Minimal realisations of odd transfer functions for first-degree nD systems

ABSTRACT

Minimal realisation problems of odd transfer functions for first-degree (multi-linear) nD single-input single-output discrete systems have been studied, but it has not been well solved. This paper provides a new, different method to solve absolutely minimal realisation problems. By methods of limits and algebraic techniques, without using the symbolic approach by Gröbner basis, the requirements of absolutely minimal realisation are transformed into a system of equations represented by the determinants. Since the equations for first-degree 2D systems are solvable by quadratic equations and the conditions for higher-dimensional realisations can be expressed by the results of 2D systems, the absolutely minimal realisations for nD systems can be found by using the realisations of n(n − 1)/2 2D systems. Furthermore, the conditions for existence and construction of the absolutely minimal realisation for the lack of items and not missing two cases are derived from the Pfaffian function of the skew-symmetric matrix. Finally, two numerical examples for 3D and 4D systems are presented to illustrate the basic ideas as well as the effectiveness of the proposed procedure.

KEYWORDS:

• Multidimensional systems
• state-space realisation
• absolutely minimal realisation
• odd transfer function
• skew-symmetric matrix

Acknowledgments

The author thanks to the referees for their suggestions which have improved the original manuscript to the present version.

Disclosure statement

No potential conflict of interest was reported by the author.