Identification of linear and nonlinear systems.

Project Details


Linear Modelling in the Presence of Nonlinear Distortions

J. Schoukens, R. Pintelon, and T. Dobrowiecki

In this project we set up a framework to deal with nonlinear distortions in a linear modelling framework. The nonlinear system is replaced by a linear system plus a noise source. The properties of this representation are given, and a measurement technique to extract the linear model and the noise variance is developed. Eventually the different aspects of this approach are studied from measurement, identification and design point of view.
Linear models are successfully applied to a wide range of modelling problems although they are based on very restrictive assumptions. The real world is not linear, and hence intrinsically the theory is not applicable. However, in practice, the linear approximations offer important advantages:
They result in useful models that give the user a lot of intuitive insight in his problem.
Many design techniques are available that can not be easily generalised to nonlinear models.
Nonlinear model building is mostly difficult and time consuming, while the additional performance that is obtained is rather small.
For these reasons we want to offer the user a generalised linear framework that can be used in a nonlinear environment. Using this framework he will get:
A better understanding of the impact of nonlinear distortions on his model.
Optimized measurement techniques that reduce the required measurement time significantly.
Generalised uncertainty bounds that describe the model variations due to the nonlinear distortions. This allows for a better balanced design that accounts for the model limitations.
A lower risk to be fooled by the classical linear identification methods that are widely used in commercial packages.
Simple design rules that help to reduce the undesired impact of nonlinear distortions on practical designs.
Linear representation of nonlinear systems
We have shown that for random excitations, a nonlinear system can be represented under very mild conditions by a linear system plus an additive noise source (see FIGURE 14.). The linear system gives the best linear approximation of the output for the considered class of excitation signals, while the noise source represents all nonlinear effects that are not captured by the model. For the class of normally distributed random excitations, or random multisines (this are periodic signals with a user specified amplitude spectrum and uniformly distributed random phases), the properties of the noise source are completely known: it is asymptotically normally and independently (over the frequencies) distributed. Moreover, although it depends non linearly on the input, the noise source spectral components are asymptoticaly independent of the input spectral components.
Conclusion: The FRF at frequency can be written as:
where is called the related dynamic system or the best linear approximation. It consists of the underlying linear system (if it exists) + the systematic contributions of the nonlinear distortions to the best linear approximation, called .
Remark: Under mild conditions, nonlinear distortions can be split in even and odd distortions (e.g. and ). For zero mean signals, even nonlinearities do not contribute to , they only create stochastic contributions. Odd nonlinearities always contribute to and to .
In this project we focus on a number of measurement aspects to characterize this model. The following questions are addressed:
How to measure the linear system?
How to measure the noise characteristics
How to detect, quantify and qualify the nonlinear contributions to the linear model?
In a next phase we study the impact of these new insights on
Design aspects
Intuitive insights
On modelling and Identification

Qualification of the nonlinear errors in FRF measurements

T. Dobrowiecki and J. Schoukens

The topic of the research was the qualification of the nonlinear errors in the FRF measurements in case of minimal or totally missing a prior information with respect to the level or the order of the nonlinearity. It was found out that for a particular class of weakly nonlinear systems the nonlinear variance experienced in this kind of measurements indeed bears information about the level of the bias of the FRF, and this level can be estimated in the worst case without any additional knowledge about the nonlinearity itself.
The findings were elaborated in a joint paper submitted and accepted for the IMTC 2001 Conference to be held in Budapest, May 2001.

An Efficient Nonlinear Least Square Multisine Fitting Algorithm

Gyula Simon, Rik Pintelon, Johan Schoukens

The estimation of the harmonic content of periodic signals is a very common problem, which can be solved by using different techniques, e.g. Fast Fourier Transformation (FFT), least squares (LS) estimation, nonlinear least squares (NLS) estimation, based on the nature of the problem and the required accuracy. The most general problem, when the frequency of the signal is unknown, can accurately be solved by NLS estimation. In typical cases, when the record contains only a reasonable number of points and frequency lines to be estimated (up to a few thousand data points and a few hundred frequencies), the problem can be solved using the methods available in the literature [1]. Based upon this method, the solution of the NLS problem requires roughly O(NM2) operations and O(NM) storage space, where M represents the number of harmonics to be estimated, and N is the length of the time record. If the number of samples and/or the harmonics to be estimated exceeds a certain limit, these methods cannot be used in conventional computers any more because of the slow down of these algorithms, and especially of the extreme memory requirements.
In this project a new model based recursive calculation method was developed to solve the NLS problem [2]. The algorithm uses a built-in signal model represented by a tuned resonator-based filter bank. The new computation method requires only O(MN) operations and O(N) storage space. The low memory requirements and the recursive nature of the computation make the hardware implementation also possible.
1. R. Pintelon and J. Schoukens, An Improved Sine-Wave Fitting Procedure for Characterizing Data Acquisition Channels, IEEE Trans. Instrum. Meas., Vol. 45, pp. 588-593, Apr. 1996.
2. Gy. Simon, R. Pintelon, L. Sujbert and J. Schoukens, An Efficient Nonlinear Least Square Multisine Fitting Algorithm, Accepted for publication in IEEE Instrumentation and Measurement Technology Conference, IMTC/2001, Budapest.
Effective start/end date20/12/9920/12/02

Flemish discipline codes

  • Electrical and electronic engineering
  • Computer engineering, information technology and mathematical engineering
  • (Bio)medical engineering


  • non-linear systems
  • System Identification
  • Linear systems