Fractional-order Low-pass Filter Research Articles

Improving small-signal stability of inverter-based microgrids using fractional-order control

Augmenting the range of the active droop gain in inverter-based microgrids (IBMGs) is required for faster power sharing and improved control at the secondary level. However, IBMG stability is degraded when active droop gain is increased, which reveals an obvious tradeoff. Therefore, fractional-order control (FOC) is used in this paper to increase the inverter-based (IB) MGs’ stability margin by increasing the stable range of the active droop gain. So, a thorough small-signal model of IBMGs is developed to assess the MG stability and tune the control parameters for the case of using FOCs. Also, a fractional-order low-pass filter (FO-LPF) is proposed for the outer control loop which increases the damping factor and reduces the power oscillations at the same active droop gain. Moreover, a further boost in stability margin is achieved by combining the proposed FO-LPF in the power control loop with FO-PI controllers in the inner control loops. The stability results of the developed FO small-signal model are validated on a MATLAB/SIMULINK MG stability benchmark system. Also, a comparative case study has shown the superiority of the proposed approach over recent ones in the case of a reconfigured and meshed network. Moreover, the discretization scheme of FOCs is outlined and tested under a load disturbance event.

Power-Efficient Electronically Tunable Fractional-Order Filter

This article describes a low-voltage, low-power fractional-order low-pass filter (FO-LPF) of order 1 + α, which is implemented using a voltage differencing differential difference amplifier (VDDDA). The VDDDA structure is implemented using the bulk-driven metal oxide semiconductor transistor technique. The transistors operate in the subthreshold region to maintain low-supply voltage and low-power consumption. The FO-LPF structure implemented using this VDDDA structure is compact. It includes three VDDDAs and three grounded capacitors along with two active resistors implemented using MOS transistors. In addition, this filter structure provides electronic tuning of its order and cut-off frequency through the bias current of the active component used. The effects of tracking error and parasitics on the functionality of the proposed FO-LPF were investigated. The VDDDA and filter operate at ±300 mV and dissipate only 207 nW and 663 nW of power, respectively. Thus, the VDDDA structure and filter are suitable for low-voltage and low-power operation. Layouts of the proposed VDDDA as well as the FO-LPF were designed in the Cadence Virtuoso environment. Post-layout simulation results of the designed circuits imply that they are suitable for fabrication. Noise, total harmonic distortion, Monte-Carlo, and PVT analyses were also performed.

Realization of fractional order lowpass filter using different approximation techniques

Many research groups are starting to pay serious attention to the problem of fractional-order circuits. In this paper, a new approach to designing fractional order low-pass filter (FOLPF) is presented. Finding a rational approximation of the fractional Laplace operator sα is a crucial step in the design of fractional order filters. A comparative study of the most widely used approximation techniques named continued fraction expansion (CFE) method and Biquadratic Approximation (RE) method is performed. Then the transfer function of the proposed FOLPF is calculated. Using operational amplifier, the proposed filter is synthesized. The proposed circuit is simulated using Texas instruments TINA software. The results obtained outperform the existing methods.

Design of a Fractional Order Low-pass Filter Using a Differential Voltage Current Conveyor

In this paper, an active implementation of a differential voltage current conveyor (DVCC) based on a low-pass filter operating in the fractional order domain is presented. The transfer function for a fractional order system is dependent on the rational approximation of sα. Different methods used for calculating the rational approximation, including Carlson, Elkhazalil, and curve fitting, are evaluated here. Finally, to validate the theoretical results, a fractional order Butterworth filter is simulated in the Pspice environment using the 0.5 micrometer CMOS technology with an R-C network-based fractional order capacitor. Additionally, using the Monte Carlo analysis, the impact of current and voltage faults on DVCC response is investigated. It has been inferred that realization with a wider bandwidth is possible.

Synchronous Position Control of Gantry Table Using Adaptive Fraction-Order Dynamic Surface Control With Dendritic Neuron Model

This article introduces a three-degree-of-freedom (3DOF) model-based adaptive fractional-order dynamic surface control (AFODSC) system for a synchronous position control of a gantry table. First, the 3DOF model of the gantry table is investigated considering the rotational dynamics caused by inter-axis mechanical coupling and synchronous error. Then, a new fractional-order dynamic surface control (FODSC) approach is introduced using a fractional-order low-pass filter to provide an extra degree of freedom for the existing dynamic surface control (DSC) method. Because it is arduous to know the system uncertainty, the AFODSC is further elaborated by incorporating an adaptive mechanism and an uncertainty observation into the FODSC. In the proposed AFODSC, a modified dendritic neuron model observer (DNMO) is proposed to identify uncertainties, whereas an exponential compensator is designed to compensate for the observation error. Multiple adaptation laws are derived to adapt the parameters of DNMO on the basis of the Lyapunov stability theorem. Experimental results confirm that the proposed AFODSC system demonstrates favorable position tracking and synchronization accuracies for the gantry table compared with the conventional DSC system.

Electronically Tunable Multifunction Transadmittance-Mode Fractional-Order Filter

In this paper a multifunction transadmittance-mode fractional-order filter is proposed for the very first time. The proposed structure provides fractional-order low-pass filter, fractional-order high-pass filter and fractional-order all-pass filter. It uses two operational transconductance amplifier and one fractional capacitor (FC). The FC is realized using the twelfth-order continued fraction expansion-based RC ladder. Functional verification of proposed fractional-order filter is verified through PSPICE using 180-nm CMOS technology model parameters. The transfer functions sensitivities’ with respect to various circuit parameters are examined through simulations. The effect of parameter variation on filter responses is studied using Monte Carlo analysis. Furthermore, the proposed filter possess electronic tunability through bias currents. Moreover, the circuit is tested experimentally using discrete components to realize the fractional-order capacitor showing great matching with the theoretical data.

Experimental validation of fractional order internal model controller design on buck and boost converter

In this paper, fractional order internal model control technique is formulated for non-ideal dc–dc buck and boost converter. The fractional order internal model control approach integrates the concept of Commande Robuste d’Ordre Non Entier principle for tuning a fractional order filter with internal model control scheme. The final controller can be expressed as a series combination of proportional integral derivative controller and a fractional order low pass filter. To assess the robustness of the proposed fractional order internal model control scheme, both the servo response and regulatory response of the dc–dc converters are investigated in the presence of disturbances. The efficacy of fractional order internal model control technique is demonstrated via comparison with 2 degrees of freedom internal model control scheme. Furthermore, an experimental validation of fractional order internal model control is conducted on laboratory setup, and a dSPACE 1104 microcontroller is used for hardware implementation. The simulation results and the hardware validation are a testimony to the effectiveness of fractional order internal model control technique.

Optimal integer‐order rational approximation of α and α + β fractional‐order generalised analogue filters

A new approach to determine the integer-order transfer function approximation for α and α + β, where 0 <; α , β <; 1, fractional-order analogue filters is presented. Reported works deal with this design problem by using techniques which are non-optimal. The proposed designs with guaranteed stability are obtained by employing a global search optimisation algorithm called colliding bodies optimisation (CBO). The modelling error is iteratively minimised by CBO in the least-squares sense directly in the frequency domain. The improved accuracy in the magnitude response of the proposed fractional-order low pass filters, fractional-order high pass filters, and fractional-order band pass filters as compared to the designs based on the Matsuda and the continued fractions expansion-based techniques is demonstrated. The efficacies of the proposed filters are also validated in SIMULINK platform.

Approximation of fractional‐order low‐pass filter

Optimal integer-order transfer function approximations to model the single fractance element-based fractional-order low-pass filter (FLF) for any arbitrary order α , where, 0 <; α <; 1, is proposed here. First of all, the integer-order filter coefficients for FLFs, with α varying from 0.01 to 0.99 in steps of 0.01, are directly obtained by using a metaheuristic algorithm called colliding bodies optimisation. For practical usability, the approximated FLF coefficients are explicitly provided in the form of analytical equations by employing a curve fitting on the optimised coefficients in the second step. The proposed approach provides a simpler design procedure in comparison to the reported literature which approximates the FLF by substituting an integer-order rational approximation of the s α operator in the transfer function of the ideal FLF. Simulations confirm the superior modelling accuracy of the proposed design over the recent literature.

Fractional-order low-pass filter with electronic tunability of its order and pole frequency

Abstract This paper presents novel solution of a fractional-order low-pass filter (FLPF). The proposed filter operates in the current mode and it is designed using third-order inverse follow-the-leader feedback topology and operational transconductance amplifiers (OTAs), adjustable current amplifiers (ACAs), auxiliary multiple-output current follower (MO-CF) as simple active elements. The filter offers the beneficial ability of the electronic control of its order and also the pole frequency thanks to electronically controlled internal parameters of OTAs and ACAs. As an example, five particular values of fractional order of the FLPF were chosen and parameters of the filter were calculated. Similarly, also electronic control of the pole frequency of the filter was studied. The design correctness and proper function of the filter are supported by simulations with CMOS models and also by experimental laboratory measurements. Comparison of the simulation results of the proposed filter for two different approximations of the parameter s α is also included.

Synthesis and Analysis of Electronically Adjustable Fractional-Order Low-Pass Filter

A proposal of a fractional ([Formula: see text])-order low-pass filter is presented in this paper. The proposed filter operates in the current-mode and it is designed using Multi-Output Current Followers (MO-CFs), Dual-Output Current Follower (DO-CF), Dual-Output Adjustable Current Amplifier (DO-ACA) and Adjustable Current Amplifiers (ACAs) as active elements within the presented topology of the filter. The filter possesses ability to electronically control its order and also the pole frequency by changing the current gain of current amplifiers (ACAs) already present in the structure. Three different values of the order and pole frequency of the proposed low-pass filter were tested as an example. Design of the proposed filter is supported by simulation and experimental results. Simulations of the circuit are carried out in PSPICE simulator with behavioral models of used active elements. The experimental laboratory measurements are performed with the help of available devices forming equivalent circuits. Simulations and experimental results of the electronical control of the order and pole frequency are compared in this contribution.

Electronically Tunable Fractional Order Filter

In this paper, an electronically tunable resistorless fractional order filter (FOF) based on operational transconductance amplifier (OTA) is presented. It uses two fractional capacitors (FC) of same order and provides fractional order low-pass filter and fractional order band-pass filter responses simultaneously. Mathematical formulations are outlined for various critical frequencies and transfer function sensitivities for presented FOF. The FCs of orders 0.5 and 0.9 are considered for illustrating the proposal. The FCs are realized using the fourth-order continued fraction expansion-based RC ladder and are characterized using SPICE simulations. Functional verification of presented FOF with FC of orders 0.5 and 0.9 is exhibited through SPICE simulations. The OTA is implemented using $$0.5\,\upmu \hbox {m}$$ CMOS technology model parameters. Electronic tunability of half power and right-phase frequencies of presented FOF is achieved through bias current variation of OTA. The transfer functions’ sensitivity with respect to various circuit parameters is also examined through simulations, and it is found that the values remain well within unity for most of the circuit parameters. Furthermore, the presented FOF is attractive from integration viewpoint as it achieves tunability via bias current variation in contrast to tuning through resistor variation in existing FOFs.

Reconfigurable Fractional-Order Filter with Electronically Controllable Slope of Attenuation, Pole Frequency and Type of Approximation

This paper presents design of electronically reconfigurable fractional-order filter that is able to be configured to operate as fractional-order low-pass filter (FLPF) or fractional-order high-pass filter (FHPF). Its slope of attenuation between pass band and stop band, i.e., order of the filter, is electronically adjustable in the range between 1 and 2. Also, pole frequency can be electronically controlled independently with respect to other tuned parameters. Moreover, particular type of approximation can be also controlled electronically. This feature set is available both for FLPF and FHPF-type of response. Presented structure of the filter is based on well-known follow-the-leader feedback (FLF) topology adjusted in our case for utilization with just simple active elements operational transconductance amplifiers (OTAs) and adjustable current amplifiers (ACAs), both providing possibility to control its key parameter electronically. This paper explains how reconfigurable third-order FLF topology is used in order to approximate both FLPF and FHPF in concerned frequency band of interest. Design is supported by PSpice simulations for three particular values of order of the filter (1.25, 1.5, 1.75), for several values of pole frequency and for two particular types of approximation forming the shape of both the magnitude and phase response. Moreover, theoretical presumptions are successfully confirmed by laboratory measurements with prepared prototype based on behavioral modeling.

On The Optimization of Fractional Order Low-Pass Filters

This paper presents three different optimization cases for normalized fractional order low-pass filters (LPFs) with numerical, circuit and experimental results. A multi-objective optimization technique is used for controlling some filter specifications, which are the transition bandwidth, the stop band frequency gain and the maximum allowable peak in the filter pass band. The extra degree of freedom provided by the fractional order parameter allows the full manipulation of the filter specifications to obtain the desired response required by any application. The proposed mathematical model is further applied to a case study of a practical second- generation current conveyor (CCII)-based fractional low-pass filter. Circuit simulations are performed for two different fractional order filters, with orders 1.6 and 3.6, with cutoff frequencies 200 and 500 Hz, respectively. Experimental results are also presented for LPF of 4.46 kHz cutoff frequency using a fabricated fractional capacitor of order 0.8, proving the validity of the proposed design approach.

CCII based fractional filters of different orders

This paper aims to generalize the design of continuous-time filters to the fractional domain with different orders and validates the theoretical results with two different CCII based filters. In particular, the proposed study introduces the generalized formulas for the previous fractional-order analysis of equal orders. The fractional-order filters enhance the design flexibility and prove that the integer-order performance is a very narrow subset from the fractional-order behavior due to the extra degrees of freedom. The general fundamentals of these filters are presented by calculating the maximum and minimum frequencies, the half power frequency and the right phase frequency which are considered a critical issue for the filter design. Different numerical solutions for the generalized fractional order low pass filters with two different fractional order elements are introduced and verified by the circuit simulations of two fractional-order filters: Kerwin–Huelsman–Newcomb (KHN) and Tow-Tomas CCII-based filters, showing great matching.

Synchronization of integer order and fractional order Chua’s systems using robust observer

In this paper we present a fractional order Chua’s circuit that behaves chaotically based on the use of a fractional order low pass filter. Next, an integer order robust observer will be designed to synchronize the fractional order Chua’s circuit as well as integer order Chua’s circuit with unknown nonlinearity. This method consists in designing a Luenberger like observer appended with an estimator of the unknown nonlinear function. The estimator assumes that the nonlinear function is slowly varying and that the observer converges quickly and uses the backward difference formula to approximate the state derivative. The efficiency of the proposed method is confirmed using numerical simulations.

Fractional order and BICO disturbance observers for a run-of-mine ore milling circuit

Grinding mill circuits are hard to control due to poor plant models, large external disturbances, uncertainties from internal couplings, and process variables that are difficult to measure. This paper proposes a novel fractional order disturbance observer (FO-DOB) for a run-of-mine (ROM) ore milling circuit. A fractional order low pass filter ( Q-filer) is used in the DOB to offer an additional degree of freedom in tuning for set-point tracking performance and disturbance rejection performance. Another disturbance observer is introduced in which a Bode ideal cut-off (BICO) filter is used for the Q-filter. A full non-linear plant model is used for evaluation of the performance gained over the ubiquitous PI controller. The simulation results show that the FO-DOB and BICO-DOB schemes are useful additional tools for ROM ore milling circuit control implementations.

Digital Fractional Order Savitzky-Golay Differentiator

This brief proposes a design method for a digital fractional order Savitzky-Golay differentiator (DFOSGD), which generalizes the Savitzky-Golay filter from the integer order to the fractional order for estimating the fractional order derivative of the contaminated signal. The proposed method calculates the moving window's weights using the polynomial least-squares method and the Riemann-Liouville fractional order derivative definition, and then computes the fractional order derivative of the given signal using the convolution between the weights and the signal, instead of the complex mathematical deduction. Hence, the computation time is greatly improved. Frequency-domain analysis reveals that the proposed differentiator is essentially a fractional order low-pass filter. Experiments demonstrate that the proposed DFOSGD can accurately estimate the fractional order derivatives of both noise-free signal and contaminated signal.

Noise analysis of single stage fractional-order low-pass filter using stochastic and fractional Calculus

In this paper, we present the noise analysis of a simple single stage low-pass ¯lter (SSLPF) with the fractional-order capacitor, using stochastic differential equations (SDE). The input noise is considered to be white and various solution statistics of output namely mean, variance, auto-correlation and power spectral density (PSD) are obtained using tools from both stochastic and fractional calculus. We investigate the change in these statistics with the change in the capacitor order. The closed form solutions of the step response of the fractional filter are also provided and it has been found that filters with capacitor order greater than unity have a faster step response but suffer from higher output noise and although, the filters with capacitor order less than unity enjoy advantage of less output noise, but they have a sluggish step response. And hence, an appropriate fractional capacitor can be chosen for the desired circuit behavior. A brief study of more generic class of single stage fractional-order high-pass and all-pass ¯ltering functions has been included. The idea can be extended to more complex and practical fractional order circuits.