Nonlinear optimal control for the dynamics of the underactuated slosh-container system

Abstract

The problem of stabilization of the dynamics of the slosh-container system is important for the safe execution of several industrial tasks and for the safe navigation of vessels, satellites and spacecrafts that incorporate fuel containers (tanks). Actually, the oscillations of fuel in partially filled containers (tanks) can affect the motion of such vessels and aerospace systems and can result in significant deviations from their targeted paths. The control problem of the slosh-container system is non-trivial because this system is highly nonlinear and underactuated. The present article proposes a nonlinear optimal control approach for the dynamics of the slosh-container system. It is proven that the dynamics of the slosh-container system can be substituted by the equivalent dynamics of a cart and pendulum system. Using Euler-Lagrange analysis the state-space model of the slosh-container system is obtained. This state-space description undergoes approximate linearization with the use of Taylor series expansion around a temporary operating point which is recomputed at each iteration of the control method. For the approximately linearized model, an H-infinity feedback controller is designed. The linearization procedure relies on the computation of the Jacobian matrices of the state-space model of the slosh-container system. The proposed control method stands for the solution of the optimal control problem for the nonlinear and multivariable dynamics of the slosh-container system, under model uncertainties and external perturbations. To find the controller’s feedback gains an algebraic Riccati equation is solved at each time-step of the control method. The new nonlinear optimal control approach achieves fast and accurate tracking of setpoints for all state variables of the slosh-container system, under moderate variations of the control inputs. The stability properties of the control scheme are proven through Lyapunov analysis.