Variance error, interpolation and experiment design

• Published in: Automatica (Oxford) Vol. 49; no. 5; pp. 1117 - 1125
• Main Author: Mahata, Kaushik
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01-05-2013; Elsevier
• Subjects: Applied sciences; Computer science; control theory; systems; Control theory. Systems; Exact sciences and technology; Experiment design; Experimental design; Mathematics; Modelling and identification; Nevanlinna–Pick interpolation; Probability and statistics; Sciences and techniques of general use; Statistics; System identification; Variance error; Experiment design; System identification; Variance error; Nevanlinna–Pick interpolation; Additive noise; Error estimation; Power spectral density; Nevanlinna Pick interpolation; Nevanlinna―Pick interpolation; Parameterization; Optimization; Optimal design; Experimental design; Optimal control; Optimal input
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2013.01.021

We investigate how the variance error associated with the prediction error identification is related to the power spectral densities of the input and the additive noise at the output. Let Φ(eiω) be the ratio of the input power spectral density (PSD) to the output-noise PSD. We characterize the set of all functions Φ for which the variance error remains constant. This analysis results a minimal, finite-dimensional, affine parameterization of the variance error. This parameterization connects our analysis with the theory of Nevanlinna–Pick interpolation. It is shown that the set of all Φ for which the variance error remains constant can be characterized by the solutions of a Nevanlinna–Pick interpolation problem. This insight has interesting consequences in optimal input design, where it is possible to use some recent tools in analytic interpolation theory to tune shape of the input PSD to suit certain needs.