Approximation of high-dimensional linear dynamic models

The use of mathematical models to represent physical phenomena or systems has become a standard in engineering. Indeed, the opportunities offered by such models for simulation, control, optimization and analysis seem inexhaustible. This trend is accompanied by an ever-increasing need for increasingly complete and accurate models, enabling reality to be represented with a high degree of fidelity. This trend is accentuated by the development of computer technologies which, through dedicated calculation software, enable the generation of highly accurate models.

However, such precision is usually accompanied by a high level of complexity. In the case of dynamic systems, this complexity either translates into a very large number of states, in which case we speak of high-dimensional models, or models with an inappropriate mathematical structure. As a result, the tasks made possible by mathematical models, such as simulation, are greatly complicated, if not rendered impossible, by the technological constraints of computers in terms of memory and computing capacity.

This is where model approximation or reduction comes into play. The aim is to simplify a high-dimensional dynamic model while preserving, as far as possible, its behavior and main characteristics. This article focuses on the case of linear dynamic models. Mathematical tools useful for understanding the problem and a varied panel of existing methods for dealing with it are detailed.