Singular value distribution of non-minimum phase systems with application to iterative learning control

This paper provides a rigorous mathematical analysis on the singular value distributions of input-output matrices for discrete time non-minimum phase (NMP) systems. It is shown that when the time scale considered is sufficiently long, the input-output matrix of a NMP system has m infinitesimally small singular values, the rest of which are significantly large with a non-zero lower bound, where m is the number of NMP zeros in the NMP systems. It is the existence of these m nearly zero singular values that causes various difficulties in analysis and design for NMP systems. The corresponding singular vector spaces can also be characterised. The analysis results are further applied to a gradient-based iterative learning control algorithm to analyse a well-known problematic slow convergence phenomenon and numerical simulations are presented to verify the theoretical predictions.