Central extensions in closed-loop optimal experiment design

We consider optimal experiment design for parametric prediction error system identification of linear time-invariant multi-input multi-output systems in closed-loop when the true system is in the model set. The optimization is performed jointly over the controller and the spectrum of the external input. Previously we tackled this problem by parametrizing the set of admissible controller - external input pairs by a finite set of matrix-valued trigonometric moments and derived a description of the set of admissible finite-dimensional moment vectors by a linear matrix inequality. Here we present a way to recover the controller and the power spectrum of the external input from the optimal moment vector. To this end we prove that the central extension of the finite moment sequence yields a feasible solution. This yields the joint power spectrum of the input and the noise vector as an explicit rational function and allows to construct the optimal “controller - external input pair” directly from the optimal moment vector.