Finite Sample MIMO System Identification with Multisine Excitation: Nonparametric, Direct, and Two-step Parametric Estimators

TL;DR

This work develops a rigorous finite-sample framework for MIMO continuous-time system identification under multisine excitation. By constructing a multisine equivalent discrete-time FIR model, it derives exact distributional properties and covariance of the nonparametric FRF estimator, accounts for aliasing under slow sampling, and establishes unbiasedness and weak consistency conditions. It then shows that the FRF estimate is a sufficient statistic for any parametric model with Gaussian noise, yielding an exact equivalence between optimal two-step frequency-domain PEM and time-domain prediction-error/ML methods for finite data and enabling closed-form ML solutions in favorable cases. The paper further provides finite-sample concentration bounds for both FRF and parameter estimates, analyzes LMFD parametrizations for direct and two-step estimators, and demonstrates the results through a representative case study with a finite-sample lens.

Abstract

Multisine excitations are widely used for identifying multi-input multi-output systems due to their periodicity, data compression properties, and control over the input spectrum. Despite their popularity, the finite sample statistical properties of frequency-domain estimators under multisine excitation, for both nonparametric and parametric settings, remain insufficiently understood. This paper develops a finite-sample statistical framework for least-squares estimation of the frequency response function (FRF) and its implications for parametric modeling. First, we derive exact distributional and covariance properties of the FRF estimator, explicitly accounting for aliasing effects under slow sampling regimes, and establish conditions for unbiasedness, uncorrelatedness, and consistency across multiple experiments. Second, we show that the FRF estimate is a sufficient statistic for any parametric model under Gaussian noise, leading to an exact equivalence between optimal two stage frequency-domain methods and time-domain prediction error and maximum likelihood estimation. This equivalence is shown to yield finite-sample concentration bounds for parametric maximum likelihood estimators, enabling rigorous uncertainty quantification, and closed-form prediction error method estimators without iterative optimization. The theoretical results are demonstrated in a representative case study.

Figures (3)

• Figure 1: Input spectrum plots with sampling periods $h=4/7$ (left) and $h=1/2$ (right), representing scenarios where Assumption \ref{['assumption2']} is satisfied and violated, respectively. The black arrows indicate the magnitude spectrum of a continuous-time multisine input with frequencies $\omega_\ell = 0, \pi$, and $5\pi$. The blue stem plots show the discrete-time spectrum, and the grey boxes denote the fundamental frequency band.
• Figure 2: Interpretation of the indirect prediction error method for system identification under multisine excitation. The FRF estimate $\hat{\bm{\mathcal{G}}}_{\textnormal{MS}}$, lying in the subspace $\bm{\mathcal{G}} \subset \mathbb{C}^{n_y(2M+1)\times n_u}$, is projected onto the subspace of parametric models, yielding $\hat{\bm{\theta}}$.
• Figure 4: Empirical and theoretical $90\%$ confidence bounds for the FRF at the input frequency points (left), and for the vector $\hat{\bm{\theta}}$ describing the parametric model (right).

Theorems & Definitions (22)

• Remark 1
• Remark 2
• Definition 1
• Remark 3
• Lemma 1
• Remark 4
• Theorem 1
• Theorem 2
• Remark 5
• Corollary 1