An Arnoldi Based Method to Discrete Time Linear Optimal Multi-periodic Repetitive Control

For LTI plant, a benchmark tracking solution was recently proposed by optimal technique for discrete time linear multi-periodic repetitive control system that gives asymptotic perfect tracking, where the original tracking problem was transfered into a regulator problem by developing a new state-space representation that combines the plant and demand signal. However, in practice, the periods of the demand signals are usually very large, therefore the dimension of this new plant description increases and that naturally leads to high order computations for solving discrete Riccati equation. In order to overcome this problem, an Arnoldi based method is applied in this paper to first reduce the high order state-space representation to a low order counterpart, then a direct method is used to solve its corresponding low order riccati equation. Finally a projection between two riccati solutions is applied to retrieve an approximate high order riccati solution. This result also leads to a new multi-periodic repetitive controller in terms of low order riccati solution. A numerical example is given and asymptotic perfect tracking is guaranteed.