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.

Boundary Energy-Shaping Control of the Shallow Water Equation

“…Moreover, it is assumed that the fluid has a unitary density. For simplicity, we are in fact considering a simpler situation than the one treated in Hamroun et al [2010]. However, all the results discussed here can be easily extended to cope with more general cases.…”

“…The dynamics of the system are described by the so-called shallow water equations, whose port-Hamiltonian formulation has been extensively discussed e.g. in Pasumarthy et al [2008], Hamroun et al [2010].…”

“…Another problem, treated e.g. in Hamroun et al [2010] but in the finite dimensional case only, is to stabilise the channel around a desired water flow δ p H(q , p ) = Q , somehow independently from the hydrodynamic pressure P , which is not constant on Z in case internal friction forces are present, i.e. when D(q, p) = 0.…”

“…ξ 1 = Lq , and ξ 2 = Lp , while K 1 , K 2 are two positive gains. Note the similarities with Hamroun et al [2010], where a finite-element approximation of the channel dynamics has been used.…”

“…In this respect, the finite dimensional formulation of the shallow water equation adopted in Hamroun et al [2010] follows from the general discretisation technique for distributed port-Hamiltonian systems discussed in Golo et al [2004]. It has been shown in Macchelli [2011] that for the "full-order" system, i.e.…”