Parametric Stability of Non-Linearly Elastic Composite Plates by Lyapunov Exponents

Gilat, Rivka; Aboudi, Jacob

doi:10.1006/jsvi.2000.2936

Cited by 16 publications

(9 citation statements)

References 11 publications

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“…It is an average of the contraction and expansion characteristics of the orbits in different directions in the phase space. 21 N dimensional system has n Lyapunov exponents, and each Lyapunov exponent can be regarded as the phase space in all directions of motion relative to the local deformation of the average…”

Section: Calculation Of Lyapunov Exponentsmentioning

confidence: 99%

Nonlinear dynamics modeling and simulation of two-wheeled self-balancing vehicle

Liu

Huang

Wang

et al. 2016

International Journal of Advanced Robotic Systems

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Two-wheeled self-balancing vehicle system is a kind of naturally unstable underactuated system with high-rank unstable multivariable strongly coupling complicated dynamic nonlinear property. Nonlinear dynamics modeling and simulation, as a basis of two-wheeled self-balancing vehicle dynamics research, has the guiding effect for system design of the project demonstration and design phase. Dynamics model of the two-wheeled self-balancing vehicle is established by importing a TSi ProPac package to the Mathematica software (version 8.0), which analyzes the stability and calculates the Lyapunov exponents of the system. The relationship between external force and stability of the system is analyzed by the phase trajectory. Proportional-integral-derivative control is added to the system in order to improve the stability of the twowheeled self-balancing vehicle. From the research, Lyapunov exponent can be used to research the stability of hyperchaos system. The stability of the two-wheeled self-balancing vehicle is better by inputting the proportional-integral-derivative control. The Lyapunov exponent and phase trajectory can help us analyze the stability of a system better and lay the foundation for the analysis and control of the two-wheeled self-balancing vehicle system.

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Section: Calculation Of Lyapunov Exponentsmentioning

confidence: 99%

Nonlinear dynamics modeling and simulation of two-wheeled self-balancing vehicle

Liu

Huang

Wang

et al. 2016

International Journal of Advanced Robotic Systems

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show abstract

“…In addition, Lyapunov exponents have also been used for the stability analysis of complex nonlinear systems, as discussed in detail in [7][8][9][10][11][12][13] and the references cited in our previous work [14]. However, when considering real world physical systems, those crucial differential equations are not always known.…”

Section: Introductionmentioning

confidence: 99%

A robust method on estimation of Lyapunov exponents from a noisy time series

Yang

2010

Nonlinear Dyn

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Lyapunov exponents can indicate the asymptotic behaviors of nonlinear systems, and thus can be used for stability analysis. However, it is notoriously difficult to estimate these exponents reliably from experimental data due to the measurement error (noise). In this paper, a novel method for estimating Lyapunov exponents from a time series in the presence of additive noise corruption is presented. The method combines the ideas of averaging the noisy data to form new neighbors and of nonlinear mapping to determine neighborhood mapping matrices. Two case studies of balancing control of a bipedal robot and the Lorenz systems are presented to demonstrate the efficacy of the proposed method. The bipedal robot system has two negative Lyapunov exponents while the Lorenz system has one positive, zero, and negative exponents, respectively. It is shown that, as compared with the existing methods, our proposed one is more robust to the ratio of signal to noise, and is particularly effective in estimating negative Lyapunov exponents. We believe that the work can contribute significantly to the stability analysis of nonlinear systems using a noisy time series.

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“…In addition, Lyapunov exponents have also been used for stability analysis of complex nonlinear systems. For example, using Lyapunov exponents, Asokanthan and Wang [7] studied the torsional stability of a Hooke's joint driven system, and Gilat and Aboudi [8] studied parametric stability of nonlinearly elastic composite plates. Zevin and Pinski [9] developed absolute stability criterion for a non-autonomous linear system controlled by a nonlinear feedback control with a time-varying delay based on Lyapunov exponents.…”

Section: Introductionmentioning

confidence: 99%

On stability analysis via Lyapunov exponents calculated from a time series using nonlinear mapping—a case study

Yang

2009

Nonlinear Dyn

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The concept of Lyapunov exponents has been mainly used for analyzing chaotic systems, where at least one exponent is positive. The methods for calculating Lyapunov exponents based on a time series have been considered not reliable for computing negative and zero exponents, which prohibits their applications to potentially stable systems. It is believed that the local linear mapping leads to inaccurate matrices which prevent them from calculating negative exponents. In this work, the nonlinear approximation of the local neighborhood-to-neighborhood mapping is derived for constructing more accurate matrices. To illustrate the approach, the Lyapunov exponents for a stable balancing control system of a bipedal robot during standing are calculated. The time series is generated by computer simulations. Nonlinear mapping is constructed for calculating the whole spectrum of Lyapunov exponents. It is shown that, as compared with those from the linear mapping, (1) the accuracy of the negative exponents calculated using the nonlinear mapping is significantly improved; (2) their sensitivity to the time lag and the evolution time is significantly reduced; and (3) no spurious Lyapunov exponent is generated if the dimension of the state space is known. Thus, the work can contribute significantly to stability analysis of robotic control systems. Issues on extending the concept of Lyapunov exponents to analyzing stable systems are also addressed.

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Parametric Stability of Non-Linearly Elastic Composite Plates by Lyapunov Exponents

Cited by 16 publications

References 11 publications

Nonlinear dynamics modeling and simulation of two-wheeled self-balancing vehicle

Nonlinear dynamics modeling and simulation of two-wheeled self-balancing vehicle

A robust method on estimation of Lyapunov exponents from a noisy time series

On stability analysis via Lyapunov exponents calculated from a time series using nonlinear mapping—a case study

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