Unbiased parameter estimation of continuous-time system based on modulating functions with input and output white noises

An efficient unbiased estimation method is proposed for the direct identification of linear continuous-time system with noisy input and output measurements. Using the Gaussian modulating filters, by numerical integration, an equivalent discrete identification model which is parameterized with continuous-time model parameters is developed, and the parameters can be estimated by the least-squares (LS) algorithm. Even with white noises in input and output measurement data, the LS estimate is biased, and the bias is determined by the variances of noises. According to the asymptotic analysis, the relationship between bias and noise variances is derived. One equation relating to the measurement noise variances is derived through the analysis of the LS errors. Increasing the degree of denominator of the system transfer function by one, an extended model is constructed. By comparing the true value and LS estimates of the parameters between original and extended model, another equation with input and output noise variances is formulated. So, the noise variances are resolved by the set of equations, the LS bias is eliminated and the unbiased estimates of system parameters are obtained. A simulation example by comparing the standard LS with bias eliminating LS algorithm indicates that the proposed algorithm is an efficient method with noisy input and output measurements.