This chapter deals with the additive decomposition of the forced response of a fractional order 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 system-dependent component and an input-dependent component. The system-dependent component is formed from the same modes as the system and, assuming stability, characterizes the transient behavior 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 Mikhailov 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 behavior is well approximated. The decomposition proves useful also for solving the so-called model-matching problem that is of particular interest in controller synthesis.
