Robust optimal control of one-reach open-channels

Journal of Computational Physics

Maschke

2012

International audienceA reduction method is presented for systems of conservation laws with boundary energy flow. It is stated as a generalized pseudo-spectral method which performs exact differentiation by using simultaneously several approximation spaces generated by polynomials bases and suitable choices of port-variables. The symplecticity of this spatial reduction method is proved when used for the reduction of both closed and open systems of conservation laws, for any choice of collocation points (i.e. for any polynomial bases). The symplecticity of some more usual collocation schemes is discussed and finally their accuracy on approximation of the spectrum, on the example of the ideal transmission line, is discussed in comparison with the suggested reduction scheme

Section: Restricted Bilinear Product and Stokes' Theoremmentioning

confidence: 99%

“…12 Proposition 18. The subspace (38) satisfies the isotropy condition: D r ⊂ D ⊥ r with respect to the symmetric power product (39).…”

Section: Restricted Bilinear Product and Stokes' Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws

Moulla

Journal of Computational Physics

Maschke

2012

“…These collocation (polynomial) methods are the most used in process engineering either for processes simulation or for control. Indeed, the nice properties of collocation methods make them also very effective to develop control laws on the reduced model [16].…”

Section: Introductionmentioning

confidence: 99%

Geometric pseudospectral method for spatial integration of dynamical systems

Moulla

Mathematical and Computer Modelling of Dynamical Systems

Maschke

2011

A reduction method that preserves geometric structure and energetic properties of non-linear distributed parameter systems is presented. It is stated as a general pseudospectral method using approximation spaces generated by polynomials bases. It applies to Hamiltonian formulations of distributed parameter systems that may be derived for hyperbolic systems (wave equation, beam model, shallow water model) as well as for parabolic ones (heat or diffusion equations, reaction-diffusion models). It is defined in order to preserve the geometric symplectic interconnection structure (Stokes-Dirac structure) of the infinite-dimensional model by performing exact differentiation and by a suitable choice of port variables. This leads to a reduced port-controlled Hamiltonian finite-dimensional system of ordinary differential equations. Moreover, the stored and dissipated power in the reduced model are approximations of the distributed ones. The method thus allows the direct use of thermodynamics phenomenological constitutive equations for the design of passivity-based or energy-shaping control techniques.

“…Most of them are based on reduced models of the shallow water equations. Some works are based on continuous time reduced model obtained by the orthogonal collocation method as in [7] (input-output linearization), in [8] (backstepping) or in [9] (robust optimal control). Other are developed on discrete time models obtained using the Preissmann implicit finite differences scheme [10] as in [11] (predictive control) or in [12] (optimal control).…”

Section: Introductionmentioning

confidence: 99%

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

Hamroun

Dimofte

European Journal of Control

et al. 2010

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.