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In this technical note, reachability properties of continuous-time positive systems are introduced and characterized. Specifically, first it is shown that reachability and strong reachability are equivalent properties, and thus the characterization of strong reachability derived in [8] is extended to the weaker notion of reachability. In the second part of the technical note, essential reachability is introduced, and necessary and sufficient conditions for this property to hold are provided. Index Terms--Continuous-time positive system, directed graph, essential reachability, reachability, strong reachability.

The observer design problem is investigated in the context of linear left shift invariant discrete behaviors, whose trajectories have supports on [Z.sub.+]. Necessary and sufficient conditions for the existence of a dead-beat observer of some relevant variables from some measured ones, in the presence of some unmeasured (and irrelevant) variables, are introduced, and a complete parametrization of all dead-beat observers is given. Equivalent conditions for the existence of causal dead-beat observers are then derived. Finally, several classical problems addressed for state-space models, like state estimation, the design of unknown input observers or the design of fault detectors and identifiers (possibly in the presence of disturbances), are cast in this general framework, and the aforementioned equivalent conditions and parametrizations are specialized to these cases. Index Terms--Behaviors, fault detection and isolation (FDI), nilpotent autonomous systems, observability, observers, reconstructibility, unknown input observers (UIO).

Bilinear systems in input-output form are introduced and represented by means of suitable rational functions in two indeterminates. Necessary and sufficient conditions for bounded-input/bounded-output (BIBO) stability are derived and compared with analogous results on two-dimensional systems stability. Several features of the output evolutions corresponding to finite support as well as to periodic input sequences are discussed and related to the polar structure of the rational functions describing the input-output maps. Index Terms - Bilinear systems, nonessential singularities, periodic dynamics, stability, two-dimensional systems.

This paper analyzes the observer design problem in the behavioral context. Observability and detectability notions are first introduced and fully characterized. Necessary and sufficient conditions for the existence of an observer, possibly an asymptotic or an exact one, are introduced, and a complete parameterization of all admissible observers is given. The problem of obtaining observers endowed with a (strictly) proper transfer matrix and the design of observer-based controllers are later addressed and solved. Finally, the above issues are particularized to the case of state-space systems, thus showing they naturally generalize well-known theorems of traditional system theory. Index Terms--Behaviors, latent variable models, observability/detectability, proper transfer matrices.

In the behavioral approach, a linear time-invariant (LTI) delay-differential system is naturally introduced [6] as a continuous-time system whose dynamics are governed by a set of delay-differential equations, involving both pointed and distributed delay operators. For this class of systems the notion of autonomy is introduced and characterized. Finally, asymptotic stability of autonomous delay-differential systems is investigated. Index Terms--Autonomy and asymptotic stability, behaviors, controllability, delay-differential equations.