On the Steady State Motions Control Problem for Mechanical Systems with Relay Controllers
DOI:
https://doi.org/10.52846/stccj.2021.1.1.10Keywords:
differential equation with discontinuous right-hand side, stability, mechanical system, steady state motion stabilizationAbstract
The paper presents the solution to the stabilization problem of steady state motions for a holonomic mechanical system by using relay controllers. This solution is achieved by proving new theorems on the asymptotic stability of the solution to a differential equation with a discontinuous right-hand side. The novelty of the theorems is based on the limiting inclusions construction and the use of semidefinite Lyapunov functions. As an example, the stabilization problem of steady-state motion for a five-link robot manipulator is solved by using relay controller.
References
M. A. Aizerman and E. S. Pyatnitskii, “Foundations of a theory of discontinuous systems. I,” Automat. Remote Control, vol. 35, no 7, part 1, pp. 1066–1079, 1974.
J.L. Davy, “Properties of solution set of a generalized differential equation,” Bull. Austral. Math. Soc., vol. 6, pp. 379–398, 1972.
A. F. Filippov, Differential Equations with Discontinuous Right-Hand Side. Dordrecht, the Netherlands: Kluwer, 1988.
L. Iannelli, K. H. Johansson, U. T. Jonsson, and F. Vasca, “Averaging of Nonsmooth Systems Using Dither,” Automatica, vol. 42, no 4, pp. 669–676, 2006.
V. I. Utkin, Sliding Modes in Control Optimization. Berlin: Springer-Verlag, 1992.
J. P. Aubin and A. Cellina, Differential Inclusions. Springer, 1984.
R. Kamalapurkar, W. E. Dixon, and A. R. Teel, “On reduction of differential inclusions and Lyapunov stability,” 2017 IEEE 56th Annual Conference on Decision and Control (CDC), Melbourne, VIC, pp. 5499–5504, 2017.
E. Roxin, “Stability in general control systems,” Journal of Differ. Equat, vol. 1, no 2, pp. 115–150, 1965.
E. Ryan, “An integral invariance principle for differential inclusions with applications in adaptive control,” SIAM J. Control Optim. vol. 36, no 3, pp. 960–980, 1998.
Yu. I. Alimov, “On the application of Lyapunov’s direct method to differential equations with ambiguous right sides,” Autom. Remote Control, vol. 22, no 7, pp. 713–725, 1961.
A. S. Andreev, O. G. Dmitrieva, and Yu. V. Petrovicheva, “The stability of the zero Solution of systems with discontinuous right side,” The Volga Region Scientific and Engineering Bulletin, no 1, pp. 15–20, 2011, in Russian.
A. S. Andreev and O. A. Peregudova, “Lyapunov vector-functions in the problem on motion stabilization of controlled systems,” Journal of the Middle Volga Mathematical Society, vol. 16, no 1, pp. 32–44, 2014, in Russian.
I. A. Finogenko, “Limit differential inclusions and the invariance principle for nonautonomous systems,” Sib. Math. J., vol. 55, pp. 372–386, 2014.
I. A. Finogenko, “Method of limit differential equations for nonautonomous discontinuous systems,” Dokl. Math., vol. 93, pp. 9–12, 2016.
A.S. Ahdreev, “On the asymptotic stability and instability of the zeroth solution of a non-autonomous system,” Journal of Applied Mathematics and Mechanics, vol. 48, no 2, pp. 154–160, 1984.
A. S. Andreev and O. A. Peregudova, “On the method of comparison in asymptotic-stability problems,” Doklady Physics, vol. 50, no 2, pp. 91–94, 2005.
A. S. Andreyev and O. A. Peregudova, “The comparison method in asymptotic stability problems,” Journal of Applied Mathematics and Mechanics, vol. 70, no 6, pp. 865–875, 2006.
A. S. Andreev and O. A. Peregudova, “Stabilization of the preset motions of a holonomic mechanical system without velocity measurement,” Journal of Applied Mathematics and Mechanics, vol. 81, no 2, pp. 95– 105, 2017.
A. Andreev and O. Peregudova, “On stabilization of program motions of holonomic mechanical system,” Automation and Remote Control, vol. 77, no 3, pp. 416–427, 2016.
A.S. Andreev and O.A. Peregudova, “On global trajectory tracking control of robot manipulators in cylindrical phase space,” International Journal of Control, vol. 93, no 12, pp. 3003–3015. 2020.
A. Andreev and O. Peregudova, “Relay Controllers in the Motion Stabilization Problems of Mechanical Systems with Cyclic Coordinates,” 2020 24th International Conference on System Theory, Control and Computing (ICSTCC), 2020, pp. 433–438, doi: 10.1109/ICSTCC50638.2020.9259654.
Z. Artstein, “Topological dinamics of ordinary differential equations,” J. Differ. Equations, vol. 23, no 2, pp. 216–223, 1977.
D. Wakeman, “An application of topological dynamics to obtain a new invariance property for nonautonomous ordinary differential equations,” J. Differential Eqns., vol. 17, pp. 259–295, 1975.
G. Sell, Topological Dynamics and Ordinary Differential Equations, New York: Van Nostrand Reinhold, 1971.
N. Rouche, P. Habets, and M. Laloy, Stability Theory by Liapunov’s Direct Method. New York: Springer, 1977.
A. V. Karapetyan, The Stability of Steady Motions, Moscow: Editorial URSS, 1998, in Russian.
E. J. Routh, The advanced part of a treatise on the dynamics of a system of rigid bodies. London: MacMillan and Co., 1884.
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Accepted 2021-06-24
Published 2021-06-30
