Continues.]