System Identification
System identification builds mathematical models of dynamic systems from measured input-output data rather than first principles.
Models from Data
Control design needs a model, but deriving one from physics is often impractical for complex or poorly understood plants. System identification builds the model instead from measured data: excite the system with known inputs, record the outputs, and fit a model that reproduces the observed behavior. It is the experimental complement to first-principles modeling.
The workflow
- Experiment design: choose an input that excites the relevant dynamics, such as a step, a chirp, or a pseudorandom binary sequence.
- Data collection: record input and output, ensuring good signal-to-noise and adequate sampling.
- Model structure selection: pick a model class, from simple transfer functions to state-space or nonlinear forms.
- Parameter estimation: fit the model parameters to the data, usually by minimizing prediction error.
- Validation: test the fitted model against fresh data it was not trained on.
Model structures
Common linear structures include the ARX, ARMAX, and output-error models, each making different assumptions about how noise enters. State-space models can be identified directly by subspace methods. The choice trades simplicity against fidelity: a model just complex enough to capture the dynamics of interest, and no more, generalizes best.
Persistent excitation
A model can only be identified for dynamics the input actually excites. An input too simple, such as a single constant, leaves most of the dynamics invisible and the parameters undetermined. The input must be persistently exciting, rich enough in frequency content to reveal the full model order. Designing such an input, while respecting operational limits, is a core part of the task.
Validation and the bias-variance trade
The decisive test is validation on independent data: a model that fits the training data perfectly may have absorbed noise and fail on new data, a symptom of overfitting. Cross-validation, residual analysis, and information criteria guard against choosing an over-complex model. The aim is the simplest model that predicts well, not the one with the smallest training error.
System identification connects tightly to adaptive control, where a model is estimated continuously online, and to modern data-driven and machine-learning methods that fit high-capacity models to large datasets while facing the same excitation and validation challenges.