Uniformly Exponentially Stable Summation-by-Parts Approximations of port-Hamiltonian Systems

For distributed parameter systems that are
exponentially stable, it is desirable that an approximation
scheme is not only dissipative but also preserves exponential
stability independently of approximation order. This
will lead to realistic simulations and is also needed for
early lumped controller design. In this paper, this issue
is addressed for networked port-Hamiltonian systems. It
is shown that summation-by-parts (SBP) approximation
schemes lead to such a property, provided that the boundary
conditions are also handled appropriately. The theoretical
framework developed in this paper result in schemes
with spectral convergence and provable uniform exponential
stability for higher-order approximations of port-
Hamiltonian systems. A final section illustrates the efficacy
of a higher-order polynomial scheme over a mixed finiteelement
method for approximating a piezoelectric beam
with magnetic effects. The characteristics of the proposed
discretization, high accuracy with low-order approximation
models and uniform conservation of the exponential stability
properties, are suitable for controller synthesis.