Biorthogonal Blaschke representation of transfer functions for pole identification

In this paper, we further develop the theory of a recently introduced class of algorithms designed for the identification of linear time invariant (LTI) systems. From the applications’ perspective, the key advantage of the considered non-parametric schemes compared to classical approaches (e.g. subspace identification) is that they do not require any apriori assumptions on the system order. On the other hand the considered methods operate in the frequency domain and depend on a high quality estimate of the frequency response. This is often difficult to measure and is approximated by numerically unsafe empirical transfer function estimates (ETFE). The main contribution of this work is the introduction of a novel biorthogonal system in the $H^{2}$ Hardy-Hilbert space. Particularly, we introduce a biorthogonal dual for a sequence of Blaschke-products and show that the acquired function system is a Riesz basis. We also investigate the connection between powers of Blaschke-factors and the orhtonormal Laguerre function system frequently used in the above-mentioned identification scheme. Furthermore, we show that our findings may be applied to input and output data to obtain the biorthogonal expansion coefficients of the transfer function. This can be viewed as an alternative to the ETFE, with the additional advantages that the biorthogonal expansion representation describes the behavior of the system on frequencies that were not present in the measured input, and it avoids the inaccuracies that arise from the division of noisy signals. The obtained coefficients may be directly used with the considered identification methods to find poles of the system. Finally, a novel iterative procedure is proposed to eliminate the effect of already identified poles from the measured output signals, hence the procedure may be repeated until sufficient numbers of dominant poles are found.