Robust and Saturation Control for Polynomial Nonlinear Systems

This dissertation studies the synthesis conditions and optimization of robust and saturation control for polynomial nonlinear systems. The control of nonlinear systems remains a difficult problem to solve, although the field has recently developed rapidly due to improved programming/computational tools. The purpose of this dissertation is to improve nonlinear control theory by developing new control methods that can be applied directly to polynomial nonlinear systems. The research focuses on generating easily solvable sum-of-squares (SOS) conditions that will handle both modeling uncertainties and input saturation for polynomial nonlinear systems. Although SOS programming has been applied successfully to nonlinear system analysis, the use of SOS optimization to solve nonlinear control synthesis problems is a real challenge.;First, we derive the synthesis conditions for nonlinear robust Hinfinity state-feedback control of a nonlinear uncertain systems with a polynomial vector field and norm bounded uncertainties. SOS programming is used to solve the state-dependent linear matrix inequalities (LMIs) required to synthesize a controller that provides robust stability. The nonlinear state-feedback control synthesis conditions use state-dependent scaling to provide robust optimization of the L2 gain of the disturbance/output. The convex state-dependent LMIs provide a non-iterative tool that is able to handle different nonlinear control design tasks in a simple direct manner. This research also makes a contribution to the study of nonlinear robust Hinfinity control by extending the analysis to nonlinear systems containing a polynomial vector field with input saturation. The nonlinear systems containing a polynomial vector field with input saturation are analyzed using norm bounded and polytopic differential inclusion sets by utilizing the nonlinear traits of saturation and dead-zone functions. The norm bounded and polytopic differential inclusion sets produce convex sets of state-dependent LMIs that are recast into an SOS optimization problem. Through SOS programming, the norm bounded and polytopic differential inclusion state-dependent LMIs produce nonlinear state-feedback controllers that stabilize the nonlinear systems and optimize robust performance. Parametric studies are performed on the nonlinear robust Hinfinity controllers to analyze the effect of constraints associated with stability, regional constraints, and the state realization to minimize the L2 gain of the disturbance/output.;Then in the second part of the thesis, the nonlinear robust Hinfinity state-feedback control is extended to derive a nonlinear robust output-feedback Hinfinity controller for a class of output-dependent nonlinear systems with a polynomial vector field and norm bounded measured parameters. The output-feedback control of a nonlinear system is a difficult problem to solve that often requires iterative approaches utilizing prior knowledge or estimation of the system's Lyapunov candidate. The prior knowledge or estimate of the Lyapunov candidate limits its application because the Lyapunov candidates are rarely known. Iterative synthesis conditions require additional mathematical optimization to approach a solution often resulting in suboptimal solutions. The output-dependent matrix inequalities can be recast as a convex semi-definite optimization problem which can be solved using SOS programming. The nonlinear robust output-feedback Hinfinity controller utilizes the application of gain-scheduling to a convex problem that will stabilize the nonlinear system and ensure robust performance. The proposed output-dependent synthesis conditions incorporate output-dependent scaling to provide a non-iterative optimization of the L2 gain of the disturbance/output.;Finally, the nonlinear output-feedback Hinfinity controller for a class of output-dependent nonlinear systems with a polynomial vector field and input saturation is studied. The output-feedback controllers for both norm bounded and polytopic differential inclusion use regional, saturation, and dead-zone functions to transform the nonlinear plant with the measured parameters. To ensure that the output-dependent LMIs are convex, different approaches had to be used to solve the norm bounded and polytopic differential inclusion sets. A gain-scheduled nonlinear output-feedback controller is used in the norm bounded differential inclusion set. The norm bounded differential inclusion set provided a series of output-dependent LMIs that are solved using SOS programming to optimize robust performance using a non-iterative method. The polytopic differential inclusion set utilizes a nonlinear controller stated in a quasi-LPV form to ensure that the resulting inequality constraints are convex conditions. The two different output-dependent control approaches provide optimal robust performance and ensure regional stability of nonlinear saturated control systems.