Sensitivity Metrics for Maximum Likelihood System Identification

This paper introduces a set of sensitivity metrics to be used along likelihood-based modal identification methods. In maximum likelihood (ML) estimation theory, the precision of ML point estimates can be measured by the curvature of the likelihood function. This paper presents closed-form partial derivatives, observed information, and variance expressions for discrete-time stochastic state-space model parameters as well as state matrix features that influence modal estimates. The results are derived for the observation matrix and the state matrix as well as eigenvalues and eigenvectors of the state matrix; these model entities correspond to natural vibration properties of a structural system. Confidence intervals are constructed for natural frequencies, damping ratios, and mode shapes using the derived asymptotic covariance matrices and the asymptotic normality property of ML estimators. The results are a supplement to the ML-based structural identification using expectation maximization (STRIDE) modal identification algorithm and are applicable to modal identification techniques formulated in the time-domain stochastic state-space model for linear time invariant systems. An application to structural modal identification is included to compare closed-form asymptotic parameter uncertainties to Monte Carlo bootstrap estimates.