Analysis, Modeling and Stability of Fractional Order Differential Systems 2

4State Control of Fractional Differential Systems

4.1. Introduction

Fractional control is at the origin of the interest of engineering researchers for fractional calculus. Since the seminal research works of Bode [BOD 45], Manabe [MAN 60] and Oustaloup [OUS 81, OUS 83] in the domain of robust control, using a non-integer order frequency template, applied fractional calculus has conquered the domain of automatic control. In fact, it is mainly Oustaloup’s research work, dedicated to a robust fractional CRONE controller [OUS 91, OUS 95b], which is at the origin of an important interest for fractional calculus in automatic control. This original work motivated a large number of publications, such as the concept of the fractional PID controller [POD 97, VIN 00, BAR 04, VAL 07, MAA 10, TEN 12] and many others [MAA 91, MON 10, TEN 13].

However, most of the previous control applications do not concern directly fractional systems and more specifically the state control of fractional systems.

State control is at the origin of optimal control of fractional differential systems. The objective, mainly theoretical, has been to transpose optimal control theory [BOU 67, STE 94, GEL 00, KIR 04] to fractional calculus [AGR 04, BAL 06]. This was an important and ambitious objective; however, it has been biased by the definition of the fractional state. As described previously for the observability and controllability of FDSs, the theory of integer order optimal control was applied to the pseudo-state ...