Weighted Null-Space Fitting for Identification of Cascade Networks

Abstract

For identification of systems embedded in dynamic networks, applying the prediction error method (PEM) to a correct tailor-made parametrization of the complete network provided asymptotically efficient estimates. However, the network complexity often hinders a successful application of PEM, which requires minimizing a non-convex cost function that in general becomes more difficult for more complex networks. For this reason, identification in dynamic networks often focuses in obtaining consistent estimates of particular network modules of interest. A downside of such approaches is that splitting the network in several modules for identification often costs asymptotic efficiency. In this paper, we consider the particular case of a dynamic network with the individual systems connected in a serial cascaded manner, with measurements affected by sensor noise. We propose an algorithm that estimates all the modules in the network simultaneously without requiring the minimization of a non-convex cost function. This algorithm is an extension of Weighted Null-Space Fitting (WNSF), a weighted least-squares method that provides asymptotically efficient estimates for single-input single-output systems. We illustrate the performance of the algorithm with simulation studies, which suggest that a network WNSF may also be asymptotically efficient estimates when applied to cascade networks, and discuss the possibility of extension to more general networks affected by sensor noise.

Weighted Null-Space Fitting for Cascade Networks with Arbitrary Location of Sensors and Excitation Signals

“…is also possible. As seen in paper [16], simulations support that way of the replacement ((12) to (13) or (12) to (15)) does not affect the asymptotic properties of the estimate as long as we use all equations.…”

“…In [18], the method has been proved to be consistent and asymptotically efficient for SISO systems. In [16], an approach based on WNSF has been proposed for some type of serial cascade networks, which we generalize in this paper. For this reason, we first review WNSF for SISO OE models, which will be instrumental to our extension for identification of cascade networks.…”

“…In this section, we review the WNSF-based approach for cascade networks proposed in [16]. This review will be useful to introduce notation and to identify the main limitations of the approach, which will be used as motivation for the approaches proposed in this paper.…”

“…In [16], it is proposed to use a method that provides asymptotically efficient estimates of all the cascade network modules without solving non-convex optimizations. The proposed method is based on the Weighted Null-Space Fitting (WNSF) method, a multi-step weighted least-squares method proposed in [17], which has been shown to be consistent and asymptotically efficient for single-input single-output (SISO) systems [18].…”

“…The proposed method is based on the Weighted Null-Space Fitting (WNSF) method, a multi-step weighted least-squares method proposed in [17], which has been shown to be consistent and asymptotically efficient for single-input single-output (SISO) systems [18]. However, the WNSF-based application to cascade networks in [16] has some limitations regarding the location of excitation signals and sensors. In this paper, we propose an extension to WNSF-based method for cascade networks that will drop these condition.…”