High-order Least Squares Identification

In order to ensure that the estimates of system parameters are unbiased and efficient, most identification schemes including the Prediction Error Method (PEM), and the Subspace Method (SM), are based on minimizing the residual of the Kalman filter, and not the equation error (associated with system model) as the residual is a zero mean white noise process whereas the equation error is coloured noise which may be correlated with data vector. The residual is linear in the input and the output of the system, and is nonlinear in the parameters to be estimated. The parameters enter in the expression for the residual as coefficients of rational polynomials associated with the input and the output. Similar to the PEM, which is a gold standard for comparing the performance identification schemes, the High Order Least Squares (HOLS) method is derived from the expression of the residual. In order to ensure that the equation error is a zero mean white noise process, the rational polynomials are approximated by finite high order polynomials by selecting the model order to be sufficiently high. As result, the relationship governing the residual and the parameters is linear, and the HOLS method becomes essentially a Least Squares (LS) method. A reduced order model is derived using frequency weighted LS approach. The performance of the HOLS is arbitrarily close to that of the PEM: estimates are unbiased and efficient. Unlike the PEM, the HOLS estimates as well the covariance of the estimation error have closed form expressions, that is, they are not computed iteratively. A reduced order model is derived using frequency weighted least squares approach. The proposed scheme has been successfully evaluated on a number of simulated and physical systems and favourably compared with the prediction error method (PEM).