Title: Optimal input design for system identification using spectral decomposition,
Abstract: 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. It is
assumed that the nominal system parameters are specified. The key idea is to
use the spectral decomposition theorem and write the power spectrum as
$\phi_{u}(j\omega)=\frac{1}{2}H(j\omega)H^*(j\omega)$. The matrix $H(j\omega)$
is expressed in terms of a truncated basis for
$\mathcal{L}^2\left(\left[-\omega_{\mbox{cut-off}},\omega_{\mbox{cut-off}}\right]\right)$.
With this parameterization, the elements of the Fisher Information Matrix and
the power constraints turn out to be homogeneous quadratics in the basis
coefficients. The optimality criterion used are the well-known
$\mathcal{D}-$optimality, $\mathcal{A}-$optimality, $\mathcal{T}-$optimality
and $\mathcal{E}-$optimality. The resulting optimization problem is non-convex
in general. A lower bound on the optimum is obtained through a bi-linear
formulation of the problem, while an upper bound is obtained through a convex
relaxation. These bounds can be computed efficiently as the associated problems
are convex. The lower bound is used as a sub-optimal solution, the
sub-optimality of which is determined by the difference in the bounds.
Interestingly, the bounds match in many instances and thus, the global optimum
is achieved. A discussion on the non-convexity of the optimization problem is
also presented. Simulations are provided for corroboration.