Numerical Methods for Fractional Optimal Control and Estimation

Abstract

Fractional calculus has become a valuable mathematical tool for modeling various physical phenomena exhibiting anomalous dynamics such as memory and hereditary properties. However, the fractional operators lead to difficulties in analysis, optimization, and estimation that limit the application of fractional models. This paper develops numerical methods to solve fractional optimal control and estimation problems with Caputo derivatives of arbitrary order.

First, fractional Pontryagin's maximum principle is used to formulate first-order necessary conditions for fractional optimal control problems. A fractional collocation method using polynomial basis functions is then proposed to discretize the resulting boundary value problems. This allows transforming an infinite-dimensional optimal control problem into a finite nonlinear programming problem.

Second, for fractional estimation, a novel ensemble Kalman filter is proposed based on a Monte Carlo approach to propagate the fractional state dynamics. This provides a recursive fractional state estimator analogous to the classical Kalman filter.

The capabilities of the proposed collocation and ensemble Kalman filter methods are demonstrated through applications including fractional epidemic control, thermomechanical oscillator control, and state estimation of viscoelastic mechanical systems. The results illustrate improved accuracy over prior discretization schemes along with the ability to handle complex system dynamics.

This work provides a comprehensive framework for numerical solution of fractional optimal control and estimation problems. The methods enable applying fractional calculus to address challenges in robotics, biomedicine, mechanics, and other fields where systems exhibit non-classical dynamics.