Incorporating Fixed-Pole Information in the Data-Driven Least Squares Realization Problem

Abstract

In practical least squares realization problems, partial information about the pole locations of the dynamical model may be known a priori. Existing techniques for incorporating this prior knowledge, such as prefiltering the given data, are typically heuristic and lack theoretical guarantees. We extend our previously developed globally optimal estimation approach to accommodate fixed poles in the least squares realization problem. In particular, we reformulate the problem as a (rectangular) multiparameter eigenvalue problem, the eigenvalues of which characterize all local and global minimizers of the constrained estimation problem. We present numerical examples to demonstrate the effectiveness of the proposed method and experimentally validate the paper's central hypothesis: incorporating a priori information on the poles enhances the estimation results.

Figures (3)

• Figure 1: Misfit $\left\Vert \widetilde{\bm{y}} \right\Vert_2^2$ of the least squares realization problem in \ref{['ex:motivational']} plotted against the model parameters $a_1$ and $a_2$, normalized by setting $a_0 = 1$. The surface shows the non-convex nature of the underlying optimization problem, with the critical points of the standard least squares objective function being minimizers ( \ref{['plot:std-minimum']} ), saddle points ( \ref{['plot:std-saddlepoint']} ), and maximizers ( \ref{['plot:std-maximum']} ). Given a fixed pole, the fixed-pole least squares realization problem is subject to an additional constraint (\ref{['plot:constraint']}). A different minimizer ( \ref{['plot:fp-minimum']} ) and maximizer ( \ref{['plot:fp-maximum']} ) are obtained in this example. The two heuristic solutions discussed in \ref{['ex:motivational']} are also shown ( \ref{['plot:heuristics']} ).
• Figure 2: Analysis of the model-compliant output data $\widehat{\bm{y}}_{}^{(i)}$ for the different realization experiments, $i = 1, 2, \ldots, 50$, in \ref{['ex:statistics']}. One pole of a third-order model is estimated by the standard (S-GOR) and fixed-pole (FP-GOR) realization approach from given output data $\bm{y}_{}\in \mathbb{R}^{15.0}$, which is generated using \ref{['eq:givendata']} for noise levels $\sigma \in \{0.05, 0.15, \ldots, 0.45\}$. From \ref{['fig:statistics:costvalues']} it seems that the S-GOR technique (\ref{['plot:sgor']}) results in a smaller misfit $\Vert\bm{y}_{}^{(i)} - \widehat{\bm{y}}_{}^{(i)}\Vert_2^2$ than the FP-GOR technique (\ref{['plot:fpgor']}). However, the latter obtains better results when comparing $\widehat{\bm{y}}_{}^{(i)}$ with the exact data $\bm{x}_{}$ generated by the underlying model, as shown in \ref{['fig:statistics:exactvalues']}.
• Figure 3: Cross-validation for noise level $\sigma = 0.25$. It visualizes the difference between S-GOR and FP-GOR, $d = \Vert\widetilde{\bm{y}}^{(i,j)}_\text{S-GOR}\Vert_2^2 - \Vert\widetilde{\bm{y}}^{(i,j)}_\text{FP-GOR}\Vert_2^2$, of the misfit $\Vert\widetilde{\bm{y}}^{(i,j)}\Vert_2^2 = \Vert\bm{y}_{}^{(i)} - \widehat{\bm{y}}_{}^{(j)}\Vert_2^2$ for all values $i,j = 1, 2, \ldots, 50$. This qualitative measure visualizes when the model obtained by FP-GOR describes the given data better ($d > 0$) or worse ($d < 0$) than S-GOR for other realizations ($i \neq j$). On the off-diagonal, the FP-GOR misfit is generally smaller than the S-GOR misfit, indicating a better generalization.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Example 1: Motivational example
• Lemma 1
• proof
• Theorem 1
• proof
• Remark 1
• Remark 2
• Example 2: \ref{['ex:motivational']} continued