Nonlinear filters for chaotic oscillatory systems

Abstract

This paper examines and contrasts the feasibility of joint state and parameter estimation of noisedriven chaotic systems using the extended Kalman filter (EKF), ensemble Kalman filter (EnKF), and particle filter (PF). In particular, we consider the chaotic vibration of a noisy Duffing oscillator perturbed by combined harmonic and random inputs ensuing a transition probability density function (pdf) of motion which displays strongly non-Gaussian features. This system offers computational simplicity while exhibiting a kaleidoscope of dynamical behavior with a slight change of input and system parameters. An extensive numerical study is undertaken to contrast the performance of various nonlinear filtering algorithms with respect to sparsity of observational data and strength of model and measurement noise. In general, the performance of EnKF is better than PF for smaller ensemble size, while for larger ensembles PF outperforms EnKF. For moderate measurement noise and frequent measurement data, EKF is able to correctly track the dynamics of the system. However, EKF performance is unsatisfactory in the presence of sparse observational data or strong measurement noise.

Application of He’s homotopy perturbation method to nonlinear shock damper dynamics

“…The widely applied techniques (i.e., perturbation method) are of great interest to be used in engineering systems [2,3]. To eliminate the limitation of "small parameter", which is assumed in the perturbation method, a new technique based on the homotopy terminology [4][5][6] has been proposed.…”

Mixed ℋ︁2/ℋ︁∞ nonlinear filtering

“…By the zero-state detectability of { f, h 1 }, we have lim t→∞x (t) = 0, and we can conclude asymptotic stability by LaSalle's invariance principle [24]. On the other hand, if we have strict inequality,˙Y <− 1 2 z 2 , asymptotic stability follows immediately from Lyapunov's theorem, and lim t→∞ z(t) = 0.…”

“…Consider the LTI system l defined by (24) and the MH2HINLFP for this system. Suppose that C 1 is full column rank and (A, C 1 ) is detectable.…”

Joint maximum a posteriori state path and parameter estimation in stochastic differential equations

“…To illustrate the proposed estimator, we apply it on simulated data of the Duffing oscillator, a benchmark model for modeling nonlinear dynamics and chaos [Aguirre and Letellier, 2009] and state estimation in SDEs [Ghosh et al, 2008, Khalil et al, 2009, Namdeo and Manohar, 2007. We note that the standard Duffing oscillator does not satisfy Assum.…”