Optimal analog‐to‐digital transformation of fractional‐order Butterworth filter using binomial series expansion with Al‐Alaoui operator

This paper deals with the optimal analog‐to‐digital transformation of fractional‐order Butterworth filter (FOBF) in terms of infinite impulse response templates. The fractional‐order transfer function of the analog FOBF is transformed into its digital counterpart by employing the Binomial series expansion of different truncation orders, based on the Al‐Alaoui operator. This nonoptimal solution is then treated as an initial point for a local search optimizer such as the Nelder–Mead simplex (NMS) algorithm and also injected as a super‐fit individual in the initial population of a global search constrained evolutionary optimization algorithm (CEOA). Design stability and minimum‐phase response constraints are formulated for the super‐fit scheme. Both the techniques demonstrate good modeling performance; however, the super‐fit CEOA can markedly outperform the NMS method as the problem dimensionality increases.     

Design and performance study of phase‐locked loop using fractional‐order loop filter

The present work reports the realization of an analog fractional‐order phase‐locked loop (FPLL) using a fractional capacitor. The expressions for bandwidth, capture range, and lock range of the FPLL have been derived analytically and then compared with the experimental observations using LM565 IC. It has been observed that bandwidth and capture range can be extended by using FPLL. It has also been found that FPLL can provide faster response and lower phase error at the time of switching compared to its integer‐order counterpart. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal analog active filter design using craziness‐based particle swarm optimization algorithm

Because of the manufacturing constraints, the optimal selection of passive component values for the design of analog active filter is very critical. As the search on possible combinations in preferred values for capacitors and resistors is an exhaustive process, it has to be automated with high accuracy within short computation time. Evolutionary computation may be an attractive alternative for automatic selection of optimal discrete component values such as resistors and capacitors for analog active filter design. This paper presents an efficient evolutionary optimization approach for optimal analog filter design considering different topologies and manufacturing series by selecting their component values. The evolutionary optimization technique employed is craziness‐based particle swarm optimization (CRPSO). PSO is very simple in concept, easy to implement and computationally efficient algorithm with two main advantages: fast convergence and only a few control parameters. However, the performance of PSO depends on its control parameters and may be influenced by premature convergence and stagnation problem. To overcome these problems, the PSO algorithm has been modified to CRPSO and is used for the selection of optimal passive component values of fourth‐order Butterworth low‐pass analog active filter and second‐order state variable low‐pass filter, respectively. CRPSO performs the dual task of efficiently selecting the component values as well as minimizing the total design errors of low‐pass active filters. The component values of the filters are selected in such a way so that they become E12/E24/E96 series compatible. The simulation results prove that CRPSO efficiently minimizes the total design error with respect to previously used optimization techniques. Copyright © 2014 John Wiley & Sons, Ltd.     

Experimental studies on realization of fractional inductors and fractional‐order bandpass filters

This paper presents the hardware realization and performance study of fractional inductors of order 0 < α < 2. The fractional inductors used in this work have been realized with the help of general impedance converter circuit and fractional capacitors. Impedance characterization of fractional inductors with different exponents has been carried out experimentally. Also a generalized approach to design a fractional‐order bandpass filter is discussed in this work. The fractional‐order bandpass filter consists of a series combination of a resistor, a fractional inductor of order 1 < α < 2, and a fractional capacitor of order 0 < β < 1. The performance of fractional‐order bandpass filters has been studied and compared with corresponding integer‐order filters through both experimentation and simulation. Copyright © 2014 John Wiley & Sons, Ltd.     

一种FIR高阶多通带数字滤波器的优化设计方法

为有效提高有限冲激响应FIR(FiniteImpulseResponse)高阶数字滤波器优化设计速度和精度,根据FIR线性相位滤波器的幅频特性,提出了一种基于激励矩阵为Hd-CTW的神经网络算法。该算法的主要思想是用神经网络算法优化设计的FIR滤波器的幅频特性与理想滤波器的幅频特性在整个通带和阻带范围内的误差平方和为最小,算法不涉及逆矩阵运算。为了保证该算法的收敛性,提出并证明了神经网络算法的收敛性定理,为神经网络学习率的选择提供了理论依据。该算法的主要特点是可实现样本集数据的并行训练,有效提高了计算速度。为了验证该算法的有效性,给出了多通带FIR高阶数字滤波器优化设计实例,仿真结果表明了该算法具有高的计算精度和快的计算速度。     

0.5‐V fractional‐order companding filters

Novel configurations of fractional‐order filter topologies, realized through the employment of the concept of companding filtering, are introduced in this paper. As a first step, the design procedure is presented in a systematic algorithmic way, while in the next step, the basic building blocks of sinh‐domain and log‐domain integrators are presented. Because of the employment of metal–oxide–semiconductor (MOS) transistors operated in the subthreshold region, the derived filter structures offer the capability for operation in an ultra‐low‐voltage environment. In addition, because of the offered resistorless realizations, the proposed topologies are reconfigurable, in the sense that the order of the filter could be chosen through appropriate bias current sources. The performance of the derived fractional‐order filters has been evaluated through simulation and comparison results using the Analog Design Environment of the Cadence software and MOS transistor parameters provided by the Taiwan Semiconductor Manufacturing Company (TSMC) 180‐nm complementary MOS (CMOS) process. Copyright © 2014 John Wiley & Sons, Ltd.     

Design of unknown‐input reduced‐order observers for a class of nonlinear fractional‐order time‐delay systems

This paper considers the design of reduced‐order state observers for fractional‐order time‐delay systems with Lipschitz nonlinearities and unknown inputs. By using the Razumikhin stability theorem and a recent result on the Caputo fractional derivative of a quadratic function, a sufficient condition for the asymptotic stability of the observer error dynamic system is presented. The stability condition is obtained in terms of linear matrix inequalities, which can be effectively solved by using existing convex algorithms. Numerical examples and simulation results are given to illustrate the effectiveness of the proposed design approach.     

Optimal approximation of asymmetric type fractional-order bandpass Butterworth filter using decomposition technique

In this letter, the optimal rational approximation of fractional-order bandpass Butterworth filter (FOBBF) is presented. The transfer function of the FOBBF is decomposed into a multiplication of first-order and second-order terms. As a result, the design stability conditions can be easily satisfied using only the variable boundary constraints. The proposed technique generalizes the symmetric fractional-order roll-off characteristic as only a special case of the asymmetric one. Several examples are presented to validate the modeling efficacy.     

Design equations for fractional‐order sinusoidal oscillators: Four practical circuit examples

Four practical sinusoidal oscillators are studied in the general form where fractional‐order energy storage elements are considered. A fractional‐order element is one whose complex impedance is given by Z = a(jω)^±α, where a is a constant and α is not necessarily an integer. As a result, these oscillators are described by sets of fractional‐order differential equations. The integer‐order oscillation condition and oscillation frequency formulae are verified as special cases. Numerical and PSpice simulation results are given. Experimental results are also reported for a selected Wien‐bridge oscillator. Copyright © 2007 John Wiley & Sons, Ltd.     

Oscillation and convergence behaviours exhibited in an ‘unstable’ second‐order digital filter with saturation‐type non‐linearity

This letter explains the oscillatory behaviours exhibited in a second‐order digital filter with saturation‐type non‐linearity via the Hopf bifurcation theorem. It is shown that depending on the bifurcation parameter, the state variables may converge to zero even when the eigenvalues of the system matrix are outside the unit circle. Copyright © 2004 John Wiley & Sons, Ltd. abstract     

On the design of multiband transmission functions synthesized by one wave digital lattice structure

In this contribution, the design of multiband transmission functions is considered. Independent and arbitrary number of bands can be designed. Moreover, the whole transmission function is synthesized by one wave digital lattice structure. The approximation process starts by extracting the scattering matrix properties of multiband reference lattice structures. Consequently, the approximation problem reduces to generating a polynomial Q of degree n, which is the degree of the filter. The degree n is depending on the number of the designed bands. Hence, if the number of bands is even, n will be odd, and if the number of bands is odd, n will be even. The polynomial Q will approximate the difference phase function of the two branch polynomials. It is composed of two subpolynomials, one of them is Hurwitz and the other is anti‐Hurwitz. The degrees of these subpolynomials differ by odd number if the number of bands is even and differ by even number if the number of bands is odd. Q is generated according to iterative interpolation and using explicit recursive formulas. After obtaining Q, the two subpolynomials are calculated and the two branch all‐pass functions are constructed. Consequently, the filter is synthesized in the digital frequency domain. The method is applied through an illustrative example. Copyright © 2011 John Wiley & Sons, Ltd.     

Autonomous response of a third‐order digital filter with two's complement arithmetic realized in cascade form

In this letter, results on the autonomous response of a third‐order digital filter with two's complement arithmetic realized as a first‐order subsystem cascaded by a second‐order subsystem are reported. The behaviour of the second‐order subsystem depends on the pole location and the initial condition of the first‐order subsystem, because the transient behaviour is affected by the first‐order subsystem and this transient response can be viewed as an excitation of the original initial state to another state. New results on the set of necessary and sufficient conditions relating the trajectory equations, the behaviours of the symbolic sequences and the sets of the initial conditions are derived. The effects of the pole location and the initial condition of first‐order subsystem on the overall system are discussed. Some interesting differences between the autonomous response of the second‐order subsystem and the response due to the exponentially decaying input are reported. Some simulation results are given to illustrate the analytical results. Copyright © 2004 John Wiley & Sons, Ltd.     

Ultra‐low voltage fractional‐order circuits using current mirrors

Fractional‐order blocks, including differentiators, lossy and lossless integrators as well as filters of order 1 + a (0 < a < 1), are presented in this paper. The proposed topologies offer the benefit of ultra low‐voltage operation; in addition, reduced circuit complexity is achieved compared to the corresponding companding schemes, which have been already introduced in the literature. The ultra‐low voltage operation is performed through the employment of metal oxide semiconductor transistors biased in the subthreshold region. The reduction of circuit complexity is achieved through the utilization of current mirrors as active elements for realizing the required building blocks. The performance of the proposed fractional‐order circuits has been evaluated through the Analog Design Environment of the Cadence software and the design kit provided by the Taiwan Semiconductor Manufacturing Company (TSMC) 180 nm complementary metal oxide semiconductor process. Copyright © 2015 John Wiley & Sons, Ltd.     

Adaptive synchronization of fractional‐order complex networks via pinning control

Adaptive synchronization of a class of fractional‐order complex networks is investigated in this paper. On the basis of the fractional‐order system stability theory, adaptive synchronization criteria of fractional‐order complex networks with 0 < q < 1 is achieved. Furthermore, pinning control method is then suggested to control the networks, and adaptive strategy is employed to tune the control gains and coupling strength. Because the nodes with high degree may not be the center of the networks, a new attempt to choose the pinned nodes on the basis of the closeness centrality scheme is proposed. Finally, numerical simulations are given to verify the effectiveness of the proposed approach based on the closeness centrality scheme. Copyright © 2012 John Wiley & Sons, Ltd.     

Pinning synchronization of fractional‐order complex networks with adaptive coupling weights

This work studies the issue of synchronization control for a type of fractional‐order complex networks, in which the adaptive coupling matrix is considered under the directed topology structure. A pinning control strategy, with the free selection of pinning nodes, is adopted for the synchronization goal. Then, by absorbing the information of eigenvectors and adaptive laws for the coupling matrix, a new Lyapunov function is constructed, by which, and with the assistance of Gronwall inequality and network features, the sufficient condition for Mittag‐Leffler synchronization of the fractional‐order network is established. Accordingly, an easy verifiable algebraic criterion is further derived by means of some matrix inequalities. Besides, we also discuss the effect of outer coupling strength on the achievement of network synchronization. Finally, a numerical experiment is performed to show the evidence of the correctness and effectiveness of the proposed results.     

An expression for the voltage response of a current‐excited fractance device based on fractional‐order trigonometric identities

We report a closed‐form expression of the voltage response of a current‐excited fractance device. The derived simple expression is made possible following the introduction of the generalized sine and cosine functions (rmsin_α(t) and cos_α(t)), which are valid on any fractional‐order surface and tend to the normal sin(t) and cos(t) at α = 1 or asymptotically as t→∞. Copyright © 2011 John Wiley & Sons, Ltd.     

Mittag‐Leffler state estimator design and synchronization analysis for fractional‐order BAM neural networks with time delays

This paper deals with the extended design of Mittag‐Leffler state estimator and adaptive synchronization for fractional‐order bidirectional associative memory neural networks with time delays. By the aid of Lyapunov direct approach and Razumikhin‐type method, a suitable fractional‐order Lyapunov functional is constructed and a new set of novel sufficient condition are derived to estimate the neuron states via available output measurements such that the ensuring estimator error system is globally Mittag‐Leffler stable. Then, the adaptive feedback control rule is designed, under which the considered FBNNs can achieve Mittag‐Leffler adaptive synchronization by means of some fractional‐order inequality techniques. Moreover, the adaptive feedback control may be utilized even when there is no ideal information from the system parameters. Finally, two numerical simulations are given to reveal the effectiveness of the theoretical consequences.     

Composite learning fuzzy synchronization for incommensurate fractional‐order chaotic systems with time‐varying delays

This paper presents a composite learning fuzzy control to synchronize two different uncertain incommensurate fractional‐order time‐varying delayed chaotic systems with unknown external disturbances and mismatched parametric uncertainties via the Takagi‐Sugeno fuzzy method. An adaptive controller together with fractional‐order composite learning laws is designed based on both a parallel distributed compensation technology and a fractional Lyapunov criterion. The boundedness of all variables in the closed‐loop system and the Mittag‐Leffler stability of tracking error can be guaranteed. T‐S fuzzy systems are provided to tackle unknown nonlinear functions. The distinctive features of the proposed approach consist in the following: (1) a supervisory control law is designed to compensate the lumped disturbances; (2) both the prediction error and the tracking error are used to estimate the unknown fuzzy system parameters; (3) parameter convergence can be ensured by an interval excitation condition. Finally, the feasibility of the proposed control strategy is demonstrated throughout an illustrative example.     

Optimal design of symmetric fractional delay filter using firefly algorithm

A fractional delay filter is used to increase the accuracy, preciseness, time synchronization, and stability of signal processing system. However, designing a fractional delay filter for a specified delay, without affecting spectral characteristics of the signal is challenging because of nondifferentiability and multimodal nature of its objective function. In this paper, a more accurate design technique has been proposed for designing fractional delay filters, based on a recently developed firefly algorithm and its improved version. The designed filters offer variable fractional delay. A novel symmetric structure of implementation has been used to design filters. The efficacy of the proposed technique is evaluated by considering a filter design example. The performance of the proposed technique is compared with the other exiting algorithm. The comparative analysis of finite impulse response (FIR) fractional delay filter design proves that the proposed algorithm has a smaller design error and an implementation complexity than the other reported existing algorithms. In addition to this, the designed FIR fractional delay filter is implemented on Xilinx Virtex-7 for experimental validation.