Invariant cones and stability of linear dynamical systems

Алілуйко, Андрій Миколайович; Mazko, A. G.

doi:10.1007/s11253-006-0159-5

Cited by 7 publications

(31 citation statements)

References 11 publications

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“…In particular, we establish sufficient conditions for the invariance of a time-varying ellipsoidal cone for a certain class of nonlinear differential systems. Analogous results for linear systems were established in [3,4].…”

Section: Introductionmentioning

confidence: 62%

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Invariant sets and comparison of dynamical systems

Алілуйко

Mazko

2007

Nonlinear Oscill

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We propose a method for the construction and investigation of invariant sets of differential systems described by cone inequalities with the use of the operator of differentiation along the trajectories of the system. Well-known conditions for the positivity of linear and nonlinear differential systems with respect to typical classes of cones are generalized. A method for comparison and ordering is developed for a family of dynamical systems.

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Section: Introductionmentioning

confidence: 62%

“…Inequality (2.11) is a generalization of known conditions for the invariance of an ellipsoidal cone for linear systems [3,4].…”

Section: Example 23 For the Nonlinear Systeṁmentioning

confidence: 99%

Invariant sets and comparison of dynamical systems

Алілуйко

Mazko

2007

Nonlinear Oscill

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“…Theorem 3.1. Let a polyhedron P = {x | Gx ≤ b}, where G ∈ R m×n and b ∈ R m , and the discrete system be given as in (1). Assume that b i − G T i f d (x) are convex functions for all i ∈ I(m).…”

Section: Invariance Conditions For Discrete Systemsmentioning

confidence: 99%

“…They also provide alternative ways to design efficient algorithms to construct invariant sets. Linear discrete and continuous dynamical systems have been extensively studied in recent decades, since such systems have a wide range of applications in control theory, see, e.g., [1,4,15]. Invariance condition for linear systems are relatively easy to derive while analogous conditions for nonlinear systems are more difficult to derive.…”

Section: Introductionmentioning

confidence: 99%

Invariance Conditions for Nonlinear Dynamical Systems

Horváth

Song

Terlaky

2016

Optimization and Its Applications in Control and Data Sciences

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Recently, Horváth, Song, and Terlaky [A novel unified approach to invariance condition of dynamical system, submitted to Applied Mathematics and Computation] proposed a novel unified approach to study, i.e., invariance conditions, sufficient and necessary conditions, under which some convex sets are invariant sets for linear dynamical systems.In this paper, by utilizing analogous methodology, we generalize the results for nonlinear dynamical systems. First, the Theorems of Alternatives, i.e., the nonlinear Farkas lemma and the S -lemma, together with Nagumo's Theorem are utilized to derive invariance conditions for discrete and continuous systems. Only standard assumptions are needed to establish invariance of broadly used convex sets, including polyhedral and ellipsoidal sets. Second, we establish an optimization framework to computationally verify the derived invariance conditions. Finally, we derive analogous invariance conditions without any conditions.

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“…The proposed approach can be used for the investigation of the stability of Takagi -Sugeno systems [7] and systems in semiordered spaces (see [8,9]). …”

Section: Corollary 3 If All Conditions Of Theorem 3 Are Satisfied Fomentioning

confidence: 99%