Integer-order Transfer Function Research Articles

Inverse Estimation of Dynamic Heat Flux Using Proportional Integral Derivative Control

To improve long-term dynamic heat flux prediction using an embedded thermocouple, a novel inverse estimation approach using a proportional integral derivative (PID) control scheme is proposed. A feedforward model of the discrete integer-order transfer function was identified from a one-dimensional dynamic heat-flux sensor using the calibrated heat flux with varying amplitudes of the pseudo-random binary sequence and its single-temperature response data. Using the model and unit-step temperature response as the controlled plant and reference input signal, the PID controller with a low-pass filter was designed. Using the on-site temperature measurement data as the reference signal instead of the unit-step temperature, the time-history control output values, that is, the inverse estimated dynamic heat flux of the designed PID controller, were obtained. The comparison of the test results and the discussion indicate that 1) the proposed inverse method avoids the impact of the thermal inertia of the thermocouple and significantly alleviates the effects of measurement noise, and 2) in addition to high identification efficiency and measurement fidelity, further improvement in the effective measuring time of the dynamic heat flux was observed.

Field Programmable Analog Array Based Non-Integer Filter Designs

The approximation of the frequency behavior of fractional-order, power-law, and double-order filters can be performed by the same rational integer-order transfer function. This can be achieved through the utilization of a curve fitting based approximation. Moreover, their implementation can be performed by the same core, by only changing the corresponding time constants and scaling factors. The aforementioned findings are experimentally verified using a Field Programmable Analog Array device.

Implementation of cable-driven waist rehabilitation robotic system using fractional-order controller

This paper presents the design and implementation of a fractional-order controller for a cable-driven waist rehabilitation robotic system (CDWRRS) based on an improved Oustaloup numerical approximation method. First, the mechanical structure and hardware design of the CDWRRS are established based on the demands of waist rehabilitation training. Next, a fractional-order controller is designed and implemented. The function of the proposed controller is realised using integer-order systems for approximation. Additionally, an improved numerical approximation method is proposed based on the Oustaloup numerical approximation method. In the Oustaloup method, the zeros and poles of the integer-order transfer functions are different, and the approximation errors near the endpoints of the frequency range are not considered. To address these problems, the proposed numerical approximation method improves the approximation accuracy and reduces the calculation cost. Finally, numerical examples and experimental results are presented to verify the feasibility and effectiveness of the improved Oustaloup numerical approximation method and the designed fractional-order controller for the developed CDWRRS.

An Approximation Method for Fractional-Order Models Using Quadratic Systems and Equilibrium Optimizer

System identification is an important methodology used in control theory and constitutes the first step of control design. It is known that many real systems can be better characterized by fractional-order models. However, it is often quite complex and difficult to apply classical control theory methods analytically for fractional-order models. For this reason, integer-order models are generally considered in classical control theory. In this study, an alternative approximation method is proposed for fractional-order models. The proposed method converts a fractional-order transfer function directly into an integer-order transfer function. The proposed method is based on curve fitting that uses a quadratic system model and Equilibrium Optimizer (EO) algorithm. The curve fitting is implemented based on the unit step response signal. The EO algorithm aims to determine the optimal coefficients of integer-order transfer functions by minimizing the error between general parametric quadratic model and objective data. The objective data are unit step response of fractional-order transfer functions and obtained by using the Grünwald-Letnikov (GL) method in the Fractional-Order Modeling and Control (FOMCON) toolbox. Thus, the coefficients of an integer-order transfer function most properly can be determined. Some examples are provided based on different fractional-order transfer functions to evaluate the performance of the proposed method. The proposed method is compared with studies from the literature in terms of time and frequency responses. It is seen that the proposed method exhibits better model approximation performance and provides a lower order model.

Design and Robust Performance Analysis of Low-Order Approximation of Fractional PID Controller Based on an IABC Algorithm for an Automatic Voltage Regulator System

In this paper, a low-order approximation (LOA) of fractional order PID (FOPID) for an automatic voltage regulator (AVR) based on the modified artificial bee colony (ABC) is proposed. The improved artificial bee colony (IABC) high-order approximation (HOA)-based fractional order PID (IABC/HOA-FOPID) controller, which is distinguished by a significant order approximation and by an integer order transfer function, requires the use of a large number of parameters. To improve the AVR system’s performance in terms of transient and frequency response analysis, the memory capacity of the IABC/HOA-FOPID controller was lowered so that it could fit better in the corrective loop. The new robust controller is named the improved artificial bee colony (IABC) low-order approximation (LOA)-based fractional order PID (IABC/LOA-FOPID). The performance of the proposed IABC/LOA-FOPID controller was compared not only to the original ABC algorithm-tuned PID controller, but also to other controllers tuned by state-of-the-art meta-heuristic algorithms such as the improved whale optimization algorithm (IWOA), particle swarm optimization (PSO), cuckoo search (CS), many optimizing liaisons (MOL), genetic algorithm (GA), local unimodal sampling (LUS), and the tree seed algorithm (TSA). Step response, root locus, frequency response, robustness test, and disturbance rejection abilities are all compared. The simulation results and comparisons with the proposed IABC/LOA-FOPID controller and other existing controllers clearly show that the proposed IABC/LOA-FOPID controller outperforms the optimal PID controllers found by other algorithms in all the aforementioned performance tests.

Approximation and realization of power-law all-pass filters

Non-integer power-law all-pass transfer functions, are approximated by suitable integer-order transfer functions in this work. The derivation of the integer-order transfer functions is based on the analytic expansion of the (non-integer) power-law transfer functions through the utilization of the binomial theorem. The offered benefit is the derivation of stable integer-order transfer functions. This study is supported by experimental results, obtained using a Field Programmable Analog Array device, after the employment of curve fitting based approximation techniques.

Versatile Field-Programmable Analog Array Realizations of Power-Law Filters

A structure suitable for implementing power-law low-pass and high-pass filter transfer functions is presented in this work. Through the utilization of a field-programmable analog array device, full programmability of the characteristics of the intermediate stages, as is required for realizing the rational integer-order transfer function that approximates the corresponding power-law function, was achieved, making the structure versatile. In addition, a comparison between power-law and fractional-order filters regarding the effect of the non-integer order was performed. The presented design examples are fully supported by experimental results.

Design of Low-Voltage FO-[PD] Controller for Motion Systems

Fractional-order controllers have gained significant research interest in various practical applications due to the additional degrees of freedom offered in their tuning process. The main contribution of this work is the analog implementation, for the first time in the literature, of a fractional-order controller with a transfer function that is not directly constructed from terms of the fractional-order Laplacian operator. This is achieved using Padé approximation, and the resulting integer-order transfer function is implemented using operational transconductance amplifiers as active elements. Post-layout simulation results verify the validity of the introduced procedure.

Power-Law Compensator Design for Plants with Uncertainties: Experimental Verification

A power-law compensator scheme for achieving robust frequency compensation in control systems including plants with an uncertain pole, is introduced in this work. This is achieved through an appropriate selection of the compensator parameters, which guarantee that the Nyquist diagram of the open-loop system compensator-plant crosses a fixed point independent of the plant pole variations. The implementation of the fractional-order compensator is performed through the utilization of a curve-fitting-based technique and the derived rational integer-order transfer function is realized on a Field-Programmable Analog Array device. The experimental results confirm that the the phase margin is well preserved, even for ±40% variation in the pole location of the plant.

Fractional-Order and Power-Law Shelving Filters: Analysis and Design Examples

Low-pass and high-pass non-integer shelving filter designs, which are suitable for acoustic applications, are presented in this work. A first design is based on a standard fractional-order bilinear transfer function, while a second one is based on the transposition of the integer-order transfer function into its power-law counterpart. Both transfer functions are approximated using the Oustaloup approximation tool, while the implementation in the case of the power-law filters is performed through the employment of the concept of driving-point impedance synthesis. An attractive benefit is the extra degree of freedom, resulting from the variable order of both fractional-order and power-law filters, which allows improved design flexibility compared to the case of integer-order filters. From the implementation point of view, only one building block is required to realize both filter types, thanks to the employment of the Voltage Conveyor.

Power law filters: A new class of fractional-order filters without a fractional-order Laplacian operator

A new category of fractional-order filters, realized without employing a fractional-order Laplacian operator is introduced in this work. This can be achieved through the utilization of an efficient curve fitting method which approximates the frequency-domain behavior of the filter and transposes the fractional-order transfer function into the integer-order domain. Thus, the procedure results in a rational integer-order transfer function and its implementation is possible using conventional integer-order realization techniques. Therefore, there is no need for fractional-order elements to realize this class of filters. Design examples of this new kind of filters are presented with the derived simulation and experimental results confirming their correct performance.

Analysis and investigation of CDBA based fractional-order filters

In this work, a new design of continuous time current differencing buffered amplifier based low-pass, high-pass, band-pass, all-pass and notch fractional-order filters is reported. The design of proposed filters is based on the approximation of fractional-order filters by using an appropriate integer order transfer function. Signal flow graph approach is used for the realization of fractional-order filters of order (1 + α). The frequency responses of the proposed circuits are verified using MATLAB in conjunction with SPICE. The evaluation of the realized fractional-order filters are also performed through the AC analysis and corner analysis. Furthermore, stability and sensitivity investigations are also investigated. It is observed from the simulation results that the fractional-order filters are appropriate for IC implementation.

An Integer-Order Transfer Function Estimation Algorithm for Fractional-Order PID Controllers

Fractional-order systems and controllers have been extensively used in many control applications to achieve robust modeling and controlling performance. To implement these systems, curve fitting based integer-order transfer function estimation techniques namely Oustaloup and Matsuda are most widely used. However, these methods are failed to achieve the best approximation due to the limitation of the desired frequency range. Thus, this article presents a simple curve fitting based integer-order transfer function estimation method for fractional-order differentiator/integrator using frequency response. The advantage of this technique is that it is simple and can fit the entire desired frequency range. Using the approach, an approximation table for fractional-order differentiator has also been obtained which can be used directly to obtain the approximation of fractional-order systems. A simulation study on fractional systems shows that the proposed approach produced better parameter approximation for the desired frequency as compared to Oustaloup, refined Oustaloup and Matsuda techniques.

Employment of the Padé approximation for implementing fractional-order lead/lag compensators

Fractional-order lead/lag compensator realizations, using Operational Transconductance Amplifiers as active blocks, are presented in this paper. Two different types of fractional-order transfer functions, derived from the integer-order lead/lag compensator transfer function, are used to describe the behavior of the fractional-order compensator. Both types are approximated using the Padé approximation tool, and are expressed by the same form of rational integer-order transfer function. This means that both types can be implemented using the same active core, which is an important advantage from the design versatility point of view. From the design flexibility point of view, the attractive feature is that both types offer electronic adjustment of their characteristics through the appropriate tuning of the DC bias currents. Simulation results, derived using the Cadence IC design suite and the design kit of the Austria Mikro Systeme 0.35μm CMOS process, are used for evaluating the performance of the presented designs.

Comparative study of nature-inspired algorithms to design (1+α) and (2+α)-order filters using a frequency-domain approach

A precise control in the stopband attenuation characteristics can be achieved by using the fractional-step filters instead of the traditional integer-order filters. In this paper, nine nature-inspired optimization algorithms, such as five advanced variants of differential evolution (DE), three advanced variants of particle swarm optimization (PSO), and an efficient evolutionary strategy method (CMA-ES-RIS) are employed to design the fractional-step low pass Butterworth filter (FLBF). The proposed ( 1 + α ) and ( 2 + α ) order models, where α ∈ ( 0 , ​ 1 ) , are optimally approximated as an integer-order transfer function by using the magnitude-frequency information of the ideal FLBF. Comparisons regarding the solution quality and robustness reveal an improved accuracy for the DE variants and CMA-ES-RIS over all the PSO variants. Results from the pair-wise Wilcoxon rank-sum test demonstrate the superiority of enhanced fitness-adaptive differential evolution (EFADE) algorithm as the most efficient optimization tool for solving this problem. Comparisons with the state-of-the-art approaches also confirm the superior modelling accuracy of the proposed FLBFs. The canonical structure circuit realization of the FLBFs using current feedback operational amplifiers is presented. Simulations carried out in OrCAD PSPICE platform suggest proximity in the magnitude responses between the proposed and the theoretical models. The optimal design of stable, minimum-phase ( 2 + α ) order FLBFs is also presented for the first time without employing the cascading concept involving the integer-order Butterworth polynomials.

Optimal rational approximation of bandpass Butterworth filter with symmetric fractional-order roll-off

In this paper, a fractional-order bandpass Butterworth filter (FBPBF) exhibiting symmetric roll-off characteristic is approximated as an integer-order transfer function using a metaheuristic optimization approach. A novel cost function is proposed by considering the exact FBPBF magnitude characteristic obtained through cascading the (n+α), where n is an integer and 0<α<1, order lowpass and highpass Butterworth filter responses. Design examples considering various values of n and α are presented to illustrate the performance efficacy of the proposed approach. PSPICE simulations for the models realized using the current feedback operational amplifiers confirm proximity to the theoretical characteristics.

Optimal approximation of fractional‐order systems with model validation using CFOA

Optimal integer-order transfer function approximation of linear time-invariant fractional-order systems (FOS) using a straight-forward approach is proposed here. The error between the rational approximant and the FOS (commensurate/non-commensurate/oscillatory types etc.) is minimised directly in the frequency domain, subject to design stability constraints, by using the colliding bodies optimisation (CBO) algorithm. Comparisons with other well-known metaheuristics regarding the computational time and the design effort expended in parameter tuning justify the efficiency of the proposed method. The CBO-optimised models achieve a superior solution quality regarding both the magnitude and phase responses in comparison to the reported literature. Analogue circuit implementations of the models using the current feedback operational amplifier are also presented. Simulations carried out in OrCAD PSPICE environment validate the practical efficacy of the proposed models.

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.

Asymptotic magnitude Bode plots of fractional-order transfer functions

Development of asymptotic magnitude Bode plots for integer-order transfer functions is a well-established topic in the control theory. However, construction of such plots for the fractional-order transfer functions has not received much attention in the existing literature. In the present paper, we investigate in this direction and derive the procedures for sketching asymptotic magnitude Bode plots for some of the popular fractional-order controllers. In addition, we deduce these plots for general fractional commensurate-order transfer functions as well. As applications of this work, we illustrate the analysis of the designed fractional-control loop, and the identification of fractional-order transfer function from a given plot.

Optimal designs of analogue and digital fractional order filters for signal processing applications

Our initial research work was focussed on employing metaheuristic optimization techniques to design optimal digital signal processing (DSP) systems such as the full band, conventional infinite impulse response differentiators and integrators meeting the accurate magnitude responses with a smaller average group delay. Since, integer order systems are a tight subset of the fractional order (FO) systems the research has been extended towards the optimal design of FO differentiators and integrators in the discrete domain with improved frequency response performances. Specific emphasis was laid on the feasibility of the designed FO differentiators in controlling a double-integrator plant. Analogue Butterworth filter with fractional stepping in the transition band has also been realized using an optimal integer-order rational transfer function. In future, we intend to design and implement generalized fractional-order filters on FPGA/DSP kit and FPAAs.