System Identification

System identification fundamentally relies on measured excitation signals and corresponding system responses rather than physical laws. The quality of the identified model is directly bounded by the information content of the data.

• Requires persistently exciting inputs that stimulate all system modes
• Outputs must be sampled at rates satisfying the Nyquist-Shannon theorem
• Data pre-processing steps include detrending, filtering, and outlier removal
• The signal-to-noise ratio of measurements critically impacts parameter variance

The practitioner must choose a model set — a parameterized family of candidate models — before estimation. This structural choice encodes prior knowledge about system dynamics and represents the most critical engineering decision in the workflow.

• Linear structures: ARX, ARMAX, Output-Error (OE), Box-Jenkins (BJ)
• Nonlinear structures: NARX, Hammerstein-Wiener, neural state-space models
• Grey-box structures: Partially known physics with unknown parameters
• The bias-variance tradeoff governs the selection of model order and complexity

Once a model structure is fixed, numerical optimization routines compute the parameter values that minimize the discrepancy between predicted output and measured output. The choice of algorithm depends on model linearity and computational constraints.

• Prediction Error Minimization (PEM) is the gold standard for linear systems
• Subspace methods (N4SID, MOESP) estimate state-space matrices directly from data
• Nonlinear least squares with Levenberg-Marquardt for nonlinear grey-box models
• Maximum Likelihood Estimation when measurement noise statistics are known

A model is not complete until it passes rigorous cross-validation against data not used during estimation. Residual analysis tests whether the model has captured all systematic information, leaving only white noise in the prediction errors.

• Whiteness test: Autocorrelation of residuals should be zero for all non-zero lags
• Independence test: Cross-correlation between residuals and past inputs must be zero
• Fit metrics: Normalized Root Mean Square Error (NRMSE) on validation datasets
• Frequency-domain validation: Compare Bode plots of estimated vs. empirical transfer functions

A model structure is identifiable if different parameter vectors produce distinct input-output behaviors. Without identifiability, estimation algorithms may converge to arbitrary or non-unique solutions, rendering the model useless for prediction or control.

• Structural identifiability: A theoretical property of the model structure alone
• Practical identifiability: Whether parameters can be estimated given finite, noisy data
• Over-parameterization leads to flat loss landscapes and numerical instability
• Canonical forms (e.g., observable canonical state-space) enforce unique parameterizations

For time-varying systems or digital twin synchronization, identification must run recursively as new data streams arrive. Recursive algorithms update parameter estimates incrementally without reprocessing the entire dataset.

• Recursive Least Squares (RLS) with exponential forgetting for gradual drift
• Kalman filter-based parameter estimation for joint state and parameter tracking
• Forgetting factor controls the memory horizon and adaptation speed
• Essential for closed-loop digital twins that must track asset degradation over time