NONE 20260713120531179 wseas/wseas content-registration-form@crossref.org content-registration-form International Journal of Electrical Engineering and Computer Science 2769-2507 Kartik Chandra Patra Department of Electrical Engineering C. V. Raman Global University, Bhubaneswar, Odisha 752054, INDIA https://orcid.org/0000-0002-4693-4883 Asutosh Patnaik Department of Electrical Engineering C. V. Raman Global University, Bhubaneswar, Odisha 752054, INDIA The paper presents an innovative graphical technique that uses computer graphics in conjunction with geometric tools in the estimation of limit cycles (LC) in 3×3 nonlinear (NL) systems with rectangular hysteresis and backlash Nonlinearities. In the event the system exhibits LCs under autonomous conditions, the work explores the possibility of mitigating/quenching limit cycling oscillations through signal stabilization using high-frequency deterministic/Gaussian signals. In addition, another attempt is taken to suppress LCs by the pole placement (PP) technique through state feedback. The state feedback gain matrix K is determined either by arbitrary selection ensuring a complete state controllability condition or by optimal selection from the Riccati equation. To avoid the mathematical complexity of formulation for 3×3 memory-type Nonlinearities, in particular for estimation/prediction of LCs, a novel graphical method is developed using the most common and popular describing function (DF), assuming harmonic balance with proper justification that the linear components of systems possess low-pass characteristics. The complexity and criticality in the mathematical expression (which diminish problem insight) are mitigated when the system exhibits LC predominantly at a single frequency. The proposed methods are demonstrated through practical examples and validated using digital simulations, implemented via a program developed in MATLAB. These results are further substantiated by those obtained using the SIMULINK toolbox in the MATLAB environment. 07 13 2026 07 13 2026 38 4 https://creativecommons.org/licenses/by/4.0/deed.en_US 10.37394/232027.2026.8.4 https://wseas.com/journals/eeacs/2026/a08eeacs-004(2026).pdf Gelb, A. (1963) ‘Limit cycles in symmetric multiple NL systems’, IEEE Trans. Autumn. Control: AC-8, pp. 177-178. Gran, R. and Rimer, M. (1965) ‘Stability analysis of systems with multiple Nonlinearities’, IEEE Trans. Autumn. Control: 10, pp. 94-97. Jud, H.G. (1964) ‘Limit cycle determination of parallel linear and non- linear elements’, IEEE Trans. Autumn. Control: AC-9, pp. 183-184. Davison, E.J. and Constantinescu, D. (1971) ‘Describing function technique for multiple NLity in a single feedback system’, IEEE Trans Autumn. Control: AC-16, pp. 50-60. Oldenburger, R. and Nakada, T. (1961) ‘Signal stabilisation of self - oscillating system’, IRE Trans. Automat Control. USA, 6, pp: 319-325. Viswandham, N., and Deekshatulu, B.L. (1966) ‘Stability analysis of NL multivariable systems’, Int. J. Control, 5, pp. 369-375. Gelb, A. and Vader-Velde, W.E. (1968) ‘Multipleinput describing functions and NL system design’, McGraw- Hill, New York. Nikiforuk, P.N., and Wintonyk, B.L.M. (1968) ‘Frequency response analysis of two-dimensional NL symmetrical and non-symmetrical control systems’, Int. J. Control, 7, pp.49- 62. Raju, G.S., and Josselson, R. (1971) ‘Stability of reactor control systems in coupled core reactors’, IEEE Trans. Nuclear Science, NS-18, pp. 388-394. Atherton, D.P. (1975) ‘Non-linear control engineering - Describing function analysis and design’, Van Noslrand Reinhold, London. Atherton, D.P., and Dorrah, H.T. (1980) ‘A survey on NL oscillations’, Int. J. Control, 31. (6), pp. 1041- 1105. Gray, J. O. And Nakhala, N.B. (1981) ‘Prediction of limit cycles in multivariable NL systems’, Proc. IEE, Part-D, 128, pp.233-241. Mees, A.I. (1984) ‘Describing function: Ten years on’, IMA J. Appl. Math., 34, 1984 pp. 221-233. Sebastian, L. (1985) ‘The self-oscillation determination to a category of NL closed loop systems, IEEE Trans. Autumn. Control, AC -30, (7), pp. 700-704. Cook, P.A. (1986) ‘NL dynamical systems’, Prentice-Hall, Englewood ClilTs, NJ. Chang, H.C., Pan, C.T., Huang, C.L., and Wei, C.C. (1987) ‘A general approach for constructing the limit cycle loci of multiple NLity systems’, IEEE Trans. Autumn. Control, AC-32, (9), pp. 845-848. Parlos, A.G., Henry, A.F., Schweppe, F.C., Gould, L.A., and Lanning, D.D. (1988) ‘NL multivariable control of nuclear power plants based on the unknown but bounded disturbance model’, IEEE Trans. Autumn.Control, AC-33, (2), pp.130-134. Pillai, V.K., and Nelson, H.D. (1988) ‘A new algorithm for limit cycle analysis of NL systems’, Trans. ASME, J. Dyn. Syst. Meas. Control, 110, pp. 272-277. Genesio, R., and Tesi. (1988) ‘A On limit cycles of feedback polynomial systems’, IEEE Trans. Circuits Syst., 35, (12), pp. 1523-1528. Fendrich, O.R. (1992) ‘Describing functions and limit cycles’, IEEE Trans. Autom. Control, AC -31, (4), pp. 486487. Zhuang, M., and Artherton, D.P. (1994) ‘PID controller design lor TITO system’, TEE Proc. Control Theory Appl. 141, (2), pp. 111-120. Loh, A.P., and Vasanu, V.V. (1994) ‘Necessary conditions for limit cycles in multi loop relay systems’, IEE Proc., Control Theory Appl., 141, 31, pp. 163-168. Tesi, A, et al. (1996) ‘Harmonic balance analysis of periodic doubling bifurcations with implications for control of NL dynamics’, Automatic, 32 (9), 1255, 1271. Lin, C.H., Han, K.W. (1999) ‘Prediction of Limit cycle in NL two input two output control system’, ‘IEE Proc.-Control Theory Appl. Vol.146, No.3. Hori, Y. et al. (1999) ‘Slow resonance ratio control for vibration suppression and disturbance rejection in torsional system’, IEEE Trans. Ind. Electron., vol. 46, (1), pp.162-168. Nordin, M. and Gutman, P. O. (2002) ‘Controlling mechanical systems with backlash- a survey’, Automatica, vol. 38, (10), pp.1633-1649. Chidambaram, I.A, and Velusami, S. (2005) ‘Decentralized biased controllers for load-frequency control of inter connected power systems considering governor dead band non-linearity, INDICON, Annual IEEE, pp.521-525. Eftekhari, M and Katebi, S. D. (2006) ‘Evolutionary Search for Limit Cycle and Controller Design in Multivariable nonlinear systems’, Asian Journal of Control, , Vol. 8, No. 4, pp. 345 – 358. Katebi, M, et al. (2009) ‘Limit Cycle Prediction Based on Evolutionary Multi objective Formulation’, Hindawi Publishing Corporation, Mathematical Problems in engineering Volume, Article ID 816707, 17pgs. Garrido, J, et al. (2009) ‘Centralized PID control by Decoupling of a Boiler-Turbine Unit’, Proceedings of the European Control Conference, Budapest, Hungary, 23-26. Tsay, T.S. (2011) ‘Limit Cycle prediction of NL multivariable feedback control systems with large transportation lags’, Hindawi Publishing corporation journal of control science and Engineering, article id 169848. Tsay, T.S. (2011) ‘Load Frequency control of interconnected power system with governor backlash Nonlinearities’, Electrical Power and Energy, vol. 33, pp.1542-1549. Sujatha, V., Panda, R. C. (2012) ‘Relay Feedback Based Time domain modelling of Linear 3-by-3 MIMO System’, American Journal of System Science, Scientific & Academic Publishing, 1(2), pp. 17-22. Wang, C, et al. (2015) ‘Vibration suppression with shaft torque limitation using explicit MPC-PI switching control in elastic drive systems, IEEE Trans. Ind. Electron, vol. 62, (11), pp. 6855-6867. Yang, M, et al. (2015) ‘Suppression of mechanical resonance using torque disturbance observer for two inertia system with backlash’, Proc. IEEE 9th Int. Conf. Power Electron., ECCE Asia, pp. 1860 - 1866. Shi, Z, and Zuo, Z. (2015) Back stepping control for gear transmission servo systems with backlash NLity’, IEEE Trans. Autumn. Sci. Eng., vol. 12, (2), pp. 752-757. Wang, C., et al. (2017) ‘Analysis and suppression of limit cycle oscillation for Transmission System with backlash NLity’, IEEE Transactions on Industrial Electronics, vol. 62, (12), pp. 9261-9270. Patra, K. C, and Dakua, B. K. (2018) ‘Investigation of limit cycles and signal stabilisation of two dimensional systems with memory type NL elements’, Archives of Control Sciences, vol. 28, (2), pp. 285-330. Patra, K. C, Kar, N. (2020) ‘Signal Stabilization of Limit cycling two Dimensional Memory Type NL Systems by Gaussian Random Signal’, International Journal of Emerging Trends & Technology in Computer Science, Vol. 9 Issue 1, pp. 10-17. Patra, K. C, Kar, N. (2022) ‘Suppression Limit cycles in 2 x 2 NL systems with memory type Nonlinearities’, International Journal of Dynamics and Control, Springer Nature’,34, vol.10 Issue 3, pp 721-733. Iurashev, D., Campa. G., Anishimov V. V. and Cosatto, E. (2017) ‘Three step approach for prediction of limit cycles pressure oscillations in combustion chambers of gas turbines’, Taylor and Francis, Combustion Theory and Modelling, 21(6), 1- 28. Tsay, T.S. (2012) ‘Stability Analysis of NL Multivariable feedback Control systems’, WSEAS Transactions on systems, Volume 11, Issue 4, pp. 140 – 151. Tripathy, S.C., Hop, G.S. and Malik, O. P. (1982) ‘Optimisation of Load-frequency control parameters for power system with read steam turbines and governor dead band NLity’, PROC IEE, Ptc. 129(1), 10-16. Patra, K.C., Patnaik, A. (2023) ‘Possibility of Quenching of Limit Cycles in Multivariable NL systems with special attention to 3 x 3 systems’, WSEAS Transactions on Systems and Control, Vol. 18. Oldenburger, R., Sridhar, R., “Signal Stabilization of a Control System with Random Input” Institute of Electrical Engineers. Part II: Applications and Industry; Vol.80, Issue 5, 1961, pp 260-268 Patra, K.C., Patnaik, A. (2023) ‘Investigation of the Existence of Limit Cycles in Multi Variable NL Systems with Special Attention to 3x3 Systems’, Int. Journal of Applied Mathematics, Computational Science and System Engineering, Vol. 5, pp. 93-114. Sridhar, R., Oldenburger, R. (1962) ‘Stability of a NL feedback system in the presence of Gaussian Noise’, Journal of Basic Engineering, 84, pp. 61-70. Gibson, J. E., Sridhar, R., (1964) ‘The Response of NL closed-loop Systems to Random Inputs’, ASME Trans., Journal of Basic Engineering, pp. 132- 138. Ogata K, Modern control engineering, 5th Edn. P H I Learning, pp.723-724 and 2012. Raymond, T, et al. (2002) ‘Design of Feedback Control Systems’, Oxford University Press, 4th edition, pp. 677-678 Singh,Y.P: Some aspects of forced oscillations of non-linear systems (Doctoral dissertation, Ph.D. Thesis, Electrical Engg, IIT Kharagpur, India (1968)