In this thesis, we investigate the control of Port-Controlled Hamiltonian (PCH) systems, which are a generalization of Euler-Lagrange (EL) Systems (also known as Hamiltonian Systems). Currently, methods exist for tracking control of EL systems and certain PCH systems when full state feedback is available. We propose a new control structure, which includes as special cases some well-known controllers for EL systems, but which generalizes to non-passive systems and to PCH systems with only output feedback available. For the case of EL systems, we modify our control to include an adaptive component to improve robustness and attenuate disturbances. We prove the stability of our proposed controls and investigate their performance via simulation.

Our basic approach is to develop a new matching equation for PCH systems based on the concept of energy-balancing. This allows us to extend passivity-based controls such as Interconnection Damping Assignment Passivity-Based Control (IDA-PBC) to the control of underactuated PCH systems and to the problem of trajectory tracking of PCH systems. The resulting control is a static controller, whose stabilization properties can be improved by introducing some controller dynamics. To this end, we introduce a new function C (called the Casimir tracking function), which depends on the reference trajectory and the system state. The dynamic control is then designed as a PCH system, whose Hamiltonian depends on C. Stable tracking is then achieved through the design of C.

The significance of our control methods is that they achieve asymptotic tracking for very general nonlinear systems, namely PCH systems and non-passive systems that can be converted to PCH via feedback. PCH systems include not only nonlinear holonomic systems, such as robotic manipulators, but also nonholonomic systems. Also, our general control structure provides a re-interpretation of established con- trols, such as computed torque control, the PD+ tracking controller, and the Slotine- Li control. In addition to our proofs of asymptotic tracking, simulations further show that the proposed switching adaptive control achieves faster convergence and better transient performance than the Slotine-Li adaptive control.