Geometric Pseudo-Spectral Method for Spatial Integration of Dynamical Systems

Abstract : A reduction method which preserves geometric structure and energetic properties of non linear distributed parameter systems is presented. It is stated as a general pseudo-spectral method using approximation spaces generated by polynomials bases. It applies to Hamil-tonian formulations of distributed parameter systems which may be derived for hyperbolic systems (wave equation, beam model, shallow water model) as well as for parabolic ones (heat or diusion equations, reaction-diusion models). It is dened in order to preserve the geometric symplectic interconnection structure (Stokes-Dirac structure) of the innite dimensional model by performing exact dierentiation and by a suitable choice of port-variables. This leads to a reduced port-controlled Hamiltonian nite-dimensional system of ordinary dierential 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 thermody-namics phenomenological constitutive equations for the design of passivity-based or energy shaping control techniques.
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Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis, 2011, 17 (1), pp.85 - 104. 〈10.1080/13873954.2010.537524〉
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Redha Moulla, Laurent Lefevre, Bernhard Maschke. Geometric Pseudo-Spectral Method for Spatial Integration of Dynamical Systems. Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis, 2011, 17 (1), pp.85 - 104. 〈10.1080/13873954.2010.537524〉. 〈hal-01625016〉

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