Time-Optimal Control of Systems with Fractional Dynamics

Christophe Tricaud,Yangquan Chen

doi:10.1155/2010/461048

Christophe Tricaud, Yangquan Chen

Open Access

https://doi.org/10.1155/2010/461048

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Abstract

We introduce a formulation for the time-optimal control problems of systems displaying fractional dynamics in the sense of the Riemann-Liouville fractional derivatives operator. To propose a solution to the general time-optimal problem, a rational approximation based on the Hankel data matrix of the impulse response is considered to emulate the behavior of the fractional differentiation operator. The original problem is then reformulated according to the new model which can be solved by traditional optimal control problem solvers. The time-optimal problem is extensively investigated for a double fractional integrator and its solution is obtained using either numerical optimization time-domain analysis.

Highlights

In the world surrounding us, the physical laws of dynamics are not always followed by all systems
The field of IOOCs has been investigated for a long time and a large collection of numerical techniques has been developed to solve this category of problems 4
We introduce a formulation to a special class of FOCP: the Fractional Time-Optimal Control Problem FTOCP

Summary

Introduction

In the world surrounding us, the physical laws of dynamics are not always followed by all systems. Frederico and Torres 13–15 , using similar definitions of the FOCPs, formulated a Noether-type theorem in the general context of the fractional optimal control in the sense of Caputo and studied fractional conservation laws in FOCPs. In 6 , a rational approximation of the fractional derivatives operator is used to link FOCPs and the traditional IOOCs. A new solution scheme is proposed in 16 , based on a different expansion formula for fractional derivatives. These different denominations define the same kind of optimal control problem in which the purpose is to transfer a system from a given initial state to a specified final state in minimum time This special class of FOCPs has been disregarded in the literature.

The Fractional Derivative Operator

Fractional Optimal Control Problem Formulation

Solution Using Rational Approximation of the Fractional Operator

Solution Using Bang-Bang Control Theory

Results

Conclusions