Fractional Derivatives, Fractional Integrals, and Fractional Differential Equations in Matlab

In book: Engineering Education and Research Using MATLAB
Source: InTech
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    ABSTRACT: A novel digital integrator and a novel digital differentiator are presented. Both the integrator and the differentiator are of first order and thus eminently suitable for real-time applications. Both have an almost linear phase. The integrator is obtained by interpolating two popular digital integration techniques, the rectangular and the trapezoidal rules. The resulting integrator outperforms both the rectangular and the trapezoidal integrators in range and accuracy. The new differentiator is obtained by taking the inverse of the transfer function of the integrator. The effective range of the differentiator is about 0.8 of the Nyquist frequency.
    Electronics Letters 03/1993; 29(4-29):376 - 378. DOI:10.1049/el:19930253 · 1.07 Impact Factor
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    ABSTRACT: Many real dynamic systems are better characterized using a non-integer order dynamic model based on fractional calculus or, differentiation or integration of non-integer order. Traditional calculus is based on integer order differentiation and integration. The concept of fractional calculus has tremendous potential to change the way we see, model, and control the nature around us. Denying fractional derivatives is like saying that zero, fractional, or irrational numbers do not exist. In this paper, we offer a tutorial on fractional calculus in controls. Basic definitions of fractional calculus, fractional order dynamic systems and controls are presented first. Then, fractional order PID controllers are introduced which may make fractional order controllers ubiquitous in industry. Additionally, several typical known fractional order controllers are introduced and commented. Numerical methods for simulating fractional order systems are given in detail so that a beginner can get started quickly. Discretization techniques for fractional order operators are introduced in some details too. Both digital and analog realization methods of fractional order operators are introduced. Finally, remarks on future research efforts in fractional order control are given.
    American Control Conference, 2009. ACC '09.; 07/2009
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    ABSTRACT: For fractional-order differentiator s<sup>r</sup> 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
    IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications 04/2002; 49(3-49):363 - 367. DOI:10.1109/81.989172


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