Model-order reduction of lumped parameter systems via fractional calculus

Abstract

This study investigates the use of fractional order differential models to simulate the dynamic response of non-homogeneous discrete systems and to achieve efficient and accurate model order reduction. The traditional integer order approach to the simulation of non-homogeneous systems dictates the use of numerical solutions and often imposes stringent compromises between accuracy and computational performance. Fractional calculus provides an alternative approach where complex dynamical systems can be modeled with compact fractional equations that not only can still guarantee analytical solutions, but can also enable high levels of order reduction without compromising on accuracy. Different approaches are explored in order to transform the integer order model into a reduced order fractional model able to match the dynamic response of the initial system. Analytical and numerical results show that, under certain conditions, an exact match is possible and the resulting fractional differential models have both a complex and frequency-dependent order of the differential operator. The implications of this type of approach for both model order reduction and model synthesis are discussed.

Applications of variable-order fractional operators: a review

“…where G(s) is the transfer function and s is the transformed variable; s = iν, where i = √ −1 and ν is frequency. A matching procedure, similar to the methodologies in [8,23], was then performed in the transformed domain between a VO framework and the CO system. The VO framework replaces the CO parameters γ n and μ n with a VO q m and a generalized coefficient ζ m .…”

“…The many characteristics of fractional operators have sparked, in recent years, much interest in fractional calculus and produced a plethora of applications with particular attention to the simulation of physical problems. Areas that have seen the largest number of applications include the formulation of constitutive equations for viscoelastic materials [1][2][3][4], transport processes in complex media [4][5][6][7][8][9][10][11], mechanics [12][13][14][15], non-local elasticity [16][17][18][19], plasticity [20][21][22], modelorder reduction of lumped parameter systems [23] and biomedical engineering [24][25][26]. These studies have typically used constant-order (CO) fractional operators.…”

Analysis of dispersion and propagation properties in a periodic rod using a space-fractional wave equation

“…This paper extends the methodology developed in [43] for discrete systems to modeling continuous systems. In [43], the authors developed fractional order differential models to represent the dynamic response of non-homogeneous discrete systems and to achieve an accurate model order reduction methodology.…”

“…This paper extends the methodology developed in [43] for discrete systems to modeling continuous systems. In [43], the authors developed fractional order differential models to represent the dynamic response of non-homogeneous discrete systems and to achieve an accurate model order reduction methodology. By extending this methodology to non-homogeneous continuous systems we obtain an approach that can be interpreted as a simplified fractional homogenization technique applicable to dynamic problems.…”

“…However, applications involving complex fractional orders are very limited. Authors including Atanackovic [41], Makris [42], and Hollkamp [43] have used complex fractional calculus to study various engineering systems. Despite the use of complex order fractional operators in these papers, the actual physical significance of a complex fractional order derivative is still not completely evident and continues to garner attention today.…”