Continuous Time and Discrete Time Fractional Order Adaptive Control for a Class of Nonlinear Systems

Date of Award


Degree Name

Ph.D. in Engineering


Department of Electrical and Computer Engineering


Advisor: Raúl Ordóñez


This research was motivated by the generalization of the derivative and difference orders. It presents a new adaptive control technique using fractional-order derivative based on Caputo's definition. This is the first time the fractional-order adaptive laws have been used with an integer order stable manifold for approximating the uncertainty. We have utilized direct and indirect adaptive control techniques in both Continuous-time (CT) and Discrete-time. In CT, the DAC approach is used to approximate the controller, while the IAC is used to approximate the plant, using universal function approximators such as fuzzy systems or multi-layer perceptrons with one hidden layer. A new Lemma is introduced to generalize the equality of the Caputo and Riemann-Liouville derivatives which is useful for developing a Lyapunov candidate to prove the stability of the fractional-order adaptive law for direct and indirect cases. Furthermore, this study demonstrates that traditional integer-order adaptive control is a special case of the more general fractional-order adaptive control scheme introduced here. A magnetic levitation example is provided to show the practicality of using the fractional-order derivative and difference for updating the approximator parameters. In DT, we implement a stand-alone function approximation, in structured uncertainty form, via generalization of the normalized gradient descent adaptive law to include the fractional orders. The designed fractional order is further modified to identify the true ideal value of the parameter vector in spite of the lack of persistence of excitation. We prove the stability using two different approaches(i.e. Lyapunov direct method and matrix stability analysis). Moreover, this study is extended to control a class of discrete nonlinear systems. Moreover, and crucially, the technique introduced here in both CT and DT does not make unrealistic assumptions about the plant, since the fractional-order differential equations are used to increase the degrees of freedom of the adaptive technique and thus provide extra flexibility to the control designer. For both CT and DT, this research will open the door for utilizing fractional calculus in the control system.


Electrical Engineering, Computer Engineering, Continuous-Time, Discrete-Time, Control System, Function Approximation, Discrete Fractional Calculus, Adaptive Control

Rights Statement

Copyright 2019, author