Dynamics and Control of Non-smooth Systems with Applications to Supercavitating Vehicles

Abstract

The subject matter of this dissertation relates to the dynamics of non-smooth vehicle systems, and in particular, supercavitating vehicles. These high-speed underwater vehicles are designed to have sustained vaporous or ventilated gas cavities that form over the entire vehicle. In terms of the modeling, the system non-smoothness is caused by the interaction forces generated when the vehicle contacts the cavity. These planing interactions can cause stable and unstable dynamics, some of which could be limit-cycle dynamics. Here, planing forces are considered on the basis of non-cylindrical cavity shapes that include shifts induced by the cavitator angle of attack. Incorporating these realistic physical effects into a vehicle system model generates a unique hydrodynamic non-smoothness that is characterized by non-constant switching boundaries and non-constant switched dynamics. Nonlinear stability analyses are carried out, Hopf bifurcations of equilibrium solutions are identified, and stabilizing control is investigated. Also considered is partially cavitating system dynamics, where active fin forces are used to support the vehicle. Non-steady planing is also considered, which accounts for vehicle motions into the cavity, and this planing provides a damping-like component in the planing force formulation. Modeled with non-steady planing is a physical time delay relating to the fact that the cavity, where planing occurs, is based on the previous cavitator position and orientation data. This delay is found to be stabilizing for certain values of speed. Maneuvering is considered by using inner-loop and outer-loop control schemes. A feedback inner-loop scheme helps reject fast planing instabilities, while a numeric optimal control approach is used to generate outer-loop commands to guide the vehicle through desired maneuvers. The maneuvers are considered for operations with tight body to cavity clearance, and in which planing is prevalent. Simple search algorithms along with a penalty method for handling the constraints are found to work the best due to the complexity of the non-smooth system dynamics.