Fractional Lyapunov inequalities provide an effective tool for stability analysis of fractional-order systems, since classical chain and product rules are not directly applicable to fractional operators. This paper investigates admissible convexity conditions for time-varying Lyapunov functions in continuous Caputo and discrete nabla fractional settings. It is shown that state convexity alone is insufficient: a historical ordering condition is needed to ensure the proper sign of the memory terms. The continuous inequality is characterized by a weighted integral of historical residuals, whereas the discrete nabla counterpart is characterized by a weighted sum over historical grid points. Reverse inequalities for concave functions are also derived under reversed ordering conditions. Product-form inequalities with nonnegative convex state factors and nonnegative nonincreasing time factors are recovered as natural admissible cases. Numerical examples and Lyapunov applications are presented to verify the results.

We present a Lyapunov-based framework for analyzing continuous-time accelerated optimization dynamics with time-dependent inertia and damping. By explicitly designing Lyapunov functions that account for varying inertia, we rigorously characterize convergence rates of the objective function, achieving exponential or polynomial acceleration beyond the classical O(1/t2), even in the absence of strong convexity. Building on this foundation, we introduce a variational extension using conformable (fractional) derivatives in the Lagrangian formulation, replacing the classical velocity term with a time-weighted fractional velocity. This approach systematically modulates the system’s effective inertia and damping, providing a principled mechanism to balance acceleration and stability, reduce oscillations, and interpolate smoothly between strongly damped gradient flows and momentum-driven dynamics. The resulting framework unifies Lyapunov analysis and fractional variational modeling, offering flexible, theoretically grounded design principles for fast and stable accelerated optimization.

Fixed-point dynamical systems provide a powerful framework for representing and analyzing a wide variety of problems in control, optimization, and learning. Many fundamental tasks—such as solving nonlinear equations, minimizing objective functions, enforcing constraints, and coordinating distributed agents—can be formulated as fixed-point problems. Traditional methods often rely on asymptotic convergence and may not guarantee performance within a finite horizon. In this work, we first establish a framework for analyzing prescribed-time convergence of fixed-point dynamical systems, ensuring that trajectories reach a fixed point within a user-defined and uniformly bounded time, independent of initial conditions. Building on this foundation, we extend the framework to proximal dynamics for equilibrium problems, thereby unifying prescribed-time fixed-point analysis with proximal operator-based methods. Rigorous analysis confirms prescribed-time stability under smooth assumptions, and numerical experiments illustrate the effectiveness of the proposed approach.

From the perspective of control theory, the gradient descent optimization methods can be regarded as a dynamic system where various control techniques can be designed to enhance the performance of the optimization method. In this paper, we propose a prescribed finite-time convergent gradient flow that uses time-varying gain nonlinear feedback that can drive the states smoothly towards the minimum. This idea is different from the traditional finite-time convergence algorithms that relies on fractional-power or signed gradient as a nonlinear feedback, that is proved to have finite/fixed time convergence satisfying strongly convex or the Polyak-Łojasiewicz (PŁ) inequality, where due to its nature, the proposed approach was shown to achieve this property for both strongly convex function, and for those satisfies Polyak-Łojasiewic inequality. Our method is proved to converge in a prescribed finite time via Lyapunov theory. Numerical experiments were presented to illustrate our results.

The implementation of digital twins can leverage the cloud computing paradigm to enable smartness capabilities on a physical asset, making the system self-aware of changes on the asset, and analytics capabilities to keep the system operating under a set of desired conditions. This paper presents a practical implementation of cloud computing to enhance digital twin capabilities for monitoring and controlling power electronics applications. The proposed architecture uses three layers: embedded, edge and cloud. The embedded Digital Twin, based on FPGA, controls a flyback converter that operates under voltage regulation tasks. Likewise, the edge layer synchronizes the embedded digital twin with the physical asset information and behavior. Finally, the cloud layer implemented on Amazon Web Services and communicated via MQTT protocol uses the information acquired by the edge layer to perform real-time analytics of the physical asset and digital twin. This application demonstrates the potential of combining edge and cloud-based Digital Twins to enhance real-time process management towards the implementation of a smart control engineering system (SCE).

Continuous-time optimization has emerged as a dynamic and growing field of research within optimization theory, providing powerful analytical tools and conceptual frameworks for rigorously characterizing the stability and convergence properties of optimization algorithms represented in continuous-time settings. In this study, we propose a finite-time convergent scheme for the accelerated gradient methods; the presented method shows fast performance where the convergence occurs in a finite time. The proposed method uses a combined back-stepping/sliding mode framework. The finite-time convergence of the proposed approach is proved via Lyapunov theory. The performance of the designed finite-time accelerated algorithm are validated through a variety of numerical optimization problems.

Developing accurate mathematical or data-driven models for effective controller design under dynamic variable conditions becomes increasingly challenging. For this reason, the concept of a digital twin (DT) as a virtual representation of a physical asset has been introduced as a tool for process modelling, design, and control implementation while providing additional knowledge of the system that can be used to enable awareness capabilities on the asset. However, digital twin models used to be complex, requiring expensive computational times depending on the application to provide the most accurate system representation, limiting its application in edge, embedded, and register transfer level computing domains. Therefore, using reduced-order digital twin models is an alternative to get DT closer to the physical asset. Considering these challenges, we propose a reduced-order FPGA-based digital twin implementation that directly sources data from the real system, operates in parallel with the virtual system, and enables awareness mechanisms to improve the systems operation. This setup removes large data transfers, cloud interfaces and expensive computational times deriving into a faster and more efficient DT. To illustrate the capabilities of this embedded digital twin, we present a case study focused on monitoring a power converter. The study involves establishing and enforcing a safe operating area (SOA) for the power converter, implementing error awareness mechanisms, and enabling machine learning models to predict converter load conditions and fault events detection. Thus, we aim to showcase the effectiveness of our proposed FPGA-based digital twin approach in addressing real-time control challenges towards smart control engineering.

This paper presents a new design methodology for robust fractional-order controllers with more than three parameters for first-order plus dead time systems using the synthesis scheme of the 'more flat phase' idea for a fractional-order controller with [proportional integral derivative] (FO[PID]) structure. The stability region of the FO[PID] controller and the synthesis scheme with the 'more flat phase' idea are illustrated through a simulation example. The corresponding pseudo-codes of the synthesis scheme are also summarised. The implementation and approximation error of the FO[PID] controller are also discussed. Likewise, the superiority of the FO[PID] controller designed with the 'more flat phase' idea is verified by additional simulations and experimental results where the closed-loop system with FO[PID] controller is not sensitive to the variations of the loop gain, ensuring satisfactory control performance. Obtained results show high potential in practical industrial applications.

This study investigates the use of fractional order double integral compensators to track ramp-like disturbances without steady state offset. The reason causes these kind of disturbances; especially in plantwide processes; refers back to the presence of time scale difference between two interacting systems. The effectiveness of fractional order low-frequency compensation in improving the performance of PI controllers is demonstrated first through a rigorous simulation involving the Continuously Stirred Tank Reactor (CSTR) as a chemical plantwide system, and secondly through a real time Peltier thermal process. The fractional order double integral compensator shows superiority in tracking ramp-like disturbances.

The plasma process plays a pivotal role in the semiconductor industry, facilitating the creation of transistors and memory storage cells. This fourth state of matter is achieved by energizing a gas with radio-frequency electrical power, initiating and maintaining a stable plasma during the process cycles. Given that plasma behaves as an impedance component, an impedance-matching network becomes essential for optimizing power transfer from the source to the load (plasma). While various control strategies have been proposed for different network configurations, such as L, T, and Π networks, our work focuses on the L-type network due to its simplicity and extensive application in this domain. Several significant challenges have been identified in the existing literature, including slow dynamics, a non-monotonic decline in the reflected power, and substantial deviation in the capacitors’ path. These issues collectively impact the overall performance of the matching control system. In this article, we present a new methodology to obtain a nonlinear state-space model of the matching network for its analysis and design a proportional-integral combined with feedforward control and a control Lyapunov-barrier function to assess their effectiveness in achieving convergence to the desired matching value and guiding the path of the capacitors. These approaches aim to mitigate the recurring issues caused by capacitors moving in the wrong direction, thus improving the stability and efficiency of the impedance-matching process over time.

An accurate component modeling of power electronics converters is crucial to understanding their real performance. Thus, a fractional-order representation of energy storage components like inductors and capacitors could be helpful to reflect the hidden dynamics of a power converter in voltage regulation tasks. This paper presents the fractional-order non-ideal modeling and behavioral matching of an isolation transformer used on a flyback converter. A multiphysics simulation model is built to replicate the flyback converter non-ideal dynamics. It is optimized to precisely reflect the real converter response by testing various transformer configurations, including ideal, non-ideal, integer and fractional-order. The transformer parameters are adjusted using raw data from the physical power converter. Likewise, the fractional-order energy storage elements are represented using discrete circuit elements for its realization. The results obtained reveal that using a non-ideal fractional-order transformer on the flyback converter with a non-integer inductance provides the maximum fit for the power converter operation. Copyright (C) 2024 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

The relay feedback auto-tuning method, which was an early commercialized approach, has maintained its popularity due to its simplicity and robustness. However, the classical proportional integral derivative (PID) controller auto-tuning method often results in unacceptable overshoot, especially for integrating and higher-order processes. By integrating the TID controller with the relay feedback technique, we significantly enhance dynamic control performance without relying on prior knowledge of the system. However, the direct application of the classical auto-tuning method to the TID controller encounters challenges due to the additional 1 fractional-order transfer function s - n . Therefore, we have developed a fractional-order Ziegler-Nichols (FOZ-N) approach, specifically designed to adjust the parameters of the TID controller. The proposed FOZ-N method is fast, simple, and capable of achieving the desired performance. In contrast to the previous Ziegler-Nichols tuning method, the TID controller parameters K t and K i are determined to shift the critical point (-1/ K u , j 0) to the FOZ-N point (-0 . 5 , - j 0 . 7) on the Nyquist curve, ensuring system robustness and dynamic performance. 1 The impact of the fractional order parameter s - n and ratio r is explored through time-domain analysis, where these parameters are determined to ensure optimal dynamic performance. Additionally, we provide a detailed tuning procedure along with a helpful example. To demonstrate the advantages of the proposed auto-tuning TID controller over the Ziegler-Nichols PID controller, Optimal PID controller, simple internal model control (SIMC) PID controller, and Ziegler-Nichols FOPID controller, we present a simulation illustration involving multiple different systems. To validate the practical outcomes of this paper, we present experimental results on the temperature control of a Peltier cell.

The concept of self-optimizing control (SOC) can be classified as a type of learning control (LC) where a system is designed to adjust itself automatically to control an arbitrary dynamic process. Classic feedback control techniques and traditional optimal control techniques are also capable of automatically adjusting a system to a desired target but only if all the a priori information about the controlled process (plant environment) is known and can be described deterministically. Hence, learning control techniques becomes essential for systems where the a priori knowledge is unknown or incompletely known. Given this challenge, researchers through out the years and across discipline have developed and refined two possible approaches to address this challenge. One approach which is more conservative involves using the available known information to design the controller with an acceptable margin of error based on the designer’s experience. The other approach which is akin to learning control involves designing a controller which is capable of estimating the unknown information during its operation and determining an optimal control action based on the estimated information. In this work, we present the different classes of learning control which inspired the concept of self-optimizing control and the trend towards digital twin and parallel intelligence to improve system knowledge acquisition for better control.

Data-driven controller synthesis methods allow finding controller parameters based only on the system input-output response. Among these methods, Virtual Reference Feedback Tuning (VRFT) is accepted thanks to its reference-model based direct parameters synthesis, including robustness and noise rejection considerations for integer and fractional-order controller design. However, for fractional-order VRFT controller synthesis, a direct solution is harder to reach involving a complex optimization problem to solve. This paper presents a recursive VRFT method for synthesizing arbitrary degree fractional-order controllers with feedforward based on a integer or fractional-order reference model. This methodology leverages existing optimization tools for fractional-order systems identification to obtain integer and fractional-order controller gains when integer and fractional-order reference models are used for the virtual error computation. The proposed method synthesizes a set of feedforward VRFT controllers implemented to control a uniform temperature control system. The results show that combining a fractional-order model or compensator improves the energy consumption of the controller and the transient temperature response in tracking and regulation tasks. Copyright (C) 2024 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

A self-optimizing controller (SOC) adds an upper optimization layer to closed-loop systems, enhancing self-awareness and improving performance during repetitive tasks. However, most SOC optimization algorithms use only the current and previous cost function evaluations to determine the next optimal parameters (first-order optimization methods). This paper introduces a high-order self-optimizing controller (HOSOC) that accelerates the optimization algorithm's convergence by utilizing the history of optimal parameters within a sliding window. A modified version of the Globalized Constrained Nelder Mead optimization method is implemented to account for the past centroids within a memory window to compute the upcoming solution. The algorithm is tested on a general delay-aware First-Order plus Dead Time (FOPDT) system with varying lag, balanced, and delay time dynamics. Results show that the high-order SOC controller speeds up convergence and meets the desired performance specifications effectively.

In this study, a fractional order equation of motion as a continuous limit model for a family of gradient descent algorithms is discussed based on fractional order Euler-Lagrange equation. The aim of this proposed scheme is to search the ability to go beyond Nesterov scheme by introducing the potential of fractional calculus. The discretized version of the "designed" fractional order equation of motion (FO- EOM) forms new gradient descent algorithm that has been tested on some optimization benchmark functions to fairly assess the performance in comparison with the standard and accelerated form of gradient descent algorithms. Promising results have been obtained with more rigorous mathematical analysis has to be carried out as our future work. Copyright (C) 2024 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

First-order plus delay time systems are commonly used to approximate thermal or temperature process control systems, covering approximately 80% of all industrial controls. These systems are characterized by a "relative time delay," which represents the delay (L) about the time constant (tau). However, this relative time delay can fluctuate in real-time control scenarios, potentially compromising system stability. To address this issue, control experts have introduced a supervisory layer called self-optimizing control (SOC) on top of the control layer. SOC aims to determine optimal feedback control (PID) parameters considering the varying relative time delay. Despite efforts to optimize parameters, it has been observed that the best results often lead to longer rise time and convergence time. This paper explores using fractional-order and integer-order reference models as feedforward compensation alongside the supervisory layer to enhance the SOC controller performance. The globalized constrained Nelder-Mead (GCNM) algorithm finds optimal parameters for the feedforward and feedback controllers. Performance indices, including root mean square (RMS), root mean square error (RMSE), integral square error (ISE), integral absolute error (IAE), SOC convergence time, settling time, and overshoot, are computed to evaluate the overall controller performance. The results indicate that the fractional-order feedforward setup slightly outperforms the integer-order feedforward setup with variable relative time delays.

The use of a digital twin as an enabling technology for industry 4.0 provides control systems engineers with novel tools for modelling, designing, and controlling complex systems, providing a deep understanding of the physical asset based not only on its physics but also the real system’s response. It is particularly critical for uniformity temperature control applications, where providing a reasonable model of the system’s diffusion is always affected by the physical behavior of the system’s components required for heating, cooling, or power distribution. In this paper, a digital twin is used to represent a multivariable thermoelectric system employed for temperature uniformity distribution control with potential applications in semiconductor manufacturing. The modelling employs a five-step methodological framework consisting of the stages: target system definition, system description, multiphysics and data-driven simulation, behavioral matching, and implementation to represent the system’s temperature distribution accurately. The temperature distribution is measured using an infrared thermal camera to perform model behavioral matching on heating and cooling temperature uniformity applications. The obtained results indicated that using digital twins not only increases the accuracy of the system’s representation but can also provide the system with novel information that can be leveraged for the design and implementation of smart control systems.

The actual industrial processes are always accompanies by many non-Gaussian behaviors due to the systems complexity. These behaviors are fractional order characteristics, which are very difficult to analyze by traditional analysis methods. This paper presents a detail fractional order theory analyses based on the fractional order characteristics present in industrial process. Initially, the -stable distribution is employed to fit the probability density distribution of the data and the auto correlation function is applied to find the long range dependence characteristic hidden in the process. Next, a re-scaled range method and multifractal detrended fluctuation analysis method is applied to analyze the fractional order features of the process in detail. Then, a fractional auto-regressive integrated moving average model (FARIMA) is proposed to predict accurately of the time series based on the fractional order characteristic of the system. Experimental results show that the superiority for prediction model with fractional order thinking.