Optimal input design for system identification using spectral decomposition

The aim of this paper is to design a band-limited optimal input with power constraints for identifying a linear multi-input multi-output system, with nominal parameter values specified. Using spectral decomposition theorem, the power spectrum is written as (Formula presented.). The matrix (Formula presented.) is expressed in terms of a truncated basis for (Formula presented.), where (Formula presented.) is the cut-off frequency. The elements of the Fisher Information Matrix and the power constraints become homogeneous quadratics in basis coefficients. The optimality criterion used are (Formula presented.) -optimality, (Formula presented.) -optimality, (Formula presented.) -optimality and (Formula presented.) -optimality. This optimization problem is not known to be convex. A bi-linear formulation gives a lower bound on the optimum, while an upper bound is obtained through a convex relaxation. These bounds can be computed efficiently. The lower bound is used as a suboptimal solution, its sub-optimality determined by the difference between the bounds. Simulations reveal that the bounds match in many instances, implying global optimality.