Integer Order Approximation Research Articles

Loewner integer-order approximation of MIMO fractional-order systems

A state–space integer–order approximation of commensurate–order systems is obtained using a data–driven interpolation approach based on Loewner matrices. Precisely, given the values of the original fractional–order transfer function at a number of generalised frequencies, a descriptor–form state–space model matching these frequency response values is constructed from a suitable Loewner matrix pencil, as already suggested for the reduction of high–dimensional integer–order systems. Even if the stability of the resulting integer–order system cannot be guaranteed, such an approach is particularly suitable for approximating (infinite–dimensional) fractional–order systems because: (i) the order of the approximation is bounded by half the number of interpolation points, (ii) the procedure is more robust and simple than alternative approximation methods, and (iii) the procedure is fairly flexible and often leads to satisfactory results, as shown by some examples discussed at the end of the article. Clearly, the approximation depends on the location, spacing and number of the generalised interpolation frequencies but there is no particular reason to choose the interpolation frequencies on the imaginary axis, which is a natural choice in integer–order model reduction, since this axis does not correspond to the stability boundary of the original fractional–order system.

Control design, analysis, and optimization of fractional-order delayed resonator for complete vibration absorption

The active vibration absorber can eliminate the vibration energy entirely from a primary system through its anti-resonance effect, which is widely applied to vibration-sensitive scenarios. This article focuses on the control design, analysis, and optimization problems for a novel vibration-absorption technique named fractional-order delayed resonator. Specifically, we develop a parameterized form for such a fractional delayed control logic and provide a detailed analysis concerning the control parameters. By the frequency-domain integer-order approximations, we design a control framework that applies to any of the position, velocity, and acceleration signal sources. Then, the particle swarm optimization algorithm is utilized to minimize the transmissibility error at the excitation frequency and determine the optimal approximation frequency band. Additionally, we discuss the stability preservation concern brought by different approximation configurations. Finally, we treat the order of the fraction as an additional tunable parameter to enhance the frequency sensitivity under delay discretization. The overall concept and analysis are validated by numerical simulations.

One active element implementation of fractional-order Butterworth and Chebyshev filters

Fractional-order low-pass and high-pass Butterworth and Chebyshev filters with cut-off frequencies determined independently of their order are approximated in this work. The already available schemes in the literature have a cut-off frequency pre-determined by the pole frequency value, and hence the filter’s order. The derivation of rational integer-order approximation functions, which approximate the frequency behavior of the filters, is performed through the utilization of a magnitude curve-fitting based method. The form of the transfer function is common for all filter functions, providing design versatility. Implementation is performed using just one active element, leading to a structure with minimized active component count. The evaluation of the accuracy of the employed approximation method is performed through numerical and experimental results, while the evaluation of the behavior of the active circuit topology is done using circuit simulations.

Dynamic prediction modelling and equilibrium stability of a fractional discrete biophysical neuron model

Here, we contemplate discrete-time fractional-order neural connectivity using the discrete nabla operator. Taking into account significant advances in the analysis of discrete fractional calculus, as well as the assertion that the complexities of discrete-time neural networks in fractional-order contexts have not yet been adequately reported. Considering a dynamic fast–slow FitzHugh–Rinzel (FHR) framework for elliptic eruptions with a fixed number of features and a consistent power flow to identify such behavioural traits. In an attempt to determine the effect of a biological neuron, the extension of this integer-order framework offers a variety of neurogenesis reactions (frequent spiking, swift diluting, erupting, blended vibrations, etc.). It is still unclear exactly how much the fractional-order complexities may alter the fring attributes of excitatory structures. We investigate how the implosion of the integer-order reaction varies with perturbation, with predictability and bifurcation interpretation dependent on the fractional-order β∈(0,1]. The memory kernel of the fractional-order interactions is responsible for this. Despite the fact that an initial impulse delay is present, the fractional-order FHR framework has a lower fring incidence than the integer-order approximation. We also look at the responses of associated FHR receptors that synchronize at distinctive fractional orders and have weak interfacial expertise. This fractional-order structure can be formed to exhibit a variety of neurocomputational functionalities, thanks to its intriguing transient properties, which strengthen the responsive neurogenesis structures.

A Hybrid Approximation Method for Integer-Order Approximate Realization of Fractional-Order Derivative Operators

The use of fractional-order (FO) calculus for the solution of different problems in many fields has increased recently. However, the usage of FO system models in practice brings some difficulties. The FO operator, fractance device, is usually realized via several integer-order approximation methods, which have pros and cons in the aspect of operation frequency, time response and stability region. These methods may not meet all performance expectations. In this regard, author proposes an efficient hybrid integer-order approximation method for FO derivative operator without causing any additional difficulty in realization. The proposed method combines Matsuda and modified stability boundary locus (M-SBL) approximation methods. The advantage of each method is combined in a single hybrid function by considering root mean square error (RMSE) rates for step response. The performance of hybrid transfer function is analyzed in comparison with Matsuda, Oustaloup, continued fraction expansion (CFE) and M-SBL transfer functions for both frequency and time response. Analog realization of the proposed model is performed experimentally via partial fraction expansion method. Analog design is verified via both Multisim simulations and experimental results. The improvements due to the hybrid behavior and the consistency of experimental results with theoretical and simulation results demonstrate the practicality and usefulness of the hybrid model.

Power mapping-based stability analysis and order adjustment control for fractional-order multiple delayed systems

This article investigates the asymptotic stability of a general class of fractional-order multiple delayed systems to evaluate the delay robustness. We establish a one-to-one spectral connection between the original fractional-order system and the transformed one under the power mapping. The applicability of the Cluster Treatment of Characteristic Roots paradigm to the transformed dynamics is proved by this connection. Then, we utilize the Dixon resultant-based frequency sweeping framework to create the complete stability map. The results demonstrate that the order adjustment control significantly enhances the control flexibility and brings unlimited possibilities for the improvement of the delay robustness. Finally, we inspect the stability preservation problem when using the integer-order approximations for practical implementation.

Modeling and tracking control of dielectric elastomer actuators based on fractional calculus

Dielectric elastomer actuators (DEAs) show broad application prospects in the area of soft robots since they offer merits of fast response, large deformation, light weight and high energy conversion efficiency. Practical soft robot applications would usually require the study on the modeling and control of the DEA. However, the DEA has a memory property, which results in a highly nonlinear characteristics, bringing difficulties to the subject. To address this issue, an effective strategy is the utilization of the fractional order model, which is a type of modeling approach that can accurately characterize the memory property of a material with a small amount of parameters. Meanwhile, the fractional order controller can better handler the memory property, and it owns a better flexibility than traditional integer order controllers. With the above considerations, this paper proposes a modeling strategy and a tracking control strategy for the DEA on the basis of the fractional calculus. In the proposed strategy, a fractional order model is established to characterize the complicated nonlinear characteristics of the DEA. Then, to facilitate the computer simulation, the Oustaloup filter is used to construct an integer order approximation model (IOAM) of the fractional order model. Since the IOAM is difficult to be employed in the system controller design due to its high order, the IOAM is further simplified into a reduced order model based on the square root balance truncation algorithm. To realize the high accuracy control of the DEA, a feedforward-feedback combined controller is devised, which is composed of a feedforward controller and a fractional order proportional integral feedback controller (FOPIFC). Among which, the feedforward controller is devised based on the analytical inverse of the reduced order model to compensate the complicated nonlinear characteristics of the DEA, and the FOPIFC is devised to handle the bad influence from the modeling error and uncertainties on the control performance. Based on the proposed strategy, control experiment was conducted, and the root-mean-square errors in experiment are all below 0.7%, indicating the superiority of the presented modeling and tracking control strategies.

Data-driven fractional order phase-lead and proportional–integral feedback control strategy with application to a reluctance-actuated compliant micropositioning system

This paper aims to overcome complex dynamic problems such as low damping resonant frequency, resonant point drift, and uncertainty of dynamic model of the large stroke micropositioning stage, based on Maxwell reluctance actuator. A composite control method of data-driven fractional phase-lead proportional-integral feedback controller with series hysteresis compensator is proposed. Firstly, the working principle of the Maxwell reluctance actuator is briefly explained, a large stroke flexure micropositioning system is built, and its motor constant and stiffness characteristics are experimentally measured. Secondly, aiming at the rapid phase drop caused by low damping resonance, a data-driven fractional order phase-lead and proportional-integral feedback controller is designed, and the parameters of the fractional order feedback controller are experimentally adjusted based on the iterative feedback tuning algorithm. And the intrinsic hysteresis nonlinearity of the reluctance actuator is firstly compensated by a Prandtl–Ishlinskii model hysteresis compensator which is always placed in the feedback loop of the control system. Then, the proposed controller is compared with other controllers. The experimental results show that this method has a significant suppression effect on high-frequency noise, and the tracking error is significantly reduced, which proves the effectiveness of this stage and control method. • Based on the Maxwell reluctance actuator, a large stroke flexure micropositioning system is built. • A fractional order phase-lead proportional integral controller is proposed, and the integer order approximation of the controller is completed by using oustaloup approximation method • The data-driven method of iterative feedback tuning is used to obtain the parameters of the fractional order controller. • Through the analysis and comparison of linearity, sensitivity function and open-loop frequency response, the advantages of the proposed controller are fully proved.

Time response optimal rational approximation: Improvement of time responses of MSBL based approximate fractional order derivative operators by using gradient descent optimization

Fractional calculus has become a popular topic in recent years. Its applications have been seen frequently in the analysis and solution of many engineering problems. Since, the exact realization of fractional order (FO) operators is not computationally feasible, several integer order approximations methods such as Oustaloup, Matsuda, Curve Fitting, modified stability boundary locus (MSBL) are utilized for frequency domain approximation of FO operators. These methods present advantages in frequency response approximation. However, real-world engineering applications require more accurate time responses to implement FO operators. Thus, any improvement in the approximate transfer function of FO systems will positively contribute to realization performance of FO system models in real-world applications. In this work, time responses of MSBL based approximate FO derivative models are improved by using gradient descent optimization (GDO) technique. Frequency domain approximation methods do not guarantee optimal approximation performance for the step response of FO elements. To deal with this weakness, author proposes a new GDO based rational approximation scheme that can considerably enhance the step response performance of MSBL method without deteriorating frequency response approximation performance. Results for the improved approximate FO model obtained by GDO (GDO-MSBL) are compared with the results of approximate FO models for popular FO approximation methods. The analog circuit realization is performed for the derivative operator, and the performance improvements for the proposed method are shown on the FO-PID controller and FO filter.

Design of a Fuzzy Fractional Order Adaptive Impedance Controller with Integer Order Approximation for Stable Robotic Contact Force Tracking in Uncertain Environment

Abstract Current research in robot compliance control is unable to take both transient contact force overshoots and steady-state force tracking problems into account. To address this problem, we propose a fuzzy fractional order (FO) adaptive impedance controller to avoid the force overshoots in the contact stage while keeping force error in the dynamic tracking stage, where traditional control algorithms are not competent. A percentage gain is adopted to map FO parameters to integer order (IO) parameters by their natural properties, and a fuzzy logical controller is introduced to improve the system stability. The simulation results indicate that the proposed controller can be made more stable than and superior to the general impedance controller, and the force tracking results also have been compared with the previous control methods.

μ-Synthesis FO-PID for Twin Rotor Aerodynamic System

μ-synthesis is a NP-hard optimization problem based on the generalized Robust Control framework which manages to find a controller which fulfills both robust stability and robust performance. In order to solve such problems, nonsmooth optimization techniques are employed to find nearly-optimal parameters values. However, the free parameters available for tuning must be involved only in classical arithmetic operations, which leads to a problem for the fractional-order operator or for its integer-order approximation, exponential operations being involved. The main goal of the current article consists of presenting a possibility to integrate a fixed-structure multiple-input-multiple-output (MIMO) fractional-order proportional-integral-derivative (FO-PID) controller in the μ-synthesis optimization problem. The solution consists in a possibility to find a set of tunable parameters isomorphic with the fractional-order such that the coefficients involved in the approximation of the fractional element, along with the formulation of a fixed-structure mixed-sensitivity loop shaping μ-synthesis control problem. The proposed design procedure is applied to a twin rotor aerodynamic system (TRAS) using both MATLAB numerical simulation and practical experiments on laboratory scale equipment. Moreover, a comparison with the unstructured μ-synthesis is performed, highlighting the advantages of the proposed solution: simpler form and guaranteed robust stability and performance.

Adaptive second‐order terminal PID sliding mode control design for integer‐order approximation of wind turbine system for maximum power extraction

The main purpose of this paper is to extract the maximum power from the variable speed wind turbine (VSWT), which is modelled using fractional calculus. The use of the fractional-order operator can harden the practical implementation and numerical simulation of the fractional-order systems. To apply the fractional calculus theory, we assume that the orders of all fractional derivatives of the system have a small deviation from the nearest integers. By defining this small deviation as a perturbation parameter, an integer-order approximation method is presented for approximating the fractional-order system into a perturbed integer-order system in the time domain. Then, a new adaptive second-order sliding mode control (SOSMC) method is proposed using the terminal proportional integral derivative (TPID) sliding surface to achieve maximum power extraction and reduce mechanical loads and chattering. A TPID sliding surface is employed to ensure better tracking, have a zero steady-state error, decrease the mechanical stresses, and eliminate the chattering phenomenon. The stability of the adaptive SOSMC based on the obtained integer-order approximation is analyzed using the Lyapunov stability criterion. The simulation results of the proposed method are presented and compared with some designed controllers for wind turbines (WTs) modelled with fractional and integer derivatives. The obtained results show the effectiveness of the proposed strategy.

Realizations of fractional-order PID loop-shaping controller for mechatronic applications

A novel procedure for the realization of a fractional-order PID loop-shaping controller, suitable for precision control of mechatronic systems, is introduced in this work. Exploiting appropriate tools, the controller function is approximated as a whole, leading to a simple form of integer-order approximation, when compared to the case where each intermediate part of the PID transfer function is approximated. This leads to a direct implementation, composed of conventional active and passive elements. Simulation and experimental results, derived from the OrCAD PSpice simulator and a Field-Programmable Analog Array respectively, verify the efficient functionality of the proposed implementation procedure.

Fractional Order Graph Filters: Design and Implementation

Existing graph filters, polynomial or rational, are mainly of integer order forms. However, there are some frequency responses which are not easily achieved by integer order approximation. It will substantially increase the flexibility of the filters if we relax the integer order to fractional ones. Motivated by fractional order models, we introduce the fractional order graph filters (FOGF), and propose to design the filter coefficients by genetic algorithm. In order to implement distributed computation on a graph, an FOGF can be approximated by the continued fraction expansion and transformed to an infinite impulse response graph filter.

Minimal realization and approximation of commensurate linear fractional-order systems via Loewner matrix method.

In this paper we propose a data driven realization and model order reduction (MOR) for linear fractional-order system (FoS) by applying the Loewner-matrix method. Given the interpolation data which obtained by sampling the transfer function of a FoS, the minimal fractional-order state space descriptor model that matching the interpolation data is constructed with low computational cost. Based on the framework, the commensurate order α of the fractional-order system is estimated by solving a least squares optimization in terms of sample data in case of unknown order-α. In addition, we present an integer-order approximation model using the interpolation method in the Loewner framework for FoS with delay. Finally, several numerical examples demonstrate the validity of our approach.

Direct approximation of fractional order systems as a reduced integer/fractional-order model by genetic algorithm

In this paper, a new method is proposed for the reduced-order model approximation of commensurate/incommensurate fractional order (FO) systems. For integer order approximation, the model order is determined via Hankel singular values of the original system; while the order of FO approximations is determined via optimization. Unknown parameters of the reduced model are obtained by minimizing a fitness function via the genetic algorithm (GA). This fitness function is the weighted sum of differences of Integral Square Error (ISE), steady-state errors, maximum overshoots, and ISE of the magnitude of the frequency response of the FO system and the reduced-order model. Therefore, both time and frequency domain characteristics of the system considered in obtaining the reduced-order model. The stability criteria of the reduced-order systems were obtained in various cases and added to the cost function as constraints. Three fractional order systems were approximated by the proposed method and their properties were compared with famous approximation methods to show the out-performance of the proposed method.

Emulation of a constant phase element by utilizing a lattice structure based fractional-order differentiator

In this paper, we propose a modular Constant Phase Element (CPE) emulator, the approximation order of which is electronically adjustable. In this way, the emulator provides a limited approximation error over a range of the parameter alpha, the order of the CPE. A fractional-order differentiator has been utilized in the emulator, which is realized by rational integer-order approximations implemented based on a lattice-type structure. The advantage of lattice-type implementation is the possibility of easily changing the order of approximation. It also provides the flexibility of compromising the approximation accuracy with power consumption. The emulator is accurate over five decades of frequency [0.1 Hz, 10 kHz] and provides electronic adjustability of the CPE capacitance, order, and operation frequency. The simulation results confirm the performance of the emulator. Two fractional band-pass filters (FBPF) are presented as an application of the CPE emulator, which are employed in electroencephalography (EEG) signal processing. One of the FBPFs is employed in extracting θ and α waves in the frequency range of 3–13 Hz; while the other filter is used for extracting the β waves in the range of 13–30 Hz. The power consumption of the simulated FBPFs are 2.871 μW and 1.836 μW, respectively.

Voltage differencing transconductance amplifier based fractional order multiple input single output universal filter

In this paper a Voltage Differencing Transconductance Amplifier (VDTA) based Voltage Mode (VM) Multi Input Single Output (MISO) Fractional Order filter (FOF) providing all five filter responses is presented. The proposed FOF is designed using single VDTA and two fractional Capacitors (FC). The FC is implemented using RC ladder network; component values of which are computed using Carlson rational approximation. For illustration, FC of 0.5 order has been implemented using first, second and third iterations of Carlson approximation, which results in different integer order approximation functions. The ladder component values for all iterations are computed using MATLAB. The effect of different iterations on FOF behavior is also studied. The proposed FOF is verified through Cadence Virtuoso simulations using 0.18 µm CMOS technology.

Frequency Response Based Curve Fitting Approximation of Fractional–Order PID Controllers

Abstract Fractional-order PID (FOPID) controllers have been used extensively in many control applications to achieve robust control performance. To implement these controllers, curve fitting approximation techniques are widely employed to obtain integer-order approximation of FOPID. The most popular and widely used approximation techniques include the Oustaloup, Matsuda and Cheraff approaches. However, these methods are unable to achieve the best approximation due to the limitation in the desired frequency range. Thus, this paper proposes a simple curve fitting based integer-order approximation method for a fractional-order integrator/differentiator using frequency response. The advantage of this technique is that it is simple and can fit the entire desired frequency range. Simulation results in the frequency domain show that the proposed approach produces better parameter approximation for the desired frequency range compared with the Oustaloup, refined Oustaloup and Matsuda techniques. Furthermore, time domain and stability analyses also validate the frequency domain results.

All Possible Topologies of the Fractional-Order Wien Oscillator Family Using Different Approximation Techniques

This paper introduces all the possible topologies of the Wien bridge oscillator family. This family has 72 topologies, 24 of them contain only RC or RL pairs, and the rest contain mixed pairs. The complete mathematical analysis of all twelve possible capacitive-based topologies is proposed in the fractional-order domain. The investigated circuits can be categorized into two groups, each with a similar characteristic equation. Three integer-order approximation techniques for the Laplacian operator $$s^\alpha $$ are employed to solve and simulate the Wien bridge system. The studied approximations are those of Matsuda, Oustaloup, and Valsa’s network. Fractional-order capacitor (FOC) emulators are built using these approximations and applied in the circuit simulation. Comparisons are made on different levels, starting with the mathematical solution of the characteristic equation, followed by PSpice simulation, which compares topologies of the Wien bridge oscillator family. Hardware implementation of the FOC emulators is presented applying passive discrete components using the Foster-I technique. Additionally, sensitivity tests of the discrete components of the FOC emulators are performed using Monte Carlo analysis. Experimental results are introduced to validate the theoretical findings.