Fractional Order System Forced-response Decomposition and Its Application

Abstract

This chapter deals with the additive decomposition of the forced response of a fractionalorder system. Precisely, it is shown how, by solving a simple polynomial Diophantine equation, this response can almost always be decomposed into the sum of a systemdependent component and an input-dependent component. The system-dependent component is formed from the same modes as the system and, assuming stabi li ty, characterises the transient behaviour of the system in the response to sustained inputs. The input-dependent component is formed from the same modes as the input, and accounts for the steady-state or long-term response of a stable system to a persistent input. Simple conditions based on the classical Routh and Mikhai lov criteria are provided to check the system input-output stability. Several examples show that the aforementioned decomposition can profitably be exploited to find simplified models in such a way that the asymptotic response is kept unchanged and, at the same time, the transient behaviour is well approximated. The decomposition proves useful also for solving the so-called model-matching problem that is of particular interest in controller synthesis.