Observer-Based Robust Controller Design for Nonlinear Fractional-Order Uncertain Systems via LMI

We discuss the observer-based robust controller design problem for a class of nonlinear fractional-order uncertain systems with admissible time-variant uncertainty in the case of the fractional-order satisfying 0<α<1. Based on direct Lyapunov approach, a sufficient condition for the robust asymptotic stability of the observer-based nonlinear fractional-order uncertain systems is presented. Employing Finsler’s Lemma, the systematic robust stabilization design algorithm is then proposed in terms of linear matrix inequalities (LMIs). The efficiency and advantage of the proposed algorithm are finally illustrated by two numerical simulations

Observer-Based Approach for Fractional-Order Chaotic Synchronization and Secure Communication

This paper presents a method based on the state observer design for constructing a chaotically synchronized systems. Fractional-order direct Lyapunov theorem is used to derive the closed-loop asymptotic stability. The gains of the observer and observer-based controller are obtained in terms of linear matrix inequalities (LMIs) formulation. The proposed approach is then applied to secure communications. The method combines chaotic masking and chaotic modulation, where the information signal is injected into the transmitter and simultaneously transmitted to the receiver. Chaotic synchronization and chaotic communication are achieved simultaneously via a state observer design technique. The fractional-order chaotic Lorenz and Lü systems are given to demonstrate the applicability of the proposed approach

Stability and Control of Caputo Fractional Order Systems

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