Abstract
This paper develops and exposes the strong relationships that exist between time-domain order-distributions and the Laplace-domain logarithmic operator. The paper presents the fundamental theory of the Laplace-domain logarithmic operator, and related operators. It is motivated by the appearance of logarithmic operators in a variety of fractional-order systems and order-distributions. Included is the development of a system theory for Laplace-domain logarithmic operator systems which includes time-domain representations, frequency domain representations, frequency response analysis, time response analysis, and stability theory. Approximation methods are included.
Highlights
The area of mathematics known as fractional calculus has been studied for over 300 years [1]
This paper develops and exposes the strong relationships that exist between timedomain order-distributions and the Laplace-domain logarithmic operator
The motivation is the appearance of logarithmic operators in a variety of fractional-order systems and orderdistributions
Summary
The area of mathematics known as fractional calculus has been studied for over 300 years [1]. Fractional-order systems, or systems described using fractional derivatives and integrals, have been studied by many in the engineering area [2,3,4,5,6,7,8,9]. It should be noted that there are a growing number of physical systems whose behavior can be compactly described using fractional-order system theory. Specific applications are viscoelastic materials [13,14,15,16], electrochemical processes [17,18], long lines [5], dielectric polarization [19], colored noise [20], soil mechanics [21], chaos [22], control systems [23], and optimal control [24]. Conferences in the area are held annually, and a interesting publication containing many applications and numerical approximations is Le Mehaute et al [25]
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