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Open AccessJournal ArticleDOI

Second-order counterexamples to the discrete-time Kalman conjecture

William P. Heath, +2 more
- 01 Oct 2015 - 
- Vol. 60, Iss: 60, pp 140-144
TLDR
A class of second-order discrete-time systems for which the Kalman conjecture is true provided the nonlinearity is odd, but false in general is discussed, which has strong implications for the analysis of saturated systems.
About: 
This article is published in Automatica.The article was published on 2015-10-01 and is currently open access. It has received 67 citations till now. The article focuses on the topics: Aizerman's conjecture & Counterexample.

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Journal ArticleDOI

Hidden attractors in dynamical systems

Dawid Dudkowski, +6 more
- 03 Jun 2016 - 
TL;DR: In this paper, the authors discuss the most representative examples of hidden attractors, discuss their theoretical properties and experimental observations, and also describe numerical methods which allow identification of the hidden attractor.
Journal ArticleDOI

Hold-In, Pull-In, and Lock-In Ranges of PLL Circuits: Rigorous Mathematical Definitions and Limitations of Classical Theory

Gennady A. Leonov, +3 more
- 16 May 2015 - 
TL;DR: In this paper, a survey of hold-in, pull-in and lock-in ranges is presented, and a solution for the unique definition of the lockin frequency, posed by Gardner is suggested.
Book ChapterDOI

Hidden attractors in fundamental problems and engineering models. A short survey

Nikolay Kuznetsov
TL;DR: In this paper, the concept of self-excited and hidden attractors is discussed, and various corresponding examples of selfexcited attractors are considered, based on surveys [1, 2, 3, 4].
Journal ArticleDOI

Hidden chaotic attractors in a class of two-dimensional maps

Haibo Jiang, +3 more
- 31 May 2016 - 
TL;DR: In this article, the hidden dynamics of a class of two-dimensional maps inspired by the Henon map are studied. But the authors focus on three typical scenarios which may generate hidden dynamics, i.e., no fixed point, single fixed point and two fixed points.
Journal ArticleDOI

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

Joaquin Carrasco, +2 more
- 01 Mar 2016 - 
TL;DR: This tutorial paper aims to provide a clear and comprehensive introduction to the topic of absolute stability from a user viewpoint, reviewing the stability theory, the properties of the multipliers (including their phase properties, phase-equivalence results and the issues associated with causality), and convex searches.
References
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Book

Nonlinear Systems Analysis

Mathukumalli Vidyasagar, +1 more
TL;DR: In this article, the authors consider non-linear differential equations with unique solutions, and prove the Kalman-Yacubovitch Lemma and the Frobenius Theorem.
Book

The stability of dynamical systems

Joseph P. LaSalle
TL;DR: In this paper, an introduction to aspects of the theory of dynamical systems based on extensions of Liapunov's direct method is presented and the main ideas and structure for the theory are presented for difference equations and for the analogous theory for ordinary differential equations and retarded functional differential equations.
Journal ArticleDOI

Nonlinear Systems Analysis

Mathukumalli Vidyasagar, +1 more
- 01 Jun 1978 - 
TL;DR: Non-linear Differential Equations with Unique Solutions, Proof of the Kalman-Yacubovitch Lemma and proof of the Frobenius Theorem.
Journal ArticleDOI

System analysis via integral quadratic constraints

Alexandre Megretski, +1 more
- 01 Jun 1997 - 
TL;DR: A stability theorem for systems described by IQCs is presented that covers classical passivity/dissipativity arguments but simplifies the use of multipliers and the treatment of causality.
Book

Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach

Wassim M. Haddad, +1 more
TL;DR: The aim of this book is to provide a Discussion of the Foundations of Discrete-Time Optimal Nonlinear Feedback Control and its Applications in Dynamical Systems and Differential Equations, as well as some suggestions for further study.
Related Papers (5)

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

Gennady A. Leonov, +2 more
- 05 Mar 2013 - 

Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion

Gennady A. Leonov, +4 more
- 27 Jul 2015 - 

Stability Conditions for Systems with Monotone and Slope-Restricted Nonlinearities

George Zames, +1 more
- 01 Feb 1968 - 

Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua's circuits

V. O. Bragin, +3 more
- 01 Aug 2011 - 

Localization of hidden Chuaʼs attractors

Gennady A. Leonov, +3 more
- 06 Jun 2011 - 
Frequently Asked Questions (6)
Q1. What are the contributions mentioned in the paper "Second-order counterexamples to the discrete-timekalman conjecture" ?

The authors show that it is false in general for second-order discrete-time systems by construction of counterexamples with stable periodic solutions. The authors discuss a class of second-order discrete-time systems for which it is true provided the nonlinearity is odd, but false in general. 

(1)The origin of the system x(i+ 1) = f(x(i)) is said to be global asymptotically stable, henceforth GAS, if f(0) = 0 and for any > 0 and x(0) there exists n such that |x(i)| < for all i > n. 

The system Σ has a non-trivial periodic solution y(i) = −y(i − 2) if ( √ 2− 1) < a < 1 and S > kp = (a 2+1)2a2+2a−1 .Corollary 1 (Carrasco et al. (2015b)) 

As a result, stability properties of discrete-time Lur’e systems with secondorder and higher plants cannot be derived from the feedback interconnection between the linear system G and a linear gain k as in the continuous-time domain. 

Such analysis requires renormalizing about steady-state values; if these are non-zero then there is no reason to assume the renormalized nonlinearity is odd even when it is an odd saturation function in the nominal case. 

As part of their wider analysis of discrete-time planar systems under odd saturation, they consider the two-state systemx(i+ 1) =[ a11x1(i) + a12x2(i)sat(a21x1(i) + a22x2(i))] , (6)where x(i) = (x1(i), x2(i)).