Application of stochastic optimal control to a hydrofoil boat problem

In this paper the solution of a stochastic optimal control problem described by linear equations of motion and a nonquadratic performance index is presented. The theory is then applied to the dynamics of a single-foil and a hydrofoil boat flying on rough water. The random disturbances caused by sea waves are represented as the response of an auxiliary system to a white noise input. The control objective is formulated as an integral performance index containing a quadratic acceleration term and a nonquadratic term of the submergence deviation of the foil from calm water submergence. The stochastic version of the maximum principle is used in the formulation of a feedback control law. The Riccati equations and the feedback gains associated with a nonquadratic performance index are non-linear functions of the state and auxiliary state variables. These equations are integrated forward with the state equations for the steady-state solution of the problem. The controller for a nonquadratic performance index contains computing elements which perform the integration of the Riccati equations to generate the instantaneous values of the feedback gains. The effect of a nonquadratic penalty on the submergence deviation and the effect of a nonquadratic control penalty on the response of the system are investigated. A comparison between an optimal nonlinear control law and a suboptimal linear control law is presented.