Feedback Stabilization of Tank-Liquid System with Robustness to Wall Friction

Abstract

We solve the feedback stabilization problem for a tank, with friction, containing a liquid modeled by the viscous Saint-Venant system of Partial Differential Equations. A spill-free exponential stabilization is achieved, with robustness to the wall friction forces. A Control Lyapunov Functional (CLF) methodology with two different Lyapunov functionals is employed. These functionals determine specific parameterized sets which approximate the state space. The feedback law is designed based only on one of the two functionals (which is the CLF) while the other functional is used for the derivation of estimates of the sup-norm of the velocity. The feedback law does not require the knowledge of the exact relation of the friction coefficient. Two main results are provided: the first deals with the special case of a velocity-independent friction coefficient, while the second deals with the general case. The obtained results are new even in the frictionless case.

Spill-Free Transfer and Stabilization of Viscous Liquid

“…Two cases have been studied in the literature (see [33,35,36]): the case where friction is present and surface tension is absent and the case where surface tension is present and friction is absent.…”

“…More specifically, in [33,34] a feedback control law is constructed by employing the Control Lyapunov Functional (CLF) methodology for the viscous Saint-Venant model without wall friction and surface tension (state feedback in [33] and output feedback in [34]). In [35,36] it is shown that the same feedback control law proposed in [33] works even if friction forces or surface tension are present. It should be noticed that [33,34,35,36] are the only works that guarantee a spill-free movement of the fluid.…”

“…In [35,36] it is shown that the same feedback control law proposed in [33] works even if friction forces or surface tension are present. It should be noticed that [33,34,35,36] are the only works that guarantee a spill-free movement of the fluid.…”

“…However, when one studies the flow in a tank there is no guarantee that the effect of viscosity is negligible because the velocity of the fluid is (expected to be) relatively small. To this purpose, viscous Saint-Venant models have been proposed and studied in [6,7,22,33,34,35,36,39,45,53]. From a mathematical point of view the effect of the viscosity is huge: in the case where surface tension is absent the system is described by two Ordinary Differential Equations (ODEs), one firstorder hyperbolic Partial Differential Equation (PDE) and one parabolic PDE, whereas in the inviscid case we have two ODEs and two first-order hyperbolic PDEs.…”

“…Thus, the solution of the spill-free and slosh-free movement problem by means of a robust feedback law becomes a significant mathematical problem with possibly important applications. The recent works [33,34,35,36] study this particular feedback stabilization problem for the viscous Saint-Venant model without linearization around an equilibrium point. More specifically, in [33,34] a feedback control law is constructed by employing the Control Lyapunov Functional (CLF) methodology for the viscous Saint-Venant model without wall friction and surface tension (state feedback in [33] and output feedback in [34]).…”