A multiinput, multioutput fractional system is described by the differential equation system involving fractional derivatives of the system input

and of the system output

:

in which
and
.
and
denote fractional differentiation operators of orders
and
, respectively. Such operators are defined in [8–11] and a detailed survey of the properties linked to these definitions can be found in [8].

If orders
and
verify relations
,
,
, then differentiation orders
and
are commensurate [12] (multiple of the same rational number
).

Using order commensurability condition and for zero initial conditions, differential equation (

2.1) admits a pseudostate space representation of the form:

where
is the pseudostate vector,
is the fractional order of the system, and
,
,
, and
are constant matrices.

As explained in [3], representation (2.2) is not strictly a state space representation and this is why it is denoted in the sequel *pseudostate space representation*. In the usual integer order system theory, the state of the system,
, known at any given time point, along with the system equations and system inputs, is sufficient to predict the response of the system. That comment can be found in [13].

As demonstrated in [3], and whatever the fractional derivative definition used (excepted Caputo's definition but this last one is not physically acceptable [14]), the value of vector
at initial time
in (2.2) is not enough to predict the future behavior of the system. Vector
in (2.2) is thus not a state vector of the system. However, as also shown in [3], a Luenberger type observer can be used to estimate pseudostate vector
.