Analysis, Modeling and Stability of Fractional Order Differential Systems 2

7Fractional Energy

7.1. Introduction

The application of Lyapunov’s method is based on the definition of a positive quadratic function, related to energy [LYA 07, LAS 61, KRA 63, NAS 68].

For linear integer order systems, the choice of this function V(t) is an elementary problem. Because V(t) should be a quadratic function of state x(t) (or ), the choice V(t) = x²(t) or as a generalization with P > 0 is straightforward [KAI 80, KHA 96].

Many researchers have used the same definition of V(t) in the fractional order case [AGU 14]. Unfortunately, x(t) (or ) is only a pseudo-state vector and V(t) = x²(t) is a positive semi-definite function which cannot be used as a Lyapunov function, as will be demonstrated using an elementary example.

In fact, the true state of the fractional integrator is the distributed variable z(ω,t) (Ẕ(ω,t) for a multidimensional system). Therefore, V(t) has to be a quadratic function based on z(ω,t).

Thus, the definition of V(t) is no longer an elementary problem as in the integer order case. We previously a priori defined V(t) as

[7.1]

in the monovariable case ...