Global stabilization of nonlinear gyroscopic systems.

We concentrate on the problems of stability and stabilization of equilibrium points for certain nonlinear dynamical systems, primarily rigid body systems with gyroscopic effects. We analyze stability of the equilibrium points of these systems, and, in the presence of control inputs, we synthesize control laws that globally asymptotically stabilize these systems to the equilibrium points. The control law synthesis method developed here is based on the Hamilton-Jacobi-Bellman theorem which guarantees stability and also provides an optimality interpretation for the control law. In particular, we attempt to achieve both the objectives of stability and optimality for the nonlinear control system by finding solutions to the Hamilton-Jacobi-Bellman equation. It is shown that the performance index obtained is not prescribed a priori, rather, there is a relationship between the nonlinear system, the stabilizing control law and the Lyapunov function for the closed-loop system through the Hamilton-Jacobi-Bellman equation. These results are then specialized to nonlinear affine control systems, Jurdjevic-Quinn type systems and minimum phase system with relative degree $\{1,\... ,1\}.$ The dynamical systems considered in this dissertation are primarily gyroscopic systems which include, in general, nonlinear coupling, centrifugal forces and trigonometric functions in the equations of motion. These inherent nonlinearities can help stabilize the nonlinear system as shown in examples in which the interplay between dynamics and control plays an important role. Hence, we connect modern nonlinear control theory and classical optimal control theory in the framework of the Hamilton-Jacobi-Bellman theorem and apply this to several nonlinear gyroscopic systems. We show that by taking advantage of the inherent nonlinearities embedded in the nonlinear system, we derive elegant and powerful results which are useful for practical applications.PhDAerospace engineeringApplied SciencesMechanicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/129323/2/9423340.pd