| dc.description.abstract | Physical systems are inherently nonlinear, and one special phenomenon of nonlinear systems is chaotic motion. The dissertation is devoted to control chaotic systems to regular motions. Numerous controlling chaos cases are studied separately in the dissertation. First, we show that one can control a chaotic system under external force excitation to arbitrary trajectories, even the desired trajectories are not located on the embedded orbits of a chaotic system. The method utilizes a newly developed sliding mode controller with a time varying manifold dynamic, to offer a feedback control in compensation with the external excitation, and drive the system orbits to any desired states. The proposed controller does not need high gain to suppress the external force, meanwhile, keeps robustness against parameter uncertainty and noise disturbance as the traditional sliding mode control. Simulations are provided to illustrate the performance of the controller.
Second, a simple and efficient method for controlling high dimensional discrete-time chaotic systems is proposed. This method is implemented similar to the OGY method, and is feasible for practical experiments. The key component is to assign the eigenvalues of a linearized map by using the well-known pole placement technique. According to the Cayley-Hamilton theorem, the trajectory will converge to the desired fixed point after iterations at most ( is the dimension of the map), if the real trajectory of the chaotic system falls within the neighborhood of the desired fixed point. The proposed approach improves the convergence rate and the robustness of the OGY method, especially for the case where the modulus of the stable eigenvalue is close to unity. The simulations illustrate the performance of our presented controller for controlling a chaotic system compared to the OGY method.
Third, a universal approach for controlling high dimensional chaotic systems is proposed, in which the controllability assumption can be relaxed and only the stabilization condition is required. The main feature of the proposed method is that all of the controllable unstable eigenvalues of the linear approximation assigned to be zero; the remained stable eigenvalues may be uncontrollable. Only small parameter perturbations are required to stabilize chaotic situation when the trajectory falls in the neighborhood of the desired fixed point, the region of attraction. We estimate the region of attraction to determine the moment of acting the controller, and this will reduce the tedious waiting time. However, to ensure zero steady state control error in the presence of uncertainty, the robustness of regulation under integral control is intuitively developed.
Finally, we present the differential geometric method to feedback linearization, which allows us to characterize the class of feedback linearizable system by geometric condition. The approach differs entirely from conventional linearization as the Jacobian linearization, in that feedback linearization is achieved by exact state transformations and feedback, rather than by linear approximation of the dynamics. As map functions are discrete, an approach for controlling discrete-time chaotic systems by feedback linearization is proposed. This method can not only stabilize unstable periodic orbits embedded in a strange attractor, but also can be applied even if the real trajectory is far from the target one. A Hénon map with different operation conditions is implemented to demonstrate the feasibility of the proposed method. | en_US |