Time domain design of fractional differintegrators using least-squares

In this paper we propose the use of the least-squares based methods for obtaining digital rational approximations (IIR filters) to fractional-order integrators and differentiators of type sα, α∈R. Adoption of the Padé, Prony and Shanks techniques is suggested. These techniques are usually applied in the signal modeling of deterministic signals. These methods yield suboptimal solutions to the problem which only requires finding the solution of a set of linear equations. The results reveal that the least-squares approach gives similar or superior approximations in comparison with other widely used methods. Their effectiveness is illustrated, both in the time and frequency domains, as well in the fractional differintegration of some standard time domain functions

Significance of Weighted-Type Fractional Fourier Transform in FIR Filters

The desired frequency response of a filter is periodic in frequency and can be expanded in Fourier series. One possible way of obtaining FIR filter is to truncate the infinite Fourier series. But abrupt truncation of the Fourier series results in oscillation in the pass band and stop band. These oscillations are due to slow convergence of the Fourier series by the Gibb's phenomenon. To reduce these oscillations the Fourier coefficients of the filter are modified by multiplying the infinite impulse response with a finite weighing sequence called a window. The Fourier transform (FT) of a window consists of a central lobe and side lobes. The central lobe contains most of the energy of the window. To get an FIR filter, the desired impulse response and window function are multiplied, which results to give finite length non-causal sequence. Since Fractional Fourier Transform (FrFT) is generalization of FT. Here an attempt is to implement filters using window by using Weighted Type Fractional Fourier Transform (WFrFt), differentiator and integrator using weighted FrFt is also present

Electronic realization of the fractional-order systems

This article is devoted to the electronic (analogue) realization of the fractional-order systems – controllers or controlled objects whose we earlier used, identified, and analyzed as a mathematical models only ��� namely a fractional-order differential equation, and solved numerically using a method based on the truncated version of the Grunwald - Letnikov formula for fractional derivative. The electronic realization of the fractional derivative is based on the continued fraction expansion of the rational approximation of the fractional differentiator from which we obtained the values of the resistors and capacitors of the electronic circuit. Along with the mathematical description are presented also simulation and measurement results

Symbolic Representation for Analog Realization of A Family of Fractional Order Controller Structures via Continued Fraction Expansion

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.This paper uses the Continued Fraction Expansion (CFE) method for analog realization of fractional order differ-integrator and few special classes of fractional order (FO) controllers viz. Fractional Order Proportional-Integral-Derivative (FOPID) controller, FO[PD] controller and FO lead-lag compensator. Contemporary researchers have given several formulations for rational approximation of fractional order elements. However, approximation of the controllers studied in this paper, due to having fractional power of a rational transfer function, is not available in analog domain; although its digital realization already exists. This motivates us for applying CFE based analog realization technique for complicated FO controller structures to get equivalent rational transfer functions in terms of the controller tuning parameters. The symbolic expressions for rationalized transfer function in terms of the controller tuning parameters are especially important as ready references, without the need of running CFE algorithm every time and also helps in the synthesis of analog circuits for such FO controllers

DESIGN OF FIRST ORDER DIFFERENTIATOR WITH PARALLEL ALL-PASS STRUCTURE

In this paper a new method for design of the first order differentiator is presented. The proposed differentiator consists of two parallel branches, i.e. direct path and IIR all-pass filter. The described design method allows one to obtain solution with minimum mean relative error at the desired region by controlling the ratio of phase response extremes. A small relative magnitude error, as well as a low phase error, at low frequencies is condition for good time domain behaviour. The obtained differentiator can be realized by means of only two multipliers, hence being a good choice for real time applications. The proposed solution provides a lower magnitude error than several known differentiators with similar phase error

Significance of Weighted-Type Fractional Fourier Transform in FIR Filters

The desired frequency response of a filter is periodic in frequency and can be expanded in Fourier series. One possible way of obtaining FIR filter is to truncate the infinite Fourier series. But abrupt truncation of the Fourier series results in oscillation in the pass band and stop band. These oscillations are due to slow convergence of the Fourier series by the Gibb’s phenomenon. To reduce these oscillations the Fourier coefficients of the filter are modified by multiplying the infinite impulse response with a finite weighing sequence called a window. The Fourier transform (FT) of a window consists of a central lobe and side lobes. The central lobe contains most of the energy of the window. To get an FIR filter, the desired impulse response and window function are multiplied, which results to give finite length non-causal sequence. Since Fractional Fourier Transform (FrFT) is generalization of FT. Here an attempt is to implement filters using window by using  Weighted Type Fractional Fourier Transform (WFrFt),  differentiator and integrator using weighted FrFt is also present

Comparison of the methods for discrete approximation of the fractional-order operator

In this paper we will present some alternative types of discretization methods (discrete approximation) for the fractional-order (FO) differentiator and their application to the FO dynamical system described by the FO differential equation (FDE). With analytical solution and numerical solution by power series expansion (PSE) method are compared two effective methods - the Muir expansion of the Tustin operator and continued fraction expansion method (CFE) with the Tustin operator and the Al-Alaoui operator. Except detailed mathematical description presented are also simulation results. From the Bode plots of the FO differentiator and FDE and from the solution in the time domain we can see, that the CFE is a more effective method according to the PSE method, but there are some restrictions for the choice of the time step. The Muir expansion is almost unusable