On a geometric characterisation of zeros for non-square linear systems with time-delay in state

The concept of invariant zeros in a linear time-invariant system with point delay in state vector is discussed in the state space framework. These zeros are treated as the triples: complex number, non-zero state-zero direction and input-zero direction. Such treatment is strictly related to the output-zeroing problem and in that spirit the zeros can be easily interpreted. As is shown, for systems with matrix CB of full row-rank, general formulas for output-zeroing inputs can be obtained as well as a characterisation of invariant zeros as the roots of a certain quasi-polynomial can be given. The question of degeneracy/non-degeneracy of the system is also addressed. Moreover, it is shown that diagonal decoupling can be achieved by constant state feedbacks and a pre-compensator. The transfer matrix of the decoupled system is square and does not contain delay. The mathematical tools used in the analysis are the Moore–Penrose pseudo-inverse and singular value decomposition of a matrix.