On system identification of complex systems from finite data

System identification deals with computation of mathematical models from an a priori chosen model-class, for an unknown system from finite noisy data. The popular maximum-likelihood principle is based on picking a model from a chosen model-parameterization that maximizes the likelihood of the data. Most other principles including set-membership identification can be thought of as extensions of this principle in so far as the concept of choosing a model to fit the data is concerned. Although these principles have been extremely successful in addressing several problems in identification and control, they have not been completely effective in addressing the question of identification in the context of uncertainty in the model class/parameterization. We introduce a new principle for identification in this paper. The principle is based on choosing a model from the model-parameterization which best approximates the unknown real system belonging to a more complex space of systems which do not lend themselves to a finite-parameterization. The principle is particularly effective for robust control as it leads to a precise notion of parametric and nonparametric error and the identification problem can be equivalently perceived as that of robust convergence of the parameters in the face of unmodeled errors. The main difficulty in its application stems from the interplay of noise and unmodeled dynamics and requires developing novel two-step algorithms that amount to annihilation of the unmodeled error followed by averaging out the noise. The principle contributions of the paper are in establishing: 1) robust convergence for a large class of systems, topologies, and unmodeled errors; 2) sample path based finite-time polynomial rate of convergence; and 3) annihilation-correlation algorithms, for linearly parameterized model structures, thus, illustrating significant improvements over prediction-error and set-membership approaches.