Stabilization of the Linearized Water Tank System

Coron, Jean‐Michel; Hayat, Amaury; Xiang, Shengquan; Zhang, Christophe

doi:10.1007/s00205-022-01778-0

Cited by 9 publications

(8 citation statements)

References 60 publications

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“…Control studies of the Saint-Venant model have focused on the inviscid Saint-Venant model (i.e., the model that ignores viscous stresses and surface tension) and its linearization around an equilibrium point (see [3,4,5,12,13,14,15,17,18,19,20,21,25,26,42,47,48]). The feedback design has been performed by employing either the backstepping methodology or the Control Lyapunov Functional (CLF) methodology.…”

Section: Tank-liquid System: Application Modeling and Controlmentioning

confidence: 99%

Spill-Free Transfer and Stabilization of Viscous Liquid

Karafyllis

Krstić

2022

2022 American Control Conference (ACC)

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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.

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Section: Tank-liquid System: Application Modeling and Controlmentioning

confidence: 99%

Spill-Free Transfer and Stabilization of Viscous Liquid

Karafyllis

Krstić

2022

2022 American Control Conference (ACC)

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“…v(t, 0) = v(t, L) = 0, for t ≥ 0 (10) where the control input f appears additively in the second equation of (7) and multiplicatively in (9). Moreover, the conditions (5) for avoiding the liquid spilling out of the tank become: max (h(t, 0), h(t, L)) < H max , for t ≥ 0 ---Condition for no spilling out (11) We consider classical solutions for the PDE-ODE system ( 7)- (10), i.e., we consider functions ξ…”

Section: Description Of the Problemmentioning

confidence: 99%

“…ξ ∈ R, w = 0 (14) where h * = m/L. We assume that the equilibrium points satisfy the conditions for no spilling out (11), i.e., h * < H max .…”

Section: Description Of the Problemmentioning

confidence: 99%

“…which achieves stabilization of the equilibrium point with ξ = 0. Moreover, we additionally require that the "spill-free condition" (11) holds for every t ≥ 0. The existence of a continuum of equilibrium points for the open-loop system given by ( 13), (14) implies that the desired equilibrium point is not asymptotically stable for the open-loop system.…”

Section: Description Of the Problemmentioning

confidence: 99%

“…In order to be able to satisfy the conditions for no spilling out (11) we need to restrict the state space. This becomes clear from the fact that the set S contains states that violate the conditions for no spilling out (11). Define…”

Section: The State Space and Useful Subsets Of The State Spacementioning

confidence: 99%

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

Karafyllis¹,

Vokos²,

Krstić³

2022

ESAIM: COCV

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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.

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