Identification of nonlinear systems using polynomial nonlinear state space models

Student thesis: Doctoral Thesis


All the work in this thesis relies on the concept of the Best Linear Approximation (BLA).
Therefore, in Chapter 2 the BLA is first introduced in an intuitive way, and then rigorously
defined for SISO and MIMO nonlinear systems. Furthermore, some interesting properties of
multisine excitation signals with respect to the qualification and quantification of nonlinear
behaviour are rehearsed. Finally, we explain how the BLA should be estimated, for both non
periodic and periodic input/output data. As will become clear in Chapter 2, periodic excitations
are preferred, since in that case more information can be extracted from the Device Under
The tools described in Chapter 2 are applied to a number of Digital Signal Processing (DSP)
algorithms in Chapter 3. A measurement technique is proposed to characterize the nonidealities
of DSP algorithms which are induced by quantization effects, overflows, or other
nonlinear effects. The main idea is to apply specially designed excitations such that a
distinction can be made between the output of the ideal system and the contributions of the
system's non-idealities. The proposed method is applied to digital filtering and to an audio
compression codec.
In Chapter 4, an identification procedure is presented for a specific kind of block-oriented
model: the Nonlinear Feedback model. By estimating the Best Linear approximation of the
system and by rearranging the model's structure, the identification of the feedback model
parameters is reduced to a linear problem. The numerical parameter values obtained by
solving the linear problem are then used as starting values for a nonlinear optimization
procedure. The proposed method is illustrated on measurements obtained from a physical
Chapter 5 introduces the Polynomial Nonlinear State Space model (PNLSS) and studies its
approximation capabilities. Next, a link is established between this model and a number of
classical block-oriented models, such as Hammerstein and Wiener models. Furthermore, by
means of two simple examples, we illustrate that the proposed model class is broader than
the Volterra framework. In the last part of Chapter 5, a general identification procedure is
presented which utilizes the Best Linear Approximation of the nonlinear system. Next,
Outline of the thesis
frequency domain subspace identification is employed to initialize the PNLSS model. The
identification of the full PNLSS model is then regarded as a nonlinear optimization problem.
In Chapter 6, the proposed identification procedure is applied to measurements from various
real-life systems. The SISO test cases comprise three electronic circuits (the Silverbox, a
Wiener-Hammerstein system and a RF crystal detector), and two mechanical set-ups (a
quarter car set-up and a robot arm). Furthermore, two mechanical MISO applications are
discussed (a combine harvester and a semi-active magneto-rheological damper).
Finally, Chapter 7 deals with the conclusions and some ideas on further research.
Date of Award23 Jan 2008
Original languageEnglish
SupervisorJoannes Schoukens (Promotor), Rik Pintelon (Co-promotor), Annick Hubin (Jury), Jean Vereecken (Jury), Steve Vanlanduit (Jury), Yves Rolain (Jury), Lennart Ljung (Jury) & Jan Swevers (Jury)


  • Identification
  • nonlinear systems

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