Global theory of nonlinear systems-chaos, knots and stability

In this paper we shall give a brief overview of nonlinear dynamical systems theory including the theory of chaos, knots, approximation theory and stability

Control of Nonlinear Distributed Parameter Systems Based on Global Approximation

We extend an iterative approximation method to nonlinear, distributed parameter systems given by partial differential and functional equations. The nonlinear system is approached by a sequence of linear time-varying systems, which globally converges in the limit to the original nonlinear systems considered. This allows many linear control techniques to be applied to nonlinear systems. Here we design a sliding mode controller for a nonlinear wave equation to demonstrate the effectiveness of this method

Iterative Self-Tuning Minimum Variance Control of a Nonlinear Autonomous Underwater Vehicle Maneuvering Model

This paper addresses the problem of control design for a nonlinear maneuvering model of an autonomous underwater vehicle. The control algorithm is based on an iteration technique that approximates the original nonlinear model by a sequence of linear time-varying equations equivalent to the original nonlinear problem and a self-tuning control method so that the controller is designed at each time point on the interval for trajectory tracking and heading angle control. This work makes use of self-tuning minimum variance principles. The benefit of this approach is that the nonlinearities and couplings of the system are preserved, unlike in the cases of control design based on linearized systems, reducing in this manner the uncertainty in the model and increasing the robustness of the controller. The simulations here presented use a torpedo-shaped underwater vehicle model and show the good performance of the controller and accurate tracking for certain maneuvering cases

Solving Optimal Control Problem Via Chebyshev Wavelet

Over the last four decades, optimal control problem are solved using direct and indirect methods. Direct methods are based on using polynomials to represent the optimal problem. Direct methods can be implemented using either discretization or parameterization. The proposed method in my thesis is considered as a direct method in which the optimal control problem is directly converted into a mathematical programming problem. A wavelet-based method is presented to solve the non-linear quadratic optimal control problem. The Chebyshev wavelets functions are used as the basis functions. The proposed method is also based on the iteration technique which replaces the nonlinear state equations by an equivalent sequence of linear time-varying state equations which is much easier to solve. Numerical examples are presented to show the effectiveness of the method, several optimal control problems were solved, and the simulation results show that the proposed method gives good and comparable results with some other methods

Nonlinear robust H∞ control.

A new theory is proposed for the full-information finite and infinite horizontime robust H∞ control that is equivalently effective for the regulation and/or tracking problems of the general class of time-varying nonlinear systems under the presence of exogenous disturbance inputs. The theory employs the sequence of linear-quadratic and time-varying approximations, that were recently introduced in the optimal control framework, to transform the nonlinear H∞ control problem into a sequence of linearquadratic robust H∞ control problems by using well-known results from the existing Riccati-based theory of the maturing classical linear robust control. The proposed method, as in the optimal control case, requires solving an approximating sequence of Riccati equations (ASRE), to find linear time-varying feedback controllers for such disturbed nonlinear systems while employing classical methods. Under very mild conditions of local Lipschitz continuity, these iterative sequences of solutions are known to converge to the unique viscosity solution of the Hamilton-lacobi-Bellman partial differential equation of the original nonlinear optimal control problem in the weak form (Cimen, 2003); and should hold for the robust control problems herein. The theory is analytically illustrated by directly applying it to some sophisticated nonlinear dynamical models of practical real-world applications. Under a r -iteration sense, such a theory gives the control engineer and designer more transparent control requirements to be incorporated a priori to fine-tune between robustness and optimality needs. It is believed, however, that the automatic state-regulation robust ASRE feedback control systems and techniques provided in this thesis yield very effective control actions in theory, in view of its computational simplicity and its validation by means of classical numerical techniques, and can straightforwardly be implemented in practice as the feedback controller is constrained to be linear with respect to its inputs