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Thesis - Campus Access Only
Master of Science (MS)
Brockett's theorem, geometric phase, nonholonomic control system, normal form equations, smooth stabilization, strongly accessible
Brockett's theorem states the three necessary conditions for the existence of a continuously differentiable closed loop control that asymptotically stabilizes the nonlinear control system to an equilibrium point. Kinematic systems are shown to fail to meet Brockett's third necessary condition. A normal form is introduced so that nonholonomic control systems are defined directly over a reduced constraint distribution. In normal form, nonholonomic control systems can then easily be shown to fail to be stabilizable to a point via a $C^1$ control. The conditions for the smooth stabilization of the nonholonomic systems to an equilibrium submanifold are then presented. For a particular case of the reduced form of mechanical control systems (Chaplygin systems), stabilization to a point can be achieved by applying the concept of geometric phase and using piecewise differentiable state controls.
Michot, Marc, "Brockett's necessary conditions and the stabilization of nonlinear control systems" (2011). Master's Theses. 3946.