l-delay input and initial state reconstruction for discrete-time linear systems

Prior results on input reconstruction for multi-input, multi-output discrete-time linear systems are extended by defining l-delay input and initial-state observability. This property provides the foundation for reconstructing both unknown inputs and unknown initial conditions, and thus is a stronger notion than l-delay left invertibility, which allows input reconstruction only when the initial state is known. These properties are linked by the main result (Theorem 4), which states that a MIMO discrete-time linear system with at least as many outputs as inputs is l-delay input and initial-state observable if and only if it is l-delay left invertible and has no invariant zeros. In addition, we prove that the minimal delay for input and state reconstruction is identical to the minimal delay for left invertibility. When transmission zeros are present, we numerically demonstrate l-delay input and state reconstruction to show how the input-reconstruction error depends on the locations of the zeros. Specifically, minimum-phase zeros give rise to decaying input reconstruction error, nonminimum-phase zeros give rise to growing reconstruction error, and zeros on the unit circle give rise to persistent reconstruction error.