Nonlinear System Identification

The purpose of this section is to introduce nonlinear system identification to a wide audience, guiding practicing engineers and newcomers to the field towards a solid data-driven solution to the nonlinear dynamical systems modeling problem. We provide a guided tour of the wide range of user choices in nonlinear system identification. A broad perspective on the topic is given, discussing the similarities and differences between linear and nonlinear problems. We keep the focus on the basic philosophy and provide an intuitive understanding of the problems and solutions. The presentation is based on the first article in the list below for further reading, and we strongly advise to study this text to get a deeper insight in the material.

Here we briefly summarize the main body of the slides. In the slides, we give at many points links to more detailed slides that cover more specialized topics.

Why is nonlinear system identification so involved?
Nonlinear system identification can be much more involved than linear identification for three reasons: i) Nonlinear models live on a complex manifold in a high-dimensional space, while linear models live on a simple hyperplane that is much easier to characterize. ii) Capturing this increased flexibility can be quite difficult, so structural model errors are often unavoidable in nonlinear system identification. This affects the three lead actors in system identification: data, model, cost. iii) Dealing with noise and disturbances entering before the nonlinearity requires new tools to get a tractable formulation of the problem.

Linear or nonlinear system identification? A user’s decision
Because nonlinear system identification is more involved than linear identification, it is also important to verify at the start of the identification process whether it is really necessary to select this more involved solution. If simpler linear models are good enough to solve the problem, a lot of time, effort, and costs can be saved. A user-friendly nonparametric distortion analysis is presented to detect the presence of nonlinearities, quantify their level and find out their nature (even or odd). With this information, a well-informed decision can be made about which approach (linear or nonlinear) to use and how much can be gained by switching from linear to nonlinear system identification.

The lead actors in nonlinear system identification
The role of the lead actors in linear system identification was already discussed in the System Identification section. This discussion is repeated here but now from a nonlinear system identification perspective keeping the presence of structural model errors in mind. This affects the user choice for the three lead actors:
Experiment design: The main issue is to guarantee that the full domain of interest is covered by the experiment, avoiding large structural model errors and extrapolation of the model in the applications. We show that it is not enough to cover the amplitude range and frequency band of interest. It is necessary to keep an eye on all the states of the system.
Choice of the cost function: In linear identification, the choice of the cost function is guided by the noise properties. However, when structural model errors dominate the disturbing noise, the latter should be kept low without inflating the impact of the noise on the estimated models.
Selection of a model structure: This is the most difficult problem, the user faces a myriad of choices and it is very difficult for the newcomers to make a good selection. Some choices are dictated by the behavior of the system (e.g. open loop or closed loop models linked to the presence of shifting resonances or varying damping, chaos, hysteresis), but others are freely selectable by the user, for example white box (physical) models – grey box models – black box models (no physical insight is used). The user must combine both aspects in the final selection, but failure to respect the behavior of the system condemns the entire modeling effort from the outset.

Linear identification in the presence of nonlinear distortions
– How to identify a linear model if we know that nonlinear distortions are present? How to select the excitation signals? What model quality can be expected? These aspects are discussed and illustrated. More information can be found in the second paper in the further reading list.

Nonlinear SI: Extensive case study
– The full nonlinear system identification process, including the nonparametric distortion analysis, the model structure selection, the identification and the validation is discussed in full detail.

Nonlinear System Identification – A User-Oriented Roadmap

Further reading
Nonlinear System Identification: A User-Oriented Road Map, J. Schoukens and L. Ljung (2019), IEEE Control Systems Magazine, vol. 39, no. 6, pp. 28-99.
Linear System Identification in a Nonlinear Setting: Nonparametric Analysis of the Nonlinear Distortions and Their Impact on the Best Linear Approximation, J. Schoukens, M. Vaes and R. Pintelon (2016), IEEE Control Systems Magazine, vol. 36, no. 3, pp. 38-69.
Nonlinear System Identification: From Classical Approaches to Neural Networks, Fuzzy Models, and Gaussian Processes, O. Nelles (2020), Springer.