Analysis, Modeling and Stability of Fractional Order Differential Systems 2

3Improved Initialization of Fractional Order Systems

3.1. Introduction

The initialization of fractional differential systems was analyzed in Chapter 1. Two approaches to this problem were compared: one proposed by Lorenzo and Hartley, based on an input/output formulation, and the other proposed by Trigeassou and Maamri, based on a state space formulation.

These two approaches are complementary and equivalent [HAR 13]; however, they do not provide a practical solution to the initialization problem. Except some particular cases, the history function approach cannot be used with any system and particularly any past history. The infinite state approach is more general; however, it is based essentially on the availability of the distributed initial state.

It has been demonstrated that the initialization of an FDS depends on the past dynamical behavior of the system, which is also called system pre-history. Theoretically, as the dimension of the distributed state is infinite, it is necessary to consider an infinite domain of the past, since t = –∞. Practically, for obvious reasons, the knowledge of the past is restricted to a finite time interval [t_p, t₀]. Consequently, the fractional system cannot be considered at rest at t = t_p.

Therefore, the initialization problem can be stated as: estimate z(ω, t₀) based on the knowledge of {u(t), y(t)} on t ∈ [t_p, t₀], with the constraint z(ω, t_p) ≠ 0.

Consequently, if we have obtained an estimation ẑ(ω, t₀), it is then easy to formulate the ...