Polynomial matrices associated with linear constant multivariable delay-differential systems

Matrices which can be identified as system matrices corresponding to (linear constant) multivariable delay-differential systems are considered. These matrices are extensions of the state-space and polynomial system matrices which are encountered in connection with multivariable ordinary differential systems. Whereas these latter matrices have elements which are polynomials in one variable, the matrices considered have elements which are polynomials in two or more variables. The matrices considered are treated in two ways. In the first approach the results available for matrices corresponding to ordinary differential systems can be readily extended to results for those corresponding to delay-differential systems. However, the main intention is to consider the extension of results without using this approach. Several results are in fact established for the matrices under consideration. Many of these results involve the new concept of zeros of matrices of the form considered. Although the second approach to these matrices is treated initially as a purely mathematical exercise, it is then shown that there is some physical justification for this approach. This is done by consideration of results concerning the controllability of delay differential systems. In fact, the question of controllability of such systems is considered in some detail, not simply with a view to justification of the preceding results. The concept of observability is also considered, but not in the same detail. In the concluding chapter another type of system matrix which can be used in the treatment of delay differential systems is considered. Such a matrix is considered in the context of the results obtained in the preceding chapters, and the connections between the results given for this form of system matrix and results previously obtained are examined. Again the concepts of controllability and observability are considered.