Research Papers

Analysis of Oscillations in Relay Feedback Systems With Fractional-Order Integrating Plants

[+] Author and Article Information
Davood Rezaei

Electrical Engineering Department,
Sharif University of Technology,
Tehran 1458889694, Iran

Mohammad Saleh Tavazoei

Electrical Engineering Department,
Sharif University of Technology,
Tehran 1458889694, Iran
e-mail: tavazoei@sharif.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 1, 2016; final manuscript received June 2, 2017; published online July 12, 2017. Assoc. Editor: Bernard Brogliato.

J. Comput. Nonlinear Dynam 12(5), 051023 (Jul 12, 2017) (7 pages) Paper No: CND-16-1315; doi: 10.1115/1.4037103 History: Received July 01, 2016; Revised June 02, 2017

Oscillatory behavior and transfer properties of relay feedback systems with a linear plant including a fractional-order integrator are studied in this paper. An expression for system response in the time domain is obtained by means of short memory principle, Poincare return map, and Mittag–Leffler functions. On the basis of this expression, the frequency of self-excited oscillations is approximated. In addition, the locus of perturbed relay system (LPRS) is derived to analyze the input–output properties of the relay system. The presented analysis is supported by a numerical example.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Tarbouriech, S. , Prieur, C. , and Queinnec, I. , 2010, “ Stability Analysis for Linear Systems With Input Backlash Through Sufficient LMI Conditions,” Automatica, 46(11), pp. 1911–1915. [CrossRef]
Shen, T. , Wang, Xi. , and Yuan, Zh. , 2010, “ Stability Analysis for a Class of Digital Filters With Single Saturation Nonlinearity,” Automatica, 46(12), pp. 2112–2115. [CrossRef]
Liua, K. Z. , and Akasaka, D. , 2014, “ A Partial Parameterization of Nonlinear Output Feedback Controllers for Saturated Linear Systems,” Automatica, 50(1), pp. 233–239. [CrossRef]
Lens, H. , Adamy, J. , and Yankulova, D. D. , 2011, “ A Fast Nonlinear Control Method for Linear Systems With Input Saturation,” Automatica, 47(4), pp. 857–860. [CrossRef]
Zou, X. , Luo, J. , and Cao, C. , 2013, “ Adaptive Control for Uncertain Hysteretic Systems,” ASME J. Dyn. Syst. Meas. Control, 136(1), p. 011011. [CrossRef]
Yu, W. , Wilson, D. I. , and Young, B. R. , 2010, “ Nonlinear Control Performance Assessment in the Presence of Valve Stiction,” J. Process Control, 20(6), pp. 754–761. [CrossRef]
Esbrook, A. , Tana, X. , and Khalil, H. , 2014, “ Self-Excited Limit Cycles in an Integral-Controlled System With Backlash,” IEEE Trans. Autom. Control, 59(4), pp. 1020–1025. [CrossRef]
Tana, X. , and Baras, J. S. , 2004, “ Modeling and Control of Hysteresis in Magnetostrictive Actuators,” Automatica, 40(9), pp. 1469–1480. [CrossRef]
Lou, A. C. , and Patel, M. T. , 2009, “ Bifurcation and Stability of Periodic Motions in a Periodically Forced Oscillator With Multiple Discontinuities,” ASME J. Comput. Nonlinear Dyn., 4(1), p. 011011. [CrossRef]
Ozdemir, N. , and Iskender, B. B. , 2010, “ Fractional Order Control of Fractional Diffusion Systems Subject to Input Hysteresis,” ASME J. Comput. Nonlinear Dyn., 5(2), p. 021002. [CrossRef]
Pascal, M. , 2008, “ Dynamics of Coupled Oscillators Excited by Dry Friction,” ASME J. Comput. Nonlinear Dyn., 3(3), p. 031009. [CrossRef]
Tsypkin, Y. Z. , 1984, Relay Control Systems, Cambridge University Press, Cambridge, UK.
Atherton, D. P. , 1975, “ Analysis and Design of Relay Control System,” CAD for Control Systems, D. A. Linkens , ed., Marcel Dekker, New York.
Yu, C. C. , 1999, Autotuning of PID Controllers, Springer, London.
Wang, Q. G. , Hang, C. C. , and Zou, B. , 2000, “ Multivariable Process Identification and Control From Decentralized Relay Feedback,” Int. J. Model. Simul., 20(4), pp. 341–348. http://www.tandfonline.com/doi/abs/10.1080/02286203.2000.11442176
Astrom, K. J. , 1995, “ Oscillations in Systems With Relay Feedback,” Adaptive Control, Filtering, and Signal Processing (IMA Volumes in Mathematics and Its Applications), Vol. 74, Springer, New York, pp. 1–25.
Wang, Q.-G. , Lee, T. H. , and Lin, C. , 2003, Relay Feedback: Analysis, Identification and Control, Springer, London.
Ye, Z. , Wang, Q. G. , Lin, C. , and Barabanov, A. E. , 2007, “ Relay Feedback Analysis for a Class of Servo Plants,” J. Math. Anal. Appl., 334(1), pp. 28–42. [CrossRef]
Xu, H. , and Wen, G. , 2014, “ Alternative Criterion for Investigation of Pitchfork Bifurcations of Limit Cycle in Relay Feedback Systems,” ASME J. Comput. Nonlinear Dyn., 9(3), p. 031004. [CrossRef]
Boiko, I. , 2005, “ Oscillations and Transfer Properties of Relay Servo Systems—The Locus of a Perturbed Relay System Approach,” Automatica, 41(4), pp. 677–683. [CrossRef]
Boiko, I. , 1999, “ Input–Output Analysis of Limit Cycling Relay Feedback Control Systems,” American Control Conference (ACC), San Diego, CA, June 2–4, Vol. 1, pp. 542–546.
Boiko, I. , 2007, “ Analysis of Closed-Loop Performance and Frequency-Domain Design of Compensating Filters for Sliding-Mode Control Systems,” IEEE Trans. Autom. Control, 52(10), pp. 1882–1891. [CrossRef]
Boiko, I. , 2008, “ Oscillations and Transfer Properties of Relay Servo Systems With Integrating Plants,” IEEE Trans. Autom. Control, 53(11), pp. 2686–2689. [CrossRef]
Petras, I. , 2011, Fractional-Order Nonlinear Systems, Springer, London.
Pudlubny, I. , 1999, Fractional Differential Equation, Academic Press, Millbrae, CA.
Tötterman, S. , and Toivonen, H. T. , 2009, “ Support Vector Method for Identification of Wiener Models,” J. Process Control, 19(7), pp. 1174–1181. [CrossRef]
Wills, A. , Schön, T. B. , Ljung, L. , and Ninness, B. , 2011, “ Blind Identification of Wiener Models,” IFAC Proc. Vol., 44(1), pp. 5597–5602. [CrossRef]
Feliu-Talegon, D. , San-Millan, A. , and Feliu-Batlle, V. , 2016, “ Fractional-Order Integral Resonant Control of Collocated Smart Structures,” Control Eng. Pract., 56, pp. 210–223. [CrossRef]
Gomez, J. C. , Jutan, A. , and Baeyens, E. , 2004, “ Wiener Model Identification and Predictive Control of a pH Neutralisation Process,” IEEE Proc. Control Theory Appl., 151(3), pp. 329–338. [CrossRef]
Wang, Q. , and Zhang, J. , 2011, “ Wiener Model Identification and Nonlinear Model Predictive Control of a pH Neutralization Process Based on Laguerre Filters and Least Squares Support Vector Machines,” J. Zhejiang Univ., Sci., C, 12(1), pp. 25–35. [CrossRef]
Mitrishkin, Y. V. , Pavlova, E. A. , Kuznetsov, E. A. , and Gaydamakac, K. I. , 2016, “ Continuous, Saturation, and Discontinuous Tokamak Plasma Vertical Position Control Systems,” Fusion Eng. Des., 108, pp. 35–47. [CrossRef]
Srinivasan, A. , and Lakshmi, P. , 2008, “ Identification and Control of Wiener Type Process Applied to Real-Time Heat Exchanger,” Asia-Pac. J. Chem. Eng., 3(6), pp. 622–629. [CrossRef]
Ferretti, G. , 2004, “ Single and Multistate Integral Friction Models,” IEEE Trans. Autom. Control, 49(12), pp. 2292–2297. [CrossRef]
Hsu, J. C. , and Meyer, A. U. , 1968, Modern Control Principles and Applications, McGraw-Hill, New York.
Diethelm, K. , Ford, N. J. , and Freed, A. D. , 2002, “ A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations,” Nonlinear Dyn., 29(1), pp. 3–22. [CrossRef]
Tavazoei, M. S. , and Haeri, M. , 2007, “ Unreliability of Frequency-Domain Approximation in Recognising Chaos in Fractional-Order Systems,” IET Signal Process., 1(4), pp. 171–181. [CrossRef]


Grahic Jump Location
Fig. 1

A relay feedback system with fractional-order integrating plant

Grahic Jump Location
Fig. 2

LPRS functions for four values of α in the complex plane

Grahic Jump Location
Fig. 3

System response to a sinusoidal input



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