System Identification (227-0689-00L)



Prof. Roy Smith
Automatic Control Laboratory
Dept. of Information Technology and Electrical Engineering
ETH Zürich

Online teaching platform: Moodle

All organisational matters will be dealt through the online platform Moodle, where you will find the lecture notes, the exercises with their solutions and the information about the office hours and the teaching assistants.

Using the Moodle platform you will be able to ask questions about the exercises or the lectures which can be answered by the assistants, the professor or even by your fellow classmates. We strongly encourage your participation in the discussions and forums in order to improve your understanding of the material. Unfortunately the use of Moodle is restricted to those registered for the class.


Familiarity with the following concepts is assumed:

  • Laplace and Fourier transforms;

  • Z-transform;

  • Differential and difference equations;

  • State-space representations;

  • Basic stochastic variable concepts.

Course Material


The slides may be revised as errors are found or I think that I have a better way of presenting the material. The date of the latest revision is shown. Those dating from last year may differ from the current lectures, so check for updates just before the lecture.

If you find errors or typographical mistakes in the slides please let Prof. Smith know so that corrections can be made and posted here.

Lecture Lecture slides Extra material Last revised
01 Introduction 10.10.2023
02 Data fitting and statistics Supplementary notes 27.09.2021
03 Least-squares estimation 03.10.2023
04 Sampled data models, parametrisations, and frequency domain analysis Supplementary notes 10.10.2023
05 Frequency-domain identification Supplementary notes 20.10.2023
06 Frequency-domain identification & input signals Supplementary notes 24.10.2023
07 Pulse response estimation and persistency of excitation Supplementary notes 31.10.2023
08 Time- & Frequency-domain methods, Prediction 07.11.2023
09 Prediction error methods & ARX models 15.11.2023
10 Regularisation (lecture by M. Abdalmoaty) 20.11.2023
11 Parametrised transfer function models 29.11.2023
12 Closed-loop identification 07.12.2021
13 Subspace Identification Supplementary notes 14.12.2021

Supporting materials

Unfortunately there are several commonly used, but different, formulae for equivalent concepts. This can lead to confusion and these notation notes discuss some of the pitfalls for beginning readers of the literature.

Primary reference

  • “System Identification; Theory for the User”, Lennart Ljung, Prentice Hall (2nd Ed), 1999.

Secondary references

  • “Dynamic system identification: Experimental design and data analysis”, GC Goodwin and RL Payne, Academic Press, 1977.

There are many texts that cover the required background in digital signals. The following are good but be aware that the notation varies a lot between texts so read carefully.

  • Fourier Transform, Digital signals processing (basics): “Signals & Systems,” A.V. Oppenheim, A.S. Willsky with S.H. Nawab, (2nd Ed.) Prentice-Hall, 1983.

  • More advanced digital signal processing: “Digital Signal Processing,” A.V. Oppenheim & R.W. Schafer, Prentice-Hall, 1975.

  • Spectral analysis: “Introduction to Spectral Analysis,” P. Stoica & R. Moses, Prentice-Hall, 1997.

Matlab functions

The following functions are provided to save you the time and trouble of coding them yourself. They are not optimised and so will not work well for very large data sets.

Matlab function Description
fdsubspaceid.m Calculate a full rank basis for the extended observability subspace and the singular values associated with each of the basis vectors.
fdsubspaceft.m Generate a state-space representation of specified rank from the extended observability subspace basis.
WfHann.m Generate a frequency domain Hann window of a specified length and width parameter.
WtHann.m Generate a time domain Hann window of a specified length and width parameter.

Related papers

Material from the following papers are discussed in the lectures. The papers are here so that you can read the details.