Optimization for Learning and Control

12System Identification

System identification is about learning models for dynamical systems. Examples include the flow of water in a river, the planetary motions, and the number of cars on a segment of a freeway. We will in this chapter limit ourselves to discrete‐time dynamical systems. However, many of the results can be generalized to continuous‐time dynamical systems using sampling. The origin of system identification goes back to the work by Karl Åström and Torsten Bolin in 1965.

We start by defining what we mean by a dynamical system in state‐space form. We define a regression problem for learning/estimating the dynamical system, and specifically, we define it as an ML problem. From this, we then derive input–output models and the corresponding ML problem. We discuss in detail how the parameters can be estimated by solving a nonlinear LS problem. Then, we discuss how to estimate the model when some of the data are missing. Nuclear norm system identification is also discussed in this context. Prior information can be incorporated into system identification easily using Gaussian processes and empirical Bayes. We show how this can be implemented using the sequential convex optimization technique based on the majorization minimization principle. Recurrent neural networks and temporal convolutional neural networks are shown to be generalizations of the linear dynamical models to nonlinear dynamical models. The chapter is finished off with a discussion on experiment design ...