Geodesic
Stabilization of steady motions for systems with redundant coordinates
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2019), pp. 46-51
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The stability and stabilization problem of steady motions for mechanical systems with nonlinear geometric constraints is considered. The steady state information is assumed to be incomplete. Redundant coordinates, Routh's variables and Shul'gin's equations of motion are used. The set of cyclical coordinates is divided into two parts for impulses (Routh variables) and controlled coordinates (Lagrange variables). The rest of coordinates is assumed to be uncontrolled. The characteristic equation for the perturbed motion has zero roots. Its number is equal to the number of impulses plus the number of redundant coordinates. The stabilization theorem is proved for three variants of the measurement vector. The control law and the observing system coefficients can be determined by solving the Krasovskiy linear-quadratic problems for the controlled subsystem. This system does not depend on the critical variables (redundant coordinates and impulses). The stability of the complete nonlinear system follows from the reduction to Lyapunov's special case and Malkin's stability theorem under time-varying perturbations. 
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A. Ya. Krasinskiy; A. N. Il'ina; È. M. Krasinskaya. Stabilization of steady motions for systems with redundant coordinates. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2019), pp. 46-51. http://geodesic.mathdoc.fr/item/VMUMM_2019_1_a7/

[1] Shulgin M. F., O nekotorykh differentsialnykh uravneniyakh analiticheskoi dinamiki i ikh integrirovanii, Nauchnye trudy Sredneaziatskogo gosudarstvennogo universiteta (SAGU), Tashkent, 1958

[2] Rumyantsev V. V., Ob ustoichivosti statsionarnykh dvizhenii sputnikov, VTs AN SSSR, M., 1967

[3] Lure A. I., Analiticheskaya mekhanika, Gos. izd-vo fiz.-mat. lit-ry, M., 1961

[4] Raus E. Dzh., Dinamika sistemy tverdykh tel, v. 2, Nauka, M., 1983

[5] Krasinskii A. Ya., Krasinskaya E. M., “Ob odnom metode stabilizatsii ustanovivshikhsya dvizhenii s nulevymi kornyami v zamknutoi sisteme”, Avtomat. i telemekhan., 2016, no. 8, 85–100 | MR | Zbl

[6] Krasinskaya E. M., Krasinskii A. Ya., Obnosov K. B., “O razvitii nauchnykh metodov shkoly M. F. Shulgina v primenenii k zadacham ustoichivosti i stabilizatsii ravnovesii mekhatronnykh sistem s izbytochnymi koordinatami”, Sb. nauchno-metodicheskikh statei po teoreticheskoi mekhanike, 28, Izd-vo MGU, M., 2012, 169–184

[7] Krasinskii A. Ya., Krasinskaya E. M., “O metode issledovaniya odnogo klassa zadach stabilizatsii pri nepolnoi informatsii o sostoyanii”, Tr. Mezhdunar. konf., posv. 90-letiyu so dnya rozhdeniya akad. N.N. Krasovskogo (Ekaterinburg, 2015), 228–235

[8] Krasinskiy A. Ya., Ilyina A. N., “The mathematical modeling of the dynamics of systems with redundant coordinates in the neighborhood of steady motions”, Vestn. YuUrGU. Ser. Matem. modelir. i program. 2017, 10:2, 38–50 | Zbl

[9] Lyapunov A. M., Sobranie sochinenii, v. 2, Izd-vo AN SSSR, M.–L., 1956 | MR

[10] Malkin I. G., Teoriya ustoichivosti dvizheniya, Nauka, M., 1952 | MR

[11] Aizerman M. A., Gantmacher F. R., “Stabilitaet der Gleichewichtslage in einem nichtholonomen System”, Z. angew. Math. und Mech., 37:1–2 (1957), 74–75 | DOI

[12] Kalman R., Falb P., Arbib M., Ocherki po matematicheskoi teorii sistem, URSS, M., 2010 | MR

[13] Krasovskii N. N., “Problemy stabilizatsii upravlyaemykh dvizhenii”, Dopolnenie IV: Malkin I. G., Teoriya ustoichivosti dvizheniya, Nauka, M., 1966, 475–515

[14] Krasinskaya E. M., Krasinskii A. Ya., “Ob odnom metode issledovaniya ustoichivosti i stabilizatsii ustanovivshikhsya dvizhenii mekhanicheskikh sistem s izbytochnymi koordinatami”, Mat-ly XII Vseros. soveschaniya po problemam upravleniya VSPU-2014 (Moskva, 1–19 iyunya 2014), M., 2014, 1766–1778

[15] Krasinskaya E. M., Krasinskii A. Ya., “O dopustimosti linearizatsii uravnenii geometricheskikh svyazei v zadachakh ustoichivosti i stabilizatsii ravnovesii”, Sb. nauchno-metodicheskikh statei po teoreticheskoi mekhanike, 29, Izd-vo MGU, M., 2015, 54–65

[16] Krasinskii A. Ya., Ilina A. N., Krasinskaya E. M., “O modelirovanii dinamiki sistemy Ball and Beam kak nelineinoi mekhatronnoi sistemy s geometricheskoi svyazyu”, Vestn. Udmurt. un-ta, 27:3 (2017), 414–430 | DOI | MR | Zbl

[17] Krasinkiy A. Ya., Krasinkaya E. M., “Modeling of dynamics of manipulators with geometrical constraints as a systems with redundant coordinates”, Int. Rob. Automat. J., 3:3 (2017) | DOI

[18] Klokov A. S., Samsonov V. A., “O stabiliziruemosti trivialnykh ustanovivshikhsya dvizhenii giroskopicheski svyazannykh sistem s psevdotsiklicheskimi koordinatami”, Prikl. matem. i mekhan. 1985, 49:2, 199–202 | MR | Zbl