Non-asymptotic fractional order differentiators via an algebraic parametric method

Recently, Mboup, Join and Fliess [27], [28] introduced non-asymptotic integer order differentiators by using an algebraic parametric estimation method [7], [8]. In this paper, in order to obtain non-asymptotic fractional order differentiators we apply this algebraic parametric method to truncated expansions of fractional Taylor series based on the Jumarie's modified Riemann-Liouville derivative [14]. Exact and simple formulae for these differentiators are given where a sliding integration window of a noisy signal involving Jacobi polynomials is used without complex mathematical deduction. The efficiency and the stability with respect to corrupting noises of the proposed fractional order differentiators are shown in numerical simulations

Fractional - order system modeling and its applications

In order to control or operate any system in a closed-loop, it is important to know its behavior in the form of mathematical models. In the last two decades, a fractional-order model has received more attention in system identification instead of classical integer-order model transfer function. Literature shows recently that some techniques on fractional calculus and fractional-order models have been presenting valuable contributions to real-world processes and achieved better results. Such new developments have impelled research into extensions of the classical identification techniques to advanced fields of science and engineering. This article surveys the recent methods in the field and other related challenges to implement the fractional-order derivatives and miss-matching with conventional science. The comprehensive discussion on available literature would help the readers to grasp the concept of fractional-order modeling and can facilitate future investigations. One can anticipate manifesting recent advances in fractional-order modeling in this paper and unlocking more opportunities for research

Convergence of Laguerre Impulse Response Approximation for Noninteger Order Systems

One of the most important issues in application of noninteger order systems concerns their implementation. One of the possible approaches is the approximation of convolution operation with the impulse response of noninteger system. In this paper, new results on the Laguerre Impulse Response Approximation method are presented. Among the others, a new proof of convergence of approximation is given, allowing less strict assumptions. Additionally, more general results are given including one regarding functions that are in the joint part of and spaces. The method was also illustrated with examples of use: analysis of “fractional order lag” system, application to noninteger order filters design, and parametric optimization of fractional controllers

Description and Realization for a Class of Irrational Transfer Functions

This paper proposes an exact description scheme which is an extension to the well-established frequency distributed model method for a class of irrational transfer functions. The method relaxes the constraints on the zero initial instant by introducing the generalized Laplace transform, which provides a wide range of applicability. With the discretization of continuous frequency band, the infinite dimensional equivalent model is approximated by a finite dimensional one. Finally, a fair comparison to the well-known Charef method is presented, demonstrating its added value with respect to the state of art.Comment: 9 pages, 9 figure

Design and practical implementation of a fractional order proportional integral controller (FOPI) for a poorly damped fractional order process with time delay

One of the most popular tuning procedures for the development of fractional order controllers is by imposing frequency domain constraints such as gain crossover frequency, phase margin and iso-damping properties. The present study extends the frequency domain tuning methodology to a generalized range of fractional order processes based on second order plus time delay (SOPDT) models. A fractional order PI controller is tuned for a real process that exhibits poorly damped dynamics characterized in terms of a fractional order transfer function with time delay. The obtained controller is validated on the experimental platform by analyzing staircase reference tracking, input disturbance rejection and robustness to process uncertainties. The paper focuses around the tuning methodology as well as the fractional order modeling of the process' dynamics

Beyond the Waterbed Effect: Development of Fractional Order CRONE Control with Non-Linear Reset

In this paper a novel reset control synthesis method is proposed: CRONE reset control, combining a robust fractional CRONE controller with non-linear reset control to overcome waterbed effect. In CRONE control, robustness is achieved by creation of constant phase behaviour around bandwidth with the use of fractional operators, also allowing more freedom in shaping the open-loop frequency response. However, being a linear controller it suffers from the inevitable trade-off between robustness and performance as a result of the waterbed effect. Here reset control is introduced in the CRONE design to overcome the fundamental limitations. In the new controller design, reset phase advantage is approximated using describing function analysis and used to achieve better open-loop shape. Sufficient quadratic stability conditions are shown for the designed CRONE reset controllers and the control design is validated on a Lorentz-actuated nanometre precision stage. It is shown that for similar phase margin, better performance in terms of reference-tracking and noise attenuation can be achieved.Comment: American Control Conference 201