Analysis, Modeling and Stability of Fractional Order Differential Systems 1

9Analytical Expressions of FDS Transients

9.1. Introduction

The determination of the analytical expression X(t) formulating the transients of the FDS:

[9.1]

is considered as a trivial problem because many researchers think that it has already been solved for several years with the initial condition X(0), where X(t) is the pseudo-state vector [MAT 96, BET 08]. Unfortunately, we have previously demonstrated that X(t) is unable to predict future system behavior, which must be replaced by the distributed state vector Z(ω,t), in the theoretical framework of the closed-loop representation. Moreover, it is generally admitted that X(t) expressions depend on the choice of the fractional derivative. We also demonstrated in Chapter 8 that it is a false problem and transients have a unique expression, which can be based on the distributed initial condition Z(ω, 0).

Therefore, the objective of this chapter is to express X(t), i.e. the free and forced responses of [9.1] using the distributed state Z(ω,t). Two solutions are proposed: the first solution is based on the Mittag-Leffler approach [MON 10, ORT 18] and the second solution is based on the new concept of distributed exponential, i.e. using all the potentialities of the infinite state approach. Moreover, beyond these theoretical expressions, another practical objective is to formulate computable solutions derived from these expressions. ...