Research Papers

Numerical Solution to the Inverse Sinusoidal Input Describing Function

[+] Author and Article Information
Fatemeh Zamanian

Mechanical Engineering Department,
University of Houston,
Houston, TX 77004
e-mail: s.zamanian@gmail.com

Matthew A. Franchek, Karolos M. Grigoriadis

Mechanical Engineering Department,
University of Houston,
Houston, TX 77004

Behrouz Ebrahimi

Mechanical Engineering Department,
University of Houston,
Houston, TX 77004

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received June 13, 2016; final manuscript received November 13, 2016; published online March 13, 2017. Assoc. Editor: Davide Spinello.

J. Dyn. Sys., Meas., Control 139(5), 051008 (Mar 13, 2017) (11 pages) Paper No: DS-16-1314; doi: 10.1115/1.4035296 History: Received June 13, 2016; Revised November 13, 2016

A computational solution to the inverse sinusoidal input describing function (SIDF) for a broad class of static nonlinearities is presented. The proposed numerical solution uses the SIDF gain and phase distortions to identify the nonlinearity. This solution, unlike existing methods in the literature, does not require a priori knowledge of the nonlinearity structure in the estimation process and is applicable to both single- and double-valued nonlinearities. The output from the algorithm is a nonparametric model of the nonlinearity from which a parametric model can be recovered by least-square estimation (LSE) method. Three examples are presented to validate the proposed algorithm.

Copyright © 2017 by ASME
Topics: Algorithms , Signals
Your Session has timed out. Please sign back in to continue.


Gelb, A. , and Vander Velde, W. E. , 1968, Multiple Input Describing Functions and Nonlinear System Design, McGraw-Hill, New York.
Hsu, J. C. , and Meyer, A. U. , 1968, Modern Control Principles and Applications, McGraw-Hill, New York.
Atherton, D. P. , 1975, Nonlinear Control Engineering, Van Nostrand Reinhold, London, UK.
Glass, J. W. , and Franchek, M. A. , 2000, “ Frequency-Based Nonlinear Controller Design for Regulating Systems Subject to Time-Domain Constraints,” Int. J. Robust Nonlinear Control, 10(1), pp. 39–57. [CrossRef]
Nassirharand, A. , and Karimi, H. , 2006, “ Nonlinear Controller Synthesis Based on Inverse Describing Function Technique in the MATLAB Environment,” Adv. Eng. Software, 37(6), pp. 370–374. [CrossRef]
Nanka-Bruce, O. , and Atherton, D. P. , 1990, “ Design of Nonlinear Controllers for Nonlinear Plants,” 11th IFAC World Congress, Tallinn, Estonia., 6, pp. 75–80.
Zhuang, M. , and Atherton, D. P. , 1996, “ CAD of Nonlinear Controllers for Nonlinear Systems,” UKACC International Conference on Control, 1, Conf. Pub. No. 427, pp. 545–550.
Taylor, J. H. , 1998, “ Robust Nonlinear Control Based on Describing Function Methods,” Proceedings of the ASME IMECE Dynamic Systems and Control Conference, Vol. 64, Anaheim, CA, Nov. 15–20.
Bergen, A. R. , and Franks, R. L. , 1971, “ Justification of the Describing Function Method,” SIAM J. Control, 9(4), pp. 568–589.
MacFarlane, A. G. J. , 1978, Frequency-Response Methods in Control Systems, IEEE Press, Cambridge, UK.
Udrea, A. , Lupu, C. , and Popescu, D. , 2011, “ Hysteresis Control of a (Ba/Sr)TiO3 Based Actuator: A Comparison of Prandtl-Ishlinskii and Nonlinear Compensator Numerical Methods,” 18th IFAC World Congress, Milano, Italy, Aug. 28–Sept. 2, 44(1), pp. 12733–12738.
Giron-Sierra, J. M. , Recas, J. , and Esteban, S. , 2011, “ Iterative Method Based on CFD Data for Assessment of Seakeeping Control Effects, Considering Amplitude and Rate Saturation,” Int. J. Robust. Nonlinear Control, 21(13), pp. 1562–1573. [CrossRef]
Tsay, T.-S. , 2014, “ Model Based Adaptive Piecewise Linear Controller for Complicated Control Systems,” J. Appl. Math., 2014, p. 120419.
Teh, S. H. , Malawaraarachci, S. , Chan, W. P. , and Nassirharand, A. , 2010, “ Design and Instrumentation of a Benchmark Multivariable Nonlinear Control Laboratory,” International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, 4(2), pp. 133–139.
Tsay, T.-S. , 2013, “ Adaptive Piecewise Linear Controller for Servo Mechanical Control Systems,” J. Appl. Math. Phys., 1(5), pp. 85–92. [CrossRef]
Li, S. C. , and Nassirharand, A. , 2012, “ Nonlinear Proportional-Integral-Derivative Synthesis for Unstable Non-Linear System Using Describing Function Inversion With Experimental Verification,” J. Syst. Control Eng., 226(2), pp. 145–153.
Atherton, D. P. , 1970, “ Comment on Describing Function Inversion,” Electron. Lett., 6(24), pp. 779–780. [CrossRef]
Atherton, D. P. , 1974, “ Inverse Random Describing Function,” Electron. Lett., 10(6), pp. 82–83. [CrossRef]
Atherton, D. P. , 1975, “ The Inverse Multiple Input Describing Function Problem,” Int. J. Control, 21(3), pp. 385–390. [CrossRef]
Nassirharand, A. , 2009, “ Matlab Software for Inversion of Describing Function,” Adv. Eng. Software, 40(8), pp. 600–606. [CrossRef]
Gibson, J. E. , DiTada, E. S. , Hill, J. C. , and Ibrahim, E. S. , 1962, Describing Function Inversion: Theory and Computational Techniques (Control & Information Systems Lab., TR-EE62-1D), Purdue University, Lafayette, IN.
Gibson, J. E. , 1963, Nonlinear Automatic Control, McGraw-Hill, New York.
Gibson, J. E. , and DiTada, E. S. , 1963, “ On the Inverse Describing Function Problem,” 2nd IFAC Conference, Basle, Switzerland.
Atherton, D. P. , 1961, “ The Evaluation of the Response of Single-Valued Nonlinearities to Several Inputs,” The Institution of Electrical Engineers, Monograph No. 474M, pp. 146–157.
Somerville, M. J. , and Atherton, D. P. , 1958, “ Multi-Gain Representation for a Single-Valued Nonlinearity With Several Inputs, and the Evaluation of Their Equivalent Gains by a Cursor Method,” The Institution of Electrical Engineers, pp. 537–549.
Douce, J. L. , 1957, “ A Note on the Evaluation of the Response of a Nonlinear Element to Sinusoidal and Random Signals,” The Institution of Electrical Engineers, pp. 88–92.


Grahic Jump Location
Fig. 1

Describing function representation

Grahic Jump Location
Fig. 3

Example of an explicit double-valued nonlinearity

Grahic Jump Location
Fig. 2

(L) Piecewise continuous nonlinearity and (R) corresponding SIDF

Grahic Jump Location
Fig. 4

Illustration of the method of cursors

Grahic Jump Location
Fig. 5

Schematic of the hysteresis nonlinearity

Grahic Jump Location
Fig. 6

Nonparametric output solution for the hysteresis nonlinearity for η1 and Δ1

Grahic Jump Location
Fig. 7

Nonparametric output solution for the hysteresis nonlinearity for η2 and Δ2

Grahic Jump Location
Fig. 9

Comparison for estimation of KP gain from two different methods (Ref. [7] and this work)

Grahic Jump Location
Fig. 10

Schematics of the asymmetric hysteresis nonlinearities: (L) with biased output and (R) with asymmetric input amplitude range and biased output

Grahic Jump Location
Fig. 11

Output of the proposed inverse SIDF algorithm for the first asymmetric hysteresis nonlinearity

Grahic Jump Location
Fig. 12

Output of the proposed inverse SIDF algorithm for the second asymmetric hysteresis nonlinearity

Grahic Jump Location
Fig. 8

Nonlinear controller gain from the proposed numerical solution, KP



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In