SISO, x Unobserved, w = 0

Abstract

With respect to the case of observed state, fewer information accrues to the control device. The missing information can be reconstructed as time goes by. We use the fact that from any initial state, a sequence of inputs uniquely determines the sequence of outputs. If these sequences are memorized, one can pose an inverse problem: find the initial state compatible with the observations. From linear system theory we know this problem is solvable when (A, C) is observable. Once the initial state is known, recalling the past inputs permits to determine the current state. This procedure can be implemented online, i.e., while the system evolves, by means of an asymptotic observer, a device generating state estimates converging to the actual state. It is remarkable that using a f/b law from the estimated rather than the actual state, all assignment methods seen in the case of observed state—stabilization in particular—keep being successful.