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

Rosenthal, Joachim; Wang, Xiaochang

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].

Reviewer: A.Akutowicz (Berlin)

MSC:

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

Keywords:

polynomial matrix; autoregressive system; gap metric; pole-zero cancellations