Stability of NonNegative Lur'e Systems

Bill, Adam; Guiver, Christopher; Logemann, Hartmut; Townley, Stuart

doi:10.1137/140994599

Cited by 14 publications

(23 citation statements)

References 38 publications

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“…Finally, for non-negative Lur'e systems, we have derived a "quasi CICS" property which applies to systems which when uncontrolled have two equilibria (one of which is the origin), a common scenario in the context of biological, ecological and chemical models. The proof of our quasi CICS result rests on a recent persistence result [6] for non-negative Lur'e systems.…”

Section: Discussionmentioning

confidence: 60%

“…In this context, we shall make contact with recent work [6] on stability properties of non-negative Lur'e systems: a certain "repeller" or "persistence" property established in [6] will play a pivotal role in the proof of Theorem 6.6. The paper is organized as follows.…”

Section: X(t) = Ax(t) + B F (C X(t) − V(t))mentioning

confidence: 98%

“…Furthermore, we consider a class of non-negative (also known as positive) Lur'e systems, cf. [6,36,37,44]. As an instance of a positive control system [15], these arise naturally in a variety of applied contexts: a common key feature is that their state variables, which may represent population abundances, chemical concentrations or economical quantities (such as prices) are, necessarily, non-negative.…”

Section: X(t) = Ax(t) + B F (C X(t) − V(t))mentioning

confidence: 99%

“…1, arise naturally in a variety of applied contexts, such as population dynamics and chemical reaction models, cf. [6,36,37,44]. We will restrict attention to models with scalar feedback f (m = p = 1 and f is a scalar function), that is, we consider forced Lur'e systems of the forṁ…”

Section: Cics Properties For Non-negative Lur'e Systemsmentioning

confidence: 99%

“…It is readily verified that A + bc T is irreducible, that is, condition (P3) is satisfied. We recall that the uncontrolled Lur'e system (6.14) has two equilibria, namely 0 and x * = (3, 6, 3) T (the latter being asymptotically stable with domain of attraction R 3 + \{0}, as follows from [6]) and that γ = 1/ G H ∞ = 1/2. We note that f (3) = 3/2 = 3γ, lim inf z→0 ( f (z)/z) = f (0) = 2, and…”

Section: Example 67mentioning

confidence: 99%

See 4 more Smart Citations

The converging-input converging-state property for Lur’e systems

Bill¹,

Guiver²,

Logemann³

et al. 2017

Math. Control Signals Syst.

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Using methods from classical absolute stability theory, combined with recent results on input-to-state stability (ISS) of Lur'e systems, we derive necessary and sufficient conditions for a class of Lur'e systems to have the converging-input converging-state (CICS) property. In particular, we provide sufficient conditions for CICS which are reminiscent of the complex Aizerman conjecture and the circle criterion and connections are also made with small gain ISS theorems. The penultimate section of the paper is devoted to non-negative Lur'e systems which arise naturally in, for example, ecological and biochemical applications: the main result in this context is a sufficient criterion for a so-called "quasi CICS" property for Lur'e systems which, when uncontrolled, admit two equilibria. The theory is illustrated with numerous examples. Keywords Absolute stability · Circle criterion · Converging-input converging-state property · Input-to-state stability · Lur'e systems · Non-negative systems Mathematics Subject Classification

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Section: Discussionmentioning

confidence: 60%

Section: X(t) = Ax(t) + B F (C X(t) − V(t))mentioning

confidence: 98%

Section: X(t) = Ax(t) + B F (C X(t) − V(t))mentioning

confidence: 99%

Section: Cics Properties For Non-negative Lur'e Systemsmentioning

confidence: 99%

Section: Example 67mentioning

confidence: 99%

See 3 more Smart Citations

The converging-input converging-state property for Lur’e systems

Bill¹,

Guiver²,

Logemann³

et al. 2017

Math. Control Signals Syst.

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On Matrix Stability and Ecological Models

McGrane-Corrigan,

Mason

2023

Entomology in Focus

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Boundedness, persistence and stability for classes of forced difference equations arising in population ecology

et al. 2019

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Boundedness, persistence and stability properties are considered for a class of nonlinear, possibly infinite-dimensional, forced difference equations which arise in a number of ecological and biological contexts. The inclusion of forcing incorporates the effects of control actions (such as harvesting or breeding programmes), disturbances induced by seasonal or environmental variation, or migration. We provide sufficient conditions under which the states of these models are bounded and persistent uniformly with respect to the forcing terms. Under mild assumptions, the models under consideration naturally admit two equilibria when unforced: the origin and a unique non-zero equilibrium. We present sufficient conditions for the non-zero equilibrium to be stable in a sense which is strongly inspired by the input-to-state stability concept well-known in mathematical control theory. In particular, our stability concept incorporates the impact of potentially persistent forcing. Since the underlying state-space may be infinite dimensional, our framework enables treatment of so-called integral projection models (IPMs). The theory is applied to a number of examples from population dynamics.

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Stability of NonNegative Lur'e Systems

Cited by 14 publications

References 38 publications

The converging-input converging-state property for Lur’e systems

The converging-input converging-state property for Lur’e systems

On Matrix Stability and Ecological Models

Boundedness, persistence and stability for classes of forced difference equations arising in population ecology

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