Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations

Abstract

Consider a rectangular tank containing an inviscid incompressible and irrotational fluid. The tank is subject to a one-dimensional horizontal move (the control) and the motion of the fluid is described by the shallow water equations. By means of a Lyapunov approach, control laws that stabilize the state of the fluid–tank system are derived. Two classes of control are considered: full-state feedback and output feedback where the output is given by the trajectory of the tank, the level of the fluid at the boundary of the tank and the time. Although global asymptotic stability is yet to be proved, stabilization is observed through numerical simulations.

Introduction

Consider a tank containing an inviscid incompressible irrotational fluid. The tank is subject to a one-dimensional horizontal move. To move such a tank we need to take the motion of the fluid into account. Several recent publications deal with this question, see, e.g., [8], [12], [24], [25]. This paper is a first attempt to study the stabilization problem with the model of the shallow water equations which are 1D-hyperbolic equations (see e.g. [3], [4]).

Our main concern is the fluid state stabilization problem (level and speed relative to the tank) and the tracking problem of the tank state (position, speed and acceleration) to a prescribed trajectory (e.g. a prescribed final position of the tank) using the acceleration as the control variable.

Stabilizing feedback laws are designed using a Lyapunov approach and backstepping (see e.g. [14] for an introduction of this technique). The design process is repeated iteratively on control problems that have increasing complexity. For each control problem, an augmented Lyapunov function is built from the previous (simpler) problem and the corresponding stabilizing control laws are deduced. More specifically, the control “sub”-problems are

• fluid state stabilization (Section 3.1),

• fluid state and tank speed stabilization (Section 3.2),

• fluid–tank state stabilization (Section 3.3) where a forward approach (see [18]) is used to design the Lyapunov function.

Two classes of stabilizing control laws are investigated: (1) time-varying full-state feedbacks and (2) output feedbacks, where the output is defined by the trajectory of the tank, the level of the fluid at the boundary of the tank and the time. Many practical and industrial motivations can be found in [11], [12], [19] for restricting ourselves to output feedbacks.

Some results can be found in [18] concerning the problem of the stabilization of a tank, but the input is defined as a flexible or a rigid wave generator and the equations are linearized around the equilibrium. Here, a different model of the control system is chosen. Moreover, the linearized shallow water are not stabilizable even locally (see [7]), thus a study of the non-linear equations is needed. For these non-linear equations, it is proved in [3] that one has a local controllability property.

Several other configurations are studied in [20] and it is proved that, for each configuration under consideration, the linear approximation is steady-state controllable. For our configuration, the linear approximation is not controllable, and thus we need to consider the non-linear equations to study the stabilization problem.

The paper is organized as follows. The shallow water equations are described in Section 2.1, the steady states in Section 2.2 and the stabilization problem in Section 2.3. The existence of Lyapunov functions and feedbacks are investigated in Section 3. At last, numerical simulation are used to check that asymptotic stabilization is achieved in Section 4.