Ettore Fornasini

Doubly-indexed dynamical systems: State-space models and structural properties

Doubly-indexed dynamical systems provide a state space realization of two-dimensional filters which includes previous state models. Algebraic criteria for testing structural properties (reachability, observability, internal stability) are introduced.

State-space realization theory of two-dimensional filters

The realization problem of two-dimensional linear filters is approached from a system theoretic point of view. The input-output behavior of such a system is defined by formal power series in two variables, and a Nerode state space is constructed. This state space is, in general, infinite dimensional. If the power series is rational, the dynamics of the filter is described by updating equations on finite-dimensional local state space. The notions of local reachability and observability are defined in a natural way and an algorithm for obtaining a reachable and observable realization is given. In general, local reachability and observability do not imply the minimality of the realization.

Stability analysis of 2-D systems

A polynomial stability criterion for 2-D systems is taken as a starting point for introducing a frequency dependent Lyapunov equation. The Fourier analysis of its matrix solution leads to an infinite dimensional quadratic form which provides a Lyapunov function for the global state of the system. The Fourier coefficients are explicitly obtained as the sum of series involving the system matrices. The convergence of these series constitutes a necessary and sufficient stability condition, which generalizes the analogous condition for 1-D systems.

Linear Copositive Lyapunov Functions for Continuous-Time Positive Switched Systems

Continuous-time positive systems, switching among p subsystems, are introduced, and a complete characterization for the existence of a common linear copositive Lyapunov function for all the subsystems is provided. When the subsystems are obtained by applying different feedback control laws to the same continuous-time single-input positive system, the above characterization leads to a very easy checking procedure.

Observability, Reconstructibility and State Observers of Boolean Control Networks

The aim of this paper is to introduce and characterize observability and reconstructibility properties for Boolean networks and Boolean control networks, described according to the algebraic approach proposed by D. Cheng and co-authors in the series of papers [3], [6], [7] and in the recent monography . A complete characterization of these properties, based both on the Boolean matrices involved in the network description and on the corresponding digraphs, is provided. Finally, the problem of state observer design for reconstructible BNs and BCNs is addressed, and two different solutions are proposed.

Stability and Stabilizability Criteria for Discrete-Time Positive Switched Systems

In this paper we consider the class of discrete-time switched systems switching between p autonomous positive subsystems. First, sufficient conditions for testing stability, based on the existence of special classes of common Lyapunov functions, are investigated, and these conditions are mutually related, thus proving that if a linear copositive common Lyapunov function can be found, then a quadratic positive definite common function can be found, too, and this latter, in turn, ensures the existence of a quadratic copositive common function. Secondly, stabilizability is introduced and characterized. It is shown that if these systems are stabilizable, they can be stabilized by means of a periodic switching sequence, which asymptotically drives to zero every positive initial state. Conditions for the existence of state-dependent stabilizing switching laws, based on the values of a copositive (linear/quadratic) Lyapunov function, are investigated and mutually related, too. Finally, some properties of the patterns of the stabilizing switching sequences are investigated, and the relationship between a sufficient condition for stabilizability (the existence of a Schur convex combination of the subsystem matrices) and an equivalent condition for stabilizability (the existence of a Schur matrix product of the subsystem matrices) is explored.

2D Markov chains

Abstract The separation property that characterizes the dynamics of Markov chains is extended to a class of discrete 2D models where the time support, given by the discrete plane Z × Z , is partially ordered by the product of the orderings. The paper analyzes the matrix representation structure of the probability transition map in a 2D Markov chain and some properties of the associated characteristic polynomial in two variables. These allow one to show how the long-term behavior depends on the intersections between the variety of the characteristic polynomial and the distinguished boundary of the unit closed bidisk.

nD Polynomial Matrices with Applications to MultidimensionalSignal Analysis

In this paper, different primeness definitions and factorizationproperties, arising in the context of nD Laurentpolynomial matrices, are investigated and applied to a detailedanalysis of nD finite support signal families producedby linear multidimensional systems. As these families are closedw.r.t. linear combinations and shifts along the coordinate axes,they are naturally viewed as Laurent polynomial modules or, ina system theoretic framework, as nD finite behaviors.Correspondingly, inclusion relations and maximality conditionsfor finite behaviors of a given rank are expressed in terms ofthe polynomial matrices involved in the algebraic module descriptions.Internal properties of a behavior, like local detectability andvarious notions of extendability, are finally introduced, andcharacterized in terms of primeness properties of the correspondinggenerator and parity check matrices.

On the periodic trajectories of Boolean control networks

Abstract In this note we first characterize the periodic trajectories (or, equivalently, the limit cycles) of a Boolean network, and their global attractiveness. We then investigate under which conditions all the trajectories of a Boolean control network may be forced to converge to the same periodic trajectory. If every trajectory can be driven to such a periodic trajectory, this is possible by means of a feedback control law.

State space realization of 2-D finite-dimensional behaviours

This paper deals with the state space realization of autonomous autoregressive two-dimensional (2-D) systems in the context of the behavioural approach. An arbitrary autoregressive 2-D system