Linear System Identification

In this section we learn how to identify a model for a single-input single-output (SISO) linear dynamic system, starting from the measured input and output signals. Also, the power spectrum of the unmodeled disturbances is identified and used to generate uncertainty bounds on the estimated model.

Linear system identification is used in many disciplines, ranging from vibrational analysis of mechanical systems, over electrical, electronic, chemical, civil, to biomedical applications. A formal system identification framework was developed from the late 1960s onwards, mainly within the control systems society. Today, the theoretical aspects are well understood, and system identification toolboxes are available that cover the complete identification process from experiment design to estimation and validation of the model. Extension of the theory towards nonlinear systems is discussed in Nonlinear System Identification.

A brief introduction to the linear system identification theory for SISO systems is given in the preliminary set of slides. The focus is on a number of topics that have a major influence on the user aspects of the identification process:
Data: what is going on between the samples? We start the identification process from discrete-time (DT) measurements u(k), y(k) of the continuous-time (CT) signals u(t), y(t), sampled at a sampling frequency fs. No information is available on how the CT signals vary between the measured samples, and an additional assumption is needed. The zero-order-hold assumption (the signal remains constant between the samples) and the band-limited assumption (the signal has no power above half the sampling frequency) are the most popular choices. Insight into the impact of the selected assumption helps a lot to arrive at reliable high-quality models that are well suited for the intended application.
Selection of a model set: Many equivalent descriptions of linear dynamic systems exist (e.g. transfer function or state space model) using a time- or frequency domain formulation, combined with a parametric or nonparametric noise model. A brief introduction to these topics is given and the relationships between the different representations are studied. Although there is complete theoretical equivalence between all these options, we show that the practical aspects can differ greatly. A better understanding is needed to make the best combination experimental setup – data – model – identification method.
Cost function: The choice of the cost function sets the statistical properties of the estimates (bias-variance). A weighted least squares framework is proposed, showing how the weighting factors can be extracted from the data.
User choices: Some choices can be made freely by the user, for example: i) Time domain or frequency domain formulation? ii) Parametric or nonparametric noise model? Again, there is a full theoretical equivalence between these options, but the actual choice you make will greatly affect the complexity of the identification process.
Validation: Does the estimated model solve our problem? Is it in conflict with the data? Can it explain new data? Some tools to answer these questions are discussed very briefly.

Preliminary slides
Identification of Linear Systems

Further reading
– An extensive discussion of system identification starting from a frequency domain formulation is given in
System Identification – A frequency Domain Approach – second edition, R. Pintelon and J. Schoukens (2012), IEEE Press, Wiley.

– A comprehensive description of system identification is given in
System Identification – Theory For the User – second edition, L. Ljung (1999), PTR Prentice Hall, Upper Saddle River, N.J.
System Identification, T. Söderström and P. Stoica (1989), Prentice Hall International, Hemel Hempstead.

– The frequency response and the impulse response functions are nonparametric models for a SISO linear system. An extensive overview, ranging from the classical approaches to the very recent methods is given in
Nonparametric Data-Driven Modeling of Linear Systems: Estimating the Frequency Response and Impulse Response Function,
J. Schoukens, K. Godfrey and M. Schoukens (2018), IEEE Control Systems Magazine, vol. 38, no. 4, pp. 49-88.