Maria Elena Valcher

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.

On the consensus and bipartite consensus in high-order multi-agent dynamical systems with antagonistic interactions

Abstract The aim of this paper is to address consensus and bipartite consensus for a group of homogeneous agents, under the assumption that their mutual interactions can be described by a weighted, signed, connected and structurally balanced communication graph. This amounts to assuming that the agents can be split into two antagonistic groups such that interactions between agents belonging to the same group are cooperative, and hence represented by nonnegative weights, while interactions between agents belonging to opposite groups are antagonistic, and hence represented by nonpositive weights. In this framework, bipartite consensus can always be reached under the stabilizability assumption on the state-space model describing the dynamics of each agent. On the other hand, (nontrivial) standard consensus may be achieved only under very demanding requirements, both on the Laplacian associated with the communication graph and on the agents’ description. In particular, consensus may be achieved only if there is a sort of “equilibrium” between the two groups, both in terms of cardinality and in terms of the weights of the “conflicting interactions” amongst agents.

Controllability and reachability criteria for discrete time positive systems

Discrete time positive systems have been the object of widespread interest in the literature. In the last decade particular attention has been devoted to the analysis and characterization of the different notions of reachability and controllability which, in this context, make their appearance. In this paper, the properties of (ordinary and essential) controllability and reachability are analysed. Based on a graph-theoretic approach, we provide practical criteria to verify whether a given positive system (F, G) is endowed with these properties, together with canonical forms for describing reachable/controllable pairs.

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.

State observers for discrete-time linear systems with unknown inputs

In the paper the problem of designing a full-order state observer for a general discrete-time linear system with unknown inputs is analyzed. Necessary and sufficient conditions for the existence either of an asymptotic or of a deadbeat observer are provided, and a constructive design procedure, together with some examples, are discussed. Finally, the equivalence of these conditions for problem solvability with those derived in previous contributions by means of different approaches is explicitly proved.

On the internal stability and asymptotic behavior of 2-D positive systems

Two-dimensional (2-D) positive systems are 2-D state-space models whose variables take only nonnegative values and, hence, are described by a family of nonnegative matrices. The free state evolution of these systems is completely determined by the set of initial conditions and by the pair of nonnegative matrices, (A.B), that represent the shift operators along the coordinate axes. In this paper, internal stability of 2-D positive systems is analyzed and related to the spectral properties of the matrix sum A+B. Also, some aspects of the asymptotic behavior are considered, and conditions guaranteeing that all local states on the same separation set C/sub t/ assume the same direction as t goes to infinity, are provided. Finally, some results on the free evolution of positive systems corresponding to homogeneous distributions of the initial local states around a finite mean value, are presented.

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.

Co-Positive Lyapunov Functions for the Stabilization of Positive Switched Systems

In this paper, exponential stabilizability of continuous-time positive switched systems is investigated. For two-dimensional systems, exponential stabilizability by means of a switching control law can be achieved if and only if there exists a Hurwitz convex combination of the (Metzler) system matrices. In the higher dimensional case, it is shown by means of an example that the existence of a Hurwitz convex combination is only sufficient for exponential stabilizability, and that such a combination can be found if and only if there exists a smooth, positively homogeneous and co-positive control Lyapunov function for the system. In the general case, exponential stabilizability ensures the existence of a concave, positively homogeneous and co-positive control Lyapunov function, but this is not always smooth. The results obtained in the first part of the paper are exploited to characterize exponential stabilizability of positive switched systems with delays, and to provide a description of all the “switched equilibrium points” of an affine positive switched system.