Prior results on input reconstruction for multi-input, multi-output discretetime 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, nonminimumphase zeros give rise to growing reconstruction error, and zeros on the unit circle give rise to persistent reconstruction error.

As an extension of existing results on input reconstruction, we define l-delay state and input reconstruction, and we characterize this property through necessary and sufficient conditions. This property is shown to be a stronger notion of left invertibility, in which the initial state is assumed to be known. We demonstrate l-delay state and input reconstruction on several numerical examples, which 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.

An approximate input reconstruction algorithm is used to reconstruct unknown inputs, which are then used for fault detection. The approximate input reconstruction algorithm is a least squares algorithm that estimates both the unknown initial state and input history. The estimated inputs are then compared to the commanded values and sensor values to assess the health of actuators and sensors. This approach is applied to the longitudinal and lateral dynamics of NASA's Generic Transport Model. The input reconstruction algorithm can be used for systems with minimum-phase or nonminimum-phase zeros; however, minimum-phase zeros entail an additional delay in reconstructing inputs, while zeros on the unit circle yield persistent estimation errors and thus poor input reconstruction regardless of the delay.