Abstract
A tool for the design of a periodic motion in underactuated systems via generating a self-excited oscillation of a desired amplitude and frequency driven by a variable structure control is reviewed. In this chapter, we overview the capabilities of the two-relay controller to induce oscillations in dynamical systems. In this chapter, we will focus on underactuated mechanical systems with degree of underactuation one, that is, n degrees-of-freedom and n – 1 actuators only. Threemethods to set the frequency and amplitude of oscillation and its application to one-degree of underactuation systems are reviewed: describing function method, Locus of the perturbed relay system design (LPRS), and Poincaré map based design. Theoretical and practical open problems are also discussed.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Aguilar, L., Boiko, I., Fridman, L., Freidovich, L.: Generating oscillations in inertia wheel pendulum via two relay controller. International Journal of Robust and Nonlinear Control 22, 318–330 (2012)
Aguilar, L., Boiko, I., Fridman, L., Iriarte, R.: Generation of periodic motions for underactuated mechanical system via second-order sliding-modes. In: Proc. of the American Control Conference, Minnesota, USA, pp. 5396–5400 (2006)
Aguilar, L., Boiko, I., Fridman, L., Iriarte, R.: Output excitation via continuous sliding-modes to generate periodic motion in underactuated systems. In: Proc. of the IEEE Conference on Decision and Control, San Diego, USA, pp. 1629–1634 (2006)
Aguilar, L., Boiko, I., Fridman, L., Iriarte, R.: Output excitation via second-order sliding-modes to generate periodic motion for underactuated systems. In: Proc. of the 9th International Workshop on Variable Structure Systems, Alghero, Italy, pp. 359–364 (2006)
Aguilar, L., Boiko, I., Fridman, L., Iriarte, R.: Generating self-excited oscillations for underactuated mechanical systems via two relay controller. International Journal of Control 82(9), 1678–1691 (2009)
Aguilar, L., Boiko, I., Fridman, L., Iriarte, R.: Generating self-excited oscillations via two-relay controller. IEEE Trans. on Automatic Control 54(2), 416–420 (2009)
Aguilar, L., Boiko, I., Iriarte, R., Fridman, L.: Periodic motion of underactuated mechanical systems self-generated by variable structure controllers: design and experiments. In: 2007 European Control Conference, Kos, Greece, pp. 3796–3801 (2007)
Atherton, D.: Nonlinear control engineering–Describing Function Analysis and Design. Van Nostrand, Workingham (1975)
Berkemeier, M., Fearing, R.: Tracking fast inverted trajectories of the underactuated acrobot. IEEE Transactions on Robotics and Automation 15(4), 740–750 (1999)
Boiko, I.: Oscillations and transfer properties of relay servo systems – the locus of a perturbed relay system approach. Automatica 41, 677–683 (2005)
Boiko, I.: Discontinuous control systems: Frequency-domain analysis and design. Birkhäuser, Boston (2009)
Chevallereau, C., Abba, G., Aoustin, Y., Plestan, E., Canudas-de-Wit, C., Grizzle, J.: Rabbit: A testbed for advanced control theory. IEEE Control Systems Magazine 23(5), 57–79 (2003)
Dranga, O., Navy, I.: Stability analysis of feedback controlled resonant DC-DC converter using Poincaré map function. In: IEEE Int. Symposium on Industrial Electronics, Pusan, Korea, pp. 2142–2147 (2001)
Estrada, A., Fridman, L.: Exact compensation of unmatched perturbation via quasi-continuous HOSM. In: 47th IEEE Conference on Decision and Control, Cancún, México, pp. 2202–2207 (2008)
Fendrich, O.: Describing functions in limit cycles. IEEE Transactions on Automatic Control 37(4), 486–488 (1992)
Fridman, L.: An averaging approach to chattering. IEEE Transactions on Automatic Control 46(8), 1260–1265 (2001)
Fridman, L.: Slow periodic motion in variable structure systems. International Journal of Systems Science 33(14), 1145–1155 (2002)
Grizzle, J., Moog, C., Chevallereau, C.: Nonlinear control of mechanical systems with an unactuated cyclic variable. IEEE Transactions on Automatic Control 50(5), 559–576 (2005)
Lai, J.: Power conditioning circuit topologies: Power conversion from low-voltage dc to high-voltage ac for single-phase grid-tie applications. IEEE Ind. Electronics Mag. 3(2), 24–34 (2009)
Levant, A.: High-order sliding modes: differentiation and output-feedback control. International Journal of Control 76, 924–941 (2003)
Martínez, S., Cortés, J., Bullo, F.: Motion Planning and Control Problems for Underactuated Robot. In: Bicchi, A., Christensen, H.I., Prattichizzo, D. (eds.) Control Problems in Robotics. STAR, vol. 4, pp. 59–74. Springer, Heidelberg (2003)
Martínez-Salamero, L., Valderrama-Blavi, H., Giral, R., Alonso, C., Estibals, B., Cid-Pastor, A.: Self-oscillating DC-to-DC switching converters with transformer characteristics. IEEE Transactions on Aerospace and Electronic Systems 41(2), 710–716 (2005)
Nakamura, Y., Suzuki, T., Koinuma, M.: Nonlinear behavior and control of a nonholonomic free-joint manipulator. IEEE Transactions on Robotics and Automation 13(6), 853–862 (1997)
Orlov, Y., Aguilar, L., Acho, L., Ortiz, A.: Asymptotic harmonic generator and its application to finite time orbital stabilization of a friction pendulum with experimental verification. International Journal of Control 81(2), 227–234 (2008)
Orlov, Y., Riachy, S., Floquet, T., Richard, J.: Stabilization of the cart-pendulum system via quasi-homogeneous switched control. In: Proc. of the 2006 Int. Workshop on Variable Structure Systems, Alghero, Italy, pp. 139–142 (2006)
Robinett, I.R., Wilson, D.: What is a limit cycle? International Journal of Control 81(12), 1886–1900 (2008)
Sanchis, P., Ursua, A., Gubia, E., Marroyo, L.: Buck-boost DC-AC inverter for a new control strategy. In: 35th Annual IEEE Power Electronics Specialist Conference, Aachen, Germany, pp. 3994–3998 (2004)
Santiesteban, R., Floquet, T., Orlov, Y., Riachy, S., Richard, J.: Second order sliding mode control for underactuated mechanical system II: orbital stabilization of an inverted pendulum with application to swing up/balancing control. International Journal of Robust Nonlinear Control 18(4–5), 544–556 (2008)
Shiriaev, A., Perram, J., Canudas-de-Wit, C.: Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual contraint approach. IEEE Transactions on Automatic Control 50(8), 1164–1176 (2005)
Utkin, V., Guldner, J., Shi, J.: Sliding Mode Control in Electromechanical Systems. CRC Press, Boca Raton (1999)
Varigonda, S., Georgiou, T.: Dynamics of relay relaxation oscillators. IEEE Transactions on Automatic Control 46(1), 65–77 (2001)
Youssef, M., Jain, P.: A novel single stage AC–DC self-oscillating series-parallel resonant converter. IEEE Transactions on Power Electronics 21(6), 1735–1744 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Aguilar, L.T., Boiko, I., Fridman, L., Iriarte, R. (2013). A Review on Self-oscillating Relay Feedback Systems and Its Application to Underactuated Systems with Degree of Underactuation One. In: Bandyopadhyay, B., Janardhanan, S., Spurgeon, S. (eds) Advances in Sliding Mode Control. Lecture Notes in Control and Information Sciences, vol 440. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36986-5_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-36986-5_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36985-8
Online ISBN: 978-3-642-36986-5
eBook Packages: EngineeringEngineering (R0)