Fractional Order Calculus Theory Research Articles

Edge Extraction Based on Memristor Cell Neural Network With Fractional Order Template

In order to extract more texture and detail information from a given image, a memristor cell neural network is proposed by replacing the state resistances of the neurons as the memristors. The characteristic of memristor could be maintained in the whole information processing of the cell neural network so that the historical state information could be used and the information processing ability of the network could be enhanced. Furthermore, based on the fractional order calculus theory, a fractional order control template is designed for the memristor cell neural network to enhance the middle- and high-frequency information and retain more low-frequency texture information in the image edge extraction. The network could be used to extract edge information from various kinds of images. The simulation results show that the edge images extracted by this method have more complete and clear contour information and more abundant texture detail information. Both the average gradient and the information entropy of the edge images could be significantly improved.

Fractional-Order Model Predictive Frequency Control of an Islanded Microgrid

Optimal frequency control of an islanded microgrid has been a challenging issue in the research field of microgrids. Recently, fractional-order calculus theory and some related control methods have attempted to handle this issue. In this paper, a novel fractional-order model predictive control (FOMPC) method is proposed to achieve the optimal frequency control of an islanded microgrid by introducing a fractional-order integral cost function into model predictive control (MPC) algorithm. Firstly, a discrete state-space model is derived for the optimal frequency control problem of an islanded microgrid. Afterward, a fractional-order integral cost function is designed to guide the FOMPC algorithm to obtain optimal control law by borrowing the Grünwald-Letnikov (GL) definition of fractional order calculus. Six simulation studies have been carried out to illustrate the superiority of FOMPC to conventional MPC under dynamical load disturbances, perturbed system parameters and random dynamical power fluctuation of wind turbines.

An Extended State Observer Based Fractional Order Sliding‐Mode Control for a Novel Electro‐Hydraulic Servo System with Iso‐Actuation Balancing and Positioning

AbstractAn extended state observer based fractional order sliding‐mode control (ESO‐FOSMC) is proposed in this study, with consideration of the strong nonlinear characteristics of a new electro‐hydraulic servo system with iso‐actuation balancing and positioning. By adopting the fractional order calculus theory, a fractional order proportional–integral–derivative (PID)‐based sliding mode surface was designed, which has the ability to obtain an equivalent positioning control with fractional order kinetic characteristics. By introducing the integral term into the sliding mode surface, it was found to be beneficial in reducing the steady‐state errors, as well as improving the precision of the control system. Also, by using the fractional order calculus to replace the integral calculus, the form of the convergence is improved; the system transfer of energy is slowed down; and the chattering of the system is greatly weakened. The extended state observer was designed to observe the real‐time disturbances, and also to generate the compensation control commands which are added to the FOSMC to achieve the dynamic compensation. By means of numerical simulations, the dynamic and static characteristics of the sliding mode control system were compared with those of the FOSMC and ESO‐FOSMC. The experimental results show that the ESO‐FOSMC system could effectively restrain the external disturbances and achieve higher control precision, as well as better control quantity without chattering. The semi‐physical simulations based experimental tests also demonstrated that the proposed ESO‐FOSMC outperformed the FOSMC in terms of system robustness and control precision, which could have a stable control of the gun system quickly and accurately.

The Effect of Fractional Orders on the Transmission Power and Efficiency of Fractional-Order Wireless Power Transmission System

To avoid the problems of integer-order magnetic resonant wireless power transmission (WPT) systems, such as low output power and transmission efficiency, high resonant frequency, frequency splitting, and parameter coupling, a novel WPT system based on the fractional-order calculus theory is proposed; the resonant frequency and coupling coefficient can be regulated by the fractional order, so that this system has completely different transmission characteristics from the integer-order WPT system. Therefore, in this paper, the circuit model based on the phasor method of fractional-order WPT system is established, and the output power and transmission efficiency of the system are analyzed. In addition, the comparative analysis of output power and transmission efficiency under different fractional orders are performed to estimate the optimal combination of fractional orders, which is beneficial to produce better characteristics of output power and transmission efficiency and provides a theoretical basis for the design and implementation of an experimental system.

Fractional-Order Modeling and Simulation of Magnetic Coupled Boost Converter in Continuous Conduction Mode

By using fractional-order calculus theory and considering the condition that capacitor and inductor are naturally fractional, we construct the fractional mathematical model of the magnetic coupled boost converter with tapped-inductor in the operation of continuous conduction mode (CCM). The fractional state average model of the magnetic coupled boost converter in CCM operation is built by exploiting state average modeling method. In these models, the effects of coupling factor, which is viewed as one generally, are directly pointed out. The DC component, the AC component, the transfer functions and the requirements of the magnetic coupled boost converter in CCM operation are obtained and investigated on the basis of the state averaged model as well as its fractional mathematical model. Using the modified Oustaloup’s method for filter approximation algorithm, the derived models are simulated and compared using Matlab/Simulink. In order to further verify the fractional model, circuit simulation is implemented. Furthermore, the differences between the fractional-order mathematical models and the corresponding integer-order mathematical models are researched. Results of the model and circuit simulations validate the effectiveness of theoretical analysis.

Lur’e Postnikov Lyapunov functional technique to global Mittag-Leffler stability of fractional-order neural networks with piecewise constant argument

Abstract In this paper, the global Mittag-Leffler stability issue of fractional-order neural networks (FNNs) with piecewise constant argument is investigated. Firstly, a new inequality with respect to the fractional derivative of integer-order variable upper limit integral is proposed, which not only is favorable to the construction of Lyapunov function but also enriches the fractional order calculus theory. Secondly, based on topological degree theory, the existence and uniqueness of equilibrium point is certified. In addition, under some suitable assumptions, by applying Picard successive approximation technique, the proof of the existence and uniqueness of solution with the initial value is given. Moreover, by introducing integral term into Lyapunov functional, Lur’e Postnikov type Lyapunov functional is constructed, and a sufficient condition is addressed in terms of linear matrix inequalities (LMIs) to guarantee that the considered FNNs are Mittag-Leffler stable. Finally, an example is given to demonstrate the validity of the obtained results.

Novel Fractional Order Calculus Extended PN for Maneuvering Targets

Based on the theory of fractional order calculus (FOC), a novel extended proportional guidance (EPN) law for intercepting the maneuvering target is proposed. In the first part, considering the memory function and filter characteristic of FOC, the novel extended PN guidance algorithm is developed based on the conventional PN after introducing the properties and operation rules of FOC. Further, with the help of FOC theory, the average load and ballistics characteristics of proposed guidance law are analyzed. Then, using the small offset kinematic model, the robustness of the new guidance law against autopilot parameters is studied theoretically by analyzing the sensitivity of the closed loop guidance system. At last, representative numerical results show that the designed guidance law obtains a better performance than the traditional PN for maneuvering target.

Dynamic Analysis of Fractional-Order Fuzzy BAM Neural Networks with Delays in the Leakage Terms

In this paper, based on the theory of fractional-order calculus, we obtain some sufficient conditions for the uniform stability of fractional-order fuzzy BAM neural networks with delays in the leakage terms. Moreover, the existence, uniqueness and stability of its equilibrium point are also proved. A numerical example is presented to demonstrate the validity and feasibility of the proposed results.

Complexity and Hopf Bifurcation Analysis on a Kind of Fractional-Order IS-LM Macroeconomic System

On the basis of our previous research, we deepen and complete a kind of macroeconomics IS-LM model with fractional-order calculus theory, which is a good reflection on the memory characteristics of economic variables, we also focus on the influence of the variables on the real system, and improve the analysis capabilities of the traditional economic models to suit the actual macroeconomic environment. The conditions of Hopf bifurcation in fractional-order system models are briefly demonstrated, and the fractional order when Hopf bifurcation occurs is calculated, showing the inherent complex dynamic characteristics of the system. With numerical simulation, bifurcation, strange attractor, limit cycle, waveform and other complex dynamic characteristics are given; and the order condition is obtained with respect to time. We find that the system order has an important influence on the running state of the system. The system has a periodic motion when the order meets the conditions of Hopf bifurcation; the fractional-order system gradually stabilizes with the change of the order and parameters while the corresponding integer-order system diverges. This study has certain significance to policy-making about macroeconomic regulation and control.

On line parameter estimation based on gradient algorithm for fractional order systems

This article investigates the problem of on line parameter estimation for fractional-order linear systems. Based on the theory of fractional-order calculus, the conventional gradient estimator is extended to the fractional-order areas. The stability of the estimator and the convergence are analysed using the continuous frequency distributed model and indirect Lyapunov method. Finally, numerical simulation examples are given to demonstrate the effectiveness of the proposed schemes.

Fractional order Pareto distributions with application to X‐band maritime radar clutter

The theory of fractional order calculus is applied to construct variations of the fundamental Pareto clutter model density, used in X-band maritime surveillance radar. Such data are characterised, at high resolution, by spiky clutter returns whose density function possesses heavy tails. By an application of fractional calculus to the Pareto distribution function, together with optimisation applied to determine the fractional derivative order, it is possible to obtain a new model for radar clutter. This is shown to fit real radar clutter returns better than a standard Pareto model. Additionally, it accounts for receiver thermal noise without the complexity of an additional noise distribution.

Design and Simulation of Fractional Order Control Systems Based on Bode's Ideal Transfer Function

The fractional order calculus theory and its modeling methods have been applied widely in control field. And design of fractional order control systems become hot point of recent years. This paper establishes fractional order systems which parameters are obtained by Bode’s ideal transfer function method to get the desired frequency response. Apply this method a new control structure of fractional order pseudo-derivative feedback (FOPDF) is suggested. Control methods are designed for integral first order system and for fractional first order system, and apply to a real high power semiconductor laser diodes constant temperature controlling system and the hydraulic servo systems and the electric drive system, the simulation results indicate that the effectiveness and validity of this method.

Research on PMSM Control for Position Sensorless Based on Fractional PD Sliding Mode Observer

According to the phase delaying question of estimated rotor position angle in traditional sliding mode observer, using fractional order calculus theory and singular perturbation two-time-scale decomposition theory, making up the parameter designing rules of fractional order PD sliding mode observer, proposed position sensorless control strategy for permanent magnet synchronous motor based on fractional order PD sliding mode observer. On one side, taking fast time-varying subsystem of singular perturbation two-time-scale decomposition theory to establish fractional order sliding mode observer; on the other side, taking slow time-varying subsystem of singular perturbation two-time-scale decomposition theory to obtain parameter of fractional order sliding mode observer. The results show that the method could not only making estimated rotor position angle follow actual rotor angle, but also establishing a global compensating control strategy of estimated rotor position angle effectively.

Modeling and simulation analysis of fractional-order Boost converter in pseudo-continuous conduction mode

Based on the fact that the inductor and the capacitor are fractional in nature, the fractional order mathematical model of the Boost converter in pseudo-continuous conduction mode is established by using fractional order calculus theory. According to the state average modeling method, the fractional order state average model of Boost converter in pseudo-continuous conduction mode is built. In view of the mathematical model, the inductor current and the output voltage are analyzed and the transfer functions are derived. Then the differences between the integer order and the fractional order mathematical models are analyzed. On the basis of the improved Oustaloup fractional order calculus for filter approximation algorithm and the model of fractional order inductance and capacitance, the simulation results have been compared between the mathematical model and circuit model with Matlab/Simulink software; the origins of model error are analyzed and the correctness of the modeling in fractional order and the theoretical analysis is verified. Finally, the differences and the relations of Boost converter among the continuous conduction mode, the discontinuous conduction mode, and the pseudo-continuous conduction mode are indicated.

Non-linear control for the fractional boost converter in pseudo continuous conduction mode

Based on the fact that the inductor and the capacitor are of fractional order in nature, the mathematical model of the fractional order boost converter in pseudo continuous conduction mode is established by using the fractional order calculus theory. Due to the affine non-linear nature in this mathematical model and a similar Lyapunov stability theorem of the fractional order system, a fractional order non-linear controller is designed. On the basis of chain fractance and the improved Oustaloup algorithm, the circuit model of fractional order inductor and capacitor are built. The fractional order controller is verified by the Matlab/Simulink software. Simulation results show that the controller enhances the dynamic and steady-state performance, so as to realize the stability and achieve good dynamic performance of the fractional order system during large fluctuation of power supply and load disturbance.