On the analysis of discrete linear time-invariant singular systems

The discrete singular equation over an interval can represent a two-point boundary-value problem or it can be considered as a dynamical relation developing forward in time. A theory is provided, by giving analytic solutions and discussing system properties in both cases, that encompasses both interpretations. The singular system fundamental matrix is used to provide analytic solutions to a time-invariant discrete singular equation defined over an interval. The two distinct cases in which the singular relation is interpreted as a two-point boundary-value problem or as a forward dynamical system on the interval are both considered. Also considered is the case in which the singular relation is considered as a backward dynamical system, and the relationship between the index of nilpotence and the Laurent series coefficients resulting from the solutions of certain state-space equations is shown. Reachability and observability are discussed, and the point is made that these properties are different, depending on how the singular relation is interpreted