Unknown-state, unknown-input reconstruction in discrete-time nonminimum-phase systems: geometric methods

The complete solution to the unknown-state, unknown-input reconstruction problem in systems with invariant zeros is inherently conditioned by the fact that, for any invariant zero, at least one initial state exists, such that the output is not affected when the mode of the invariant zero is properly injected into the system. Despite this intrinsic limitation, the problem of reconstructing the initial state and the inaccessible inputs from the available measurements has recently attracted remarkable interest, owing to its impact on the synthesis of enhanced-reliability control systems. This contribution consists of a geometric method which solves the unknown-state, unknown-input reconstruction problem in discrete-time systems with invariant zeros anywhere in the complex plane, except the unit circumference. The case of systems with the invariant zeros in the open set outside the unit disc is regarded as the basic one. The difficulties related to the presence of those invariant zeros are overcome by accepting a reconstruction delay commensurate to the invariant zero time constants and the accuracy required for reconstruction. The solution devised for that case also applies to systems without invariant zeros. However, in this case, reconstruction is exact and the delay depends on the number of iterations needed for a certain conditioned invariant algorithm to converge. Finally, the more general case of systems with invariant zeros lying anywhere in the complex plane, with the sole exception of the unit circumference, is reduced to the fundamental one through the synthesis of an appropriate filter.