Weight Selection in Interpolation with a Dimensionality Constraint

Abstract

The topic of the paper relates to a recent parametrization of analytic interpolants with a bound on their dimension, as solutions to certain weighted entropy minimization problems. The analytic interpolation problem arises in the context of shaping closed-loop transfer functions via a suitable choice of controller. Our goal is to shed light on how the choice of weights affects the shape of the corresponding closed-loop transfer functions. Further, given a desirable shape, we indicate how a suitable weight can be obtained as a solution of a certain quasi-convex problem.

Sensitivity shaping for systems with time delay by Nevanlinna-Pick interpolation

“…This theory has been applied to the unweighted sensitivity shaping problem [14], [8], [20]. The shaping of the closed loop systems is handled by tuning the spectral zeros of interpolants given by σ(s) in (15).…”

“…In fact, the achievable minimum of the H ∞ norm of the sensitivity function by feedback control was only discussed by using the singularity of the Pick matrix [6]. However, in the context of the theory of the Nevanlinna-Pick interpolation with degree constraint [1], [2], some potentials of the unweighted H ∞ sensitivity shaping have been shown in [14], [8], [20]. The sensitivity function can be shaped by tuning the spectral zeros of the sensitivity function, which completely parametrize the sensitivity function in a smooth way [1].…”

Sensitivity shaping with degree constraint via convex optimization

“…This is admittedly somewhat ad-hoc. An alternative approach presented in [19] leads to a systematic design methodology.…”