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

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

Optimal controllers with complex order derivatives

This paper studies the optimization of complex-order algorithms for the discrete-time control of linear and nonlinear systems. The fundamentals of fractional systems and genetic algorithms are introduced. Based on these concepts, complexorder control schemes and their implementation are evaluated in the perspective of evolutionary optimization. The results demonstrate not only that complex-order derivatives constitute a valuable alternative for deriving control algorithms, but also the feasibility of the adopted optimization strategy

Design of digital differentiators

A digital differentiator simply involves the derivation of an input signal. This work includes the presentation of first-degree and second-degree differentiators, which are designed as both infinite-impulse-response (IIR) filters and finite-impulse-response (FIR) filters. The proposed differentiators have low-pass magnitude response characteristics, thereby rejecting noise frequencies higher than the cut-off frequency. Both steady-state frequency-domain characteristics and Time-domain analyses are given for the proposed differentiators. It is shown that the proposed differentiators perform well when compared to previously proposed filters. When considering the time-domain characteristics of the differentiators, the processing of quantized signals proved especially enlightening, in terms of the filtering effects of the proposed differentiators. The coefficients of the proposed differentiators are obtained using an optimization algorithm, while the optimization objectives include magnitude and phase response. The low-pass characteristic of the proposed differentiators is achieved by minimizing the filter variance. The low-pass differentiators designed show the steep roll-off, as well as having highly accurate magnitude response in the pass-band. While having a history of over three hundred years, the design of fractional differentiator has become a ‘hot topic’ in recent decades. One challenging problem in this area is that there are many different definitions to describe the fractional model, such as the Riemann-Liouville and Caputo definitions. Through use of a feedback structure, based on the Riemann-Liouville definition. It is shown that the performance of the fractional differentiator can be improved in both the frequency-domain and time-domain. Two applications based on the proposed differentiators are described in the thesis. Specifically, the first of these involves the application of second degree differentiators in the estimation of the frequency components of a power system. The second example concerns for an image processing, edge detection application

Digital Filters and Signal Processing

Digital filters, together with signal processing, are being employed in the new technologies and information systems, and are implemented in different areas and applications. Digital filters and signal processing are used with no costs and they can be adapted to different cases with great flexibility and reliability. This book presents advanced developments in digital filters and signal process methods covering different cases studies. They present the main essence of the subject, with the principal approaches to the most recent mathematical models that are being employed worldwide

Iterative design of variable fractional-order IIR differintegrators

[[abstract]]In this paper, the variable fractional-order (VFO) differintegrator is designed based on IIR-typed Farrow structure. The stability of the designed VFO IIR differintegrator is achieved by incorporating a constrained function into the objective error function. But the minimization of the original objective error function is a highly nonlinear problem, so an iterative quadratic method is proposed to overcome it. Comparing with the design based on FIR-typed Farrow structure, several designed examples, including a VFO differintegrator, a pure VFO differentiator and a pure VFO integrator, are presented to demonstrate the effectiveness of the proposed method

Designs of Variable Fractional Delay Digital Filter and Fractional Order Differintegrator

[[abstract]]近年來，由於可調式數位濾波器在系統上自我調整的能力，可調式數位濾波器的設計變成在數位訊號處理裡最重要的分支之一。可調式數位濾波器通常可分成二類。一類是具有可調整振幅響應的濾波器。例如，可調變截止頻率的濾波器以及分數階可調式微分器或積分器。另一類是具有分數延遲之可調式濾波器。在本論文中，我們提出加權最小平方差逼近法來設計可調式數位濾波器。通常，目標誤差能以線性函數來表示時，那麼，我們會直接以加權最小平方差逼近法來求最佳解。相反地，當它是非線性最佳化問題時，我們則使用遞迴二次式的方法。此外，若想將最大目標誤差最小化，則可使用遞迴式的加權最小平方差逼近法的技術。 在本論文中，上述的方法將用在下列之研究： 用遞迴加權式最小平方差逼近法設計分數延遲可調式FIR數位濾波器。（第二章） 一種設計分數延遲可調式FIR數位濾波器之新準則。（第三章） 一種分數延遲可調式FIR數位濾波器之新架構及其設計。（第四章） 用遞迴加權式最小平方差逼近法設計分數延遲可調式Allpass數位濾波器。（第五章） 一種設計分數延遲可調式Allpass數位濾波器的新準則。（第六章） 一種設計分數延遲可調式IIR數位濾波器的新方法。（第七章） 以遞迴式之方法設計分數階可調式FIR數位微積分器。（第八章） 一種分數階可調式寛頻帶FIR數位微分器之新架構及其設計。（第九章） 以遞迴式之方法設計分數階可調式IIR數位微積分器。（第十章）[[abstract]]For the past decade, the design of variable digital filters became one of the most important branches in digital signal processing because of the self-adjustable ability of a variable digital filter online. The variable digital filters are generally classified into two categories. One is the filters with adjustable magnitude response such as the filters with variable cut-off frequencies and the variable fractional-order differentiators/integrators. The other is the filters with variable fractional-delay response. In this dissertation, the weighted least-squares method will be proposed to design variable digital filters. Generally, a general weighted least-squares method can be applied directly to find the optimal solution when the objective error can be formulated in a linear function. On the contrary, when the problem concerns a nonlinear optimization, an iterative quadratic method is applied. Furthermore, if it is desirable to minimize a specified maximum error, the technique of iterative weighted least-squares method will be used which constitutes the inner loop of the overall procedures while the iterative method stated above makes up the outer loop. In this dissertation, the stated method will be applied to the following topics: Minimax design of variable fractional-delay FIR digital filters by iterative weighted least- squares approach (Chapter 2). A new criterion for the design of variable fractional-delay FIR digital filters (Chapter 3). A new structure for the design of variable fractional-delay FIR digital filters (Chapter 4). Minimax phase error design of allpass variable fractional-delay digital filters by iterative weighted least-squares method (Chapter 5). A new method for least-squares and minimax group-delay error design of allpass variable fractional-delay digital filters (Chapter 6). A new method for the design of variable fractional-delay IIR digital filters (Chapter 7). An iterative method for the design of variable fractional-order FIR differintegrators (Chapter 8). A new structure for the design of wideband variable fractional-order FIR differentiators (Chapter 9). Iterative design of variable fractional-order IIR differintegrators (Chapter 10)