Stability analysis of mechanical systems with distributed delay via decomposition

A. Yu. Aleksandrov ; A. A. Tikhonov

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 17 (2021) no. 1, pp. 13-26

Résumé

The article analyzes a linear mechanical system with a large parameter at the vector of velocity forces and a distributed delay in positional forces. With the aid of the decomposition method, conditions are obtained under which the problem of stability analysis of the original system of the second-order differential equations can be reduced to studying the stability of two auxiliary first-order subsystems. It should be noted that one of the auxiliary subsystems does not contain a delay, whereas for the second subsystem containing a distributed delay, the stability conditions are formulated in terms of the feasibility of systems of linear matrix inequalities. To substantiate this decomposition, the Lyapunov direct method is used. Special constructions of Lyapunov—Krasovskii functionals are proposed. The developed approach is applied to the problem of monoaxial stabilization of a rigid body. The results of a numerical simulation are presented confirming the conclusions obtained analytically.

Détail

Keywords: mechanical system, stability, distributed delay, rigid body, Lyapunov—Krasovskii functionals.
Mots-clés : decomposition

Bibliographie

[1] V. M. Matrosov, Vector Lyapunov function method: analysis of dynamical properties of nonlinear systems, Fizmatlit, M., 2001, 384 pp. (In Russian)

[2] F. L. Chernous'ko, I. M. Anan'evskii, S. A. Reshmin, Methods of control of nonlinear mechanical systems, Fizmatlit, M., 2006, 328 pp. (In Russian)

[3] D. D. Siljak, Decentralized control of complex systems, Academic Press, New York, 1991, 525 pp. | MR

[4] V. I. Zubov, Analytical dynamics of gyroscopic systems, Sudostroenie Publ., L., 1970, 320 pp. (In Russian)

[5] E. S. Pyatnitskii, “The principle of decomposit ion in the control of mechanical systems”, Proceedings of Academy Sciences USSR, 300:2 (1988), 300–303 (In Russian) | Zbl

[6] A. Yu. Aleksandrov, A. V. Platonov, “On stability and dissipativity of some classes of complex systems”, Automation and Remote Control, 70:8 (2009), 1265–1280 | DOI | MR | Zbl

[7] V. N. Tkhai, “Model with coupled subsystems”, Automation and Remote Control, 74:6 (2013), 919–931 | DOI | MR | Zbl

[8] S. L. Podval'nyi, V. V. Provotorov, “Start control of a parabolic system with distributed parameters on a graph”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2015, no. 3, 126–142 (In Russian)

[9] V. V. Provotorov, V. I. Ryazhskikh, Yu. A. Gnilitskaya, “Unique weak solvability of a non-linear initial boundary value problem with distributed parameters in a netlike domain”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 13:3 (2017), 264–277 | MR

[10] L. A. Klimina, “Method for forming autorotations in controllable mechanical system with two degrees of freedom”, Journal of Computer and Systems Sciences International, 59:6 (2020), 817–827 | DOI | Zbl

[11] S. Yu. Kuptsov, “On a method for stability investigation of linear differential systems families”, Proceedings of the Middle Volga Mathematical Society, 8:1 (2006), 224–235 (In Russian)

[12] A. A. Kosov, “Stability investigation of singular systems via vector Lyapunov functions method”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2005, no. 4, 123–129 (In Russian)

[13] A. Yu. Aleksandrov, Y. Chen, A. A. Kosov, L. Zhang, “Stability of hybrid mechanical systems with switching linear force fields”, Non-linear Dynamics and Systems Theory, 11:1 (2011), 53–64 | MR | Zbl

[14] A. Yu. Aleksandrov, A. A. Kosov, Y. Chen, “Stability and stabilization of mechanical systems with switching”, Automation and Remote Control, 72:6 (2011), 1143–1154 | DOI | MR | Zbl

[15] A. Yu. Aleksandrov, E. B. Aleksandrova, A. P. Zhabko, “Asymptotic stability conditions for certain classes of mechanical systems with time delay”, WSEAS Transactions on Systems and Control, 9 (2014), 388–397 | MR

[16] A. Y. Aleksandrov, E. B. Aleksandrova, “Asymptotic stability conditions for a class of hybrid mechanical systems with switched non-linear positional forces”, Non-linear Dynamics, 83:4 (2016), 2427–2434 | DOI | MR | Zbl

[17] Y. X. Su, C. H. Zheng, “PID control for global finite-time regulation of robotic manipulators”, International Journal of Systems Sciences, 48 (2017), 547–558 | DOI | MR | Zbl

[18] I. M. Anan'evskii, V. B. Kolmanovskii, “On stabilization of some control systems with an aftereffect”, Automation and Remote Control, 1989, no. 9, 1174–1181 | MR

[19] X. Zhang, X. Chen, G. Zhu, C.-Y. Su, “Output feedback adaptive motion control and its experimental verification for time-delay non-linear systems with asymmetric hysteresis”, IEEE Transactions on Industrial Electronics, 67:8 (2020), 6824–6834 | DOI | MR

[20] S. V. Pavlikov, A. G. Isavnin, “On stabilization of a controlled mechanical system with delayed feedback”, Vestnik of Voronezh State University. Series Physics. Mathematics, 2014, no. 1, 139–146 (In Russian)

[21] S. V. Pavlikov, “On the stabilization of movements of controlled mechanical systems with a retarded controller”, Proceedings of Academy Sciences of Russia, 412:2 (2007), 176–178 (In Russian) | MR | Zbl

[22] E. Fridman, “Tutorial on Lyapunov-based methods for time-delay systems”, European Journal of Control, 20 (2014), 271–283 | DOI | MR | Zbl

[23] J. K. Hale, S. M. Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag, New York, 1993, 458 pp. | MR | Zbl

[24] V. I. Zubov, Dynamics of control systems, Saint Petersburg University Press, Saint Petersburg, 2004, 380 pp. (In Russian)

[25] A. Aleksandrov, O. Mason, “Diagonal Riccati stability and applications”, Linear Algebra Appl., 492 (2016), 38–51 | DOI | MR | Zbl

[26] V. A. Samsonov, M. Z. Dosaev, Y. D. Selyutskiy, “Methods of qualitative analysis in the problem of rigid body motion in medium”, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 21:10 (2011), 2955–2961 | MR | Zbl

[27] H. M. Sedighi, F. Daneshmand, “Non-linear transversely vibrating beams by the homotopy perturbation method with an auxiliary term”, Journal of Applied and Computational Mechanics, 1:1 (2015), 1–9

[28] Y. Park, C. Lee, “Dynamic investigation of non-linear behavior of hydraulic cylinder in mold oscillator using PID control process”, Journal of Applied and Computational Mechanics, 7:1 (2021), 270–276

[29] A. A. Tikhonov, “Resonance phenomena in oscillations of a gravity-oriented rigid body. Pt. 4. Multifrequency resonances”, Vestnik of Saint Petersburg University. Series 1. Mathematics. Mechanics. Astronomy, 2000, no. 1, 131–137 (In Russian)

[30] E. A. Kosjakov, A. A. Tikhonov, “Differential equations for librational motion of gravity-oriented rigid body”, International Journal of Non-linear Mechanics, 73 (2015), 51–57 | DOI