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-integer) 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. The stability and control of Caputo fractional order systems (systems of ordinary differential equations with fractional order differential operators of Caputo type) will be focused in this thesis. Our studies begin with Caputo fractional order linear systems, for which, three frequency-domain designs: pole placement, internal model principle and model matching, are developed to make the controlled systems bounded-input bounded-output stable, disturbance rejective and implementable, respectively. For these designs, fractional order polynomials are systematically defined and their root distribution, coprimeness, properness and $\rho-\kappa$ polynomials are well explored. We next move to Caputo fractional order nonlinear systems, of which the fundamental theory including the 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 to yield 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.