On Estimation of Cascade Systems with Common Dynamics

Abstract

Recent research on identification of cascade systems has revealed some intriguing variance results for the estimated transfer functions of the subsystems. Such structures are common in most engineering applications. Even so, little is known about quality properties for structured estimated models of cascade systems. The objective of this paper is to analyze the underlying mechanism for some non-intuitive variance results when the true subsystems have common dynamics. It turns out that a simple FIR example of two cascaded subsystems can be used to understand the basic issues. The cascade system identification problem for this case corresponds to solving a second order equation in a least squares sense constraining the roots to be real. The difficult case is when the second order equation has double roots (the discriminant Δ zero), which holds when the transfer functions of the subsystems are equal. In this case a more proper statistical analysis should be done conditional on the sign of Δ. If only the second output signal is used for estimation the result is that Δ > 0 gives estimates with poor statistical properties (variance of order O(1/N)), while Δ < 0 will automatically constrains the roots to be real and double and hence this case gives variance of order O (1/N). If both output signals are used for estimation the unconditional variance of the estimate of the first system does not depend, on the average, on the output from the second system. A simulation example shows that the statistical properties also in this case are much better than predicted by average variance analysis if Δ< 0. For this simple example, it is hence possible to monitor the quality of the estimate by studying the sign of Δ. Traditional variance analysis only considers the average effects and hence misses this two mode (good or bad) situation.