Fractional Order Calculus Theory Research Articles

Nonlinear Dynamic Analysis of Riemann–Liouville Fractional-Order Damping Giant Magnetostrictive Actuator

Abstract In view of the large errors in the integer-order prediction model of the current giant magnetostrictive actuator (GMA), existing studies have shown that the fractional-order theory can improve the classical integer-order error situation. To this end, the Riemann–Liouville (R–L) fractional-order calculus theory is applied to the damping part of the GMA system; based on the averaging method and the power series method, the analytical and numerical solutions of the system are obtained, respectively, the motion of the GMA system is obtained through simulation, the parameters affecting the main resonance response of the system are analyzed as well as the motion characteristics of the system under the parameters, and the bifurcation and chaotic characteristics of the system are analyzed qualitatively and quantitatively. It is shown that the fractional-order model can improve the prediction accuracy of the system, the fractional order has a significant effect on the motion of the system, and the interval of the periodical motion parameter is less than an integer when the order of the damping term is (0,1), and the system can be induced to shift to periodic motion by changing the parameters.

A Hybrid Control Framework for Chemical Processes with Long Time Delay: Theory and Experiments.

This paper proposes a hybrid control framework based on internal model concepts, sliding mode control methodology, and fractional-order calculus theory. As a result, a modified Smith predictor (SP) is proposed for nonlinear systems with significant delays. The particular predictive approach enhances the sliding mode control (SMC) controller's transient responses for dead-time processes, and the SMC gives the predictive structure robustness for model mismatches by combining the previous methods with fractional order concepts; the result is a dynamical sliding mode controller. A numerical example is considered to evaluate the performance of the proposed approach, where a step change, external disturbance, and parametric uncertainty test are performed. A real application in the TCLab Arduino kit is presented; the proposed method presented good performance with a little amount of chattering, and in the disturbance rejection case, the overshoot increased with an aggressive response; in both cases, better tuning parameters can improve the process response and the controller action.

Viscoelastic creep model and parameter inversion of bond interface in steel plate reinforced tunnel lining

The interface filler material (epoxy resin) used for reinforcing tunnel linings with steel plates exhibits time-dependent mechanical properties, commonly referred as creep behavior. The creep behavior of the filler material will significantly influence the long-term service performance of the reinforced tunnel, especially considering the design service life scale of 100 years. This paper presents an experimental test and mathematical modelling to illustrate the creep mechanical behavior of bond interfaces within shield tunnels reinforced with steel plates. It was observed that in the initial moments of stress application, the bond interface demonstrates an instantaneous response through elastic strain and on-set of creep deformation stage at maximum stress value. Three stages can be characterized in the creep deformation process: the decay creep stage, the steady-state creep stage and the accelerated creep stage. In the accelerated creep stage, the bond interface rapidly deteriorated after experiencing 90 % of the ultimate shear stress for 186 hours or 90 % of the ultimate tensile stress for 96 hours. Based on the fractional-order calculus theory and drawing inspiration from the modeling principles of classical component combination models, a constitutive equation that accurately characterizes the creep mechanical behavior of the bond interface between steel plates and concrete linings was formulated. The model parameters are inversely computed using the principle of least squares and the accuracy of this model is validated through a comparison with the results of the creep tests.

Dynamic Analysis and Sliding Mode Synchronization Control of Chaotic Systems with Conditional Symmetric Fractional-Order Memristors

In this paper, the characteristics of absolute value memristors are verified through the circuit implementation and construction of a chaotic system with a conditional symmetric fractional-order memristor. The dynamic behavior of fractional-order memristor systems is explored using fractional-order calculus theory and the Adomian Decomposition Method (ADM). Concurrently, the investigation probes into the existence of coexisting symmetric attractors, multiple coexisting bifurcation diagrams, and Lyapunov exponent spectra (LEs) utilizing system parameters as variables. Additionally, the system demonstrates an intriguing phenomenon known as offset boosting, where the embedding of an offset can adjust the position and size of the system’s attractors. To ensure the practical applicability of these findings, a fractional-order sliding mode synchronization control scheme, inspired by integer-order sliding mode theory, is designed. The rationality and feasibility of this scheme are validated through a theoretical analysis and numerical simulation.

Fractional order modelling and optimal control of dual active bridge converters

As the construction of new power systems centred around renewable energy gains traction, the installed capacity of new energy sources has experienced explosive growth. Research has shown that the actual external characteristics of inductors and capacitors in circuits exhibit fractional order characteristics. As a core device in dual active bridge converters, the non-integer order of inductors and capacitors has an important impact on their dynamic performance, frequency domain characteristics, and control system design. This paper conducts modelling and control research on dual active bridge converters based on fractional-order calculus theory. First, based on the definition of fractional-order calculus, according to the operating mode of the fractional-order dual active bridge converter and the characteristics of fractional-order components, The fractional-order state space average model was established, and the influence of the order of inductance and capacitance on the amplitude frequency characteristics, phase frequency characteristics and dynamic performance of the converter was analyzed. Secondly, the average model and circuit model were built on the MATLAB/Simulink simulation platform, and the simulation results under different fractional orders were compared. Finally, in order to further improve the control performance of the converter, a fractional-order P I λ control strategy is designed based on the established state space average model of the fractional-order dual active bridge converter and based on the transfer function from the control to the output of the fractional-order dual active bridge converter. The research results show that the fractional-order model can more accurately describe the actual characteristics of the converter. In addition, the fractional-order controller P I λ enables the fractional-order dual active bridge converter to obtain better robustness and improve the dynamic performance of the converter.

State of Charge Estimation for Lithium Battery Based on Fractional Order Square Root Cubature Kalman Filter and Adaptive Multi-innovation Unscented Kalman Filter

Accurate state of charge (SOC) estimation of batteries is of great significance for electric vehicles. A SOC estimation method based on a fractional order square root cubature Kalman filter (FOSRCKF) and an adaptive multi-innovation unscented Kalman filter (AMIUKF) is proposed. The battery is modelled using fractional order calculus theory and the model parameters are identified by adaptive genetic algorithm. The FOSRCKF estimates the battery SOC, while the AMIUKF online updates the internal resistance in the model, and there exchanges information between two filters. The experimental results under the Urban Dynamometer Driving Schedule (UDDS) and the US06 Highway Driving Schedule show that the proposed method has lower SOC estimation error and lower terminal prediction error compared with the traditional SRCKF method based on integer order models, which demonstrates the effectiveness, accuracy and robustness of the proposed method.

Backstepping Sliding Mode Control Algorithm for Unmanned Aerial Vehicles Based on Fractional-Order Theory

Aiming at the problems of slow convergence speed and low tracking accuracy in attitude control and position tracking of quadrotor unmanned aerial vehicles (UAVs). This paper combines the fractional-order calculus theory with the backstepping sliding mode control algorithm, using the backstepping control to compensate for the nonlinearity of the system and the fractional-order theory to eliminate the jitter brought about by the sliding mode control, and proposes a new fractional-order backstepping sliding mode control strategy for the trajectory tracking control of the quadrotor UAV. The proposed fractional-order sliding mode surface increases the control flexibility and improves the robustness and anti-interference ability of the system to some extent. The stability analysis of the system is carried out using the Lyapunov stability theory, and the results prove the stability of the proposed controller. Finally, the effectiveness and feasibility of the proposed method are verified by comparing it with the traditional backstepping sliding mode controller. The simulation results show that the fractional-order reverse-step sliding mode control algorithm proposed in this paper is significantly better than other control algorithms in terms of convergence speed and also has a certain degree of superiority in terms of error elimination.

Fractional order sliding mode control for linear maglev synchronous motor based on an adaptive fixed-time extended state observer

In order to improve the tracking accuracy of the linear maglev synchronous motor maglev system with non-linear, strongly coupled, and other uncertainty problems, a fixed-time fractional order terminal sliding mode control (FFTSMC) strategy based on adaptive fixed-time extended state observer (AFxESO) is proposed. Firstly, a fast AFxESO is constructed and an adaptive law is designed to dynamically adjust the gain to reduce the estimation error and measurement noise, which enables the observer to accurately estimate the perturbation of the unknown boundary in a fixed-time. Secondly, a new FFTSM controller is designed by applying the theory of fractional order calculus, which can speed up the system convergence and suppress the chattering problem in SMC. Then, using the Lyapunov stability theory, it is demonstrated that the proposed AFxESO-FFTSM control scheme can guarantee the global fixed-time convergence of the system to equilibrium and the convergence time is dependent on the selection of the controller parameters, which effectively overcomes the drawbacks of finite-time control. Finally, the feasibility and superiority of the proposed strategy are verified on the linear maglev synchronous motor system experiment platform.

Lithium-Ion Battery Modeling and State of Charge Prediction Based on Fractional-Order Calculus

Predicting lithium-ion batteries’ state of charge (SOC) is essential to electric vehicle battery management systems. Traditional lithium-ion battery models mainly include equivalent circuit models (ECMs) and electrochemical models (EMs). ECMs are based on integer-order component modeling, which cannot characterize the internal electrochemical reaction mechanism of the battery, resulting in lower SOC prediction accuracy. In contrast, due to their complex structure, EMs are limited in their application. This study takes lithium batteries as the research object and proposes a fractional-order impedance model (FOIM) that characterizes the dynamic properties of the internal behavior of lithium-ion batteries using fractional-order elements. Considering the highly nonlinear characteristics of lithium-ion batteries, this study introduces the theory of fractional-order calculus into the extended Kalman filter (EKF) algorithm, and proposes the fractional-order extended Kalman filter (FEKF) algorithm applied to the prediction of battery charge state. Comparative analysis of simulation and experimental results shows that the accuracy of the FOIM, compared to ECMs, is significantly improved. The FEKF algorithm has good robustness in estimating the SOC, and the SOC prediction accuracy achieved with the algorithm is also improved compared with that obtained using the EKF algorithm of the integer-order model.

Containment control for fractional order MASs with nonlinearity and time delay via pull-based event-triggered mechanism

The containment control for fractional order multi-agent systems with nonlinearity and time delay is investigated in this article. Firstly, a distributed control protocol with fixed time delay is proposed for the purpose of achieving containment control for nonlinear fractional order multi-agent systems. Secondly, by using the pull-based event-triggered mechanism, a distributed event-based control law with time delay is further given to accomplish containment control for fractional order multi-agent systems with nonlinearity. In this way, the controller of each agent is merely updated at its own trigger moments. Hence, it can decrease network correspondence congestion and lower computational cost. Thanks to the stability theory of fractional order calculus, matrix theory, and properties of Mittag-Leffler function, the feasibility of the designed control protocols is confirmed. Furthermore, the Zeno behavior is precluded by the strict proof. Ultimately, two simulation examples are used to illustrate the availability of above theoretical analysis respectively.

The fractional neural grey system model and its application

Recently, grey system, neural network, and fractional order calculus theory have become popular research areas, and an increasing number of scholars have joined these studies, conducted illuminating research, and produced a number of significant results. Numerous research studies have demonstrated that these three strategies are crucial to solving a wide range of practical problems. In this paper, we present a fractional order neural grey system model with a three-layer structure in which the input of the network is a fractional order cumulative sequence, and the output is a predicted value in order to maximize the benefits of each of the three elements. The purpose of this research is to present a strategy for reducing the number of conditions in order to improve the stability of parameter estimation by using QR decomposition. The order of the models is determined by an intelligent optimization algorithm. Finally, real-world examples are used to validate the model’s validity, and experimental results indicate that the newly presented model is more accurate than previous models.

Research on Manufacturing Equipment Operation State Evaluation Technology Based on Fractional Calculus

The operational status of manufacturing equipment is directly related to the reliability of the operation of manufacturing equipment and the continuity of operation of the production system. Based on the analysis of the operation status of manufacturing equipment and its characteristics, it is proposed that the concept of assessing the operation status of manufacturing equipment can be realized by applying the real-time acquisition of accurate inspection data of important parts of weak-motion units and comparing them with their motion status evaluation criteria. A differential data fusion model based on the fractional-order differential operator is established through the study of the application characteristics of fractional-order calculus theory. The advantages of Internet of Things (IoT) technology and a fractional order differential fusion algorithm are integrated to obtain real-time high-precision data of the operating parameters of manufacturing equipment, and the research objective of the operating condition assessment of manufacturing equipment is realized. The feasibility and effectiveness of the method are verified by applying the method to the machining center operation status assessment.

Dual-Loop Voltage–Current Control of a Fractional-Order Buck-Boost Converter Using a Fractional-Order PIλ Controller

Based on the fact that the inductor and capacitor are of a non-integer order by nature, to provide a more accurate theoretical basis for the optimal control of the converter, the fractional-order model of the Buck-Boost converter in the continuous mode of current is established according to the fractional-order calculus theory. The fractional-order PIλ control system of the fractional-order Buck-Boost converter is designed to compare the performance of the integer-order PI controller with the fractional-order controller. Secondly, the sparrow search algorithm is applied to the optimal design of the fractional-order PIλ control system of the fractional-order Buck-Boost converter to improve the system’s phase margin, stability, and robustness. Finally, the simulation is verified on the Matlab/Simulink simulation platform and compared with the integer-order PI controller.

Auxiliary Model-Based Multi-Innovation Fractional Stochastic Gradient Algorithm for Hammerstein Output-Error Systems

This paper focuses on the nonlinear system identification problem, which is a basic premise of control and fault diagnosis. For Hammerstein output-error nonlinear systems, we propose an auxiliary model-based multi-innovation fractional stochastic gradient method. The scalar innovation is extended to the innovation vector for increasing the data use based on the multi-innovation identification theory. By establishing appropriate auxiliary models, the unknown variables are estimated and the improvement in the performance of parameter estimation is achieved owing to the fractional-order calculus theory. Compared with the conventional multi-innovation stochastic gradient algorithm, the proposed method is validated to obtain better estimation accuracy by the simulation results.

A new memristor-based fractional-order chaotic system

In this paper, a new four-dimensional incommensurate fractional-order system is proposed by introducing an ideal flux-controlled memristor into a three-dimensional chaotic system, and combining it with fractional-order calculus theory, which is solved by using the Adomian decomposition method (ADM). Through theoretical analysis we found the system has numerous equilibrium points. Compared with the original system, the modified system exhibits richer dynamical behaviors. The main manifestations are: (i) Antimonotonicity varying with the initial value. (ii) Three kinds of transient transition behaviors: transient asymptotically-period (A-period), transient chaos, and tri-state transition (chaos-A-period-chaos). (iii) Initial offset boosting behavior. (iv) Hidden extreme multistability. (v) As the order q changes, the system is capable of generating a variety of asymptotically periodic attractors and chaotic attractors. These behaviors above are analyzed in detail by means of numerical simulations such as phase diagram, bifurcation diagram, Lyapunov exponent spectrum (LEs), time-series diagram, and attraction basin. Finally, the system is implemented with a hardware circuit based on a digital signal processor (DSP), which in turn proved the correctness of the numerical analysis simulations and the physical realizability of the system.

Design of Quad-rotor Controller Based on Fractional Order Sliding Mode Control

According to the chattering problems of traditional sliding mode index exponential reaching law, this paper proposes a fractional order sliding mode index exponential reaching law control strategy, which is applied to the quad-rotor helicopter attitude control. Combined the theory of fractional order calculus and sliding mode variable structure control theory, the fractional order sliding mode controller is designed. The method of lyapunov analysis proves that this controller can make the system asymptotically stable. Simulation and experiments show that the proposed fractional order sliding mode control system not only undermines the chattering of traditional sliding mode exponential reaching law but also reduces the adjusting time and control margin of the system.

Fractional-Order Modeling and Analysis of a Variable Structure Hybrid Energy Storage System for EVs

Hybrid energy storage system has been widely studied as an important technology for electric vehicles. Since the hybrid energy storage system is a nonlinear and complex system, the modeling of the system and the high-precision nonlinear control strategy are technical difficulties for research. The establishment of a high-precision mathematical model of the hybrid energy storage system is the basis for the study of high-quality nonlinear control algorithms. Fortunately, the theory of fractional calculus can help build accurate mathematical models of hybrid energy storage systems. In order to obtain the high-quality nonlinear control strategy of this complex system, this paper, respectively, carried out fractional-order modeling and analysis on the three basic equivalent working states of the hybrid energy storage system of electric vehicles. Among them, the fractional-order average state space model is carried out for the equivalent Buck and Boost mode. Also, the steady-state analysis of the equivalent Dual-Boost mode is carried out by combining the fractional-order calculus theory with the equivalent small parameter variable method. Finally, the effectiveness and precision of the fractional-order model are proved by simulation and experiment.

Research on a Differential Geometric Guidance Law Based on Fractional-Order Theory

In this paper, fractional-order calculus theory is used to investigate the geometric law for intercepting an agile target. In order to overcome the challenges presented by the divergence of line of sight rate (LOSR) of proportional navigation (PN), the fractional LOSR is used as a compensated term in the proposed fractional differential geometric guidance law (FDGGL). By adjusting the navigation gain of the FDGGL, the new proposed guidance law can be transformed into the traditional differential geometric guidance law (DGGL) and PN. The average overload and ballistic stability of the FDGGL are analyzed based on the fractional and control theory. Some analytical results about energy consumption and trajectory variation of the FDGGL were obtained. The simulation results shows that, compared with PN and DGGL, the FDGGL has better guidance performance when intercepting different maneuvering targets.

Chaos Synchronization of a Class of Fractional-order Economic Systems with Time-delay

In this paper, the chaos Synchronization problem of a class of economic systems with time delays was studied on Lyapunov stability theory and fractional-order calculus theory. Two methods of fractional-order active sliding mode control and single controller sliding mode control are used, and the differences between the two methods are compared. Numerical simulations verify the feasibility of the two methods. Results show that active sliding mode control scheme is highly resistant to interference.

A Simplified Fractional Order Equivalent Circuit Model and Adaptive Online Parameter Identification Method for Lithium-Ion Batteries

In order to improve the battery management system performance and enhance the adaptability of the system, a fractional order equivalent circuit model of lithium-ion battery based on electrochemical test was established. The parameters of the fractional order equivalent circuit model are identified by the least squares parameter identification method. The least squares parameter identification method needs to rely on the harsh test conditions of the laboratory, and the parameter identification result is static; it cannot adapt to the characteristics of the lithium battery under dynamic conditions. Taking into account the dynamic changes of lithium batteries, a parameter adaptive online estimation algorithm for fractional equivalent circuit model is proposed. Based on the theory of fractional order calculus and indirect Lyapunov method, the stability and convergence of the estimator are analyzed. Finally, simulation experiments show that this method can continuously estimate the parameters of the fractional order equivalent circuit under UDDS conditions.