What is the distance between two autoregressive systems? (English) Zbl 0821.93028

Bowers, Kenneth L. (ed.) et al., Computation and control III. Proceedings of the third Bozeman conference, Bozeman, MT, USA, August 5-11, 1992. Boston, MA: Birkhäuser. Prog. Syst. Control Theory. 15, 333-340 (1993).
Let us write a row reduced \(p \times q\) polynomial matrix representing an autoregressive system with observability indices \((\nu_ 1,\dots, \nu_ p)\): \(R(s) = \sum^{i = \nu_ 1}_{i = 0} R_ i s^ i\). Consider the matrix obtained by stacking the matrices \(M_ l\) defined by: \(M_ l = [0_{p \times lq} R_ 0 \dots R_{\nu_ 1} 0 \dots ]\); \((0 \leq l \leq \nu_ 1 - \nu_ p + 1)\). The rows of this matrix form a finite- dimensional subspace of the Hilbert space of square integrable sequences. The gap metric in this Hilbert space provides a metric for the associated autoregressive systems. This new metric allows to distinguish systems where pole-zero cancellations give the same transfer function.
For the entire collection see [Zbl 0809.00032].


93B28 Operator-theoretic methods
93A10 General systems
93B35 Sensitivity (robustness)
93B40 Computational methods in systems theory (MSC2010)
93C05 Linear systems in control theory