Exact Modeling

Abstract

This chapter develops methods for computing the most powerful unfalsified model for the data in the model class of linear time-invariant systems. The first problem considered is a realization of an impulse response. This is a special system identification problem when the given data is an impulse response. The method presented, known as Kung’s method, is the basis for more general exact identification methods known as subspace methods. The second problem considered is computation of the impulse response from a general trajectory of the system. We derive an algorithm that is conceptually simple—it requires only a solution of a system of linear equations. The main idea of computing a special trajectory of the system from a given general one is used also in data-driven simulation and control problems (Chap. 6). The next topic is identification of stochastic systems. We show that the problem of ARMAX identification splits into three subproblems: (1) identification of the deterministic part, (2) identification of the AR-part, and (3) identification of the MA-part. Subproblems 1 and 2 are equivalent to deterministic identification problem. The last topic considered in the chapter is computation of the most powerful unfalsified model from a trajectory with missing values. This problem can be viewed as Hankel matrix completion or, equivalently, computation of the kernel of a Hankel matrix with missing values. Methods based on matrix completion using the nuclear norm heuristic and ideas from subspace identification are presented.

A given phenomenon can be described by many models. A well known example is the Ptolemaic and the Copernican models of the solar system. Both described the observed plant movements up to the accuracy of the instruments of the 15th century.

A reasonable scientific attitude to a model to be used is that it has so far not been falsified. That is to say that the model is not in apparent contradiction to measurements and observations.

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Another valid scientific attitude is that among all the (so far) unfalsified possible models we should use the simplest one. This statement is a variant of Occam’s razor. It obviously leaves room for subjective aesthetic and pragmatic interpretations of what “simplest” should mean. It should be stressed that all unfalsified models are legitimate, and that any discussion on which one of these is the “true” one is bound to be futile.

K. Lindgren