Proceedings ArticleDOI
A Generalization of Linear Positive Systems
Eyal Weiss,Michael Margaliot +1 more
- pp 340-345
TLDR
This work yields a generalization of positive linear systems called k-positive linear systems, which reduces to positive systems for k=1 and shows an application of this new class of systems to the analysis of invariant sets in nonlinear time-varying dynamical systems.Abstract:
The dynamics of linear positive systems maps the positive orthant to itself. Namely, it maps a set of vectors with zero sign variations to itself. Hence, a natural question is: what linear systems map the set of vectors with k sign variations to itself? To address this question we use tools from the theory of cooperative dynamical systems and the theory of totally positive matrices. Our approach yields a generalization of positive linear systems called k-positive linear systems, which reduces to positive systems for k=1. We show an application of this new class of systems to the analysis of invariant sets in nonlinear time-varying dynamical systems.read more
Citations
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Posted Content
Variation diminishing linear time-invariant systems
TL;DR: In this paper, the Toeplitz and Hankel operators of finite-dimensional linear time-invariant (LTI) systems have been characterized in terms of series or parallel interconnections of first order positive systems.
Posted ContentDOI
Compact attractors of an antithetic integral feedback system have a simple structure
TL;DR: It is shown that the model is a strongly 2-cooperative system, implying that the dynamics in the omega-limit set of any precompact solution is conjugate to the dynamics of a compact invariant subset of a two-dimensional Lipschitz dynamical system, thus precluding chaotic and other strange attractors.
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Balanced truncation of $k$-positive systems
TL;DR: The results generalize the fact that balanced truncation of a relaxation system are known to be totally positive for any order and extends the preliminary work on first order approximation of internally positive systems to externally (input-output) positive systems.
Journal ArticleDOI
Balanced Truncation of $k$-Positive Systems
TL;DR: In this article , the authors considered balanced truncation of discrete-time Hankel $k$-positive systems, characterized by Hankel matrices whose minors up to order 1/k$ are nonnegative.
Journal ArticleDOI
Variation diminishing linear time-invariant systems
TL;DR: In this paper , the authors studied the variation diminishing property of k-positive linear time-invariant (LTI) systems, which diminish the number of sign changes (variation) from input to output, if the input variation is at most k−1.
References
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Book
Mathematical Control Theory: Deterministic Finite Dimensional Systems
TL;DR: This book covers what constitutes the common core of control theory and is unique in its emphasis on foundational aspects, covering a wide range of topics written in a standard theorem/proof style and develops the necessary techniques from scratch.
Book
Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems
TL;DR: Monotone dynamical systems Stability and convergence Competitive and cooperative differential equations Irreducible cooperative systems Cooperative systems of delay differential equations Nonquasimonotone delay differential equation Quasimonoteone systems of parabolic equations A competition model Appendix Bibliography as discussed by the authors
Journal ArticleDOI
Survey paper: Set invariance in control
TL;DR: An overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems is provided.
Book
Positive Linear Systems: Theory and Applications
Lorenzo Farina,Sergio Rinaldi +1 more
TL;DR: In this article, the authors define and define conditions of positivity of equilibria, including reachability and observability, and define a set of conditions for positive equilibrium. But they do not define the conditions of transparency.