On asymptotically stabilizing control of nonlinear fractional control systems using an optimization scheme

Abstract

In this paper, stabilization of the nonlinear fractional order systems with unknown control coefficients is considered where the dynamic control system depends on the Caputo fractional derivative. Related to the nonlinear fractional control (NFC) system, an infinite-horizon optimal control (OC) problem is first proposed. It is shown that the obtained OC problem can be an asymptotically stabilizing control for the NFC system. Using the help of an approximation, the Caputo derivative is replaced with the integer order derivative. The achieved infinite-horizon OC problem is then converted into an equivalent finite-horizon one. According to the Pontryagin minimum principle for OC problems and by constructing an error function, an unconstrained minimization problem is defined. In the optimization problem, trial solutions are used for state, costate and control functions where these trial solutions are constructed by using a two-layered perceptron neural network. A learning algorithm with convergence properties is also provided. Two numerical results are introduced to explain the main results.