Al-Alaoui Operator Research Articles

A new approach to optimal design of digital fractional-order PIλDμ controller

Fractional-order PIλDμ controller has been under intensive investigations for better control performance in recent years. To facilitate the implementation of digital controller, the PIλDμ controller is usually firstly optimally designed in continuous domain, and then discretized to obtain a digital PIλDμ controller. However, the aforementioned methodology may lead to performance deterioration and other undesirable results. To tackle the issues identified above, a new approach is proposed to directly design the optimal digital PIλDμ controller, which is formulated to be a controller optimization problem with six tunable parameters by using the parameterized Al-Alaoui operator for discretization. To optimize the digital PIλDμ controller, an improved PSO-SADV algorithm with switching adaptive delayed velocities has been developed to deal with some frequently encountered obstacle phenomena in the optimization problem, and the superiorities of the PSO-SADV are verified by several comparative simulation experiments. Finally, the PSO-SADV is employed to handle the parameter optimization issue of the digital PIλDμ controller based on the ITSE index of step response. Several illustrative simulation experiments are carried out upon some typical controlled objects for the comparison with the PIλDμ controller and its discretizations, and the effectiveness and superiority of the new approach can be confirmed by the simulation results.

Fractional-order proportional integral controller based on Al-Alaoui operator for resonant optical gyro

The first- and second-order fractional-order proportional integral (FOPI) controllers based on the Al-Alaoui operator and the continued fraction expansion method are designed and applied to the closed-loop control system of a resonant optical gyro. The phase margin method is used to tune the control parameters. The characteristics of the unit step responses of the integer order proportional integral (IOPI) and the FOPI controllers are calculated and compared. Responses to perturbations at different positions of the closed-loop control system are simulated. Results show that for the resonant optical gyro, the FOPI closed-loop control system has much better unit-step-response performance and noise-resistance ability than the IOPI control system, which is important for improving the dynamic response characteristics and the robustness of the gyro’s feedback system and enhancing the detection ability of the gyro. Experiments on a resonant optical gyro experimental system also verify the advantages of the FOPI controller in step response and bias stability.

Discretization of Fractional Order Differentiator and Integrator with Different Fractional Orders

This paper is concerned with the discretization of the fractional-order differentiator and integrator, which is the foundation of the digital realization of fractional order controller. Firstly, the parameterized Al-Alaoui transform is presented as a general generating function with one variable parameter, which can be adjusted to obtain the commonly used generating functions (e.g. Euler operator, Tustin operator and Al-Alaoui operator). However, the following simulation results show that the optimal variable parameters are different for different fractional orders. Then the weighted square integral index about the magtitude and phase is defined as the objective functions to achieve the optimal variable parameter for different fractional orders. Finally, the simulation results demonstrate that there are great differences on the optimal variable parameter for differential and integral operators with different fractional orders, which should be attracting more attentions in the design of digital fractional order controller.

Efficient Design of Discrete Fractional-Order Differentiators Using Nelder–Mead Simplex Algorithm

Applications of fractional-order operators are growing rapidly in various branches of science and engineering as fractional-order calculus realistically represents the complex real-world phenomena in contrast to the integer-order calculus. This paper presents a novel method to design fractional-order differentiator (FOD) operators through optimization using Nelder---Mead simplex algorithm (NMSA). For direct discretization, Al-Alaoui operator has been used. The numerator and the denominator terms of the resulting transfer function are further expanded using binomial expansion to a required order. The coefficients of z-terms in the binomial expansions are used as the starting solutions for the NMSA, and optimization is performed for a minimum magnitude root-mean-square error between the ideal and the proposed operator magnitude responses. To demonstrate the performance of the proposed technique, six simulation examples for fractional orders of half, one-third, and one-fourth, each approximated to third and fifth orders, have been presented. Significantly improved magnitude responses have been obtained as compared to the published literature, thereby making the proposed method a promising candidate for the design of discrete FOD operators.

New improved fractional order integrators using PSO optimisation

This article investigates the optimal results of new improved fractional order integrators (FOIs) of different orders. Mathematical models of FOIs have been first developed by a single-step procedure of direct linear interpolation of fractional integrators based on Al-Alaoui operator in fractional domain itself, instead of using three steps of the well-known conventional method, namely, digital interpolation, series expansion and truncation. Later, these transfer functions (TFs) are optimised for their coefficient values for finding a minimum error function by particle swarm optimisation (PSO) algorithm. Simulation results of magnitude responses, phase responses and relative magnitude errors (dB) for all the proposed half integrators have validated the effectiveness of this new technique of interpolation of fractional order operators, mixed with PSO algorithm. A parallel comparison has been also drawn between the proposed optimised half integrators and those obtained by discretisation of PSO optimised integer order digital integrators (DIs) to properly support the proposed novel combination of interpolation and PSO, both applied together in fractional domain.

Design of fractional order integrators and differentiators using novel rational approximations

This paper contributes a relatively broader realm of new improved rational approximations based on Halley’s iterative method incorporating different orders of one-half, one-third and one-fourth fractional order integrators (FOIs) and fractional order differentiators (FODs) when conceived by existing first and higher order s-to-z transformations for indirect discretisation. The proposed approach has been observed to gear up towards more efficient discretised models when compared with those of existing well established approximation techniques namely continued fraction expansion (CFE) and power series expansion (PSE). Responses of proposed models of FOIs/differentiators based on Al-Alaoui operator and new optimised four segment operator have been reported to have relative magnitude error (dB) as low as –40 dB and linear phase curves in almost full band of normalised frequency. FOIs and FODs based on Tustin operator have presented constant phase response in full range resulting in tremendous improvement over phase responses of its respective existing models.

Design of Improved Fractional Order Integrators using Indirect Discretization Method

paper has adopted rational approximation based on Regular Newton method, for frequency domain fitting of transfer functions of fractional order integrators (FOIs). Further, different discretized mathematical models of one-half, one-third and one-fourth order integrators based on Al-Alaoui operator and New optimized four segment operator have been also developed using these rational approximations by indirect discretization technique. All the proposed models of FOIs are found to be stable when investigated for stability. Simulation results of magnitude responses, phase responses and absolute magnitude errors show that the proposed FOIs obtained by approximations based on Regular Newton method, clearly outperform the other existing approximation techniques which have been used for designing fractional order operators. Results of absolute magnitude errors for all proposed fractional order integrators have been reported to be as low as 0.01, in range 0.35 ≤ ω ≤ 1 π radians of full band of normalized frequency. Among the proposed FOIs, the one-half, one-third and one- fourth order models based on Al-Alaoui operator (for 2 nd iterations) are noticeable with tremendously improved results with absolute magnitude errors of ≤ 0.004 in complete normalized frequency range.

Digital fractional‐order differentiator and integrator models based on first‐order and higher order operators

AbstractIn this paper, new discretized models for fractional‐order differentiator (FOD) (sr) and integrator (FOI) (s−r) using first‐order and higher order operators are proposed. The expansions for FOIs of the first‐order s‐to‐z transform proposed by Hsue et al. are obtained by using the Taylor series and continued fraction expansion techniques. Second‐order Schneider operator and third‐order Al‐Alaoui–Schneider–Kaneshige–Groutage (Al‐Alaoui–SKG) rule have also been fractionalized to obtain expansions of FODs by using the Taylor series. Specifically, in this paper, Hsue operator based on third‐order and fourth‐order models of FOI, Schneider operator as well as Al‐Alaoui–SKG rule based on third‐order, fourth‐order, fifth‐order and sixth‐order models of FOD have been suggested. The stability of the proposed models has been investigated and the unstable ones were stabilized by the pole reflection method. The performance of the proposed models has been compared with that of recent FOD and FOI models based on the Al‐Alaoui operator. Performance results using the proposed discrete‐time formulations are found to converge to the analytical results of FOD and FOI, in the continuous‐time domain. Copyright © 2010 John Wiley & Sons, Ltd.

Implementation of switched capacitor fractional order differentiator (PDδ) circuit

This paper proposes a novel implementation of switched capacitor (SC) fractional order differentiators (FOD) based on Tustin operator and Al-Alaoui operator. The existing models of half differentiator s 1/2 have been expanded using continued fraction expansion and implemented using PSpice. The PSpice simulation results are compared with the theoretical results in the continuous time domain. The results validate the effectiveness of the SC circuit implementation of the proposed approach. A detailed analysis of the influence of the non-idealities of the third order Al-Alaoui operator based half differentiator is performed. The analytical expressions for errors in magnitude and phase due to the above mentioned non-idealities have been derived for the SC integrator and amplifier circuits used in our realisation.

Al-Alaoui operator and the new transformation polynomials for discretization of analogue systems

The “new transformation polynomials for discretization of analogue systems” was recently introduced. The work proposes that the discretization of 1/s n should be done independently rather than by raising the discrete representation of 1/s to the power n. Several examples are given in to back this idea. In this paper it is shown that the “new transformation polynomials for discretization of analogue systems” is exactly the same as the parameterized Al-Alaoui operator. In the following sections, we will show that the same results could be obtained with the parameterized Al-Alaoui operator.

An Alternative Derivation of the Al-Alaoui Operator

It is shown in this letter that the well-known class of first-order digital differentiators and integrators can be simply derived from a classical continuous-time approximate differentiator by using the bilinear transformation. The result obtained represents a simple and alternative method of deriving the Al-Alaoui operator (also known as Al-Alaoui's differentiator, integrator, transform).

An enhanced first-order sigma-delta modulator with a controllable signal-to-noise ratio

This paper presents an enhanced first-order sigma-delta modulator. The proposed modulator, derived using the Al-Alaoui operator, outperforms the conventional modulator. A comparison is drawn showing that the conventional sigma-delta modulator is a special case of the enhanced sigma-delta modulator. The paper includes an analytical derivation as well as extensive simulations revealing the superiority of the proposed modulator reaching optimal performance

Al-Alaoui operator and the α-approximation for discretization analog systems

The a-approximation for discretization of analog systems was recently introduced. In this paper it is shown that the a-approximation is exactly the same as the parameterized Al-Alaoui operator.

Continued Fraction Expansion Approaches to Discretizing Fractional Order Derivatives?an Expository Review

This paper attempts to present an expository review of continued fraction expansion (CFE) based discretization schemes for fractional order differentiators defined in continuous time domain. The schemes reviewed are limited to infinite impulse response (IIR) type generating functions of first and second orders, although high-order IIR type generating functions are possible. For the first-order IIR case, the widely used Tustin operator and Al-Alaoui operator are considered. For the second order IIR case, the generating function is obtained by the stable inversion of the weighted sum of Simpson integration formula and the trapezoidal integration formula, which includes many previous discretization schemes as special cases. Numerical examples and sample codes are included for illustrations.

Discretization schemes for fractional-order differentiators and integrators

For fractional-order differentiator s/sup r/ where r is a real number, its discretization is a key step in digital implementation. Two discretization methods are presented. The first scheme is a direct recursive discretization of the Tustin operator. The second one is a direct discretization method using the Al-Alaoui operator via continued fraction expansion (CFE). The approximate discretization is minimum phase and stable. Detailed discretization procedures and short MATLAB scripts are given. Examples are included for illustration.