Optimal control of fractional systems based on the diffusive representation

This paper deals with optimal control of fractional order systems using control vector parameterization. The main idea consists in transforming the optimal control problem to a nonlinear optimization problem where the optimization variables are the parameters of the optimal control law to be determined. Thus, by parameterizing the control variable by unknown parameters and by substituting its expression in the diffusive representation of the fractional system, a set of ordinary differential equations is obtained. These equations are solved by the variational iteration method to get an approximate analytical expression of the optimal trajectories as a function of time and the unknown parameters of the optimal control law. Then, by substituting the expression of the control law and the optimal trajectories into the performance index, a non linear optimization problem is obtained where the control law parameters are the optimization variables. The solution of the obtained optimization problem using a global optimization method gives the optimal values of the control parameters, that is, the optimal control law. The proposed approach is illustrated by an application example.