Matched–Mismatched Uncertainty Compensation in Dynamic SMC Using Optimal Fractional Loop-Transfer-Recovery Observer

A new fractional dynamic sliding mode control (FD-SMC) framework is introduced to reduce chattering in the control of fractional-order chaotic systems. In this method, chattering is eliminated by placing a fractional integrator before the system control input. As a result, the augmented system has a higher dimension than the original system, meaning that additional states are introduced. Effective control therefore requires identifying or estimating these new states or the corresponding plant model. To address this issue, a robust optimal fractional loop-transfer-recovery observer (ROF-LTRO) is developed. Furthermore, the key advantage of sliding mode control (SMC)—its invariance to matched uncertainties—is often lost in many plants such as chaotic systems, because many of them contain mismatched uncertainties. To restore and extend the invariance property, multiple sliding surfaces combined with a virtual control input are employed. In addition, the proposed FD-SMC and ROF-LTRO do not rely on prior knowledge of uncertainty bounds, which is beneficial for practical implementation. Then, a two-stage design procedure based on two-surface definition is presented, and simulation results are provided for the extended fractional Duffing–Holmes chaotic system (EF-DHCS) under both matched and mismatched uncertainties.