186,210 research outputs found
On the Common Linear Copositive Lyapunov Functions for Compartmental Switched Systems
No full textFor a positive switched system, the existence of a common linear copositive Lyapunov function (CLCLF) for the family of the subsystem matrices represents an important sufficient condition for its asymptotic stability. The main necessary and sufficient condition for the existence of a CLCLF (Fornasini and Valcher, IEEE Trans Autom Control 55:1933–1937, 2010, [1], Knorn et al, Automatica 45:1943–1947, 2009, [2]) consists in the explicit evaluation of the Hurwitz property of a family of matrices, where p is the number of subsystems and n the size of each subsystem. In this paper we show that, when restricting our attention to compartmental switched systems, the Hurwitz property may be checked on a smaller subset of smaller matrices. Based on this result, we provide an algorithm that allows to determine whether a CLCLF exists, by simply checking the column sums of matrix sets of increasingly lower dimension and cardinality
Autonomous behaviors decomposition and modal analysis
No full textIn this paper, the notions of simple, very simple and
indecomposable autonomous behavior
are introduced and characterized. By resorting to some recent results about the direct sum decomposition
of (linear, time-invariant, differential) behaviors (Bisiacco & Valcher, 2001), as well as to the well known
primary decomposition theorem for finitely generated modules (Hartley & Hawkes, 1970), it is shown that
every autonomous behavior can be expressed as a direct sum of indecomposable components, which are just
cyclic modules of order p^ν, for some irreducible polynomial p and some positive integer ν. The nonuniqueness
of this result is also discussed. Finally, this decomposition is interpreted in terms of modal analysis, and
related to the results that can be obtained by mean of a state-space realization of the behavior
Co-positive lyapunov functions for the stabilization of positive switched systems
No full textIn 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. © 1963-2012 IEEE
Recent advances on the reachability of single-input positive switched systems
No full textIn this paper the reachability property for singleinput
continuous-time positive switched systems is investigated.
By referring to an existing (and hard to check) characterization
of the reachability of the class of positive switched systems
which commute among n single-input n-dimensional systems
[9], we develop new algebraic tools which allow us to derive
sufficient reachability conditions which are easier to evaluate
Zero patterns and dominant modes of the state evolutions of autonomous continuous-time positive systems
No full textAbstract—In this paper, the zero pattern properties and
the asymptotic evolution of the trajectories of an autonomous
continuous-time positive system are investigated. To this end, a
normal form for the exponential of a Metzler matrix is provided,
and the concept of “echelon basis” is introduced. By making
use of these two ingredients, the dominant mode of each single
block appearing in the normal form of the exponential matrix is
determined. As a result, the zero pattern as well as the dominant
mode of every state evolution, depending on the zero pattern
of the initial state, can be easily inferred
Monomial reachability and zero controllability of discrete-time positive switched systems
No full textIn this paper, monomial reachability and zero controllability properties of discrete-time positive switched systems are investigated. Necessary
and sufficient conditions for these properties to hold, together with some interesting examples and some testing algorithms, are provided
Reachability Properties of Single-Input Continuous-Time Positive Switched Systems
No full textIn this paper, two reachability properties for single input
positive switched systems are introduced: monomial reachability
and reachability. Monomial reachability, which represents a
necessary but not sufficient condition for reachability, is fully characterized.
Necessary and sufficient conditions for reachability are
provided for the class of -dimensional systems, switching among
subsystems. Several necessary or sufficient conditions are also
provided. Moreover, the definition of -switching reachability set
is given, and an equivalent condition for the existence of an upper
bound on the number of switchings required to reach any reachable
vector is given. Finally, it is shown that, for reachable systems
of low dimension (2 or 3), each vector of the positive
orthant can be reached by resorting to a switching sequence which
switches no more than times
On the reachability and weak reachability of single-input positive switched systems
No full textAbstract—In the first part of the paper the reachability
property for positive switched systems which commute among n
single-input n-dimensional systems is investigated. By referring
to a (hard to check) necessary and sufficient condition for
the reachability of this class of systems [8], we develop new
algebraic tools which allow us to derive sufficient reachability
conditions which are easy to test. In the second part of the paper
weak reachability is introduced and a sufficient condition for
this property to hold is provided
Asymptotic exponential cones of Metzler matrices and their use in the solution of an algebraic problem
No full textThe aim of this paper is that of investigating the asymptotic exponential
cone of a single Metzler matrix, introduced in [23],
and of defining and analysing the new concept of asymptotic exponential
cone of a family of Metzler matrices (along a certain
direction). These results will provide necessary and/or sufficient
conditions for the solvability of an interesting algebraic problem
that arises in the context of continuous-time positive switched
systems and, specifically, in the investigation of the reachability
property
Is stabilization of switched positive linear systems equivalent to the existence of an Hurwitz convex combination of the system matrices?
No full textAbstract|In this paper exponential stabilizability
of continuous-time positive switched systems is in-
vestigated. It is proved that, when dealing with two-
dimensional systems, exponential stabilizability can
be achieved if and only if there exists an Hurwitz
convex combination of the (Metzler) system matrices.
However, for systems of higher dimension this is not
true.
In general, exponential stabilizability corresponds
to the existence of a (positively homogeneous, concave
and co{positive) control Lyapunov function, but this
function is not necessarily smooth. The existence of
an Hurwitz convex combination is equivalent to the
stronger condition that the system is not only expo-
nentially stable, but it also admits a smooth control
Lyapunov function. These two conditions, in turn, are
equivalent to the fact that the stabilizing switching
law can always be based on a linear co{positive control
Lyapunov function. Finally, the characterization of
exponential stabilizability is exploited to provide a
description of all the \switched equilibrium points"
of a positive ane switched system
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