System identification of complex systems: problem formulation and results

We are motivated by the need to derive simple control-oriented models of complex systems. Briefly, complex systems are a collection of systems that are not approximable (arbitrarily closely) by a finitely parameterized collection of systems. The study of system identification for such classes of systems has received considerable attention in the past few years. In this paper a new formulation of the system identification problem, which balances between the set-membership and probabilistic approaches is presented. The salient feature of the formulation is that we distinguish between the two principal sources of error encountered in the input-output data-noise and unmodeled dynamics. Unmodeled dynamics arise from the fact that the finitely parameterized model that we seek does not truly characterize the real system. Therefore, unmodeled dynamics are “modeled” as the residual error between the parametric model and the real system. This viewpoint leads to a decomposition between the parametric model class and unmodeled dynamics. In contrast noise is modeled so that it is uncorrelated (in a deterministic or a stochastic sense) from the input. The identification problem deals with obtaining the appropriate finite parametric model from input-output data. The identification problem is studied for several different norms including l₁ and H_∞. One of the chief outcomes is a new notion of a persistent input and showing that there are both deterministic and stochastic inputs which meet the new criterion