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Fractional-order calculus and derivative is extended from integral-order calculus and derivative. This paper investigates a nonlinear robust control problem using fractional order and operator theory. In order to improve the tracking performance and antidisturbance ability, operator- and fractional-order-based nonlinear robust control for the spiral counter-flow heat exchanger described by the parallel fractional-order model (PFOM) is proposed. The parallel fractional-order model for the spiral counter-flow heat exchanger was identified by particle swarm optimization (PSO) and the parameters of a fractional-order PID (FOPID) controller were optimized by the PSO. First, the parallel fractional-order mathematical model for a spiral counter-flow heat exchanger plant was identified by PSO. Second, a fractional-order PID controller and operator controller for the spiral heat exchanger were designed under the identified parallel fractional-order mathematical model. Third, the parameters of the operator and fractional-order PID were optimized by PSO. Then, tracking and antidisturbance performance of the control system were analyzed. Finally, comparisons of two control schemes were performed, and the effectiveness illustrated.

This paper presents fractional order fixed-time nonsingular terminal sliding mode control for stabilization and synchronization of fractional order chaotic systems with uncertainties and disturbances. First, a novel fractional order terminal sliding mode surface is proposed to guarantee the fixed-time convergence of system states along the sliding surface. Second, a nonsingular terminal sliding mode controller is designed to force the system states to reach the sliding surface within fixed-time and remain on it forever. Furthermore, the fractional Lyapunov stability theory is used to prove the fixed-time stability and the robustness of the proposed control scheme and estimate the upper bound of convergence time. Next, the proposed control scheme is applied to the synchronization of two nonidentical fractional order Liu chaotic systems and chaos suppression of fractional order power system. Simulation results verify the effectiveness of the proposed control scheme. Finally, some application issues about the proposed scheme are discussed.

Community detection is a crucial task in network research, applicable to social systems, biology, cybersecurity, and knowledge graphs. Recent advancements in graph neural networks (GNNs) have exhibited significant representational capability; yet, they frequently experience instability and erroneous clustering, often referred to as ”hallucinations.” These artifacts stem from sensitivity to high-frequency eigenmodes, over-parameterization, and noise amplification, undermining the robustness of learned communities. To mitigate these constraints, we present F 2 -CommNet, a Fourier–Fractional neural framework that incorporates fractional-order dynamics, spectrum filtering, and Lyapunov-based stability analysis. The fractional operator implements long-memory dampening that mitigates oscillations, whereas Fourier spectral projections selectively attenuate eigenmodes susceptible to hallucination. Theoretical analysis delineates certain stability criteria under Lipschitz non-linearities and constrained disturbances, resulting in a demonstrable expansion of the Lyapunov margin. Experimental validation on synthetic and actual networks indicates that F 2 -CommNet reliably diminishes hallucination indices, enhances stability margins, and produces interpretable communities in comparison to integer-order GNN baselines. This study integrates fractional calculus, spectral graph theory, and neural network dynamics, providing a systematic method for hallucination-resistant community discovery.

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 V˙ is positive during a transient period, as long as V˙(e,ϕ) 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 V˙(e,ϕ).

Robot manipulators are widely used in many fields and play a vital role in the assembly, maintenance, and servicing of future complex in-orbit infrastructures. They are also helpful in areas where it is undesirable for humans to go, for instance, during undersea exploration, in radioactive surroundings, and other hazardous places. Robotic manipulators are highly coupled and non-linear multivariable mechanical systems designed to perform one of these specific tasks. Further, the time-varying constraints and uncertainties of robotic manipulators will adversely affect the characteristics and response of these systems. Therefore, these systems require effective modelling and robust controllers to handle such complexities, which is challenging for control engineers. To solve this problem, many researchers have used the fractional-order concept in the modelling and control of robotic manipulators; yet it remains a challenge. This review paper presents comprehensive and significant research on state-of-the-art fractional-order modelling and control strategies for robotic manipulators. It also aims to provide a control engineering community for better understanding and up-to-date knowledge of fractional-order modelling, control trends, and future directions. The main table summarises around 95 works closely related to the mentioned issue. Key areas focused on include modelling, fractional-order modelling type, model order, fractional-order control, controller parameters, comparison controllers, tuning techniques, objective function, fractional-order definitions and approximation techniques, simulation tools and validation type. Trends for existing research have been broadly studied and depicted graphically. Further, future perspective and research gaps have also been discussed comprehensively.

Industrialization has led to a huge demand for a network control system to monitor and control multi-loop processes with high effectiveness. Due to these advancements, new industrial wireless sensor network (IWSN) standards such as ZigBee, WirelessHART, ISA 100.11a wireless, and Wireless network for Industrial Automation-Process Automation (WIA-PA) have begun to emerge based on their wired conventional structure with additional developments. This advancement improved flexibility, scalability, needed fewer cables, reduced the network installation and commissioning time, increased productivity, and reduced maintenance costs compared to wired networks. On the other hand, using IWSNs for process control comes with the critical challenge of handling stochastic network delays, packet drop, and external noises which are capable of degrading the controller performance. Thus, this paper presents a detailed study focusing only on the adoption of WirelessHART in simulations and real-time applications for industrial process monitoring and control with its crucial challenges and design requirements.

In this article, an adaptive sliding mode technique based on a fractional‐order (FO) switching type control law is designed to guarantee robust stability for a class of uncertain three‐dimensional FO nonlinear systems with external disturbance. A novel FO switching type control law is proposed to ensure the existence of the sliding motion in finite time. Appropriate adaptive laws are shown to tackle the uncertainty and external disturbance. The calculation formula of the reaching time is analyzed and computed. The reachability analysis is visualized to show how to obtain a shorter reaching time. A stability criteria of the FO sliding mode dynamics is derived based on indirect approach to Lyapunov stability. Effectiveness of the proposed control scheme is illustrated through numerical simulations. © 2015 Wiley Periodicals, Inc. Complexity 21: 363–373, 2016

This paper proposes a theoretical framework for generalization of the well established first order plus dead time (FOPDT) model for linear systems. The FOPDT model has been broadly used in practice to capture essential dynamic response of real life processes for the purpose of control design systems. Recently, the model has been revisited towards a generalization of its orders, i.e., non-integer Laplace order and fractional order delay. This paper investigates the stability margins as they vary with each generalization step. The relevance of this generalization has great implications in both the identification of dynamic processes as well as in the controller parameter design of dynamic feedback closed loops. The discussion section addresses in detail each of this aspect and points the reader towards the potential unlocked by this contribution.

The goals of this paper are: (a) to investigate adaptive and fractional-order adaptive control algorithms for an automatic anesthesia process, using a closed-loop system, and (b) to develop an easy-to-use tool for MATLAB/Simulink to facilitate simulations for users with less knowledge about anesthesia and adaptive control. A model reference adaptive control structure was chosen for the entire system. First of all, to control the patient’s state during the surgery process, the patient mathematical model is useful, or even required for simulation studies. The pharmacokinetic/pharmacodynamics (PK/PD) model was determined using MATLAB’s SimBiology tool, starting from a previously available block diagram, and validated through simulation. Then, to achieve the desired control performances, two controllers are designed: a PI adaptive controller and a PI λ (PI-fractional) adaptive controller, using the MIT algorithm. The time response during anesthetic drug infusion for each patient can be plotted with the AnesthesiaGUIDE tool, which is also designed in MATLAB/Simulink. The tool was tested on data from 12 patients, subjected to general anesthesia, with successful results. Through this tool, the article provides a good opportunity for any user to experience with adaptive control for the anesthesia process.