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Abstract
This thesis uses H∞ control methods to study robust state estimation and Lagrange stabilization of sector bounded nonlinear systems. According to different stability notions and nonlinearity configurations, this thesis is divided into two parts.
Part A considers robust state estimation for uncertain stochastic systems with slopebounded nonlinearities satisfying Integral Quadratic Constraints (IQC) and multipliers are used to exploit the uncertainties and nonlinearities. A new approach for constructing a robust state estimator for the nonlinear system is developed.
Part B studies stability analysis and Lagrange stabilizing controller synthesis for pendulum-like systems. This part proposes two new versions of a Lagrange stability criterion: one is used for stability analysis of pendulum-like system with a single nonlinearity and the other one is used for pendulum-like system with multiple nonlinearities. As these new Lagrange stability criteria do not require the linear part of the pendulum-like system under consideration to be minimal, they can be applied to more general pendulum-like systems than the previous results in the literature. Also, a sufficient condition which ensures that a periodic (in state) nonlinear system is a pendulum-like system is established. These two results enable Lagrange stabilizing controller synthesis for pendulum-like systems with multiple nonlinearities.
To facilitate Lagrange stabilizing controller synthesis for pendulum-like systems, some results in standard H∞ control are generalized. An extended strict bounded real lemma is developed which only requires the pair of system matrices (A,B) to be stabilizable, rather than the Hurwitz property of the system matrix A. Also, a pseudo H∞ control theory is developed. This includes a pseudo strict bounded real lemma which relates a pseudo H∞ condition (in which the system matrix A has n − 1 eigenvalues with negative real parts and one eigenvalue with positive real part and a frequency domain norm bound condition holds) to the existence of a sign indefinite solution to an algebraic Riccati equation. These results can handle some system control design problems which the standard H∞ control theory is not applicable. This is one of the main contributions of this thesis.
The Lagrange stabilization problem for pendulum-like systems with different version of periodic (in state) nonlinearities is formulated and solved in term of the solutions to game-type Riccati equations. These equations can be solved using available software kits. In addition, some examples with simulations are given to illustrate the efficacy of the proposed methods.