Definitions

# System identification

System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data. A dynamical mathematical model in this context is a mathematical description of the dynamic behavior of a system or process in either the time or frequency domain. Examples include:

One could build a so-called white-box model based on first principles, eg. a model for a physical process from the Newton equations, but in many cases such models will be overly complex and possibly even impossible to obtain in reasonable time due to the complex nature of many systems and processes.

A much more common approach is therefore to start from measurements of the behavior of the system and the external influences (inputs to the system) and try to determine a mathematical relation between them without going into the details of what is actually happening inside the system. This approach is called system identification. Two types of models are common in the field of system identification:

• grey box model: although the peculiarities of what is going on inside the system are not entirely known, a certain model based on both insight into the system and experimental data is constructed. This model does however still have a number of unknown free parameters which can be estimated using system identification. As an example, uses the Monod saturation model for microbial growth. The model contains a simple hyperbolic relationship between substrate concentration and growth rate, but this can be justified by molecules binding to a substrate without going into detail on the types of molecules or types of binding. Grey box modeling is also known as semi-physical modeling.
• black box model: No prior model is available. Most system identification algorithms are of this type.

In the context of non-linear model identification Jin et.al. describe greybox modeling as assuming a model structure a priori and then estimating the model parameters. This model structure can be specialized or more general so that it is applicable to a larger range of systems or devices. The parameter estimation is the tricky part and Jin et.al point out that the search for a good fit to experimental data tend to lead to an increasingly complex model. Jin et.al. then define a black-box model as a model which is very general and thus containing little a priori information on the problem at hand and at the same time being combined with an efficient method for parameter estimation. But as Nielsen and Madsen points out, the choice of parameter estimation can itself be problem-dependent.

## Literature

• Daniel Graupe: Identification of Systems, Van Nostrand Reinhold, New York, 1972 (2nd ed., Krieger Publ. Co., Malabar, FL, 1979)
• Lennart Ljung: System Identification — Theory For the User, 2nd ed, PTR Prentice Hall, Upper Saddle River, N.J., 1999.
• Jer-Nan Juang: Applied System Identification, Prentice Hall, Upper Saddle River, N.J., 1994.
• T. Söderström, P. Stoica, System Identification, Prentice Hall, Upper Saddle River, N.J., 1989. ISBN 0-13-881236-5
• Oliver Nelles: Nonlinear System Identification, Springer, 2001. ISBN 3-540-67369-5