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

Abstract

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.

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

“…Note that Z r (n; T 0 ) in (14) represents the neighboring vectors, which is equivalent to Z r (n; T 0 ) described in Sect. 3.2, except that the effects of noise have been compensated using the averaging procedure.…”

“…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.…”

“…However, when considering real world physical systems, those crucial differential equations are not always known. Even if the models are available, due to their complexities and uncertainties, the estimation of Lyapunov exponents can be unfeasible [13][14][15][16].…”

“…However, it is extremely important for the analysis of stable systems, especially those of which all exponents are negative. To address this challenge, in our previous work [14], an improved method was proposed for estimating negative Lyapunov exponents from a time series using nonlinear mapping to form neighborhood matrices. The results have shown that our method has three advantages over those based on linear mapping, in that (1) the accuracy of the negative Lyapunov exponents estimated using nonlinear mapping is improved as compared with those from linear mapping; (2) the Lyapunov exponents are less sensitive to the parameters, such as the time lag and the evolve time; and (3) since Taken's theorem on embedding dimension is often required for linear mapping, spurious exponents are generated.…”

“…In the estimation of Lyapunov exponents, the selection of several parameters, such as the time delay, evolution time and embedding dimensions, can affect the estimated exponents significantly [3,35]. In this work, the time delays were determined by the first zero crossing of the autocorrelation method [14]. The selections of the embedding dimension and the evolution time are discussed later.…”