Search Results (13)

Fractional-order operators play a fundamental role in the modeling and control of complex dynamical systems; however, their infinite-dimensional nature necessitates rational approximation for practical implementation. This paper presents a unified comparative framework for evaluating widely used approximation methods, including standard and refined Oustaloup filters, continued fraction expansion (CFE), Matsuda, curve-fitting, and modified stability boundary locus (M-SBL) approaches. A systematic evaluation methodology is developed to assess these methods based on frequency-domain accuracy, time-domain performance, and robustness. Furthermore, a Pareto-based multi-objective analysis is introduced to explicitly capture the trade-offs among conflicting performance criteria, enabling the identification of non-dominated solutions without relying on weighted-sum formulations. Extensive simulations are conducted over a wide frequency range to evaluate approximation accuracy and control-oriented performance. The results reveal that different methods exhibit distinct trade-offs between accuracy, robustness, and complexity. In particular, the Oustaloup and M-SBL approaches demonstrate strong overall performance across multiple criteria, while methods such as CFE and curve-fitting show limitations under wideband conditions. The proposed framework provides a systematic and reproducible basis for selecting appropriate approximation techniques in fractional-order control applications, offering valuable insights into their practical implementation and performance trade-offs. Full article

This paper presents a novel approach to optimal input signal design for open-loop fractional-order system identification, using an integer-order approximation of the fractional operators to minimize the average input power. This is obtained by formulating the problem as an LMI (Linear Matrix Inequality) optimization problem with the limitation of achieving at least a specified model accuracy. The ORA (Oustaloup Recursive Approximation) method has been employed to model the fractional-order differentiation operator in discrete integer-order Output Error model form. The optimal input design is executed using finite-dimensional FIR (Finite Impulse Response) filter spectrum parameterization, where the decision variables are calculated through convex optimization. The A-optimality criterion has been used to examine the relationship between the input signal spectrum power and the accuracy of estimated models. Finally, numerical examples illustrate the proposed approach, confirming the method’s suitability for fractional-order system identification. Full article

To address the challenge of physically realizing fractional-order electrical networks, this study proposes an implementation method for a mechatronic inerter–spring–damper (ISD) suspension based on a fractional-order biquadratic transfer function. Building upon a previously established model of a mechatronic ISD suspension, the influence of parameter perturbations on the suspension’s dynamic performance characteristics was systematically investigated. Positive real synthesis was employed to determine the optimal five-element passive network structure for the fractional-order biquadratic electrical network. Subsequently, the Oustaloup filter approximation algorithm was utilized to realize the integer-order equivalents of the fractional-order electrical components, and the approximation effectiveness was analyzed through frequency-domain and time-domain simulations. Bench testing validated the effectiveness of the proposed method: under random road excitation at 20 m/s, the root mean square (RMS) values of the vehicle body acceleration, suspension working space, and dynamic tire load were reduced by 7.86%, 17.45%, and 2.26%, respectively, in comparison with those of the traditional passive suspension. This research provides both theoretical foundations and practical engineering solutions for implementing fractional-order transfer functions in vehicle suspensions, establishing a novel technical pathway for comprehensively enhancing suspension performance. Full article

This study introduces an innovative filter topology capable of providing simultaneous positive and negative gain outputs for one-fractional order LP, with high-pass, all-pass, and fractional-order shelving filter responses. The circuit, utilizing multi-output second-generation current-controlled conveyors, stands out as the first to deliver ten outputs, incorporating both integer and fractional-order filter responses, without requiring additional components. Its current-mode design simplifies the process, employing minimal active and grounded passive elements, making it appropriate for low-voltage/low-power applications. The filter utilizes fifth-order Oustaloup approximation and Foster type-I RC networks for fractional-order capacitors, providing enhanced control over the transition slope. PSpice simulations confirmed a 1 kHz cut-off, showcasing low power consumption, minimal noise, and a wide dynamic range, positioning the filter as suitable for sensors, control, and acoustic applications. Full article

Gantry-type dual-axis platforms can be used to move heavy loads or perform precision CNC work. Such gantry systems drive a single axis with two linear motors, and under heavy loads, a high driving force is required. This can generate a pulling force between the drive shafts in the coupling mechanism. In these situations, when a synchronization error becomes too large, mechanisms can become deformed or damaged, leading to damaged equipment, or in industrial settings, an additional power consumption. Effectively and accurately acquiring the synchronized movement of the platform is important to reduce energy consumption and optimize the system. In this study, a fractional-order fuzzy PID controller (FOFPID) using Oustaloup’s recursive filter is used to control a synchronous X–Y gantry-type platform. The optimized controller parameters are obtained by the measurement of control errors in a simulated environment. Four optimization methods are tested and compared: particle swarm optimization, invasive weed optimization, a gray wolf optimizer, and biogeography-based optimization. The systems were tested and compared in order to optimize the control parameters. Each of the four algorithms is simulated on four contour shapes: a circle, bow, heart, and star. The simulations and control scheme of the experiments are implemented using MATLAB, and the reference paths were planned using non-uniform rational B-splines (NURBS). After running the simulations to determine the optimal control parameters, each set of acquired control parameters is also tested and compared in the experiments and the results are recorded. Both the simulations and experiments show good results, and the tracking of the X–Y platform showed improved performance. Two performance indices are used to determine and validate the relative performance of the models and results. Full article

Accurate calculation of the electromagnetic force distribution of transformer windings under different loads and fault conditions is of great significance for transformer maintenance, condition evaluation and life prediction. Due to the influence of offshore wind power systems, offshore wind power transformers have high harmonic content and large changes in load rates, which can easily cause the coil destabilization, winding deformation or even damage because of the uneven distribution of the electromagnetic force. To improve the accuracy of electromagnetic force calculation, this paper proposes a fractional order numerical method. First, a three-dimensional axisymmetric transformer model and a symmetrical lumped parameter equivalent circuit model are established, respectively, based on field-circuit coupling. Second, the fractional order approximation of circuit components is realized by using the improved Oustaloup filter. In the fractional order model, the transformer is replaced by the lumped parameter equivalent circuit model. Third, as in the calculation process for integer order electromagnetic force, the integer order current has a large error, and the current waveform does not match the actual power frequency. The fractional order current and electromagnetic force at the 0.9 order are closer to the rated value. Finally, the effects of different load rates, three-phase short circuits and harmonic conditions are studied with the fractional order model. Compared with the traditional integer order finite element electromagnetic model, the fractional order equivalent circuit model established in this paper is more accurate and suitable for electromagnetic force calculation. The proposed method is significant for the structural design and state detection of transformers and also could be applied in the analysis of other dry-type transformers. Full article

The Zeta converter is an essential and widely used high-order converter. The current modeling studies on Zeta converters are based on the model that devices, such as capacitors and inductors, are of integer order. For this reason, this paper takes the Zeta converter as the research object and conducts an in-depth study on its fractional-order modeling. However, the existing modeling and analysis methods have high computational complexity, the analytical solutions of system variables are tedious, and it is difficult to describe the ripple changes of state variables. This paper combines the principle of harmonic balance with the equivalent small parameter method (ESPM); the approximate analytic steady-state solution of the state variable can be obtained in only three iterative steps in the whole solving process. The DC components and ripples of the state variables obtained by the proposed method were compared with those obtained by the Oustaloup’s filter-based approximation method; the symbolic period results obtained by ESPM had sufficient precision because they included more combinations of higher harmonics. Finally, the influence of fractional order on harmonics were analyzed. The obtained results show that the proposed method has the advantage of being less computational and easily describing changes in the ripple of the state variables. The simulation results are provided for validity of the theoretical analysis. Full article

In order to more effectively design the structure of vehicle ISD (Inerter Spring Damper) suspension system using the inerter, this paper proposed a design method using a fractional-order electrical network structure of a mechatronic inerter for fractional-order electrical network components, according to the characteristics that the external electrical network of a mechatronic inerter can simulate the corresponding mechanical network structure equivalently. First, the 1/4 dynamic model of the suspension is constructed. The improved Oustaloup filtering algorithm is used to simulate fractional calculus, and the fractional order components are simulated. Then, the simulation model of the vehicle mechatronic ISD suspension is established. In order to simplify the electrical network, one resistance, one fractional inductance and one fractional capacitance are limited in the design of the fractional electrical network at the outer end of the mechatronic inerter. The structure-immittance approach is used to obtain two general layouts of all possible structures of three elements. At the same time, the optimal fractional electrical network structure and parameters are obtained by combining the optimization algorithm. The simulation results verify the performance of the fractional ISD suspension with the optimized structure, which can provide a new idea for the structural design of a fractional-order electrical network applied in vehicle mechatronic ISD suspension. Full article

A simple and direct procedure for implementing fractional-order filters with transfer functions that contain Laplace operators of different fractional orders is presented in this work. Based on a general fractional-order transfer function that describes fractional-order low-pass, high-pass, band-pass, band-stop and all-pass filters, the introduced concept deals with the consideration of this function as a whole, with its approximation being performed using a curve-fitting-based technique. Compared to the conventional procedure, where each fractional-order Laplace operator of the transfer function is individually approximated, the main offered benefit is the significant reduction in the order of the resulting rational function. Experimental results, obtained using a field-programmable analog array device, verify the validity of this concept. Full article

The motion differential equation of hydro-pneumatic suspension is established to describe the vibration characteristics for a certain type of construction vehicle. The output force was deduced from the suspension parameters. Based on the suspension characteristics of a multi-phase medium, fractional calculus theory was introduced, and its fractional Bagley–Torvik equation was formed. The numerical computation by a low-pass filter of the Oustaloup algorithm was performed. The numerical solution of a nonlinear fractional equation was obtained to investigate the vibration characteristics of the suspension fractional system. Through the building of an equal-ratio test platform and simulation model, the fractional- integer-order model simulation and experimental data were compared. When the fractional order is 0.9, it better describes the motion characteristics of suspension system. The experiments show that the experimental data can fit the fractional-order system model well, and thereby prove the model on a hydro-pneumatic suspension system. Full article

In this paper, several modeling methods for the continuous current mode (CCM) fractional-order Cuk converter are investigated. First, the state space averaging method is used to establish the model. Based on this model, the expressions of inductors’ current and capacitor voltage as well as the transfer functions are derived. Then, the equivalent small parameter method (ESPM) is employed to model the converter. Based on the Oustaloup filter principle, the approximate models of fractional-order capacitor and inductors are constructed, which consist of integer-order components, to build the circuit model (CM) of the converter. In addition, the numerical model (NM) of the converter is established. Simulation results are provided to compare the modeling methods, which show that the ESPM has some advantages over the other methods. Finally, the hardware-in-the-loop experiment is conducted to verify the effectiveness of the circuit model. Full article

Fractional-order (FO) differential equations are more and more frequently applied to describe real-world applications or models of phenomena. Despite such models exhibiting high flexibility and good fits to experimental data, they introduce their inherent inaccuracy related to the order of approximation. This article shows that the chosen model influences the dynamic properties of signals. First, we calculated symbolically the steady-state values of an FO inertia using three variants of the Oustaloup filter approximation. Then, we showed how the models influence the Nyquist plots in the frequency domain. The unit step responses calculated using different models also have different plots. An example of FO control system evidenced different trajectories dependent on applied models. We concluded that publicized parameters of FO models should also consist of the name of the model used in calculations in order to correctly reproduce described phenomena. For this reason, the inappropriate use of FO models may lead to drawing incorrect conclusions about the described system. Full article

The growing number of operations in implementations of the non-local fractional differentiation operator is cumbersome for real applications with strict performance and memory storage requirements. This demands use of one of the available approximation methods. In this paper, the analysis of the classic integer- (IO) and fractional-order (FO) models of the brushless DC (BLDC) micromotor mounted on a steel rotating arms, and next, the discretization and efficient implementation of the models in a microcontroller (MCU) is performed. Two different methods for the FO model are examined, including the approximation of the fractional-order operator $s^{\nu }$ ( $\nu \in \mathbb{R}$ ) using the Oustaloup Recursive filter and the numerical evaluation of the fractional differintegral operator based on the Grünwald–Letnikov definition and Short Memory Principle. The models are verified against the results of several experiments conducted on an ARM Cortex-M7-based STM32F746ZG unit. Additionally, some software optimization techniques for the Cortex-M microcontroller family are discussed. The described steps are universal and can also be easily adapted to any other microcontroller. The values for integral absolute error (IAE) and integral square error (ISE) performance indices, calculated on the basis of simulations performed in MATLAB, are used to evaluate accuracy. Full article