Optimal State Control of Fractional Order Differential Systems: The Infinite State Approach

Optimal control of fractional order systems is a long established domain of fractional calculus. Nevertheless, it relies on equations expressed in terms of pseudo-state variables which raise fundamental questions. So in order remedy these problems, the authors propose in this paper a new and original approach to fractional optimal control based on a frequency distributed representation of fractional differential equations called the infinite state approach, associated with an original formulation of fractional energy, which is intended to really control the internal system state. In the first step, the fractional calculus of variations is revisited to express appropriate Euler Lagrange equations. Then, the quadratic optimal control of fractional linear systems is formulated. Thanks to a frequency discretization technique, the previous theoretical equations are converted into an equivalent large dimension integer order system which permits the implementation of a feasible optimal solution. A numerical example illustrates the validity of this new approach.

Automated summary

The paper introduces a new approach to fractional optimal control based on a frequency distributed representation of fractional differential equations combined with an original formulation of fractional energy to truly control the internal system state. By revisiting the fractional calculus of variations to express appropriate Euler Lagrange equations, the quadratic optimal control of fractional linear systems is formulated. Through a frequency discretization technique, theoretical equations are converted into an equivalent large dimension integer order system for the implementation of a feasible optimal solution, as demonstrated in a numerical example.