Optimal simultaneous state estimation and parameter identification in linear discrete-time systems

The topic of this paper is the simultaneous estimation of state and parameters in linear discrete-time dynamic systems. The system is subject to a known arbitrary input (control), a random input (additive driving noise), and the output observation is also contaminated by noise. The noises are Gaussian, zero mean, independent, and with known variances. The problem is formulated under the assumption that the system parameters are unknown constants. Previous works in the literature treated this problem either approximately or by assuming that each parameter can take values over a finite set with known a priori probabilities. An exact solution has been presented only for a scalar parameter. The proposed scheme yields the maximum likelihood estimates for the system's state and unknown parameters. They are obtained by solving the likelihood equations, a system of nonlinear equations with the state and parameters as unknowns. Use is made of the fact that the dynamical system considered is linear and the problem is separated into two interconnected linear problems: one for the state, the other for the parameters. The solution is obtained by iterating between two systems of linear equations. The estimation technique presented is optimal in the following sense. No approximations are involved and the estimates of the parameters converge to the true values at the fastest possible rate, as given by the Cramér-Rao lower bound, i.e., they are asymptotically efficient. This is proved, using a theorem which states that, under certain conditions, the maximum likelihood estimate with dependent observations is consistent and asymptotically efficient. The problem of uniqueness of the solution is discussed for the case of a scalar unknown parameter. Use is made of a theorem due to Perlman, generalized for the case of dependent observations. Due to the fact that the estimation-identification is done in the presence of input and output noise and an arbitrary known input, the procedure can be considered an on-line technique. Since estimates are available after each measurement, this estimation-identification procedure is suited for use in the adaptive control of unknown (or partially known) linear plants.