Two counterexamples to Aizerman's conjecture

Fitts, Renee

doi:10.1109/tac.1966.1098369

Cited by 97 publications

(37 citation statements)

References 2 publications

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“…9 shows the result generated in Simulink when ζ = 0.1 and k = 2000. Such phenomena were first observed by Fitts (1966), and have attracted much attention as counterexamples to the Kalman conjecture. Barabanov (1988) questioned the validity of Fitts' original counterexample; this has led to considerable discussion Kuznetsov et al, 2011;Leonov and Kuznetsov, 2013).…”

Section: O'shea's Examplementioning

confidence: 93%

“…Thus we know a priori that a third order system is absolutely stable provided φ ∈ S [0, k N ) and we can benchmark a test for stability by seeking the maximum slope value and comparing with this upper bound (e.g Safonov and Wyetzner, 1987;Carrasco et al, 2014b). But the conjecture is false in general and the fourth-order counterexamples proposed more than 40 years ago (Fitts, 1966;O'Shea, 1967;Willems, 1971;Leonov and Kuznetsov, 2013) can also be used as benchmarks as they can be very challenging for stability tests. We illustrate such a benchmark in this paper.…”

Section: The Nyquist Value and The Kalman Conjecturementioning

confidence: 99%

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Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches

Carrasco

Turner

Heath

2015

2015 European Control Conference (ECC)

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Absolute stability attracted much attention in the sixties. Several stability conditions for loops with slope-restricted nonlinearities were developed. Results such as the Circle Criterion and the Popov Criterion form part of the core curriculum for students of control. Moreover, the equivalence of results obtained by different techniques, specifically Lyapunov and Popov's stability theories, led to one of the most important results in control engineering: the KYP Lemma.For Lurye 1 systems this work culminated in the class of multipliers proposed by O'Shea in 1966 and formalised by Zames and Falb in 1968. The superiority of this class was quickly and widely accepted. Nevertheless the result was ahead of its time as graphical techniques were preferred in the absence of readily available computer optimization. Its first systematic use as a stability criterion came twenty years after the initial proposal of the class. A further twenty years have been required to develop a proper understanding of the different techniques that can be used. In this long gestation some significant knowledge has been overlooked or forgotten. Most significantly, O'Shea's contribution and insight is no longer acknowledged; his papers are barely cited despite his original parameterization of the class.This tutorial paper aims to provide a clear and comprehensive introduction to the topic from a user's viewpoint. We review the main results: the stability theory, the properties of the multipliers (including their phase properties, phase-equivalence results and the issues associated with causality), and convex searches. For clarity of exposition we restrict our attention to continuous time multipliers for single-input single-output results. Nevertheless we include several recent significant developments by the authors and others. We illustrate all these topics using an example proposed by O'Shea himself.

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Section: O'shea's Examplementioning

confidence: 93%

Section: The Nyquist Value and The Kalman Conjecturementioning

confidence: 99%

Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches

Carrasco

Turner

Heath

2015

2015 European Control Conference (ECC)

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“…[Krasovsky, 1952;Pliss, 1958;Fitts, 1966;Bernat & Llibre, 1996;Bragin et al, 2011;Leonov & Kuznetsov, 2013a]). …”

Section: Oscillation) From a Computational Point Of View This Allowsunclassified

“…[Blondel & Megretski, 2004;Hsu & Meyer, 1968;Westphal, 2001]) Fitts' counterexample [Fitts, 1966]. In [Fitts, 1966], a simulation of system (98) is given for n = 4 with the transfer function…”

Section: Hidden Oscillations In Counterexamples To Aizerman's and Kalmentioning

confidence: 99%

Hidden Attractors in Dynamical Systems. From Hidden Oscillations in Hilbert–kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits

Леонов

Kuznetsov

2013

Int. J. Bifurcation Chaos

770

414

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From a computational point of view, in nonlinear dynamical systems, attractors can be regarded as self-excited and hidden attractors. Self-excited attractors can be localized numerically by a standard computational procedure, in which after a transient process a trajectory, starting from a point of unstable manifold in a neighborhood of equilibrium, reaches a state of oscillation, therefore one can easily identify it. In contrast, for a hidden attractor, a basin of attraction does not intersect with small neighborhoods of equilibria. While classical attractors are self-excited, attractors can therefore be obtained numerically by the standard computational procedure. For localization of hidden attractors it is necessary to develop special procedures, since there are no similar transient processes leading to such attractors.At first, the problem of investigating hidden oscillations arose in the second part of Hilbert's 16th problem (1900). The first nontrivial results were obtained in Bautin's works, which were devoted to constructing nested limit cycles in quadratic systems, that showed the necessity of studying hidden oscillations for solving this problem. Later, the problem of analyzing hidden oscillations arose from engineering problems in automatic control. In the 50-60s of the last century, the investigations of widely known Markus-Yamabe's, Aizerman's, and Kalman's conjectures on absolute stability have led to the finding of hidden oscillations in automatic control systems with a unique stable stationary point. In 1961, Gubar revealed a gap in Kapranov's work on phase locked-loops (PLL) and showed the possibility of the existence of hidden oscillations in PLL. At the end of the last century, the difficulties in analyzing hidden oscillations arose in simulations of drilling systems and aircraft's control systems (anti-windup) which caused crashes.Further investigations on hidden oscillations were greatly encouraged by the present authors' discovery, in 2010 (for the first time), of chaotic hidden attractor in Chua's circuit.This survey is dedicated to efficient analytical-numerical methods for the study of hidden oscillations. Here, an attempt is made to reflect the current trends in the synthesis of analytical and numerical methods. †

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“…The describing function method is mostly used for non-linear control systems and presently is seldom used. This technique can lead to erroneous results: (1) it can fail to predict existing limit cycles (see, e.g., reference [17]), and (2) it can spuriously predict non-existing limit cycles (see, e.g., reference [18]). …”

Section: Introductionmentioning

confidence: 99%

Limit Cycle Behavior in Three- Or Higher-Dimensional Non-Linear Systems: The Lotka–volterra Example

Shahruz¹,

Kalkin²

2001

Journal of Sound and Vibration

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Two counterexamples to Aizerman's conjecture

Cited by 97 publications

References 2 publications

Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches

Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches

Hidden Attractors in Dynamical Systems. From Hidden Oscillations in Hilbert–kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits

Limit Cycle Behavior in Three- Or Higher-Dimensional Non-Linear Systems: The Lotka–volterra Example

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