Abstract
We consider the problem of ensuring the invariance property for a linear stationary system with relay controls. In the case of perfect relays, wide-class exogenous disturbances are compensated theoretically on the basis of sliding mode theory. Existing imperfections of relay elements make it impossible to implement a perfect sliding mode; thus, the closed-loop system has dynamics in a zero neighborhood of the given variety. This causes undesirable oscillations in a steady-state mode (“chattering”), leading to missed invariance with respect to exogenous and parametric disturbances. This paper solves the problem of ensuring invariance under imperfect relay elements and involves two approaches. The first one allows for making the relay characteristic close to that of the perfect counterpart (by means of increasing the feedback amplification coefficient); consequently, the chattering problem is eliminated. The second approach proceeds from the high-frequency modulation principle and performs regularization of relay switchings; using the linearization effect for the relay characteristics, the second approach guarantees the invariance property.
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References
Andreev, Yu.N., Upravlenie konechnomernymi lineinymi ob”ektami (Controlling Finite-dimensional Linear Objects), Moscow: Nauka, 1976.
Nguen Huang Hyng and Utkin, V.A., Control of DC Electric Motor, Autom. Remote Control, 2006, no. 5, pp. 767–782.
Sistemy avtomaticheskogo upravleniya s peremennoi strukturoi (Automated Control Systems with Variable Structure), Emel’yanov, S.V., Ed., Moscow: Nauka, 1967.
Utkin, V.A. and Utkin, V.I., Design of Invariant-Systems by the Method of Separation of Motions, Autom. Remote Control, 1983, vol. 44, no. 12, pp. 1559–1567.
Utkin, V.I., Skol’zyashchie rezhimy v zadachakh optimizatsii i upravleniya (Sliding Modes in Optimization and Control Problems), Moscow: Nauka, 1982.
Utkin, V.I. and Yang, K.D., Methods for Constructing Discontinuity Planes inMultidimensional Variable Structure Systems, Autom. Remote Control, 1978, vol. 39, no. 10, part 1, pp. 1466–1470.
Wonham, W.M., Linear Multivariable Control: A Geometric Approach, New York: Springer-Verlag, 1979. Translated under the title Lineinye mnogomernye sistemy upravleniya. Geometricheskii podkhod, Moscow: Nauka, 1980.
Fel’dbaum, A.A., Elektricheskie sistemy avtomaticheskogo regulirovaniya (Electric Systems of Automatic Control), Moscow: Oborongiz, 1957.
Tsypkin, Ya.Z., Releinye avtomaticheskie sistemy (Automatic Relay Systems), Moscow: Nauka, 1974.
Bartolini, G., Ferrara, A., and Usai, E., Chattering Avoidance by Second-order Sliding Mode Control, IEEE Trans. Automat. Control, 1998, vol. 43, no. 2, pp. 241–246.
Bondarev, A.G., Bondarev, S.A., Kostyleva, N.Ye., and Utkin, V.I., Sliding Modes in Systems with Asymptotic State Observers, Autom. Remote Control, 1985, vol. 46, no. 6, pp. 679–684.
Drazenovic, B., The Invariance Condition in Variable-Structure Systems, Automatica, 1969, vol. 5, no. 3, pp. 287–295.
Utkin, V.A., The Method of Dynamic Compensation in a Problem of Estimation of External Disturbances, Proc. SICPRO’06, Moscow, 2006, pp. 754–771.
Utkin, V.I., Guldner, J., and Shi, J., Sliding Mode Control in Electromechanical Systems, London: Tailor and Francis, 1999.
Young, K.K., Kokotovic, P.V., and Utkin, V.I., A Singular Perturbation Analysis of High-gain Feedback Systems, IEEE Trans., 1977, vol. AC-22, pp. 931–939.
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Original Russian Text © S.A. Kochetkov, V.A. Utkin, 2010, published in Upravlenie Bol’shimi Sistemami, 2010, No. 27, pp. 117–168.
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Kochetkov, S.A., Utkin, V.A. The invariance property of control systems with imperfect relay elements. Autom Remote Control 73, 1085–1115 (2012). https://doi.org/10.1134/S0005117912060136
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DOI: https://doi.org/10.1134/S0005117912060136