Stability analysis of systems with nonlinearities

Jönsson, Ulf; Laiou, Maria-Christina

doi:10.1109/cdc.1996.572936

Cited by 12 publications

(5 citation statements)

References 4 publications

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“…Megretski [46] shows that there exists a phase limitation for continuous-time Zames-Falb multipliers. Another phase limitation of Zames-Falb multipliers is given by Jönsson and Laiou [47], [48]. Such limitations are often ignored when new searches for multipliers are presented (see for example [45] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Phase Limitations of Zames–Falb Multipliers

Wang

Carrasco

Heath

2018

IEEE Trans. Automat. Contr.

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Abstract-Phase limitations of both continuous-time and discrete-time Zames-Falb multipliers and their relation with the Kalman conjecture are analysed. A phase limitation for continuous-time multipliers given by Megretski is generalised and its applicability is clarified; its relation to the Kalman conjecture is illustrated with a classical example from the literature. It is demonstrated that there exist fourth-order plants where the existence of a suitable Zames-Falb multiplier can be discarded and for which simulations show unstable behavior. A novel phase-limitation for discrete-time Zames-Falb multipliers is developed. Its application is demonstrated with a second-order counterexample to the Kalman conjecture. Finally, the discretetime limitation is used to show that there can be no direct counterpart of the off-axis circle criterion in the discrete-time domain.

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Section: Introductionmentioning

confidence: 99%

Phase Limitations of Zames–Falb Multipliers

Wang

Carrasco

Heath

2018

IEEE Trans. Automat. Contr.

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“…Hence substituting (35) (if h i < 0) or (36) (if h i > 0) in (37) for each i in the summation, yields…”

Section: B Duality Conditionsmentioning

confidence: 99%

“…Proof: The proof follows from Theorem 4.2 in [35]. In the spirit of [37], the duality condition is satisfied if…”

Section: Appendixmentioning

confidence: 99%

“…In Section V we discuss the application to numerical examples, and compare results with both the search for multipliers of [16] and the phase limitations of [32]. In two Appendices we provide details about the algorithm used to check conditions in [32] and provide a complementary continuous-time results to [37] for the case where the nonlinearity is non-odd.…”

Section: Introductionmentioning

confidence: 99%

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Duality bounds for discrete-time Zames-Falb multipliers

Zhang

Carrasco

Heath

2020

Preprint

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We develop phase limitations for the discrete-time Zames-Falb multipliers based on the separation theorem for Banach spaces. By contrast with their continuous-time counterparts they lead to numerically efficient results that can be computed either in closed form or via a linear program. The closed-form phase limitations are tight in the sense that we can construct multipliers that meet them with equality. We discuss numerical examples where the limitations are stronger than others in the literature. The numerical results complement searches for multipliers in the literature; they allow us to show, by construction, that the set of plants for which a suitable Zames-Falb multiplier exists is non-convex.

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“…The efficiency of the searches for discrete-time OZF multipliers has generated questions on the conservatism of the sufficient condition with OZF multipliers, e.g. investigations into phase limitations [14,19,26] and limits derived from duality theory [31]. It was conjectured by Carrasco that the existence of a suitable OZF multiplier is, in fact, both necessary and sufficient for robust stability [26].…”

Section: Introductionmentioning

confidence: 99%

On the Necessity and Sufficiency of Discrete-Time O'Shea-Zames-Falb Multipliers

Su¹,

Seiler²,

Carrasco³

et al. 2021

Preprint

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This paper considers the robust stability of a discrete-time Lurye system consisting of the feedback interconnection between a linear system and a bounded and monotone nonlinearity. It has been conjectured that the existence of a suitable linear time-invariant (LTI) O'Shea-Zames-Falb multiplier is not only sufficient but also necessary. Roughly speaking, a successful proof of the conjecture would require: (a) a conic parameterization of a set of multipliers that describes exactly the set of nonlinearities, (b) a lossless S-procedure to show that the non-existence of a multiplier implies that the Lurye system is not uniformly robustly stable over the set of nonlinearities, and (c) the existence of a multiplier in the set of multipliers used in (a) implies the existence of an LTI multiplier. We investigate these three steps, showing the current bottlenecks for proving this conjecture. In addition, we provide an extension of the class of multipliers which may be used to disprove the conjecture.

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Stability analysis of systems with nonlinearities

Cited by 12 publications

References 4 publications

Phase Limitations of Zames–Falb Multipliers

Phase Limitations of Zames–Falb Multipliers

Duality bounds for discrete-time Zames-Falb multipliers

On the Necessity and Sufficiency of Discrete-Time O'Shea-Zames-Falb Multipliers

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