Feedback stabilization: Nonlinear solutions to inherently nonlinear problems

Control strategies are developed for nonlinear systems that fail to satisfy differential geometric conditions for input-to-state linearizability under state feedback and change of coordinates.; The central part of this work is motivated primarily by a popular "ball and beam" laboratory experiment. For this example, the differential geometric conditions for input-to-state linearizability are not satisfied. Strategies have been developed previously to overcome this limitation in a neighborhood of an equilibrium manifold in order to achieve (approximate) tracking and local stabilization. However, the domains of attraction for these methods are very small.; Control strategies are presented for a general class of nonlinear systems, of which the "ball and beam" is an example, which result in arbitrarily large domains of attraction for both the small signal tracking problem and the stabilization problem. The main component of the approach is the use of saturation functions to limit the destabilizing effects that cannot be removed by geometric linearization techniques. One of the new elements of this work is the nesting of saturation functions to systematically isolate and diminish these destabilizing effects.; One can think of linear chain of integrator systems that are subject to "actuator constraints" as nonlinear systems that cannot be made to appear linear globally. The methodology of nested saturation functions provides new, simple globally stabilizing control laws for such systems.; In addition to developing methodologies for systems like the "ball and beam" and linear systems subject to "actuator constraints", asymptotically stabilizing control strategies are developed for a class of nonholonomic control systems. These systems generically do not satisfy geometric conditions for input-to-state linearization. New, smooth time-varying and locally stabilizing control laws are developed based on previous work in the literature on steering nonholonomic systems with sinusoids. Globally stabilizing strategies are then achieved by again introducing saturation functions.; Finally, results are presented that improve regions of feasibility for a recently developed nonlinear adaptive control scheme.; These different settings are used to argue for the desirability of tackling inherently nonlinear control problems with new, inherently nonlinear solutions. The case is made for continued research to develop powerful, specialized tools to add to the nonlinear control toolbox.