Determination of Limit Cycles Using Both the Slope of Correlation Integral and Dominant Lyapunov Methods

Castillo, Rogelio; Alonso, Gustavo; Palacios, Javier

doi:10.13182/nt04-a3465

Cited by 8 publications

(3 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then Lyapunov exponents λ i K are given by (11) where K is the available number of matrices, T is sampling time step, i = 1, 2, . .…”

Section: Calculation Of Lyapunov Exponents Based On a Time Seriesmentioning

confidence: 99%

“…Awrejcewicz and Kudra [10] carried out the stability analysis of a multi-body mechanical system with rigid unilateral constraints via Lyapunov exponents. Rogelio et al [11] developed a method for nonlinear analysis of boiling water sector in a nuclear generator based on Lyapunov exponents. Wu et al [12] developed a continuous feedback control law stabilizing a based-excited inverted pendulum system where the stability was proven using the largest Lyapunov exponents.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Yang

2009

Nonlinear Dyn

View full text Add to dashboard Cite

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.

show abstract

“…Then Lyapunov exponents λ i K are given by (11) where K is the available number of matrices, T is sampling time step, i = 1, 2, . .…”

Section: Calculation Of Lyapunov Exponents Based On a Time Seriesmentioning

confidence: 99%

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

View full text Add to dashboard Cite

show abstract

“…However, in general BWR signals are non-stationary and non-linear, thus Fourier-based or wavelet-based approaches might lead to a biased stability analysis. Several methods for non-linear BWR stability analysis have been applied before [12,13], to study BWR signals containing stationary and non-stationary segments. In this work, the Shannon Entropy (SE) was applied, to infer whether it can be used as a novel stability parameter for BWRs.…”

mentioning

confidence: 99%

Non-Linear Stability Analysis of Real Signals from Nuclear Power Plants (Boiling Water Reactors) Based on Noise Assisted Empirical Mode Decomposition Variants and the Shannon Entropy

Olvera-Guerrero

Prieto‐Guerrero

Espinosa-Paredes

2017

Entropy

View full text Add to dashboard Cite

There are currently around 78 nuclear power plants (NPPs) in the world based on Boiling Water Reactors (BWRs). The current parameter to assess BWR instability issues is the linear Decay Ratio (DR). However, it is well known that BWRs are complex non-linear dynamical systems that may even exhibit chaotic dynamics that normally preclude the use of the DR when the BWR is working at a specific operating point during instability. In this work a novel methodology based on an adaptive Shannon Entropy estimator and on Noise Assisted Empirical Mode Decomposition variants is presented. This methodology was developed for real-time implementation of a stability monitor. This methodology was applied to a set of signals stemming from several NPPs reactors (Ringhals-Sweden, Forsmark-Sweden and Laguna Verde-Mexico) under commercial operating conditions, that experienced instabilities events, each one of a different nature.Entropy 2017, 19, 359 2 of 33 operates at a low mass flow and at high nominal power. Another current trend is to increase the size of the core, which causes a weaker special coupling in the neutron field which increases the susceptibility of the reactor to experiencing unstable oscillations. In summary, all current tendencies related to reactor design enhance the regions where the reactor should not be operated (reactor operation at low flow and high power).Also, the DR often jumps discontinuously from the well stable to the far-unstable region [7]. The BWR stability is of primary interest from the point of view of BWR operation, due to the fact, that the stability margin may be strongly reduced during plant maneuvering and transients [8]. According to these issues, the DR might not be a reliable monitoring index after all, under certain operating conditions. Besides, in regular operating conditions, the need for stationary signals might be a handicap for DR estimation. Thus, it is relevant to explore new alternative methodologies and indexes adapted to accommodate for non-stationary and non-linear BWR signal behavior.In [9] a short time Fourier transform based technique was explored to study the time dependence of the natural frequency when the BWR signal is non-stationary. Later, the wavelet theory was applied to explore new alternatives for transient instability analysis [10,11]. However, in general BWR signals are non-stationary and non-linear, thus Fourier-based or wavelet-based approaches might lead to a biased stability analysis. Several methods for non-linear BWR stability analysis have been applied before [12,13], to study BWR signals containing stationary and non-stationary segments. In this work, the Shannon Entropy (SE) was applied, to infer whether it can be used as a novel stability parameter for BWRs. The SE is a concept that was developed by Claude E. Shannon [14] to study a discrete source through the information content of this source. The SE is a statistical index that quantifies the complexity of a signal. In this case, the BWR stability issue is assessed quantifying the complexity of BWR si...

show abstract

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

Yang

2010

Nonlinear Dyn

View full text Add to dashboard Cite

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.

show abstract

Determination of Limit Cycles Using Both the Slope of Correlation Integral and Dominant Lyapunov Methods

Cited by 8 publications

References 11 publications

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

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

Non-Linear Stability Analysis of Real Signals from Nuclear Power Plants (Boiling Water Reactors) Based on Noise Assisted Empirical Mode Decomposition Variants and the Shannon Entropy

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

Contact Info

Product

Resources

About