Exact unknown-state, unknown-input reconstruction: A geometric framework for discrete-time systems

The complete solution of the unknown-state, unknown-input reconstruction problem in systems with invariant zeros is intrinsically limited by the fact that for any invariant zero, at least one initial state exists, such that, when the mode associated to the invariant zero is suitably injected into the system, the corresponding output is zero. Although in the awareness of this restriction, the problem of reconstructing the initial state and the inaccessible inputs from the available measurements is the object of a fair amount of research activities because of its impact on a wide range of applications, specifically those dealing with the synthesis of enhanced-reliability control systems. In this context, the present paper contributes a geometric method aimed at solving the exact unknown-state, unknown-input reconstruction problem in discrete-time linear time-invariant multivariable systems with nonminimum-phase zeros. The case where all the system invariant zeros lie in the open set outside the unit disc of the complex plane is regarded as the basic one. The difficulties related to the presence of those invariant zeros are overcome by allowing a reconstruction delay commensurate to the invariant zero time constants. The same technique also applies to the case of systems without invariant zeros. In the latter circumstance, however, the reconstruction delay is related to the number of iteration required by the algorithm for the computation of a specific subspace to converge. Finally, the more general case where the problem is stated for a system whose invariant zeros lie both inside and outside the unit disc of the complex plane is reduced to the basic problem referred to a new system, derived from the original one through a procedure aimed at replacing the minimum-phase zeros with their mirror images with respect to the unit circle.