Stabilizing inverted pendulum systems remains a challenging and open control problem due to their inherent instability and relevance in a wide range of real-world applications, including robotics and aerospace systems. While PID and fractional-order PID (FOPID) controllers offer distinct advantages, they individually suffer from trade-offs between performance and control energy. This paper presents the design, implementation, and experimental validation of a switched SW FOPID-PID controller for the stabilization of an inverted pendulum (InvP) system, aiming to achieve an improved balance between system performance and control energy used. The controller was tuned offline using particle swarm optimization (PSO) and a mathematical model of the system for simulation. Additional PID and FOPID controllers were also designed, tuned and validated for comparison purposes. Their performance was assessed through key indicators, including ITAE, ISI, settling time, peak values, and variance and compared against a manufacturer-provided PID controller. The experimental results demonstrated that all three designed controllers outperformed the manufacturer’s PID under nominal conditions. The SW FOPID-PID controller achieved the best overall performance, balancing control energy efficiency and response quality. Under external disturbances, the FOPID and SW FOPID-PID controllers exhibited superior robustness, with the switched controller being the most effective, responding quickly to disturbances while minimizing positional and angular errors. Still, this research is limited to a specific plant and switching strategy; thus, further validation on other systems and switching criteria is necessary to generalize these findings.

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.

The leaching process is essential in the mining industry, because it efficiently extracts valuable minerals, such as copper. However, monitoring and controlling the leaching process presents significant challenges due to material variability, uneven distribution of the leaching solution, and the effects of environmental factors like temperature and moisture content. One of the main technological challenges is measuring variables within the leaching heap. Implementing state observers or estimators (i.e., virtual sensors) offers a promising solution, allowing for a cost-effective estimation of non-measurable process variables. To validate this approach, this paper proposes and analyzes the use of two estimation methods, the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF), to estimate the moisture content, copper in the ore, and acid consumption based on measurements of acid and copper concentrations in the heap leaching process. The results obtained from simulations demonstrate accurate estimations from both state observers. The variable best estimated with EKF was the moisture content, achieving a 0.041% Integral Absolute Error (IAE) and a 0.069% Integral Square Error (ISE) in one of the analyzed scenarios. Utilizing these state estimators improves the understanding of the internal dynamics of heap leaching, often limited by the lack of field-level instrumentation, such as sensors and transmitters. This approach can enhance the operational efficiency of heap leaching plants by enabling the real-time estimation of unmeasurable variables, ultimately improving metal recovery and reducing acid consumption.

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 $\dot{V}(e,\varphi )$.