A Fractional Order Sliding Mode Control (FOSMC) is proposed in this paper for an integer second order nonlinear system with an unknown additive perturbation term. A sufficient condition is given to assure the attractiveness to a given sliding surface where trajectory tracking is assured, despite the presence of the perturbation term. The control scheme is applied to the model of a quadrotor vehicle in order to have trajectory tracking in the space. Simulation results are presented to evaluate the performance of the control scheme.

This study introduces a novel model reference adaptive control (MRAC) framework that incorporates fractional-order gradients (FGs) to regulate the displacement of an inverted pendulum-cart system. Fractional-order gradients have been shown to significantly improve convergence rates in domains such as machine learning and neural network optimization. Nevertheless, their integration with fractional-order error models within adaptive control paradigms remains unexplored and represents a promising avenue for research. The proposed control scheme extends the classical MRAC architecture by embedding Caputo fractional derivatives into the adaptive law governing parameter updates, thereby improving both convergence dynamics and control flexibility. To ensure optimal performance across multiple criteria, the controller parameters are systematically tuned using a multi-objective Particle Swarm Optimization (PSO) algorithm. Two fractional-order error models (FOEMs) incorporating fractional gradients (FOEM2-FG, FOEM3-FG) are investigated, with their stability formally analyzed via Lyapunov-based methods under conditions of sufficient excitation. Validation is conducted through both simulation and real-time experimentation on a physical pendulum-cart setup. The results demonstrate that the proposed fractional-order MRAC (FOMRAC) outperforms conventional MRAC, proportional-integral-derivative (PID), and fractional-order PID (FOPID) controllers. Specifically, FOMRAC-FG achieved superior tracking performance, attaining the lowest Integral of Squared Error (ISE) of $2.32\times {10}^{-5}$ and the lowest Integral of Squared Input (ISI) of 6.40 in simulation studies. In real-time experiments, FOMRAC-FG maintained the lowest ISE ($5.11\times {10}^{-6}$). Under real-time experiments with disturbances, it still achieved the lowest ISE ($1.06\times {10}^{-5}$), highlighting its practical effectiveness.

A method for fault detection and isolation is developed using the concatenated variances of the continuous wavelet transform (CWT) of plant outputs. These concatenated variances are projected onto the principal component space corresponding to the covariance matrix of the concatenated variances. Fisher and quadratic discriminant analyses are then performed in this space to classify the concatenated sample CWT variances of outputs in a given time window. The sample variance is a variance estimator obtained by taking the displacement average of the squared wavelet transforms of the current outputs. This method provides an alternative to the multimodel approach used for fault detection and identification, especially when system inputs are unmeasured stochastic processes, as is assumed in the case of the mechanical system example. The performance of the method is assessed using matrices having the percentage of correct condition identification in the diagonal and the percentages misclassified conditions in the off-diagonal elements. Considerable performance improvements may be obtained due to concatenation-when two or more outputs are available-and to discriminant analysis, as compared with other wavelet variance methods.

This paper extends Lyapunov stability theory 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. The proposed approach can also be applied to other types of model reference adaptive controllers (MRACs), provided the form of the control error dynamics and the fractional order adaptive control law are similar. This paper demonstrates that the control error will converge to zero, even if the derivative of the classical Lyapunov function $\dot{V}$ is positive during a transient period, as long as $\dot{V}(e,\varphi )$ tends to zero as time approaches infinity. Finally, this paper provides application examples that illustrate both the convergence of the control error to zero and the behavior of $\dot{V}(e,\varphi )$.

In adaptive control, error models use system output error and adaptive laws to update controller parameters for control or identification tasks. Fractional-order calculus, involving non-integer-order derivatives and integrals, is increasingly important for modeling, estimation, and control due to its ability to generalize classical methods and offer improved robustness to disturbances. This paper addresses the gap in the literature where fractional-order gradient methods have not yet been extensively applied in identification and adaptive control schemes. We introduce a fractional-order error model with fractional-order gradient (FOEM1-FG), which integrates fractional gradient operators based on the Caputo fractional derivative. By using theoretical analysis and simulations, we confirm that FOEM1-FG maintains stability and ensures bounded output errors across a variety of input signals. Notably, the fractional gradient’s performance improves as the order, $\beta$, increases with $\beta >1$, leading to faster convergence. Compared to existing integer-order methods, the proposed approach provides a more flexible and efficient solution in adaptive identification and control schemes. Our results show that FOEM1-FG offers superior stability and convergence characteristics, contributing new insights to the field of fractional calculus in adaptive systems.