Noise-induced bias in last principal component modeling of linear system

Last principal component (LPC) modeling relies on principal component transformation, and utilizes the eigenvectors associated with the last (smallest) principal components. When applied to experimental data, it may be considered an alternative to least squares based estimation of model parameters. Experimental results in the literature (cited in the body of the paper) suggest that LPC modeling is inferior to LS, in terms of estimation bias, in the presence of noise. Other results show that LPC produces unbiased estimates only in a very special case. In this paper, we derive explicit expressions for noise-induced bias in LPC-based identification. We investigate static systems with input actuator and measurement noise, and discrete dynamic systems with output measurement noise. We show that, indeed, LPC-based estimates are biased even when LS-based ones are not, and when the LS estimate is also biased, the LPC estimate has the LS bias plus an additional term. The theoretical results are supported by simulation studies.