Applications of Control Error Convergence Proof Using Lyapunov Direct Method Approach: for a Class of Mixed Fractional Order Model Reference Adaptive Control

This book compiles some important theoretical advances and application examples, applied to fractional control but mainly, it is an extension of one of our papers published in 2025, in which the Lyapunov stability theory was extended to mixed fractional-order direct model reference adaptive control (FO-DMRAC), where the adaptive control parameter is of fractional order, and the control error model is of integer order. In addition, examples are shown that help to consolidate this theorical result. We have primarily focused on the use of direct adaptive control due to its ease of implementation. However, other methods are not necessarily excluded, as long as they maintain a structure similar to the case where the plant is of integer order, and the adjustment laws are of fractional order. Finally, we present illustrative examples that we hope will allow the reader to gain more confidence in using these theoretical tools, developed over time, in the practical application of these fractional-order adaptive algorithms.