Abstract. The purpose of this paper is to show that the method of controlled Lagrangians and its Hamiltonian counterpart (based on the notion of passivity) are equivalent under rather general hypotheses. We study the particular case of simple mechanical control systems (where the underlying Lagrangian is kinetic minus potential energy) subject to controls and external forces in some detail. The equivalence makes use of almost Poisson structures (Poisson brackets that may fail to satisfy the Jacobi identity) on the Hamiltonian side, which is the Hamiltonian counterpart of a class of gyroscopic forces on the Lagrangian side.Mathematics Subject Classification. 34D20, 70H03, 70H05, 93D15.

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This dissertation addresses the modeling, identification, and control of an automated planing vessel. To provide motion models for trajectory generation and to enable model-based control design for trajectory tracking, several experimentally identified models are compared over a wide range of speed and planing conditions for the Virginia Tech Ribcraft Unmanned Surface Vehicle. The modeling and identification objective is to determine a model which is sufficiently rich to enable effective model-based control design and trajectory optimization, sufficiently simple to allow parameter identification, and sufficiently general to describe a variety of hull forms and actuator configurations. Beginning with a 6 degree of freedom nonlinear dynamic model, several linear steering and speed models are obtained as well as a thruster model.The Ribcraft USV tracks trajectories generated with the selected maneuvering models by using a backstepping trajectory controller. A PD cascade trajectory control law is also developed and the performance of the two controllers is compared using aggressive trajectories. The backstepping control law compares favorably to the PD cascade controller. The backstepping control law is then further modified to account for nonlinear sternward dynamics and for a constant or slowly varying fluid flow. ACKNOWLEDGMENTSThis dissertation is the product of the culmination of years of research and field experimentation that required the collective abilities only found in a group of skilled researchers. There are many people that guided me through the higher level decision-making and advising process as well as the more laborious experimentation and implementation efforts. Without these people I would not have been able to finish a dissertation even approaching this quality.

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This paper describes a procedure for incorporating artificial gyroscopic forces and physical dissipation in the method of controlled Lagrangians. Energy-conserving gyroscopic forces provide additional freedom to expand the basin of stability and tune closed-loop system performance. We also study the effect of physical dissipation on the closed-loop dynamics and discuss conditions for stability in the presence of natural damping. We apply the technique to the inverted pendulum on a cart, a case study from previous papers. We develop a controller that asymptotically stabilizes the inverted equilibrium at a specific cart position for the conservative dynamic model. The region of attraction contains all states for which the pendulum is elevated above the horizontal plane. We also develop conditions for asymptotic stability in the presence of linear damping.