Simultaneous State Observability and Parameters Identifiability of Discrete Stochastic Linear Systems

A new representation of discrete stochastic linear time-invariant systems is presented. This representation generalizes in a rigorous way the concept of observability to parameters identifiability. The state of the augmented system is a combination of the state of the orginal system and the unknown parameters. It is shown that simultaneous state observability and parameters identifiability of linear time-invariant system is an observability problem of an augmented linear time-variant system. It is shown that the well known results derived by Least Squares(LS) algorithms evolve as a special case of the new representation. The representation yields necessary and sufficient conditions on the simultaneous state observability and parameters identifiability. Sufficient conditions derived from the necessary and sufficient conditions are weaker than the well known persistent excitation conditions in the existing least squares schemes. These conditions apply to estimation in open and closed loop without further restrictions. This reestablishes the well known results for identification in closed loop. The observability analysis enables generalization of similar results for nonlinear time-varying feedback. Simulation results demonstrate the performance of estimation with this new approach.