Prescribed-Performance Control of Variable-Order Fractional-Order Nonlinear Systems Through Adaptive Dynamic Programming

This article investigates the prescribed-performance control (P²C) of variable-order fractional-order nonlinear systems (VOFONSs) through a control barrier function (CBF)-embedded adaptive dynamic programming (ADP) method. Owing to the time dependence and the global memory characteristics of the variable-order fractional calculus, the existing analysis methods for traditional integer-order systems cannot be directly applied. By using the variable-order fractional calculus and the integration-by-parts formula, the VOFONS is approximately transformed into a time-varying integer-order nonlinear system. To address the P²C problem, a prescribed-performance function is designed to reformulate the P²C problem as a constrained control problem of the time-varying integer-order nonlinear system. Unlike conventional function transformation-based methods, the proposed CBF-embedded ADP control method incorporates the safety constraints into the ADP-based optimal control framework without requiring explicit state transformation. Since the constructed discounted cost function is time-varying due to the involvement of time-varying CBF, it is approximated by a neural network with a time-varying activation function. Then, a time-varying policy iteration algorithm is developed to solve the Hamilton-Jacobi-Bellman (HJB) equation. Hereafter, the CBF-based control policy is derived to guarantee that the tracking error satisfies the predefined constraint. Simulation results verify the effectiveness of the present CBF-embedded ADP control scheme.