Closed-loop stability analysis of discrete-time negative imaginary systems

We derive necessary and sufficient conditions for stability analysis of a positive feedback interconnection of a discrete-time negative imaginary system and a discrete-time strictly negative imaginary system. General stability analysis results for continuous-time negative imaginary systems connected in positive feedback have recently been proposed. Those recent results extend previous theorems by removing restrictive assumptions on the infinite frequency gains imposed in the earlier literature and by extending the class of negative imaginary systems for which the results are applicable to include systems with free body dynamics (i.e., poles at the origin). Here, we present the discrete-time counterparts of the aforementioned recently developed results which specialise to simple and easy-to-check conditions under specific assumptions. Last, we illustrate some of the results by several examples.