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, with nominal parameter values specified. Using spectral decomposition theorem, the power spectrum is written as ${\phi }_u\left(\mathrm{j}\omega \right)=\frac{1}{2}H\left(\mathrm{j}\omega \right)H^{\ast }\left(\mathrm{j}\omega \right)$. The matrix $H\left(\mathrm{j}\omega \right)$ is expressed in terms of a truncated basis for ${\mathcal{L}}^2\left(\right[-{\omega }_c,{\omega }_c\left]\right)$, where ${\omega }_c$ 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 $\mathcal{D}$-optimality, $\mathcal{A}$-optimality, $\mathcal{T}$-optimality and $\mathcal{E}$-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.

KEYWORDS:

• System identification
• optimal input design
• spectral decomposition
• Fisher information matrix
• convex optimisation

Disclosure statement

No potential conflict of interest was reported by the authors.