System Identification › Summary

• System identification learns the transition model from trajectories, complementing filtering, which infers state given a known model. Together they form the estimation backbone of Part II.
• Identifiability is a joint property of system, class, and data. Structural ambiguities like similarity transforms are unavoidable facts; practical unidentifiability from under-excited datasets is a solvable experiment-design problem.
• Persistent excitation formalizes input richness. With a correctly specified identifiable model, suitable exogeneity, stability and regularity conditions, sufficiently rich inputs support convergence to a unique probability limit. Under feedback, independently rich dither can restore missing excitation, but dither alone does not repair model misspecification or endogenous noise.
• Prediction-error methods generalize least squares to arbitrary model classes by optimizing predictive likelihood. Consistency with the true system requires structural containment, identifiability, informative data, and suitable regularity conditions; correct noise specification additionally supports the maximum-likelihood interpretation and asymptotic efficiency.
• Nonparametric methods estimate impulse and frequency responses directly, trading structure for robustness, and modern kernel methods regularize them with stability-aware priors.
• Subspace methods realize dynamics from data geometry, reading state-space matrices out of a low-rank Hankel factorization with no iterative optimization; rank discovery reveals system order en route.
• Neural identification replaces function-class choice with learned networks, buying expressive power for nonlinear partially observed worlds while paying in compounding rollout error, exploitation risk, and lost interpretability.
• Grading differs from using: one-step loss, recursive rollout error, off-distribution probes, closed-loop success, and calibration each certify a different claim about a world model.

You now possess the tools to turn trajectories into models, and the vocabulary to ask the uncomfortable question: what has this dataset certified about them? The next chapter, Classical Planning and Optimal Control, puts those models to work. Given a roughly correct, non-exploitable, usefully excited identified model, we will ask how to extract decisions from it: dynamic programming over states, trajectory optimization through continuous spaces, and the stability and optimality bounds that classical control attaches to model-derived policies. Those qualifiers are not throwaway hedging; each corresponds to a concrete risk cataloged in this chapter. Knowing which qualifier the evidence supports is often the difference between a planner that ships and a planner that surprises.