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

A method for obtaining an unbiased estimate of the finite
q
-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
r \geq q
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
r > q
unknown parameters provides a necessary condition for their unique identification. From it,
r - 1
additional independent equations are generated. The resulting
r
linear independent equations provide the unbiased estimate of the parameter vector in which the excess
r - q
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.