Applied Mathematics, University of Waterloo
Stability and Control of Caputo Fractional Order Systems
As pointed out by many researchers in the last few decades, differential equations with fractional (non-interger) order differential operators, in comparison with classical integer-order ones, have apparent advantages in modelling mechanical and electrical properties of various real materials, e.g. polymers, and in some other fields. Caputo fractional order systems (systems of ordinary differential equations with fractional order differential operators of Caputo type) and stability will be focused in this thesis.
Our studies begin with Caputo fractional order linear systems, for which, frequency-domain designs: pole placement, internal model principle and model matching are developed to make the controlled systems bounded-input bounded-output stable, disturbance-rejection and implementable,
respectively. In order for these designs, fractional order polynomials are systematically defined and related properties including root distribution, coprimeness, properness and $\rho-\kappa$ polynomials are well explored. Then move to Caputo fractional order nonlinear systems, of which the fundamental theory including continuation and smoothness of
solutions is developed; the diffusive realizations are shown to be equivalent with the systems; and the Lyapunov-like functions based on the realizations prove to be well-defined.This paves the way to stability analysis. The smoothness property of solutions suffices a simple estimation for the Caputo fractional order derivative of any quadratic Lyapunov function, which together with the continuation leads to our results on Lyapunov stability, while the Lyapunov-like function contributes to our results on external stability. These stability results are then applied to $H_\infty$ control, and finally extended to Caputo fractional order hybrid systems.