Skip to main content
Top
Published in:
Cover of the book

2015 | OriginalPaper | Chapter

1. Introduction

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Nonlinear analysis and design are hot topics in the literature. This chapter gives a brief introduction to frequency domain methods for nonlinear analysis and design focusing on the Volterra series approach, its main attractive features, and its associated methods/results.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literature
go back to reference Atherton DP (1975) Nonlinear control engineering. Van Nostrand Reinhold, London, Full edition 1975, student edition 1982 Atherton DP (1975) Nonlinear control engineering. Van Nostrand Reinhold, London, Full edition 1975, student edition 1982
go back to reference Bendat JS (1990) Nonlinear system analysis and identification from random data. Wiley, New York, NYMATH Bendat JS (1990) Nonlinear system analysis and identification from random data. Wiley, New York, NYMATH
go back to reference Billings SA, Lang ZQ (1996) A bound of the magnitude characteristics of nonlinear output frequency response functions. Int J Control, Part 1, 65(2):309–328 and Part 2, 65(3):365–384 Billings SA, Lang ZQ (1996) A bound of the magnitude characteristics of nonlinear output frequency response functions. Int J Control, Part 1, 65(2):309–328 and Part 2, 65(3):365–384
go back to reference Billings SA, Peyton-Jones JC (1990) Mapping nonlinear integro-differential equation into the frequency domain. Int J Control 54:863–879CrossRef Billings SA, Peyton-Jones JC (1990) Mapping nonlinear integro-differential equation into the frequency domain. Int J Control 54:863–879CrossRef
go back to reference Boutabba N, Hassine L, Loussaief N, Kouki F, Bouchriha H (2003) Volterra series analysis of the photocurrent in an Al/6T/ITO photovoltaic device. Org Electron 4:1–8CrossRef Boutabba N, Hassine L, Loussaief N, Kouki F, Bouchriha H (2003) Volterra series analysis of the photocurrent in an Al/6T/ITO photovoltaic device. Org Electron 4:1–8CrossRef
go back to reference Boyd S, Chua L (1985) Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Trans Circuits Syst CAS-32(11):1150–1160CrossRefMathSciNet Boyd S, Chua L (1985) Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Trans Circuits Syst CAS-32(11):1150–1160CrossRefMathSciNet
go back to reference Brilliant MB (1958) Theory of the analysis of non-linear systems. Technical Report 345, MIT, Research Laboratory of Electronics, Cambridge, MA, 3 March 1958 Brilliant MB (1958) Theory of the analysis of non-linear systems. Technical Report 345, MIT, Research Laboratory of Electronics, Cambridge, MA, 3 March 1958
go back to reference Bussgang JJ, Ehrman L, Graham JW (1974) Analysis of nonlinear systems with multiple inputs. Proc IEEE 62(8):1088–1119CrossRef Bussgang JJ, Ehrman L, Graham JW (1974) Analysis of nonlinear systems with multiple inputs. Proc IEEE 62(8):1088–1119CrossRef
go back to reference Doyle FJ III, Pearson RK, Ogunnaike BA (2002) Identification and control using Volterra models. Springer, BerlinCrossRef Doyle FJ III, Pearson RK, Ogunnaike BA (2002) Identification and control using Volterra models. Springer, BerlinCrossRef
go back to reference Elizalde H, Imregun M (2006) An explicit frequency response function formulation for multi-degree-of-freedom non-linear systems. Mech Syst Signal Process 20:1867–1882CrossRef Elizalde H, Imregun M (2006) An explicit frequency response function formulation for multi-degree-of-freedom non-linear systems. Mech Syst Signal Process 20:1867–1882CrossRef
go back to reference Engelberg S (2002) Limitations of the describing function for limit cycle prediction. IEEE Trans Autom Control 47(11):1887–1890CrossRefMathSciNet Engelberg S (2002) Limitations of the describing function for limit cycle prediction. IEEE Trans Autom Control 47(11):1887–1890CrossRefMathSciNet
go back to reference Fard RD, Karrari M, Malik OP (2005) Synchronous generator model identification for control application using Volterra series. IEEE Trans Energy Convers 20(4):852–858CrossRef Fard RD, Karrari M, Malik OP (2005) Synchronous generator model identification for control application using Volterra series. IEEE Trans Energy Convers 20(4):852–858CrossRef
go back to reference French S (1976) Practical nonlinear system analysis by Wiener Kernel estimation in the frequency domain. Biol Cybern 24:111–119CrossRefMATH French S (1976) Practical nonlinear system analysis by Wiener Kernel estimation in the frequency domain. Biol Cybern 24:111–119CrossRefMATH
go back to reference Friston KJ, Mechelli A, Turner R, Price CJ (2000) Nonlinear responses in fMRI: the Balloon model, Volterra kernels, and other hemodynamics. Neuroimage 12:466–477CrossRef Friston KJ, Mechelli A, Turner R, Price CJ (2000) Nonlinear responses in fMRI: the Balloon model, Volterra kernels, and other hemodynamics. Neuroimage 12:466–477CrossRef
go back to reference Gelb A, Vander Velde WE (1968) Multiple-input describing functions and nonlinear system design. McGraw-Hill, New York, NYMATH Gelb A, Vander Velde WE (1968) Multiple-input describing functions and nonlinear system design. McGraw-Hill, New York, NYMATH
go back to reference George DA (1959) Continuous nonlinear systems. Technical Report 355, MIT Research Laboratory of Electronics, Cambridge, MA, July 24 George DA (1959) Continuous nonlinear systems. Technical Report 355, MIT Research Laboratory of Electronics, Cambridge, MA, July 24
go back to reference Gilmore RJ, Steer MB (1991) Nonlinear circuit analysis using the method of harmonic balance—a review of the art. Part I. Introductory concepts. Int J Microw Millimet Wave Comput Aided Eng 1:22–27CrossRef Gilmore RJ, Steer MB (1991) Nonlinear circuit analysis using the method of harmonic balance—a review of the art. Part I. Introductory concepts. Int J Microw Millimet Wave Comput Aided Eng 1:22–27CrossRef
go back to reference Glass JW, Franchek MA (1999) Frequency based nonlinear controller design of regulating systems subjected to time domain constraints. Proceedings of the American control conference, San Diego, CA, June 1999, pp 2082–2086 Glass JW, Franchek MA (1999) Frequency based nonlinear controller design of regulating systems subjected to time domain constraints. Proceedings of the American control conference, San Diego, CA, June 1999, pp 2082–2086
go back to reference Jing XJ (2011) Frequency domain analysis and identification of block-oriented nonlinear systems. J Sound Vib 330(22):5427–5442CrossRef Jing XJ (2011) Frequency domain analysis and identification of block-oriented nonlinear systems. J Sound Vib 330(22):5427–5442CrossRef
go back to reference Jing XJ, Lang ZQ, Billings SA (2007a) New bound characteristics of NARX model in the frequency domain. Int J Control 80(1):140–149CrossRefMATHMathSciNet Jing XJ, Lang ZQ, Billings SA (2007a) New bound characteristics of NARX model in the frequency domain. Int J Control 80(1):140–149CrossRefMATHMathSciNet
go back to reference Jing XJ, Lang ZQ, Billings SA (2007b) Some typoes in ‘New Bound Characteristics of NARX Model in the Frequency Domain’. Int J Control 80(3):492–494CrossRefMATHMathSciNet Jing XJ, Lang ZQ, Billings SA (2007b) Some typoes in ‘New Bound Characteristics of NARX Model in the Frequency Domain’. Int J Control 80(3):492–494CrossRefMATHMathSciNet
go back to reference Jing XJ, Lang ZQ, Billings SA (2008a) Frequency domain analysis for suppression of output vibration from periodic disturbance using nonlinearities. J Sound Vib 314:536–557CrossRef Jing XJ, Lang ZQ, Billings SA (2008a) Frequency domain analysis for suppression of output vibration from periodic disturbance using nonlinearities. J Sound Vib 314:536–557CrossRef
go back to reference Jing XJ, Lang ZQ, Billings SA (2008b) Magnitude bounds of generalized frequency response functions for nonlinear Volterra systems described by NARX model. Automatica 44:838–845CrossRefMATHMathSciNet Jing XJ, Lang ZQ, Billings SA (2008b) Magnitude bounds of generalized frequency response functions for nonlinear Volterra systems described by NARX model. Automatica 44:838–845CrossRefMATHMathSciNet
go back to reference Jing XJ, Lang ZQ, Billings SA (2008c) Frequency domain analysis for nonlinear Volterra systems with a general nonlinear output function. Int J Control 81(2):235–251CrossRefMATHMathSciNet Jing XJ, Lang ZQ, Billings SA (2008c) Frequency domain analysis for nonlinear Volterra systems with a general nonlinear output function. Int J Control 81(2):235–251CrossRefMATHMathSciNet
go back to reference Jing XJ, Lang ZQ, Billings SA (2008d) Output frequency response function based analysis for nonlinear Volterra systems. Mech Syst Signal Process 22:102–120CrossRef Jing XJ, Lang ZQ, Billings SA (2008d) Output frequency response function based analysis for nonlinear Volterra systems. Mech Syst Signal Process 22:102–120CrossRef
go back to reference Jing XJ, Lang ZQ, Billings SA (2008e) Mapping from parametric characteristics to generalized frequency response functions of nonlinear systems. Int J Control 81(7):1071–1088CrossRefMATHMathSciNet Jing XJ, Lang ZQ, Billings SA (2008e) Mapping from parametric characteristics to generalized frequency response functions of nonlinear systems. Int J Control 81(7):1071–1088CrossRefMATHMathSciNet
go back to reference Judd KL (1998) Numerical methods in economics. MIT Press, Cambridge, MAMATH Judd KL (1998) Numerical methods in economics. MIT Press, Cambridge, MAMATH
go back to reference Kim KI, Powers EJ (1988) A digital method of modelling quadratically nonlinear systems with a general random input. IEEE Trans Acoust Speech Signal Process 36:1758–1769CrossRefMATH Kim KI, Powers EJ (1988) A digital method of modelling quadratically nonlinear systems with a general random input. IEEE Trans Acoust Speech Signal Process 36:1758–1769CrossRefMATH
go back to reference Lang Z.Q. and Billings S.A., Output frequency characteristics of nonlinear systems, International Journal of Control, Vol 64, 1049–1067, 1996CrossRefMATHMathSciNet Lang Z.Q. and Billings S.A., Output frequency characteristics of nonlinear systems, International Journal of Control, Vol 64, 1049–1067, 1996CrossRefMATHMathSciNet
go back to reference Lang ZQ, Billings SA, Tomlinson GR, Yue R (2006) Analytical description of the effects of system nonlinearities on output frequency responses: a case study. J Sound Vib 295:584–601CrossRef Lang ZQ, Billings SA, Tomlinson GR, Yue R (2006) Analytical description of the effects of system nonlinearities on output frequency responses: a case study. J Sound Vib 295:584–601CrossRef
go back to reference Lang ZQ, Billings SA, Yue R, Li J (2007) Output frequency response functions of nonlinear Volterra systems. Automatica 43:805–816CrossRefMATHMathSciNet Lang ZQ, Billings SA, Yue R, Li J (2007) Output frequency response functions of nonlinear Volterra systems. Automatica 43:805–816CrossRefMATHMathSciNet
go back to reference Leonov GA, Ponomarenko DV, Smirnova VB (1996) Frequency-domain methods for nonlinear analysis, theory and applications. World Scientific, SingaporeMATH Leonov GA, Ponomarenko DV, Smirnova VB (1996) Frequency-domain methods for nonlinear analysis, theory and applications. World Scientific, SingaporeMATH
go back to reference Ljung L (1999) System identification: theory for the user, 2nd edn. Prentice Hall, Upper Saddle River, NJ Ljung L (1999) System identification: theory for the user, 2nd edn. Prentice Hall, Upper Saddle River, NJ
go back to reference Mees AI (1981) Dynamics of feedback systems. Wiley, New York, NYMATH Mees AI (1981) Dynamics of feedback systems. Wiley, New York, NYMATH
go back to reference Nam SW, Powers EJ (1994) Application of higher-order spectral analysis to cubically nonlinear-system identification. IEEE Trans Signal Process 42(7):1746–1765CrossRef Nam SW, Powers EJ (1994) Application of higher-order spectral analysis to cubically nonlinear-system identification. IEEE Trans Signal Process 42(7):1746–1765CrossRef
go back to reference Nuij PWJM, Bosgra OH, Steinbuch M (2006) Higher-order sinusoidal input describing functions for the analysis of non-linear systems with harmonic responses. Mech Syst Signal Process 20:1883–1904CrossRef Nuij PWJM, Bosgra OH, Steinbuch M (2006) Higher-order sinusoidal input describing functions for the analysis of non-linear systems with harmonic responses. Mech Syst Signal Process 20:1883–1904CrossRef
go back to reference Ogota K (1996) Modern control engineering, 3rd edn. Prentice-Hall, Upper Saddle River, NJ Ogota K (1996) Modern control engineering, 3rd edn. Prentice-Hall, Upper Saddle River, NJ
go back to reference Orlowski P (2007) Frequency domain analysis of uncertain time-varying discrete-time systems. Circuits Syst Signal Process 26(3):293–310CrossRefMATHMathSciNet Orlowski P (2007) Frequency domain analysis of uncertain time-varying discrete-time systems. Circuits Syst Signal Process 26(3):293–310CrossRefMATHMathSciNet
go back to reference Pavlov A, Van De Wouw N, Nijmeijer H (2007) Frequency response functions for nonlinear convergent systems. IEEE Trans Autom Control 52(6):1159–1165CrossRef Pavlov A, Van De Wouw N, Nijmeijer H (2007) Frequency response functions for nonlinear convergent systems. IEEE Trans Autom Control 52(6):1159–1165CrossRef
go back to reference Peyton Jones JC (2003) Automatic computation of polyharmonic balance equations for non-linear differential systems. Int J Control 76(4):355–365CrossRefMATHMathSciNet Peyton Jones JC (2003) Automatic computation of polyharmonic balance equations for non-linear differential systems. Int J Control 76(4):355–365CrossRefMATHMathSciNet
go back to reference Peyton Jones JC, Billings SA (1989) Recursive algorithm for computing the frequency response of a class of nonlinear difference equation models. Int J Control 50(5):1925–1940CrossRefMATHMathSciNet Peyton Jones JC, Billings SA (1989) Recursive algorithm for computing the frequency response of a class of nonlinear difference equation models. Int J Control 50(5):1925–1940CrossRefMATHMathSciNet
go back to reference Peyton Jones JC, Billings SA (1990) Interpretation of non-linear frequency response functions. Int J Control 52:319–346CrossRefMATH Peyton Jones JC, Billings SA (1990) Interpretation of non-linear frequency response functions. Int J Control 52:319–346CrossRefMATH
go back to reference Pintelon R, Schoukens J (2001) System identification: a frequency domain approach. IEEE Press, Piscataway, NJCrossRef Pintelon R, Schoukens J (2001) System identification: a frequency domain approach. IEEE Press, Piscataway, NJCrossRef
go back to reference Raz GM, Van Veen BD (1998) Baseband Volterra filters for implementing carrier based nonlinearities. IEEE Trans Signal Process 46(1):103–114CrossRef Raz GM, Van Veen BD (1998) Baseband Volterra filters for implementing carrier based nonlinearities. IEEE Trans Signal Process 46(1):103–114CrossRef
go back to reference Rijlaarsdam D, Nuij P, Schoukens J, Steinbuch M (2011) Spectral analysis of block structured nonlinear systems and higher order sinusoidal input describing functions. Automatica 47(12):2684–2688CrossRefMATHMathSciNet Rijlaarsdam D, Nuij P, Schoukens J, Steinbuch M (2011) Spectral analysis of block structured nonlinear systems and higher order sinusoidal input describing functions. Automatica 47(12):2684–2688CrossRefMATHMathSciNet
go back to reference Rugh WJ (1981) Nonlinear system theory: the Volterra/Wiener approach. Johns Hopkins University Press, Baltimore, MDMATH Rugh WJ (1981) Nonlinear system theory: the Volterra/Wiener approach. Johns Hopkins University Press, Baltimore, MDMATH
go back to reference Sanders SR (1993) On limit cycles and the describing function method in periodically switched circuits. IEEE Trans Circuits Syst I Fund Theor Appl 40(9):564–572CrossRefMATH Sanders SR (1993) On limit cycles and the describing function method in periodically switched circuits. IEEE Trans Circuits Syst I Fund Theor Appl 40(9):564–572CrossRefMATH
go back to reference Schetzen M (1980) The Volterra and wiener theories of nonlinear systems. Wiley, New York, NYMATH Schetzen M (1980) The Volterra and wiener theories of nonlinear systems. Wiley, New York, NYMATH
go back to reference Schoukens J, Nemeth J, Crama P, Rolain Y, Pintelon R (2003) Fast approximate identification of nonlinear systems. In: 13th IFAC symposium on system identification, Rotterdam, The Netherlands, 27–29 August 2003, pp 61–66 Schoukens J, Nemeth J, Crama P, Rolain Y, Pintelon R (2003) Fast approximate identification of nonlinear systems. In: 13th IFAC symposium on system identification, Rotterdam, The Netherlands, 27–29 August 2003, pp 61–66
go back to reference Shah MA, Franchek MA (1999) Frequency-based controller design for a class of nonlinear systems. Int J Robust Nonlinear Control 9(12):825–840CrossRefMATHMathSciNet Shah MA, Franchek MA (1999) Frequency-based controller design for a class of nonlinear systems. Int J Robust Nonlinear Control 9(12):825–840CrossRefMATHMathSciNet
go back to reference Solomou M, Evans C, Rees D, Chiras N (2002) Frequency domain analysis of nonlinear systems driven by multiharmonic signals. In: Proceedings of the 19th IEEE conference on instrumentation and measurement technology, vol 1, pp 799–804 Solomou M, Evans C, Rees D, Chiras N (2002) Frequency domain analysis of nonlinear systems driven by multiharmonic signals. In: Proceedings of the 19th IEEE conference on instrumentation and measurement technology, vol 1, pp 799–804
go back to reference Swain AK, Billings SA (2001) Generalized frequency response function matrix for MIMO nonlinear systems. Int J Control 74(8):829–844CrossRefMATHMathSciNet Swain AK, Billings SA (2001) Generalized frequency response function matrix for MIMO nonlinear systems. Int J Control 74(8):829–844CrossRefMATHMathSciNet
go back to reference Taylor JH, Strobel KL (1985) Nonlinear compensator synthesis via sinusoidal-input describing functions. In: Proceedings of the American control conference, Boston, MA, June 1985, pp 1242–1247 Taylor JH, Strobel KL (1985) Nonlinear compensator synthesis via sinusoidal-input describing functions. In: Proceedings of the American control conference, Boston, MA, June 1985, pp 1242–1247
go back to reference Van De Wouw N, Nijmeijer H, Van Campen DH (2002) A Volterra series approach to the approximation of stochastic nonlinear dynamics. Nonlinear Dyn 27:397–409CrossRefMATH Van De Wouw N, Nijmeijer H, Van Campen DH (2002) A Volterra series approach to the approximation of stochastic nonlinear dynamics. Nonlinear Dyn 27:397–409CrossRefMATH
go back to reference Volterra V (1959) Theory of functionals and of integral and integro-differential equations. Dover, New York, NYMATH Volterra V (1959) Theory of functionals and of integral and integro-differential equations. Dover, New York, NYMATH
go back to reference Worden K, Tomlinson GR (2001) Non-linearity in structural dynamics: detection, identification and modeling. Institute of Physics Publishing, BristolCrossRef Worden K, Tomlinson GR (2001) Non-linearity in structural dynamics: detection, identification and modeling. Institute of Physics Publishing, BristolCrossRef
go back to reference Yang J, Tan SX-D (2006) Nonlinear transient and distortion analysis via frequency domain Volterra series. Circuits Syst Signal Process 25(3):295–314CrossRefMATHMathSciNet Yang J, Tan SX-D (2006) Nonlinear transient and distortion analysis via frequency domain Volterra series. Circuits Syst Signal Process 25(3):295–314CrossRefMATHMathSciNet
go back to reference Yue R, Billings SA, Lang Z-Q (2005) An investigation into the characteristics of non-linear frequency response functions. Part 1: understanding the higher dimensional frequency spaces. Int J Control, 78(13):1031–1044; and Part 2: new analysis methods based on symbolic expansions and graphical techniques. Int J Control 78:1130–1149 Yue R, Billings SA, Lang Z-Q (2005) An investigation into the characteristics of non-linear frequency response functions. Part 1: understanding the higher dimensional frequency spaces. Int J Control, 78(13):1031–1044; and Part 2: new analysis methods based on symbolic expansions and graphical techniques. Int J Control 78:1130–1149
Metadata
Title
Introduction
Authors
Xingjian Jing
Ziqiang Lang
Copyright Year
2015
DOI
https://doi.org/10.1007/978-3-319-12391-2_1