Small Noise Analysis of Non-Parametric Closed-Loop Identification

Abstract

We revisit the problem of non-parametric closed-loop identification in frequency domain; we give a brief survey of the literature and provide a small noise analysis of the direct, indirect, and joint input-output methods when two independent experiments with identical excitation are used. The analysis is asymptotic in the noise variance (i.e., as the standard deviation of the innovations $σ\to 0$), for a finite data record of length $N$. We highlight the relationship between the estimators accuracy and the loop shape via asymptotic variance expressions given in terms of the sensitivity function. The results are illustrated using a numerical simulation example.

Figures (4)

• Figure 1: Closed-loop configuration
• Figure 2: Variance comparison, together with $|G|$, $|S|$ and the noise spectrum. Sample variances are computed using $10^4$ MC simulations.
• Figure 3: The absolute value of the difference between the actual (non-asymptotic) variances as approximated via Monte Carlo simulations, and the asymptotic variances. We also show the noise spectrum.
• Figure 4: $\Re[GC]$ versus frequency: The sign of $\Re[GC]$ switches to negative at $\omega = 0.64$ rad/sample. This is also the frequency around which the variance of $\overline{ \widehat{\newline { \widehat{G} }} _\text{io}}$ goes above that of $\overline{\widehat{G}_\text{dir}}$.

Theorems & Definitions (4)

• proof
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