Analysis, Modeling and Stability of Fractional Order Differential Systems 2

9Lyapunov Stability of Non-commensurate Order Fractional Systems

9.1. Introduction

In the previous chapter, it has been demonstrated that the Lyapunov stability of commensurate fractional order systems is closely related to the stability of integer order systems, as they are commensurate order systems with n = 1, nevertheless with important specificities. In spite of these constraints, it has been possible to express an LMI stability condition, thanks to the duality between closed-loop and open-loop representations.

The situation is completely different with non-commensurate fractional order systems corresponding to the general model:

[9.1]

Let us recall that matrix transformation or modal representation does not exist for these systems (see Chapter 2).

Therefore, it is not possible to formulate a Lyapunov function equivalent to

[9.2]

for at least three main reasons:

• – the weighting function µ_ni((ω) depends on each fractional order n_i;
• – it is not possible to formulate a weighting matrix P = (M^–1)^T M^–1 > 0 since the matrix transformation does not exist;
• – we cannot use the duality closed-loop/open-loop representation because partial fraction expansion cannot be used.

Thus, ...