Closed-loop optimal experiment design: The partial correlation approach

We consider optimal experiment design for parametric prediction error system identification of linear time-invariant systems in closed loop. The optimisation is performed jointly over the controller and the external input. We use a partial correlation approach, i.e. we parameterize the set of “admissible controller”-“external input” pairs by a finite set of matrix-valued trigonometric moments. Our main contribution is twofold. First we derive a description of the set of admissible finite-dimensional moments by a linear matrix inequality. Optimal input design problems with semi-definite constraints and criteria which are linear in these moments can then be cast as semi-definite programs and solved by standard semi-definite programming packages. Secondly, we develop algorithms to recover the controller and the power spectrum of the external input from the optimal moment vector. This furnishes the user a complete and very general procedure to solve the input design problems of the considered class. Our results can be applied to multi-input multi-output systems, but for pedagogical reasons we present here the single-input single-output case. We also assume that the true system is in the model set.