Passivity-preserving model reduction by analytic interpolation

Model Reduction by Moment Matching for Linear and Nonlinear Systems

“…The interpolation problem has been also extensively studied by Byrnes, Georgiou and Lindquist (and co-workers), see e.g. [55], [56], [57], [58], [59], [60], that have developed a complete theory for analytic interpolation with degree constraint. Note that, the interpolation problem does not have direct counterpart for nonlinear systems.…”

Dissipativity preserving model reduction by retention of trajectories of minimal dissipation

“…Of course, the results in our paper are valid for general supply rates. In our paper we show, a fortiori, that the transfer matrices of our reduced order behaviors are solutions of certain tangential Nevanlinna interpolation problems, with interpolation points at the spectral zeros of the original behavior (see also [11]). …”

“…Since, by constructionLX = L X andLŶ = LY , the latter is indeed equal to B given by (11). Thus we have shown that the method of Sorensen computes for the given k-dimensional subbehavior B (corresponding to his k-dimensional H -invariant subspace im X Y ) a reduced order systemB that satisfies the four conditions of our Main Problem.…”

“…This already appeared to be the case in Example 3.9. It will turn out that any of the approximantsB that we will obtain as solution to our problem allows an interpretation as a solution of a rational interpolation problem, with as interpolation points some of the antistable spectral zeros together with their mirror images in the imaginary axis, see also [2] and [11].…”

“…Subsequently, we will show that a certain transfer matrix associated with our reduced order behavior is in fact a solution of a Nevanlinna-type tangential interpolation problem (see also [11]). In fact, the transfer matrix of the reduced order behavior will turn out to interpolate the transfer matrix of the original behavior in certain directions, with interpolation points at some of the spectral zeros and their mirror images in the imaginary axis of the original behavior.…”