  Using Hermite's formulation of polynomial stability conditions, static output
feedback (SOF) controller design can be formulated as a polynomial matrix
inequality (PMI), a (generally nonconvex) nonlinear semidefinite programming
problem that can be solved (locally) with PENNON, an implementation of a
penalty method. Typically, Hermite SOF PMI problems are badly scaled and
experiments reveal that this has a negative impact on the overall performance
of the solver. In this note we recall the algebraic interpretation of Hermite's
quadratic form as a particular Bezoutian and we use results on polynomial
interpolation to express the Hermite PMI in a Lagrange polynomial basis, as an
alternative to the conventional power basis. Numerical experiments on benchmark
problem instances show the substantial improvement brought by the approach, in
terms of problem scaling, number of iterations and convergence behavior of
PENNON.
