MC 6460
Candidate
Cong Wu
Applied Mathematics, University of À¶Ý®ÊÓƵ
Title
Stability and Control of Caputo Fractional Order Systems
Abstract
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 areÌýdeveloped to make the controlled systems bounded-input bounded-output stable, disturbance-rejection and implementable,Ìý
respectively.ÌýIn order for these designs, fractional order polynomials are systematically defined and related propertiesÌýincluding root distribution, coprimeness, properness and $\rho-\kappa$ polynomials are well explored. Then move toÌýCaputo fractional order nonlinear systems, of which the fundamental theory including continuation and smoothness ofÌý
solutions is developed; the diffusive realizations are shown to be equivalent with the systems; and the Lyapunov-likeÌýfunctions based on the realizations prove to be well-defined.This paves the way to stability analysis. The smoothnessÌýproperty of solutions suffices a simple estimation for the Caputo fractional order derivative of any quadratic LyapunovÌýfunction, which together with the continuation leads to our results on Lyapunov stability, while the Lyapunov-likeÌýfunction contributes to our results on external stability. These stability results are then applied to $H_\infty$ control,Ìýand finally extended to Caputo fractional order hybrid systems.