Spill-Free Transfer and Stabilization of Viscous Liquid

Abstract

Flow control occupies a special place in the fields of partial differential equations (PDEs) and control theory, where the complex behavior of solutions of nonlinear dynamics in very high dimension is not just to be understood but also to be assigned specific desired properties, by feedback control. Among several benchmark problems in flow control, the liquid-tank problem is particularly attractive as a research topic. It is among the few hard problems in PDE control where the solutions are readily comprehensible-with feedback laws that are relatively simple in form, with a clear physical meaning.In the liquid-tank problem the objective is to move a tank filled with liquid, suppress the nonlinear oscillations of the liquid in the process, bring the tank and liquid to rest, and avoid liquid spillage in the process. In other words, this is a problem of nonlinear PDE stabilization subject to state constraints.This review article focuses only on recent results on liquid-tank stabilization for viscous (incompressible, Newtonian) liquids, as viscosity makes the earlier control designs for inviscid Saint-Venant models inapplicable. All possible cases are studied: with and without friction from the tank walls, with and without surface tension. Moreover, results that have not been previously published are provided for the linearization of the tank-liquid system. The linearization of the tank-liquid system gives a high-order PDE which is a combination of a wave equation with Kelvin-Voigt damping and an Euler-Bernoulli beam equation.The feedback design methodology presented in the article is based on Control Lyapunov Functionals (CLFs), suitably extended from the CLF methodology for ODEs to the infinite-dimensional case. The CLFs proposed are modifications and augmentations of the total energy functionals for the tank-liquid system, so that the dissipative effects of viscosity, friction, and surface tension are captured and additional dissipation by feedback is made relatively easy.The article closes with an extensive list of open problems.

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

“…The problem studied in the present work is similar to the problem added in [22] and consequently there are some similarities between the present work and [22] (there are less similarities with [23] since [23] focuses on output feedback laws and numerical aspects). However, there are important differences between the present work and [22]. Indeed, the absence of wall friction in [22] makes the analysis simpler.…”

“…The feedback design is performed by employing either the backstepping methodology (see [13,14]) or the Control Lyapunov Functional (CLF) methodology (see [2-4, 9-12, 15, 18, 19, 29]). The only works that study the feedback stabilization problem for the viscous Saint-Venant model without linearization around an equilibrium point are [22,23]. More specifically, in [22,23] a feedback control law is constructed by employing the CLF methodology for the viscous Saint-Venant model without wall friction and surface tension (state feedback in [22] and output feedback in [23]).…”

“…The only works that study the feedback stabilization problem for the viscous Saint-Venant model without linearization around an equilibrium point are [22,23]. More specifically, in [22,23] a feedback control law is constructed by employing the CLF methodology for the viscous Saint-Venant model without wall friction and surface tension (state feedback in [22] and output feedback in [23]). In addition [22,23] are the only works that guarantee a spill-free movement of the fluid.…”

“…More specifically, in [22,23] a feedback control law is constructed by employing the CLF methodology for the viscous Saint-Venant model without wall friction and surface tension (state feedback in [22] and output feedback in [23]). In addition [22,23] are the only works that guarantee a spill-free movement of the fluid.…”

Event-Triggered Safe Stabilizing Boundary Control for the Stefan PDE System with Actuator Dynamics

“…While most of the research in safe/constrained control has focused on the systems described by Ordinary Differential Equations (ODEs), a few recent works have been developing and analyzing the safety or state constraints in the systems described by Partial Differential Equations (PDEs), where the safety in the infinite-dimensional state needs to be satisfied, such as for distributed concentration [8], gas density [9], or the liquid level [10]. Our recent work [11] has incorporated the concept of CBF into the boundary control of a PDE system, the so-called "Stefan system" [12], [13], which is a representative model for the thermal melting process [14] and biological growth process [15].…”