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## Archive of Issues

Russia Irkutsk
Year
2013
Issue
3
Pages
65-78
 Section Mathematics Title Pulse-sliding modes of controlled mechanical systems Author(-s) Ponomarev D.V.a Affiliations Irkutsk State Universitya Abstract We consider a controlled mechanical system with dry friction and positional pulse or positional discontinuous control. It can be presented in a form of Lagrange equations of the second kind $$A(t,q)\ddot{q} = g(t,q,\dot{q}) + Q^A(t,q,\dot{q}) + Q^{T}(t,q,\dot{q}) + u,\quad t\in I=[t_0,t_0+T]. \qquad {(1)}$$ The goal of the control is the motion of the system $(1)$ in set $S=\bigl\{(t,q,\dot{q})\in I\times R^n\times R^n : \sigma(t,q,\dot{q})=0\bigl\}$ (problem of stabilization) or in the neighborhood of set $S$ (approach problem). The first problem is solved with discontinuous positional control of relay type with limited resources, for which a decomposition mode is a stable sliding mode of system $(1)$. In case of insufficiency of resources of discontinuous control the motion of the controlled system in the neighborhood of set $S$ can be implemented under high-frequency impacts on the system in discrete time moments in the pulse-sliding mode, the uniform limit of which (an ideal pulse-sliding mode) is equal to the decomposition mode. The distinctive feature of the assigned problems is dry friction in the system $(1)$, and said dry fiction, generally speaking, can be considered as uncontrollable discontinuous or multivalued perturbations. Main definitions are given in the introduction of the article. In the first section the connection between ideal pulse-sliding modes of inclusion $$A(t,x)\dot{x}\in F(t,x) + u,$$ where $u$ is a positional pulse control, and sliding modes of system $$A(t,x) \dot{x} \in F(t,x) + B(t,x)\tilde{u}(t,x)$$ with a positional discontinuous control is considered. The second section is devoted to systems of type $(1)$. In the third section we consider set $S$, which is important in relation to applications and is defined by the vector function $\sigma(t,q,\dot{q})=\dot{q}-\varphi(t,q)$. For the last case more simple and informative conditions of the existence of sliding modes for a system with discontinuous controls were used. An example was considered in conclusion. Keywords differential inclusion, positional pulse control, pulse-sliding mode, sliding mode UDC 517.911.5 MSC 34A37, 34A60 DOI 10.20537/vm130306 Received 1 April 2013 Language Russian Citation Ponomarev D.V. Pulse-sliding modes of controlled mechanical systems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 3, pp. 65-78. References Zavalishchin S.T., Sesekin A.N., Drozdenko S.E. Dinamicheskie sistemy s impul'snoi strukturoi (Dynamic systems with impulse structure), Sverdlovsk: Sredn. Ural. Knizh. Izd., 1983, 112 p. Zavalishchin S.T., Sesekin A.N. Pulse-sliding modes in non-linear dynamic systems, Differentsial'nye Uravneniya, 1983, vol. 19, no. 5, pp. 790-799. Finogenko I.A., Ponomarev D.V. On differential inclusions with positional discontinuous and pulse controls, Trudy Inst. Mat. Mekh. Ural Otd. Ross. Akad. Nauk, 2013, vol. 19, no. 1, pp. 284-299. Filippov A.F. Differentsial'nye uravneniya s razryvnoi pravoi chast'yu (Differential equations with discontinuous righthand side), Moscow: Nauka, 1985, 224 p. Matrosov V.M., Finogenko I.A. Analitical dynamics of systems of rigid bodies with friction, Nonlinear mechanics, Moscow: Fizmatlit, 2001, pp. 39-61. Finogenko I.A. On stability of mechanical systems with dry friction and discontinuous positional controls, Trudy X Mezhdunarodnoi Chetaevskoi konferentsii. Tom 1. Analiticheskaya mekhanika (Proceedings of X International Chetaev Conference. Volume 1. Analitical Mechanics), Kazan State Technical University named after A.N. Tupolev, Kazan, 2012, pp. 488-498. Pyatnitskii E.S. Synthesis of hierarchical systems of control of mechanical and electromechanical objects on the principle of decomposition I, Avtomatika i Telemekhanika, 1989, no. 1, pp. 87-98. Pyatnitskii E.S. Synthesis of hierarchical systems of control of mechanical and electromechanical objects on the principle of decomposition II, Avtomatika i Telemekhanika, 1989, no. 2, pp. 57-71. Full text