Markovian approximation of linear systems with fractional viscoelastic term

Abstract

It is well known that the response of a linear system enforced by a Gaussian white noise is Markovian. The order of Markovianity is n-1 being n the maximum order of the derivative of the equation ruling the evolution of the system. However when a fractional operator appears, the order of Markovianity of the system becomes infinite. Then the main aim developed in the proposed paper, consists of rewriting the system with fractional term of order r with an "equivalent" one, in which the fractional operator is substituted by two classical differential terms with integer order of derivative int(r) and int(r + 1) (for a real r). In this way the fractional differential equation reverts into a classical differential equation and then the Markovianity of the system is restored. The proposed technique for evaluating the equivalent coefficient of two terms involving the derivatives of order int(r) and int(r + 1) are evaluated by means of the classical stochastic linearisation technique, that is by performing the minimisation of the error made in passing for the original system with fractional order derivative into equivalent linear system with classical derivative terms. Since the original system is linear (fractional operators are linear) the response to a Gaussian input is Gaussian too as for the response of the equivalent one.