Analysis, Modeling and Stability of Fractional Order Differential Systems 2

1Initialization of Fractional Order Systems

1.1. Introduction

The initialization problem, or the initial value problem, is considered an elementary topic in the integer order case. On the contrary, for a long time it has been considered a very complex problem in the fractional order case; certainly because fractional derivatives, mainly Caputo’s derivative, have perpetuated a deep confusion about initial conditions (see Chapter 8 of Volume 1).

The initialization of an ODE is an obvious problem. Initial conditions are well defined because this concept directly refers (or indirectly) to the system state [KAI 80, CHE 84, ZAD 08]. In the fractional order case, the definition of system state has been confused for a long time because it was not possible to directly generalize the concepts of integer order state space to fractional systems. As demonstrated previously, the variable x(t), output of the fractional order integrator, is only a pseudo-state variable, unable to represent the true internal state of the fractional system (see Chapter 7 of Volume 1). Moreover, since the so-called initial conditions of Caputo’s derivative were considered as the value of the system state at t =0, it was impossible to predict the future system dynamics based on these pseudo-initial conditions, as in the integer order case (see Chapter 8 of Volume 1). Thus, the initialization of fractional systems ...