System Identification in the Behavioral Setting - A Structured Low-Rank Approximation Approach

System identification is a fast growing research area that encompasses a broad range of problems and solution methods. It is desirable to have a unifying setting and a few common principles that are sufficient to understand the currently existing identification methods. The behavioral approach to system and control, put forward in the mid 80’s, is such a unifying setting. Till recently, however, the behavioral approach lacked supporting numerical solution methods. In the last 10 yeas, the structured low-rank approximation setting was used to fulfill this gap. In this paper, we summarize recent progress on methods for system identification in the behavioral setting and pose some open problems. First, we show that errors-in-variables and output error system identification problems are equivalent to Hankel structured low-rank approximation. Then, we outline three generic solution approaches: 1) methods based on local optimization, 2) methods based on convex relaxations, and 3) subspace methods. A specific example of a subspace identification method—data-driven impulse response computation—is presented in full details. In order to achieve the desired unification, the classical ARMAX identification problem should also be formulated as a structured lowrank approximation problem. This is an outstanding open problem.