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 .