c42c6659-b62c-4b46-9c5a-def23c6a976b20230719045147511wseas:wseasmdt@crossref.orgMDT DepositInternational Journal of Applied Mathematics, Computational Science and Systems Engineering2766-982310.37394/232026https://wseas.com/journals/amcse/452023452023510.37394/232026.2023.5https://wseas.com/journals/amcse/2023.phpKartik ChandraPatraDepartment of Electrical Engineering, C. V. Raman Global University, Bhubaneswar, Odisha 752054, INDIAAsutoshPatnaikDepartment of Electrical Engineering, C. V. Raman Global University, Bhubaneswar, Odisha 752054, INDIAThe proposed work addresses the dynamics of a general system and explores the existence of limit cycles (LC) in multi-variable Non-linear systems with special attention to 3x3 nonlinear systems. It presents a simple, systematic analytical procedure as well as a graphical technique that uses geometric tools and computer graphics for the prediction of limit cycling oscillations in three-dimensional systems having both explicit and implicit nonlinear functions. The developed graphical method uses the harmonic balance/harmonic linearization for simplicity of discussion which provides a clear and lucid understanding of the problem and considers all constraints, especially the simultaneous intersection of two straight lines & one circle for determination of limit cycling conditions. The method of analysis is made simpler by assuming the whole system exhibits the limit cycling oscillations predominantly at a single frequency. The discussions made either analytically/graphically are substantiated by digital simulation by a developed program as well as by the use of the SIMULINK Toolbox of MATLAB Software.71920237192023931149https://wseas.com/journals/amcse/2023/a185103-1253.pdf10.37394/232026.2023.5.9https://wseas.com/journals/amcse/2023/a185103-1253.pdf10.1049/ip-cta:19960520PATRA, K. C, SINGH, Y.P, Graphical method of prediction of limit cycle for multivariable nonlinear system. IEE Proc. Control Theory Appl.: 143, 1996, pp. 423- 428. 10.1109/tac.1963.1105543GELB, A, Limit cycles in symmetric multiple nonlinear systems. IEEE Trans. Autumn. Control: AC-8, 1963, pp. 177-178. 10.1109/tac.1964.1105675JUD, H.G Limit cycle determination of parallel linear and non- linear elements. IEEE Trans. Autumn. Control: AC-9, 1964, pp. 183-184. 10.1109/tac.1965.1098078GRAN, R., and RIMER, M Stability analysis of systems with multiple nonlinearities. IEEE Trans. Autumn. Control: 10, 1965, pp. 94-97. 10.1109/tac.1971.1099625DAVISON, E.J., and CONSTANTINESCU, D Describing function technique for multiple nonlinearity in a single feedback system IEEE Trans Autumn. Control: AC-16: 1971, pp. 50-60. 10.1109/tac.1961.1105218OLDENBURGER, R., T. NAKADA T Signal stabilisation of self - oscillating system IRE Trans. Automat Control. USA, 6, 1961, pp: 319-325. VISWANDHAM, N., and DEEKSHATULU, B.L Stability analysis of nonlinear multivariable systems. Int. J. Control, 5, 1966, pp. 369-375. GELB, A. and VADER-VELDE, W.E Multiple-input describing functions and nonlinear system design, McGraw- Hill, New York, 1968 10.1080/00207176808905580NIKIFORUK, P.N., and WINTONYK, B.L.M Frequency response analysis of twodimensional nonlinear symmetrical and non symmetrical control systems. Int. J. Control, 7, 1968, pp.49- 62. 10.1109/tns.1971.4325897RAJU, G.S., and JOSSELSON, R Stability of reactor control systems in coupled core reactors, IEEE Trans. Nuclear Science, NS18, 1971, pp. 388-394. ATHERTON, D.P Non-linear control engineering - Describing function analysis and design. Van Noslrand Reinhold, London, 1975 ATHERTON, D.P., and DORRAH, H.T A survey on nonlinear oscillations, Int. J. Control, 31. (6), 1980, pp. 1041-1 105. 10.1049/ip-d.1981.0050GRAY, J. O. And NAKHALA, N.B Prediction of limit cycles in multivariable nonlinear systems. Proc. IEE, Part-D, 128, 1981 pp. 233-241. 10.1093/imamat/32.1-3.221MEES, A.I Describing function: Ten years on. IMA J. Appl. Math., 34, 1984 pp. 221- 233. 10.1109/tac.1985.1104028SEBASTIAN, L the self-oscillation determination to a category of nonlinear closed loop systems, IEEE Trans. Autumn. Control, AC-30, (7), 1985 pp. 700-704. PATRA, K.C Analysis of self oscillations and signal stabilisation of two-dimensional nonlinear systems. Ph. D Thesis (Dissertation), IIT, Kharagpur. India, 1986 COOK, P.A Nonlinear dynamical systems, Prentice-Hall, Englewood ClilTs, NJ, 1986 10.1109/tac.1987.1104717CHANG, H.C., PAN, C.T., HUANG, C.L., and WEI, C.C A general approach for constructing the limit cycle loci of multiple nonlinearity systems, IEEE Trans. Autumn. Control, AC-32, (9), 1987, pp. 845-848. 10.1109/9.380PARLOS, A.G., HENRY, A.F., SCHWEPPE, F.C., GOULD, L.A., and LANNING, D.D Nonlinear multivariable control of nuclear power plants based on the unknown but bounded disturbance model, IEEE Trans. Autumn. Control, AC-33, (2), 1988 pp. 130-134. 10.1115/1.3152681PILLAI, V.K., and NELSON, H.D A new algorithm for limit cycle analysis of nonlinear systems, Trans. ASME, J. Dyn. Syst. Meas. Control, 110, 1988, pp. 272-277. 10.1109/31.9915GENESIO, R., and TESI, A On limit cycles of feedback polynomial systems, IEEE Trans. Circuits Syst., 35, (12), 1988, pp. 1523-1528. FENDRICH, O.R Describing functions and limit cycles, IEEE Trans. Autom. Control, AC -31, (4), 1992, pp. 486487. PATRA, K.C., SWAIN, A.K., and MAJHI, S Application of neural network in the prediction of self-oscillations and signal stabilisation in nonlinear multivariable systems. Proceedings of ANZIIS-93, Western Australia, 1993, pp. 585-589. 10.1080/03772063.1994.11437206PATRA, K.C., and SINGH, Y.P Structural formulation and prediction of limit cycle for multivariable nonlinear system. IETE, Tech. Rev. India, 40, (5 & 6), 1994, pp. 253-260. 10.1049/ip-cta:19949977ZHUANG, M., and ARTHERTON, D.P PID controller design lor TITO system, TEE Proc. Control Theory Appl. 141, (2), 1994, pp. 111-120. 10.1049/ip-cta:19941091LOH, A.P., and VASANU, V.V Necessary conditions for limit cycles in multi loop relay systems, IEE Proc., Control Theory Appl., 141, 31, 1994, pp. 163-168 PATRA, K.C., et al Prediction of limit cycles in nonlinear multivariable systems, Arch. Control Sci. Poland, 4(XL) 1995, pp 281- 297. 10.1016/0005-1098(96)00065-9TESI, A, et al Harmonic balance analysis of periodic doubling bifurcations with implications for control of nonlinear dynamics, Automatic, 32 (9), 1996, 1255, 1271. PATRA, K.C, et al Signal Stabilization of two Dimensional Non Linear relay Control Systems Archives of Control Sciences, Volume 6(XLII), No. 1 – 2, 1997, pages 89 – 101. 10.1016/s0167-6911(98)00056-5PATRA, K. C, PATI, B.B An investigation of forced oscillation for signal stabilization of two dimensional non linear system, Systems & Control Letters, 35, 1998, pp. 229 – 236. 10.1049/ip-cta:19990660LIN, C.H., HAN, K.W Prediction of Limit cycle in Nonlinear two input two output control system, ‘IEE Proc.-Control Theory Appl. Vol.146, No.3 may. 1999. PATRA, K.C, et al Structural Formulation and Self-Oscillation Prediction in Multidimensional Nonlinear Closed-Loop Autonomous Systems, Int. J. App. Math. And Comp. Sci., Vol. 9, No. 2, 1999, pp. 327 - 346. 10.1109/41.744407HORI, Y. et al Slow resonance ratio control for vibration suppression and disturbance rejection in torsional system, IEEE Trans. Ind. Electron., vol. 46, (1), 1999, pp.162-168. 10.1016/s0005-1098(02)00047-xNORDIN, M. and Gutman, P. O Controlling mechanical systems with backlash- a survey, Automatica , vol. 38, (10), 2002, pp.1633- 1649. T. RAYMOND, T, et al Design of Feedback Control Systems, Oxford University Press, 4th edition, 2002, pp. 677-678. STANISLAW, H. ak Systems and Control’ Oxford University Press, 2003, pp. 77 – 83. 10.1109/indcon.2005.1590225CHIDAMBARAM, I.A, and VELUSAMI, S Decentralized biased controllers for loadfrequency control of inter connected power systems considering governor dead band nonlinearity, INDICON, Annual IEEE, 2005, pp.521-525. 10.1111/j.1934-6093.2006.tb00286.xEFTEKHARI, M and KATEBI, S. D Evolutionary Search for Limit Cycle and Controller Design in Multivariable non linear systems, Asian Journal of Control, , Vol. 8, No. 4, 2006, pp. 345 – 358. 10.1155/2009/816707KATEBI, M, et al Limit Cycle Prediction Based on Evolutionary Multi objective Formulation, Hindawi Publishing Corporation, , Mathematical Problems in engineering Volume, Article ID 816707, 2009, 17pgs. 10.23919/ecc.2009.7075027GARRIDO, J, et al Centralized PID control by Decoupling of a Boiler-Turbine Unit, Proceedings of the European Control Conference, Budapest, Hungary, Aug. 2009, 23-26. 10.1016/j.ijepes.2011.06.005TSAY, T.S Load Frequency control of interconnected power system with governor backlash nonlinearities, Electrical Power and Energy, vol. 33, 2011, pp.1542-1549. 10.1155/2011/169848TSAY, T.S Limit Cycle prediction of nonlinear multivariable feedback control systems with large transportation lags, Hindawi Publishing corporation journal of control science and Engineering, Vol., article id 169848, 2011. TSAY, T.S Stability Analysis of Non linear Multivariable feedback Control systems, WSEAS Transactions on systems, Volume 11, Issue 4, 2012, pp. 140 – 151. SUJATHA, V., PANDA, R. C Relay Feedback Based Time domain modelling of Linear 3-by-3 MIMO System, American Journal of System Science, Scientific & Academic Publishing, 1(2) 2012, pp. 17-22. 10.1109/tie.2015.2438055WANG, C, et al Vibration suppression with shaft torque limitation using explicit MPC-PI switching control in elastic drive systems, IEEE Trans. Ind. Electron, vol. 62,(11), 2015, pp. 6855-6867. 10.1109/icpe.2015.7168032YANG, M, et al Suppression of mechanical resonance using torque disturbance observer for two inertia system with backlash Proc. IEEE 9th Int. Conf. Power Electron., ECCE Asia, 2015, pp. 1860 - 1866. 10.1109/tase.2014.2369430SHI, Z, and ZUO, Z back stepping control for gear transmission servo systems with backlash nonlinearity IEEE Trans. Autumn. Sci. Eng., vol. 12, (2), 2015, pp. 752-757. 10.1109/tie.2017.2711564WANG, C, et al, Analysis and suppression of limit cycle oscillation for Transmission System with backlash Nonlinearity, IEEE Transactions on Industrial Electronics, vol. 62, (12), 2017, pp. 9261-9270. PATRA, K. C, and DAKUA, B. K, Investigation of limit cycles and signal stabilisation of two dimensional systems with memory type nonlinear elements, Archives of Control Sciences, vol. 28, (2), 2018, pp. 285- 330. 10.22541/au.158022324.44338046PATRA, K. C, KAR, N Signal Stabilization of Limit cycling two Dimensional Memory Type Nonlinear Systems by Gaussian Random Signal, International Journal of Emerging Trends & Technology in Computer Science, Vol. 9 Issue 1, 2020, pp. 10-17. 10.1007/s40435-021-00860-xPATRA, K. C, KAR, N Suppression Limit cycles in 2 x 2 nonlinear systems with memory type nonlinearities, International Journal of Dynamics and Control, Springer Nature’,34,95€, vol.10 Issue 3, 2022, pp 721- 733. 10.11159/enfht23.133LOPEZ, D.S, VEGA, A.P, Fuzzy Control of a Toroidal Thermosyphon for Known Heat Flux Heating Conditions, Proceeding of the 8 th World Congress on Momentum, Heat and Mass Transfer (MHMT’23), Lisbon PortugalMarch 26-28, 2023. DOI:10.11159/enfht23.133 CORRADO. C, et. al, Quantifying the impact of shape uncertainty on predict arrhythmias, Computers in Biology and Medicine, Elsevier Ltd., 153, 2023, 106528. 10.1016/j.egyr.2022.11.184CHEN, W., et. al, Oscillation characteristics and trajectory stability region analysis method of hierarchical control microgrids, Energy Reports, 9, 2023, pp 315-324. Kumar, U., et.al. The effect of sub diffusion on the stability of autocatalytic systems, Chemical Engineering Science, Elsevier Ltd., 265, 2023, 118230. 10.1038/s41598-022-27047-4Marrone, J.I., et.al. A nested bistable module within a negative feedback loop ensures different types of oscillations in signalling systems, Scientific reports| Nature portfolio, 2023, 13:529. MUNCH, S, B., et.al. Recent developments in empirical dynamic modelling, Methods in Ecology and Evolution, 2022, 14, pp 732- 745.