On the normal forms of linear time-delay systems with application to output feedback stabilization

This paper mainly investigates the normal form of linear systems with non-commensurate state, input and output delays. This work begins with introducing delay operators, so that the linear time-delay system can be associated with a system over a multivariate polynomial ring. A design procedure for constructing a null-space matrix and its left inverse matrix is proposed based on the equivalence of polynomial matrices, such that under the assumption of a pure relative degree, a novel construction method of the state transformation matrix, referred to as the unimodular completion method, can be given. Then the system can be transformed into a normal form, which contributes to exploring some properties of the system, including invariant zeros, spectral controllability, spectral observability and system inversion. As an application of the normal form, the output feedback stabilization problem is addressed. Numerical examples are given to illustrate the validity of the obtained results.