Multidimensional Discrete Stability by Serre Categories and the Construction and Parametrization of Observers via Gabriel Localizations

Immediate predecessors of this work were a paper on two-dimensional deadbeat observers by Bisiacco and Valcher [Multidimens. Systems Signal Process., 19 (2008), pp. 287–306] and one on one-dimensional functional observers by Blumthaler [Linear Algebra Appl., 432 (2010), pp. 1560–1577] (compare also Fuhrmann’s comprehensive paper [Linear Algebra Appl., 428 (2008), pp. 44– 136]). The present paper extends Blumthaler’s results to continuous or discrete multidimensional behaviors, i.e., constructs and parametrizes all controllable observers of a given multidimensional behavior, and for this purpose also discusses the required multidimensional stability. Such an observer produces a signal that approximates or estimates a desired component of the behavior such that the signal difference is negligible in a suitable sense. This definition thus presupposes that of negligible or stable autonomous systems. In the standard one-dimensional case these are the asymptotically stable behaviors. We define and investigate the characteristic variety of an autonomous behavior in the needed generality of this paper and define stability, as in the one-dimensional case, by the spectral condition that the characteristic variety is contained in a preselected stability region of an appropriate multidimensional affine space. This stability is equivalent to the property that all polynomial exponential trajectories in the behavior have frequencies in the stability region only. The stability region gives rise to a Serre category or class of modules over the relevant ring of operators that, by definition, is closed under isomorphisms, submodules, factor modules, extensions, and direct sums and that determines the stability region. The spectral condition for stability is equivalent to the algebraic condition that the system module belongs to the associated Serre category. This category, in turn, gives rise to an associated Gabriel localization that is indispensable for the construction and parametrization of controllable observers.