A method for unbiased parameter estimation by means of the equation error input covariance

A method for obtaining an unbiased estimate of the finite -dimensional parameter vector defining a time-invariant linear dynamical system in the presence of noise is described. The system is excited by a stationary mean-square bounded process. The method is based on an parameter "equation error" and is presented in continuous time. The equation error input covariance (EEIC) is equated to zero, and the resulting single linear equation having unknown parameters provides a necessary condition for their unique identification. From it, additional independent equations are generated. The resulting linear independent equations provide the unbiased estimate of the parameter vector in which the excess components vanish. The method does not require the identification of the noise statistics, and it can be applied without a priori assumption of the order of the system\´s numerator and denominator. Performance of the method is illustrated by simulated examples demonstrating the convergence of the parameter estimate in on-line recursive identification both in open and closed loop.