Elsevier

Automatica

Volume 47, Issue 12, December 2011, Pages 2684-2688
Automatica

Brief paper
Spectral analysis of block structured nonlinear systems and higher order sinusoidal input describing functions

https://doi.org/10.1016/j.automatica.2011.08.049Get rights and content

Abstract

When analyzing and modeling dynamical systems in the frequency domain, the effects of nonlinearities need to be taken into account. This paper contributes to the analysis of the effects of nonlinearities in the frequency domain by supplying new analytical tools and results that allow spectral analysis of the output of a class of nonlinear systems. A mapping from the parameters defining the nonlinear and LTI dynamics to the output spectrum is derived, which allows analytic description and analysis of the corresponding higher order sinusoidal input describing functions. The theoretical results are illustrated by examples that show both the use and efficiency of the proposed algorithms.

Introduction

Dynamical systems are often modeled and analyzed in the frequency domain by identifying their frequency response function (FRF). Modeling systems in the frequency domain, rather than the time domain, offers important advantages such as easy interpretation of the system dynamics and has led to significant progress in linear controller design (Bode, 1945). However, when nonlinearities are present, the FRF fails to model the complete system dynamics. Use of such models without taking nonlinearities into account can lead to unexpected and undesired results. Amplitude dependent gains can for example render a designed controller unstable and harmonics generated by nonlinearities may excite unwanted dynamics. Therefore, when extending frequency domain techniques to nonlinear systems, the effects of nonlinearities in the frequency domain have to be taken into account. Numerous studies have been performed to investigate the frequency domain modeling of nonlinear systems. Next, four approaches relevant to the presented work are discussed.

Generalized FRF The Generalized Frequency Response Function (GFRF) provides a generalization of the FRF for linear time invariant systems to nonlinear systems. The GFRF was first introduced in George (1959) and methods to measure and interpret the GFRF were developed in Billings and Tsang (1989). This research is continued in for example Li and Billings (2011) and in Yue, Billings, and Lang (2005) where the GFRFs are interpreted in terms of multidimensional frequency spaces and visualization techniques are developed. In Shanmugam and Jong (1975) the GFRFs are related to the parameters of block structured nonlinear systems similar to the systems analyzed in this paper. Application of the GFRFs to force a required output spectrum from a nonlinear system is presented in Jing, Lang, and Billings (2010) (see also Nuij, Steinbuch, and Bosgra (2008c)). Furthermore, the analysis of the effects of time domain model parameters on the frequency domain behavior of the nonlinear system (Jing, Lang, Billings, & Tomlinson, 2006) led to the definition of the output frequency response function (OFRF) in Lang, Billings, Yue, and Li (2007).

FRF for nonlinear systems In Pavlov, Van de Wouw, and Nijmeijer (2007) the notion of an FRF for uniformly convergent nonlinear systems (Pavlov, Pogromsky, Van De Wouw, & Nijmeijer, 2004) is introduced. This function maps a harmonic input signal to the output of the system by means of a state or output FRF. The authors also propose an extension of the concept of the Bode plot, to visualize part of the nonlinear dynamics. These ideas lead to a well defined notion of performance for a class of (controlled) nonlinear systems as demonstrated in Van de Wouw, Pastink, Heertjes, Pavlov, and Nijmeijer (2008) in an application to variable gain control design for optical storage drives.

BLA and detection of nonlinearities When identifying frequency domain models, the quality of the model with respect to linearity needs to be assessed. In Pintelon and Schoukens (2001) frequency domain identification methods are discussed which provide a quantitative measure for the level and type of nonlinear influences. The authors utilize the properties of a special class of multisine excitation signals (Pintelon, Vandersteen, De Locht, Rolain, & Schoukens, 2004) and averaging techniques to derive a best linear approximation (BLA) (Schoukens et al., 2005, Schoukens et al., 2009) of the systems dynamics and to quantify nonlinear influences.

HOSIDF The Higher Order Sinusoidal Input Describing Functions (HOSIDFs) are introduced in Nuij, Bosgra, and Steinbuch (2006) and are an extension of the Sinusoidal Input Describing Function (SIDF) (Gelb & Vander Velde, 1968). Other than the SIDF, the HOSIDFs describe the systems response (gain and phase) to a sinusoidal input signal, at harmonics of the excitation frequency. Open and closed loop identification of the HOSIDFs are discussed in Nuij et al. (2006) and Nuij, Steinbuch, and Bosgra (2008a). The HOSIDFs are compared to the best linear approximation in Rijlaarsdam, van Loon, Nuij, and Steinbuch (2010) and used to derive physical (friction) parameters in Nuij, Steinbuch, and Bosgra (2008b). Finally, the application of HOSIDFs to (nonlinear) controller design for nonlinear systems yields significant advantages over conventional time domain tuning (Rijlaarsdam, Nuij, Schoukens, & Steinbuch, 2011).

Contribution This paper contributes to the analysis of the effects of nonlinearities in the frequency domain by supplying new analytical tools and results that allow for the analysis of the output spectra of nonlinear systems and the corresponding Higher Order Sinusoidal Input Describing Functions (HOSIDFs). Compared to the GFRF, the ‘FRF for nonlinear systems’ and the BLA, the main disadvantage of the results presented in this paper is the limited class of excitation signal considered. Although the class of systems considered is significant, the methods presented in this paper are applicable for sinusoidal inputs only. However, the response of a nonlinear system to a sinusoidal input signal already yields valuable information in, for example, optimal nonlinear control design (Rijlaarsdam et al., 2011).

The main advantage of the results presented in the following, compared to the GFRF, is their simplicity and intuitive insight in the effects of nonlinearities in the frequency domain. This yields a clear extension to identification purposes as recently demonstrated in Rijlaarsdam (2011) where novel techniques for broadband identification of the HOSIDFs are presented. An important advantage of the results presented here over the FRF for nonlinear systems and the BLA is that they provide detailed phase information about nonlinear influences. This is found to be crucial in control applications (Rijlaarsdam et al., 2011).

Structure This paper deals with the spectral analysis of systems with polynomial nonlinearities. First, in Section 3, an analytical expression for the output spectra of a class of block structured nonlinear systems is derived. Next, in Section 4, the Higher Order Sinusoidal Input Describing Functions (HOSIDFs) are introduced and analytical expressions for the HOSIDFs of a class of nonlinear systems are derived. Finally, Section 5 presents illustrative examples. Matlab tools to apply the presented results are available online.2

Section snippets

Nomenclature and preliminaries

In the following, the continuous time signals considered are denoted by non-capitalized roman letters x(t)R. The corresponding continuous spectra X(ω)C are denoted in capitalized, calligraphic font. Unless specified otherwise, single sided spectra are considered. Frequent use is made of vectors containing only specific spectral components X[]=X((1)ω0) denoted in capitalized roman letters. Hence, X[]C contains the spectral component at the k=(1)th harmonic of the excitation frequency ω0

Spectral analysis of nonlinear systems

In this section new efficient analytical results are introduced which allow us to model the input–output behavior of a class of nonlinear systems with polynomial nonlinearities. These results are applied in the analysis of block structured nonlinear systems and allow frequency domain analysis of such systems by means of the higher order sinusoidal input describing functions.

Higher Order Sinusoidal Input Describing Functions

The Higher Order Sinusoidal Input Describing Functions (HOSIDFs) are introduced in Nuij et al. (2006). In this section a new definition of the HOSIDF is provided and the HOSIDFs of LPL¯ systems are analyzed.

Examples

The following examples illustrate the presented results.

Example 1 Output Spectrum

Consider a polynomial mapping (1) with P=3 subject to (2). Then, the the output spectrum is readily computed using Theorem 1. Y=Φ(φ0)ΩΓ(γ)α=[10000e1iφ00000e2iφ00000e3iφ0][020206020002][(γ2)1000(γ2)2000(γ2)3][α1α2α3].

Example 2

Spectrum & HOSIDFs of LPL¯ Systems

Consider the two branch LPL¯ system depicted in Fig. 2, subject to (2). Application of Lemma 1 immediately yields an analytic expression for the output spectrum: Y=Δ(ω0)G1+(ω)Φ(φ0)ΩΓ(γ)α[1]+Δ(ω0)Φ(φ0+G2)ΩΓ(γ|G2|)α[2]=[b2γ22|G

Conclusion

This paper presents new, efficient analytical tools and results that allow spectral analysis of the output of a class of nonlinear systems. This provides insight into the dynamics of block structured dynamical systems and allows analytic description and analysis of the corresponding Higher Order Sinusoidal Input Describing Functions (HOSIDFs). Given the systems linear dynamics, the output spectra and HOSIDFs can be described as a simple polynomial function of the parameters defining the

David Rijlaarsdam received the M.Sc. degree from the Eindhoven University of Technology (TU/e) in 2008. During his studies he worked on synchronization in nonlinear systems with an application to modeling neuronal behavior at the RIKEN Brain Science Institute (Tokyo, Japan). Currently, he is a Ph.D. candidate at the Control Systems Technology group at the TU/e and at the Department Fundamental Electricity and Instrumentation at the Vrije Universiteit Brussel, Brussels. His work focuses on

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      However, as the excitation signal is a single sinusoid, both desensitization and intermodulation are not captured by the HOSIDFs. The HOSIDFs yield valuable information about nonlinearities that can be used for modeling and control purposes [47,48,71,72]. Finally, related studies as presented in [49] use the concept of the HOSIDF to find an input signal that yields a sinusoidal output signal, similar to Volterra based approaches presented in [32,34].

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    David Rijlaarsdam received the M.Sc. degree from the Eindhoven University of Technology (TU/e) in 2008. During his studies he worked on synchronization in nonlinear systems with an application to modeling neuronal behavior at the RIKEN Brain Science Institute (Tokyo, Japan). Currently, he is a Ph.D. candidate at the Control Systems Technology group at the TU/e and at the Department Fundamental Electricity and Instrumentation at the Vrije Universiteit Brussel, Brussels. His work focuses on modeling and control of nanoscale positioning systems in close cooperation with industry. His research interests are modeling and control of nonlinear systems and synchronization.

    Pieter Nuij received the M.Sc. degree in Mechanical Engineering from Eindhoven University of Technology, Eindhoven, The Netherlands, in 1982. From 1982 until 2000 he worked in industry both in Holland (Philips) and abroad (Brüel & Kjær Denmark) focusing on signal analysis and system analysis applications in mechatronics development. He is assistant professor in the Control Systems Technology group of the Mechanical Engineering Department of Eindhoven University of Technology. He gained his Ph.D. in 2007. His research interests are focused on identification of nonlinear systems and control of fusion plasmas.

    Johan Schoukens received the Engineer and the Doctor degrees in applied sciences from Vrije Universiteit Brussel (VUB), Brussels, Belgium, in 1980 and 1985, respectively. He is currently a Professor with the Department of Fundamental Electricity and Instrumentation (ELEC), VUB. The main interests of his research are in the field of system identification for linear and nonlinear systems. Dr. Schoukens received the Best Paper Award in 2002 and the Distinguished Service Award in 2003 from the IEEE Instrumentation and Measurement Society. In 2011, he received an Honorary Degree (Doctor Honoris Causa) from the Budapest University of Technology and Economics.

    Maarten Steinbuch received the M.Sc. and Ph.D. degrees from Delft University of Technology. From 1987 to 1999 he was with Philips, Eindhoven. Since 1999 he has been full professor in Systems and Control, and head of the Control Systems Technology group of the Mechanical Engineering Department of Eindhoven University of Technology. He is editor-in-chief of IFAC Mechatronics. Since July 2006 he has also been Scientific Director of the Centre of Competence High Tech Systems of the Federation of Dutch Technical Universities. His research interests are modeling, design and control of motion systems, robotics, automotive powertrains and control of fusion plasmas.

    This work is carried out as part of the Condor project, a project under the supervision of the Embedded Systems Institute (ESI) and with the FEI company as the industrial partner. This project is partially supported by the Dutch Ministry of Economic Affairs under the BSIK program. This work was supported in part by the Fund for Scientific Research (FWO-Vlaanderen), by the Flemish Government (Methusalem), and by the Belgian Government through the Interuniversity Poles of Attraction (IAP VI/4) Program. This paper has not been presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Er-Wei Bai under the direction of Editor Torsten Söderström.

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