Control by Interconnection and Energy-Shaping Methods of Port Hamiltonian Models. Application to the Shallow Water Equations

Abstract

Abstract-In this paper a control algorithm for the reduced port-Controlled Hamiltonian model (PCH) of the shallow water equations (PDEs) is developed. This control is developed using the Interconnection and Damping Assignment Passivity Based Control (IDA-PBC) method on the reduced PCH model without the natural dissipation. It allows to assign desired structure and energy function to the closed loop system. The same control law is then derived using an energy shaping method based on Casimir's invariants, associated with a particular conservative interconnection between the boundary variables. This gives a physical interpretation for the designed controller. Finally, a stability analysis of the dissipative system in closed loop with the designed control is done using LaSalle's invariance principle. Simulation results and an experimental validation of the control algorithm on an a micro-canal platform are presented showing the effectiveness of the control law.

A partitioned finite element method for power-preserving discretization of open systems of conservation laws

“…The SWEs are sets of partial differential equations that can be used to represent an incompressible fluid with free-surface motion. These equations are typically used to model fluid motion in water channels (Hamroun et al, 2010), wave propagations in oceans and lakes and sloshing in fluid tanks (Alemi Ardakani, 2016;Cardoso-Ribeiro et al, 2017, 2020c. When one considers the frictionless flow in a horizontal channel with uniform rectangular cross-section, the one-dimensional mass and momentum balance equations may be written as…”

An operator theoretic approach to infinite‐dimensional control systems

“…For our class of PDEs we have a natural energy function, see (5). The domain should be a part of the state space X, which we identify next.…”

“…The domain should be a part of the state space X, which we identify next. For our class of PDEs we have a natural energy function, see (5). Hence, it is quite natural to consider only states which have a finite energy.…”

“…This led to the class of port-Hamiltonian systems, see van der Schaft [1] and van der Schaft and Maschke. [2] The Hamiltonian/energy has been used to control a port-Hamiltonian system, see, for example, Baaiu et al, [3] Cervera et al, [4] Hamroun et al [5] and Ortega et al [6] For port-Hamiltonian systems described by ordinary differential equations this approach is very successful, see the references mentioned above. Port-Hamiltonian systems described by PDE is a subject of current research, see, for example, Eberard et al, [7] Jeltsema and van der Schaft [8] , Kurula et al, [9] Macchelli and Macchelli.…”

An Improvement of Port-Hamiltonian Model of Fluid Sloshing Coupled by Structure Motion

“…Such simple models are also used for the control problems for water-tank systems [26], dynamic coupling between shallow-water sloshing and horizontal vehicle motion [35], sloshing in vessels undergoing prescribed rigid-body motion in three dimensions [36], and stabilization of a one-dimensional tank containing a fluid modeled by the shallow water equations [29]. The shallow water equation is also recently studied in the port-Hamiltonian framework but with a different goal: modeling and controlling the flow on open channel irrigation systems [37,38]. In this case, there is neither rotation nor translation of the channels.…”