Periodic Behaviors

SIAM Journal on Control and Optimization (Impact Factor: 1.46). 01/2010; 48(7):4652-4663. DOI: 10.1137/100782577
Source: DBLP


This paper studies behaviors that are defined on a torus, or equivalently, behaviors defined in spaces of periodic functions, and establishes their basic properties analogous to classical results of Malgrange, Palamodov, Oberst et al. for behaviors on R^n. These properties - in particular the Nullstellensatz describing the Willems closure - are closely related to integral and rational points on affine algebraic varieties.

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    • "These are the spaces of principal concern in this paper. Later, other spaces are briefly considered, such as the space S ′ of temperate distributions and spaces of periodic functions, where necessary and sufficient conditions for controllability are different (Theorem 3.1 in [15] and Theorem 4.2 in [8], quoted in Section 5 below). All these conditions are actually necessary and sufficient conditions that a behavior, given as the kernel of a map P (∂) ∶ F k → F ℓ , admit an image representation, i.e. also be equal to the image of some map M (∂) ∶ F k 1 → F k . "
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    ABSTRACT: The computational effectiveness of Kalman's state space controllability rests on the well-known Hautus test, which describes a rank condition of the matrix (d/dt I - A,B). This paper generalizes this test to a generic class of behaviors (belonging to a Zariski open set) defined by systems of PDE (i.e., systems which arise as kernels of operators given by matrices (p(ij) (partial derivative)) whose entries are in C[partial derivative(1), ... , partial derivative(n)]) and studies its implications, especially to issues of genericity. The paper distinguishes two classes of systems, underdetermined and overdetermined. The Hautus test developed here implies that a generic strictly underdetermined system is controllable, whereas a generic overdetermined system is uncontrollable.
    Full-text · Article · Oct 2013 · SIAM Journal on Control and Optimization
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    ABSTRACT: Immediate predecessors of this work were a paper on two-dimensional deadbeat observers by M. Bisiacco and M. E. Valcher [“Dead-beat estimation problems for 2D behaviors”, Multidimensional Syst. Signal Process. 19, No. 3–4, 287–306 (2008; Zbl 1213.93020)] and one on one-dimensional functional observers by I. Blumthaler [“Functional T-observers”, Linear Algebra Appl. 432, No. 6, 1560–1577 (2010; Zbl 1181.93019)] (compare also the comprehensive paper [P. A. Fuhrmann, “Observer theory”, Linear Algebra Appl. 428, No. 1, 44–136 (2008; Zbl 1143.93009)]). 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.
    No preview · Article · Jan 2013 · SIAM Journal on Control and Optimization