The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce ...
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ISBN:
(纸本)9783642028939
The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling technique, based on tropical algebra, which applies in particular to this companion form. This scaling, which is inspired by all earlier work of Akian, Bapat, and Gaubert, relics oil the computation of "tropical roots". We give explicit hounds, in a typical case, indicating that these roots provide accurate estimates of the order of magnitude of the different eigenvalues, and we show by experiments that this scaling improves the accuracy (measured by normwise backward error) of the computations, particularly in situations in which the data have various orders of magnitude. In the case of quadratic polynomial matrices, we recover in this way a scaling due to Fail, Lin, and Van Dooren, which coincides with the tropical scaling when the two tropical roots are equal. If not the eigenvalues generally split in two groups, and the tropical method leads to making one specific scaling for each of the groups.
In the paper, the k-switching reachability set R-k of a continuous-tune positive switched system is introduced, and a necessary and sufficient condition for the chain of this sets {R-k,k is an element of N} to stop in...
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ISBN:
(纸本)9783642028939
In the paper, the k-switching reachability set R-k of a continuous-tune positive switched system is introduced, and a necessary and sufficient condition for the chain of this sets {R-k,k is an element of N} to stop increasing after some finite index k is given. It is shown that, for special classes of (multiple-input) positive switched systems, reachability always ensures that R-n = R-+(n), n being the system dimension.
We give a survey of delectability, observability and reconstructability concepts for positive systems and sketch some applications to the analysis of stochastic equations.
ISBN:
(纸本)9783642028939
We give a survey of delectability, observability and reconstructability concepts for positive systems and sketch some applications to the analysis of stochastic equations.
Identifiability is a fundamental prerequisite for model identification. Differential algebra tools have been applied to Study identifiability of dynamic systems described by nonlinear polynomial equations. In a previo...
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ISBN:
(纸本)9783642028939
Identifiability is a fundamental prerequisite for model identification. Differential algebra tools have been applied to Study identifiability of dynamic systems described by nonlinear polynomial equations. In a previous paper a differential algebra method for testing identifiability for locally and globally non accessible systems has been proposed. In this paper we describe a strategy to simplify the above differential algebra method to test identifiability of systems which are non accessible from everywhere. In particular we make the method more efficient and thus of snore general applicability. A strategy for testing identifiability also of nonlinear models described by non polynomial equations is proposed.
In [Jain, Tynan, Linear Algebra and its Applications 379, 381-394, 2004], the authors shown that a nonnegative square matrix A satisfies that AA(#) >= O, being A(#) the group inverse of A, if and only if A is permu...
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ISBN:
(纸本)9783642028939
In [Jain, Tynan, Linear Algebra and its Applications 379, 381-394, 2004], the authors shown that a nonnegative square matrix A satisfies that AA(#) >= O, being A(#) the group inverse of A, if and only if A is permutationally similar to a matrix with a special Structure. In this paper, a similar Structure for this kind of matrices, slightly simplified, is presented, where the restriction of the nonnegativity of the matrix A is omitted. In addition, this result to characterize the {k}-group involutory matrices is applied.
A nonsingular real matrix A is said to be inverse-positive if all the elements of its inverse are nonnegative. This class of matrices contains the M-matrices, from which inherit some of their properties and applicatio...
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ISBN:
(纸本)9783642028939
A nonsingular real matrix A is said to be inverse-positive if all the elements of its inverse are nonnegative. This class of matrices contains the M-matrices, from which inherit some of their properties and applications, especially in Economy. In this work we analyze the inverse-positive concept for a particular type of pattern: the checkerboard pattern. In addition, we study the Hadamard product of certain classes of inverse-positive matrices whose entries have a particular sign pattern.
The problem of synthesizing stabilizing state-feedback controllers is solved when the closed-loop system is required to remain positive, for the class of 2-D linear systems described by the Fornasini-Marchesini second...
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ISBN:
(纸本)9783642028939
The problem of synthesizing stabilizing state-feedback controllers is solved when the closed-loop system is required to remain positive, for the class of 2-D linear systems described by the Fornasini-Marchesini second model. First, a constructive necessary and sufficient condition expressed as a Linear Programming problem is provided for stabilization of these systems when the states must be nonnegative (assuming that the boundary conditions are nonnegative). It is shown how it is simple to include additional constraints (such as positive controls). Moreover, this result is also extended to include uncertainty in the model, making possible to synthesize robust state-feedback controllers, solving Linear Programming problems. Some numerical examples are included to illustrate the proposed approach for different design problems.
We investigate sufficient conditions for a discrete nonlinear non-homogeneous dynamical system to converge to consensus. We formulate a theorem which is based on the notion of averaging maps. Further on, we give examp...
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ISBN:
(纸本)9783642028939
We investigate sufficient conditions for a discrete nonlinear non-homogeneous dynamical system to converge to consensus. We formulate a theorem which is based on the notion of averaging maps. Further on, we give examples that demonstrate that the theory of convergence to consensus is still not complete.
A family of multi-point iterative methods for solving systems of nonlinear equations is described. Some classical methods are included in the mentioned family. Under certain conditions, convergence order is proved to ...
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ISBN:
(纸本)9783642028939
A family of multi-point iterative methods for solving systems of nonlinear equations is described. Some classical methods are included in the mentioned family. Under certain conditions, convergence order is proved to be 2d+1, where d is the order of the partial derivatives required to be zero in the solution. Moreover, different numerical tests confirm the theoretical results and allow us to compare these variants with Newton's method.
Let M be a compact invariant set contained in a boundary hyperplane of the positive orthant of R-n for a discrete or continuous time dynamical system defined on the positive orthant. Using elementary arguments, we sho...
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ISBN:
(纸本)9783642028939
Let M be a compact invariant set contained in a boundary hyperplane of the positive orthant of R-n for a discrete or continuous time dynamical system defined on the positive orthant. Using elementary arguments, we show that M is uniformly weakly repelling in directions normal to the boundary in which M resides provided all normal Lyapunov exponents are positive. This result is useful in establishing uniform persistence of the dynamics.
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