Fractional-Order Online Policy Iteration-Based Approximate Optimal Control for Fractional-Order Nonlinear Systems

In this article, a fractional-order online policy iteration (FOOPI) algorithm-based approximate optimal control scheme is developed for fractional-order nonlinear systems (FONSs). Using the fractional Taylor series and the property of Hadamard product, the fractional Hamilton–Jacobi–Bellman (FHJB) equation corresponding to FONSs is formulated. Motivated by the procedure of solving the optimal control problem of integer-order nonlinear systems (IONSs), the design of the fractional-order (FO) controller for FONSs is transformed into solving the FHJB equation utilizing a FOOPI algorithm. By constructing a critic neural network (NN) and using the fractional calculus theory, the FO derivatives of the value function are established, followed by the approximate fractional Hamiltonian, which is obtained via the FOOPI algorithm. Hereafter, the FO control policy is obtained approximately to guarantee the stability of closed-loop FONSs via the FO Lyapunov’s direct method. Simulation results guarantee the effectiveness of the proposed FOOPI-based approximate optimal control scheme.