Robust adaptive backstepping control of uncertain fractional-order nonlinear systems with input time delay

In the present work, a fractional adaptive backstepping control approach is proposed for controlling a class of uncertain fractional-order nonlinear systems subjected to unknown external disturbance and input delay. In control design, by introducing fractional command-filter, the key issue of the common backstepping method called the “explosion of complexity” is addressed. Furthermore, the Szász–Mirakyan operator is used as a simple but efficient universal approximator to approximate uncertain or unknown terms and reduce the design complexity. In addition, to enhance the capability of the proposed controller in dealing with the approximation error and the unknown disturbance, an efficient robust control term is also incorporated into it. One advantage of the suggested control approach is that the input-delay fractional-order system is effectively treated by introducing a new variable as a non-delayed fractional-order system. The other advantage is that the number of tuning variables is considerably low; specifically, for an n-order fractional nonlinear system, only n adaptive parameters are adjusted online. This makes the implementation of the proposed approach simpler. The Lyapunov stability theorem is used to prove that all of the closed-loop system’s signals are semi-globally uniformly ultimately bounded. The feasibility of the suggested control approach is shown by comparing the simulation results obtained with the fuzzy system.