Proceedings of the International Conference of Control, Dynamic Systems, and Robotics Ottawa, Ontario, Canada, May 15-16 2014 Paper No. 57 57-1 High-order Least Squares Identification Rajamani Doraiswami The University of New Brunswick Fredericton, New Brunswick, Canada dorai@unb.ca Lahouari Cheded Systems Eng. Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, KSA cheded@kfupm.edu.sa Abstract - 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). Keywords: High order least squares method, least squares method, prediction error method, subspace method, Kalman filter, residual, equation error. 1. Introduction The LS method is widely used to identify a system as it is simple, numerically efficient, and yields a closed-form solution to the parameter estimate. If the equation error is a zero- mean white noise process, the estimate will be unbiased and efficient. However if the equation error is a colored noise, the estimate will be biased as the colored noise will be correlated with the data vector. To overcome this problem, approaches such as the PEM, the SM and the HOLS method have been proposed. The PEM is iterative and does not provide a closed-form solution to the parameter estimation problem. A high-order model is used in various applications including the non-parameteric identification of impulse response, estimation of Markov parameters in the SM, in model predictive control, identification of a signal model and in system identification. The use of HOLS for identification of a signal model is inspired by the seminal paper by (Kumaresan & Tufts, 1982) for an accurate estimation of the parameters of an impulse response from measurements in an additive white noise. It is shown via simulation that the variance of the parameter estimation error approaches the Cramer-Rao lower bound. Further it is shown analytically that using a high-order model (with an order several times larger than the true order) improves significantly the accuracy of the parameter estimates. The HOLS has not received much