Hypersonic State Estimation Using the Frobenius-Perron Operator

Abstract

This paper presents a nonlinear state-estimation algorithm that combines the Frobenius-Perron operator theory with the Bayesian estimation theory. The Frobenius―Perron operator is used to predict evolution of uncertainty in the nonlinear system and obtain the prior probability density function in the estimation process. The Bayesian update rule is used to determine the posterior density function from the available measurements. The framework for this filter is similar to particle filters where the density function is sampled using a cloud of points and the system dynamics are integrated with these points as the initial condition. The key issue in particle filters is that the weight for the sample points are typically determined using histograms to obtain the prior density function, and thus requires many samples for acceptable accuracy. Moreover, the weights of the majority of the particles converge to zero after a few iterations, rendering them useless for state-estimation purposes. This issue can be resolved with the application of the Frobenius―Perron operator, which determines the time evolution of the weights along sample paths. This greatly simplifies the determination of the prior density function and can be achieved with fewer sample points. Consequently, the associated computational time is also greatly reduced. The presented algorithm is demonstrated on a hypersonic reentry problem with uncertain initial states, with given initial probability density functions. The performance is compared with particle filters, and it is observed that the proposed algorithm is computationally superior as expected.