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Such systems are said to have fractional dynamics. Derivatives and integrals of fractional orders are used to describe objects that can be characterized by power-law nonlocality, [2] power-law long-range dependence or fractal properties. Fractional-order systems are useful in studying the anomalous behavior of dynamical systems in physics ...
A fractional-order integrator or just simply fractional integrator is an integrator device that calculates the fractional-order integral or derivative (usually called a differintegral) of an input. Differentiation or integration is a real or complex parameter.
To find the α th order derivative, the n th order derivative of the integral of order (n − α) is computed, where n is the smallest integer greater than α (that is, n = ⌈α⌉). The Riemann–Liouville fractional derivative and integral has multiple applications such as in case of solutions to the equation in the case of multiple systems ...
Fractional-order control (FOC) is a field of control theory that uses the fractional-order integrator as part of the control system design toolkit. The use of fractional calculus can improve and generalize well-established control methods and strategies. [1] The fundamental advantage of FOC is that the fractional-order integrator weights ...
An alternative fractional derivative was introduced by Caputo in 1967, [7] and produces a derivative that has different properties: it produces zero from constant functions and, more importantly, the initial value terms of the Laplace Transform are expressed by means of the values of that function and of its derivative of integer order rather ...
In mathematics, the Caputo fractional derivative, also called Caputo-type fractional derivative, is a generalization of derivatives for non-integer orders named after Michele Caputo. Caputo first defined this form of fractional derivative in 1967.
Consider the following second-order problem, ′ + + = () =, where = {,, <is the Heaviside step function.The Laplace transform is defined by, = {()} = ().Upon taking term-by-term Laplace transforms, and utilising the rules for derivatives and integrals, the integro-differential equation is converted into the following algebraic equation,
First-order means that only the first derivative of y appears in the equation, and higher derivatives are absent. Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations, because a higher-order ODE can be converted into a larger system of first-order equations by introducing extra variables.