Combined state and parameter estimation and identifiability of state space realizations

The objective of combined state and parameter estimation is to estimate both unmeasured states and unknown entries of the dynamics matrix. Since the dynamics involve products of states and parameters, this is a nonlinear estimation problem. The classical approach to this problem is to use the extended Kalman filter, although more recent techniques, such as the unscented Kalman filter, can be used. The goal of this paper is to determine conditions under which the combined state and parameter estimation problem is feasible. To do this, we recast this problem as an identifiability problem and, for several special cases, we develop necessary and sufficient conditions for identifiability, which provides necessary and sufficient conditions for feasibility of the combined state and parameter estimation problem.