CAPÍTULO IV: MARCO PROPOSITIVO
Grafico 50: Flujograma de selección
The content of this chapter is based on Tan, Wong and Godfrey (2012). The chapter shows a utilisation of one property of the Gaussian Bla—the Bla is proportional to the combined linear dynamics of the system. This allows a brute-force exhaustive search of all poles and zero combinations resulting in two physically realisable filters for the first and second linearities of a Wiener-Hammerstein system. The iterative algorithm proposed also models the static nonlinearity using a polynomial or a piecewise polynomial function. The parameters are tuned in multiple stages, first for linearity assignment, then to nonlinearity, to minimise the number of parameters the optimisation algorithm has to tackle at any stage. The polynomial degrees are increased one degree at a time to minimise the chance of the optimisation being stuck in a local minimum.
Chapter 8 documents a modelling exercise on a ‘hyperfast’ Peltier cooling system assembly (Cham, Tan & Tan, 2010). The solution encompasses a mixture of blackbox and whitebox modelling approaches, incorporating as much physical and structural knowledge of the system as possible and without relying on numer- ical non-parametric optimisation methods. The solution shows that even a simple, naïve approach that does not use established sophisticated system identification methods, can yield models with acceptable performance given enough prior physical knowledge. This piece of work was based on Wong and Godfrey (2010).
1.3
Contributions
Much of the research documented in this thesis has previously been published in journal or conference publications; a list of which is available on p. xiv. Unless otherwise specified or referenced, all the work are performed by the author under the guidance of the supervisors named on p. ii with the exception of Chapter 7
which was part of a collaboration (see p.xvi). Major contributions arising from the research work performed over the course of this Ph.D. degree are highlighted in this section as follows:
The objective of this thesis is mainly concerned with the use of non-Gaussian signal in identifying theBlaof the system. While the dependence of theBlaon the input amplitude distribution is well known in the literature, the work in Chapter3, first published in Wong, Schoukens and Godfrey (2012a), tackles some questions rel- evant to the system identification community. To the best of the author’s knowledge, the effect of non-Gaussianity of inputs to the Blaof a time-invariant discrete-time nonlinear system has not been investigated in detail, and this thesis is able to provide closed-form algebraic expressions for the Bla of generalised discrete-time Volterra
1.3. Contributions systems with arbitrary input distributions. This answers the question made in the beginning of the chapter: ‘What happens if a binary excitation is used instead of a Gaussian noise excitation?’. Another question is: ‘Will the results significantly change, or will the differences be sufficiently small that, for practical purposes, they can be ignored?’. The simulation experiments performed for a Wiener system with a) cubic (Section 3.3.1) and b) quintic (Section3.3.2) nonlinearity have shown that the Gaussian Bla and the Bla obtained from other input signals can have a sig- nificant difference between them. Finally, to answer: ‘Is it possible to quantify the differences between non-Gaussian and Gaussian excitations on the basis of the non- Gaussian measurement results?’, a measure called theDiscrepancy Factor(Df) was developed in Section3.5 to quantify the difference.
The first part of Chapter 4 discusses the behaviour of noise and nonlinear distortions and how it affects the choice of input signal. This piece of work was also published in Wong, Schoukens and Godfrey (2012c). In the second part, significant improvement to the quality of estimate of theBlaof a nonlinear system can be made when the input sequences used were Maximum Length Binary Sequences(Mlbs’s), giving rise to nonlinear distortions having a certain structure unique to Mlbs’s. While this structured behaviour is known in the literature, see for example, Godfrey and Moore (1974), Vanderkooy (1994), the structured nonlinear noise has been treated as a negative aspect rather than an opportunity. In this thesis, it was found that the use of alternative averaging schemes rather than the traditional mean-based averaging can offer significant improvement in the Blaestimate under moderate to high signal-to-noise ratio (Snr) due to the structured nonlinear distortions. The findings were submitted as Wong, Schoukens and Godfrey (2013b), which has been accepted for publication.
Another notable contribution involves the physical experiment performed to verify the theory developed in Chapter3, documented in Chapter6of this thesis and published in Wong, Schoukens and Godfrey (2012b). Here the experiment showed good agreement between the theory and practice.
Furthermore, noting the dependence of theBlaon Gaussianityof the input signal, the author investigated the merits in designing discrete-time discrete-level sequences up to 5 levels (quinary) to match the higher order statistics to a discrete- time Gaussian sequence. Chapter 5 has shown that it is possible to, depending on the degree of nonlinearity, minimise or completely eliminate the bias of the estimated Bla with respect to the Gaussian Bla. This piece of research was also published to Wong, Schoukens and Godfrey (2013a) and was presented at the 31st Benelux Meeting on Systems and Control, held on the 27th–29th, March, 2012 in Heijen, the
1.3. Contributions Netherlands.
Lastly, the benchmark Chapters of7and8document solutions to benchmark problems previously proposed in two separate conferences. The benchmark studies provided valuable training opportunities for the author in system identification.
C
2
Periodic Input Sequences
P
eriodic sequences are widely used in system identification. They offer several advantages over aperiodic sequences in, for instance, eliminating spectral leakage (see Section 2.2), the ability to perform inter-period averaging to drive down the effect of exogenous environment noise (see Section 4.2) and allowing the user to compare the level of nonlinear distortions to that of noise (Pintelon & Schoukens, 2012). Several types of periodic input sequences used throughout this thesis are described in this chapter. Before these sequences are introduced in Section2.3, two concepts first explained: periodicity-invariant (Pi) property in Section 2.1 and the variousFourier transforms (Fts) in Section2.2.2.1
Periodicity-invariance (Pi)
Definition A nonlinear system is designated as periodicity-invariant (Pi) when excited by any periodic input𝑢with period𝑁; the noise-free output𝑦would always share the same period𝑁.
This is the case if the spectral harmonic frequencies of the discrete output spectrum are all divisible (in whole) by the fundamental frequency of the periodic input. For linear dynamical systems, the Pi property always hold true, as the existence of transfer functions dictates that the output frequency grid to be equal to that of the input. In some literature, e.g. Marconato, Van Mulders, Pintelon, Rolain and Schoukens (2010), Pi systems are denoted period-in-same-period-out (Pispo) systems.