Motion planning and nonlinear simulations for a tank containing a fluid

We consider a tank containing a fluid. The tank is subjected to a one-dimensional horizontal move and the motion of the fluid is described by Saint-Venant´s equations. We show how to parameterize the trajectories of the linearized system thanks to the horizontal coordinate of a particular point in the system - the “flat output”, see figure 2- and a periodic function. The motion planning problem of the linearized model is solved in the general case of joining two steady states. Next we provide an algorithm, based on Godunov scheme, with a dedicated way of dealing with boundary conditions, to numerically simulate the evolution of the nonlinear system. Nonlinear simulations provide a way of checking the accuracy of the motion planning based on the tangent linear system.