Model identification in state space

Determining the model of a dynamic process is an essential problem in many scientific disciplines, such as automation, economics and medicine.

The layout of such a model enables, among other things, a better understanding of the process under study, an analysis of the interactions and causal relationships between different variables and quantities relating to the process, and the observation and prediction of some of these variables.

Generically, there are two distinct procedures for building a model. The first procedure consists of breaking down the process into elementary subsystems and then, through the addition of elementary laws of physics, finance, life, etc., establishing a dynamic model of the entire process. This type of modeling is known as "white-box modeling". It has two major drawbacks. Firstly, it requires in-depth knowledge of these elementary laws and of the system's internal behavior. Secondly, it often leads to the determination of a complex model, based on partial derivatives or unknown parameters for example, which is difficult to exploit.

The second procedure for developing a model consists of carrying out one or more experiments on the process and extracting a coherent dynamic model. This second procedure involves acting specifically on the process (variables through which it is possible to act on the process are called "inputs") and making descriptive measurements of process behavior (variables through which it is possible to observe process behavior are called "outputs"). This type of modeling is known as black-box or identification modeling. This is the type of modeling we're interested in here.

The literature describes numerous techniques for implementing such a procedure. The major expansion of these techniques took place in the 1960s, when two main approaches were developed. The first was model identification in the form of transfer functions (differential equations for continuous-time models, or difference equations for discrete-time models). These methods allow only the external behavior of the process to be represented, i.e. only the input-output behavior to be described. They have been widely studied, and the literature includes a number of reference works on the subject.

In this article, we focus on the second approach. It corresponds to methods for estimating a model in the form of a state representation, a model structure characterizing the internal workings of the process. These methods took a back seat in the 1970s and 1980s, only to be revived in the 1990s under the name of "subspace methods". This name comes from the fact that these methods are based on a reconstruction of the vector subspace characterizing the internal workings of the process. These...