Stabilization of positive linear systems

Leenheer, Patrick De; Aeyels, Dirk

doi:10.1016/s0167-6911(01)00146-3

Cited by 227 publications

(140 citation statements)

References 5 publications

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“…Positive linear systems (in continuous as well as discrete time) have attracted much attention in the control literature, see for instance [8,9,19,21,22,31]. We will say that a finite dimensional linear system, possibly time-varying,ẋ = A(t)x + B(t)u…”

Section: Relations To Positivitymentioning

confidence: 99%

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Monotone control systems

Angeli

Sontag

2003

IEEE Trans. Automat. Contr.

590

675

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Monotone systems constitute one of the most important classes of dynamical systems used in mathematical biology modeling. The objective of this paper is to extend the notion of monotonicity to systems with inputs and outputs, a necessary first step in trying to understand interconnections, especially including feedback loops, built up out of monotone components. Basic definitions and theorems are provided, as well as an application to the study of a model of one of the cell's most important subsystems.

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Section: Relations To Positivitymentioning

confidence: 99%

“…It is well known for time-invariant systems (A and B constant), see for instance [19], Chapter 6, or [8] for a recent reference, that a system is positive if and only if it is a Metzler system. This also holds for the general case, and we provide the proof here for completeness.…”

Section: Relations To Positivitymentioning

confidence: 99%

Monotone control systems

Angeli

Sontag

2003

IEEE Trans. Automat. Contr.

590

675

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“…In fact, the origin is typically not very interesting as an equilibrium: for example, a power control algorithm that converges to zero power is meaningless, and similarly for the other applications. A common trick to move the equilibrium point from the origin to the interior of the positive orthant is to add a non-vanishing positive input, usually a constant, to the system dynamics (a current in a neural network, a noise power in the wireless interference function, a constant mRNA synthesis rate in the gene networks) [15], [18]. This trick is standard (and necessary) in linear positive systems [4], but not strictly necessary in the nonlinear case.…”

Section: Introductionmentioning

confidence: 99%

Existence, uniqueness and stability properties of positive equilibria for a class of nonlinear cooperative systems

Abara

Ticozzi

Altafini

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Abstract-We provide conditions that guarantee existence, uniqueness and stability of strictly positive equilibria for nonlinear cooperative systems associated to vector fields that are concave or subhomogeneous. This class of positive systems describes well interconnected dynamics that are of key interest for communication, biological, economical and neural network applications. These conditions can be formulated directly in terms of the spectral radius of the Jacobian of the system, and do not require to use constant inputs to move the equilibrium point from the origin to the interior of the positive orthant.

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“…Due to their importance and wide applicability, the analysis and control of positive systems has attracted considerable attention from the control community (see, e.g., [2]- [9] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Asymptotic stability and decay rates of positive linear systems with unbounded delays

Feyzmahdavian

Charalambous

Johansson

2013

52nd IEEE Conference on Decision and Control

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Abstract-There are several results on the stability analysis of positive linear systems in the presence of constant or timevarying delays. However, most existing results assume that the delays are bounded. This paper studies the stability of discrete-time positive linear systems with unbounded delays. We provide a set of easily verifiable necessary and sufficient conditions for delay-independent stability of positive linear systems subject to a general class of heterogeneous timevarying delays. For two particular classes of unbounded delays, explicit expressions that bound the decay rate of the system are presented. We demonstrate that the best bound on the decay rate that our results can guarantee can be found via convex optimization. Finally, the validity of the results is demonstrated via a numerical example.

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Stabilization of positive linear systems

Cited by 227 publications

References 5 publications

Monotone control systems

Monotone control systems

Existence, uniqueness and stability properties of positive equilibria for a class of nonlinear cooperative systems

Asymptotic stability and decay rates of positive linear systems with unbounded delays

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