Matrix-valued Impedances with Fractional Derivatives and Integrals in Boundary Feedback Control: a port-Hamiltonian approach

This paper discusses the passivity of the port-Hamiltonian formulation of a multivariable impedance matching boundary feedback of fractional order, expressed through diffusive representation. It is first shown in the 1D-wave equation case that the impedance matching boundary feedback can be written as a passive feedback on the boundary port variables. In the Euler-Bernoulli case, the impedance matching feedback matrix involves fractional derivatives and integrals. It is shown that the usual diffusive representation of such feedback is not formally a dissipative port-Hamiltonian system, even if from a frequency point of view this feedback proves passive.