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Generation of stable oscillations in uncertain nonlinear systems with matched and unmatched uncertainties. (English) Zbl 1415.93155

Summary: In this paper, a robust limit cycle control technique is proposed for generation of stable oscillations in a class of uncertain nonlinear systems with both matched and unmatched uncertainties. For this purpose, first, the modified Lyapunov function is introduced which is appropriate for stability analysis of invariant sets (instead of equilibrium points). The structure of the proposed Lyapunov function is related to the shape of the desirable limit cycle. Next, in order to design the robust limit cycle control input, the backstepping and Lyapunov redesign methods are employed, simultaneously. The classical Lyapunov redesign controller is discontinuous and robust with respect to matched uncertainties. To overcome unmatched uncertainties, a modified version of the Lyapunov redesign controller is suggested in each step of backstepping which results in a continuous robust control law. Furthermore, the convergence of the phase trajectories of the uncertain closed-loop system to the target limit cycle is proved using the extended Lyapunov stability theorem. Finally, computer simulations are performed to show the applicability of the given approach. In this regard, two uncertain nonlinear practical systems are considered and robust stable oscillations are generated in these systems via the proposed controller. Simulation results confirm the effectiveness of the proposed technique.

MSC:

93C41 Control/observation systems with incomplete information
93C10 Nonlinear systems in control theory
93B35 Sensitivity (robustness)
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C20 Control/observation systems governed by partial differential equations
93C15 Control/observation systems governed by ordinary differential equations
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[1] Adachi, M.; Ushio, T.; Yamamoto, S., Synthesis of hybrid systems with limit cycles satisfying piecewise smooth constraint equations, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 87, 4, 837-842 (2004)
[2] Aguilar, L. T.; Boiko, I.; Fridman, L.; Iriarte, R., Generating self-excited oscillations via two-relay controller, IEEE Transactions on Automatic Control, 54, 2, 416-420 (2009) · Zbl 1367.93384
[3] Aguilar-Ibánez, C.; Martinez, J. C.; De Jesus Rubio, J.; Suarez-Castanon, M. S., Inducing sustained oscillations in feedback-linearizable single-input nonlinear systems, ISA Transactions, 54, 117-124 (2015)
[4] Albea, C.; Gordillo, F.; Aracil, J., Control of the boost DC-AC converter by energy shaping, 754-759 (2006)
[5] Aracil, J.; Gordillo, F.; Ponce, E., Stabilization of oscillations through backstepping in high-dimensional systems, IEEE Transactions on Automatic Control, 50, 5, 705-710 (2005) · Zbl 1365.93385
[6] Benmiloud, M.; Benalia, A., Finite-time stabilization of the limit cycle of two-cell DC/DC converter: Hybrid approach, Nonlinear Dynamics, 83, 1-2, 319-332 (2016)
[7] Biel, D.; Fossas, E.; Guinjoan, F.; Alarcón, E.; Poveda, A., Application of sliding-mode control to the design of a buck-based sinusoidal generator, IEEE Transactions on Industrial Electronics, 48, 3, 563-571 (2001)
[8] Binazadeh, T., Finite-time tracker design for uncertain nonlinear fractional-order systems, Journal of Computational and Nonlinear Dynamics, 11, 4, 041028 (2016)
[9] Binazadeh, T.; Bahmani, M., Design of robust controller for a class of uncertain discrete‐time systems subject to actuator saturation, IEEE Transactions on Automatic Control, 62, 3, 1505-1510 (2017) · Zbl 1366.93324
[10] Binazadeh, T.; Shafiei, M. H., Output tracking of uncertain fractional-order nonlinear systems via a novel fractional-order sliding mode approach, Mechatronics, 23, 888-892 (2013)
[11] Cardenas-Maciel, S. L.; Castillo, O.; Aguilar, L. T., Generation of walking periodic motions for a biped robot via genetic algorithms, Applied Soft Computing, 11, 8, 5306-5314 (2011)
[12] Chemori, A.; Alamir, M., Limit cycle generation for a class of non-linear systems with jumps using a low dimensional predictive control, International Journal of Control, 78, 15, 1206-1217 (2005) · Zbl 1097.93016
[13] Chenarani, H.; Binazadeh, T., Flexible structure control of unmatched uncertain nonlinear systems via passivity-based sliding mode technique, Iranian Journal of Science & Technology, Transactions of Electrical Engineering, 41, 1, 1-11 (2017)
[14] Favela-Contreras, A.; Carbajal, F. B.; Piñón, A.; Raimondi, A., Limit cycle analysis in a class of hybrid systems, Mathematical Problems in Engineering, 2016, 2981952 (2016) · Zbl 1400.34043
[15] Golestani, M.; Mobayen, S.; Tchier, F., Adaptive finite-time tracking control of uncertain non-linear n-order systems with unmatched uncertainties, IET Control Theory & Applications, 10, 14, 1675-1683 (2016)
[16] Green, D., Synthesis of systems with periodic solutions satisfying V (x) = 0, IEEE Transactions on Circuits and Systems, 31, 4, 317-326 (1984) · Zbl 0537.93031
[17] Guermouche, M.; Ali, S. A.; Langlois, N., Super-twisting algorithm for DC motor position control via disturbance observer, IFAC-Papers On Line, 48, 30, 43-48 (2015)
[18] Haddad, W. M.; Chellaboina, V., Nonlinear dynamical systems and control: A Lyapunov-based approach (2008), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 1142.34001
[19] Hakimi, A. R.; Binazadeh, T., Stable limit cycles generating in a class of uncertain nonlinear systems : Application in inertia pendulum, Modares Journal of Electrical Engineering, 12, 3, 1-6 (2015)
[20] Hakimi, A. R.; Binazadeh, T., Inducing sustained oscillations in a class of nonlinear discrete time systems, Journal of Vibration and Control (2016) · Zbl 1400.39011 · doi:10/77546316659223
[21] Hakimi, A. R.; Binazadeh, T., Robust generation of limit cycles in nonlinear systems: Application on two mechanical systems, Journal of Computational and Nonlinear Dynamics, 12, 4, 041013 (2017)
[22] Hashimoto, S.; Naka, S.; Sosorhang, U.; Honjo, N., Generation of optimal voltage reference for limit cycle oscillation in digital control-based switching power supply, Journal of Energy and Power Engineering,, 6, 4, 623-628 (2012)
[23] Hobbelen, D.; De Boer, T.; Wisse, M., System overview of bipedal robots flame and tulip: Tailor-made for limit cycle walking, Intelligent Robots and Systems, IROS 2008, IEEE/RSJ International Conference on Sept, 22-26, 2008, 2486-2491 (2008), IEEE
[24] Kai, T., Limit-cycle-like control for planar space robot models with initial angular momenta, Acta Astronautica, 74, 20-28 (2012)
[25] Kai, T., Limit-cycle-like control for 2-dimensional discrete-time nonlinear control systems and its application to the Hénon map, Communications in Nonlinear Science and Numerical Simulation, 18, 1, 171-183 (2013) · Zbl 1253.93079
[26] Kai, T.; Masuda, R., Limit cycle synthesis of multi-modal and 2-dimensional piecewise affine systems, Mathematical and Computer Modelling, 55, 3-4, 505-516 (2012) · Zbl 1255.34031
[27] Khalil, H. K., Nonlinear control (2014), Upper Saddle River, NJ:: Prentice Hall, Upper Saddle River, NJ:
[28] Ramirez-Rodriguez, H.; Parra-Vega, V.; Sanchez-Orta, A.; Garcia-Salazar, O., Robust backstepping control based on integral sliding modes for tracking of quadrotors, Journal of Intelligent & Robotic Systems, 73, 1-4, 51-66 (2014)
[29] Saublet, L. M.; Ghoachani, R. G.; Martin, J. P.; Mobarakeh, B. N.; Pierfederici, S., Asymptotic stability analysis of the limit cycle of a cascaded DC-DC converter using sampled discrete-time modeling, IEEE Transactions on Industrial Electronics, 63, 4, 2477-2487 (2016)
[30] Shiriaev, A.; Perram, J. W.; Canudas-De-Wit, C., Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach, IEEE Transactions on Automatic Control, 50, 8, 1164-1176 (2005) · Zbl 1365.93416
[31] Solomon, J. H.; Wisse, M.; Hartmann, M. J., Fully interconnected, linear control for limit cycle walking, Adaptive Behavior,, 18, 6, 492-506 (2010)
[32] Teplinsky, A.; Feely, O., Limit cycles in a MEMS oscillator, IEEE Transactions on Circuits and Systems II: Express Briefs,, 55, 9, 882-886 (2008)
[33] Yang, Y.; Feng, G.; Ren, J., A combined backstepping and small-gain approach to robust adaptive fuzzy control for strict-feedback nonlinear systems, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans,, 34, 3, 406-420 (2004)
[34] Zhang, A.; Ma, S.; Li, B.; Wang, M.; Guo, X.; Wang, Y., Adaptive controller design for underwater snake robot with unmatched uncertainties, Science China Information Sciences, 59, 5, 1-15 (2016)
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