Problems of the partial stability and detectability of dynamical systems / Alekseyeva C.A., Vorotnikov V.I., Feofanova V.A. // Journal of Applied Mathematics and Mechanics. - 2007. - V. 71, l. 6. - P. 869-879.

ISSN:
00218928
Type:
Article
Abstract:
The conditions under which uniform stability (uniform asymptotic stability) with respect to a part of the variables of the zero equilibrium position of a non-linear non-stationary system of ordinary differential equations signifies uniform stability (uniform asymptotic stability) of this equilibrium position with respect the other, larger part of the variables, which include an additional group of coordinates of the phase vector, are established. These conditions include the condition for uniform asymptotic stability of the zero equilibrium position of the "reduced" subsystem of the original system with respect to the additional group of variables. Since within the conditions obtained the stability with respect to the remaining unmeasured coordinates of the phase vector remains undetermined or is investigated additionally, partial zero-detectability of the original system occurs in this case, and the conditions obtained supplement the series of known results from partial stability theory. The application of the results obtained to problems of the partial stabilization of non-linear controlled systems, particularly to the problem of stabilizing an asymmetric rigid body relative to an assigned direction in an inertial space, is considered. The partial detectability of linear systems with constant coefficients is also investigated. © 2008 Elsevier Ltd. All rights reserved.
Author keywords:
Index keywords:
Asymptotic stability; Nonlinear control systems; Problem solving; Vectors; Constant coefficients; Partial stability theory; Zero equilibrium position; Dynamical systems
DOI:
10.1016/j.jappmathmech.2007.12
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Affiliations Nizhnii Tagil, Russian Federation
References Lyapunov, A.M., Investigation of a special case of the problem of the stability of motion (1956) Collected Papers, 2, pp. 272-331. , Lyapunov A.M. (Ed), Izd Akad Nauk SSSR, Moscow, Leningrad; Rumyantsev, V.V., The stability of motion with respect to a part of the variables (1957) Vestn Mosk Univ Ser Mat Mekh Astron Fiz Khim, 4, pp. 9-16; Corduneanu, S., Sur la stabilité partielle (1964) Rev Roum Math Pure Appl, 9 (3), pp. 229-236; Halanay, A., (1966) Differential Equations: Stability, Oscillations, Time Lags, , Academic Press, New York; Pieffer, K., Rouche, N., Liapounov's second method applied to partial stability (1969) J Mech, 8 (2), pp. 323-334; Rumyantsev, V.V., Oziraner, A.S., (1987) Stability and Stabilization of Motion with Respect to a Part of the Variables, , Nauka, Moscow; Savchenko, A.Ya., Ignat'yev, A.O., (1989) Some Stability Problems of Non-Autonomous Systems, , Naukova Dumka, Kiev; Hatvani, L., On the stability of the solutions of ordinary differential equations with mechanical applications (1990) Alkalm Mat Lap, 1-2, pp. 1-90; Andreyev, A.S., Investigation of partial asymptotic stability (1991) Prikl Mat Mekh, 55 (4), pp. 539-547; Vorotnikov, V.I., (1991) Stability of Dynamical Systems with Respect to a Part of the Variables, , Nauka, Moscow; Khapaev, M.M., (1993) Averaging in Stability Theory, , Kluwer, Dordrecht; Vorotnikov, V.I., (1998) Partial Stability and Control, , Birkhauser, Boston; Fradkov, A.L., Miroshnik, I.V., Nikiforov, V.O., (1999) Nonlinear and Adaptive Control of Complex Systems, , Kluwer, Dordrecht; Vorotnikov, V.I., Problems of stability with respect to a part of the variables (1999) Prikl Mat Mekh, 63 (5), pp. 736-745; Vorotnikov, V.I., Rumyantsev, V.V., (2001) Stability and Control with Respect to a Part of the Coordinates of the Phase Vector of Dynamical Systems: Theory, Methods and Applications, , Nauch Mir, Moscow; Chellaboina, V., Haddad, W.M., A unification between partial stability and stability theory for time-varying systems (2002) IEEE Control Syst Magazine, 22 (6), pp. 66-75. , (Erratum. IEEE Control Syst Magazine 2003;23(1):103); Michel, A.N., Molchanov, A.P., Sun, Y., Partial stability and boundedness of general dynamical systems on metric spaces (2003) Nonlinear Anal TMA, 52 (4), pp. 1295-1316; Jian, J.G., Liao, X.X., Partial exponential stability of nonlinear time-varying large-scale systems (2004) Nonlinear Anal TMA, 59 (5), pp. 789-800; Zuyev, A.L., Partial asymptotic stability of abstract differential equations (2006) Ukr Mat Zh, 58 (1), pp. 629-637; Vorotnikov, V.I., Partial stability and control: state of the problem and prospects for development (2006) Avtomat Telemekh, 4, pp. 3-59; Sontag, E.D., Wang, Y., Output-to-state stability and detectability of nonlinear systems (1997) Syst Control Lett, 29 (4-5), pp. 279-290; Sontag, E.D., (1998) Mathematical Control Theory; Deterministic Finite Dimensional Systems, , Springer, New York; Sontag, E.D., Input to state stability: Basic concepts and results (2006) Nonlinear and Optimal Control Theory, , Springer, Berlin; Shiriaev, A.S., Fradkov, A.L., Stabilization of invariant sets for nonlinear non-affine systems (2000) Automatica, 36 (11), pp. 163-220; Shiriaev, A.S., The notion of V-detectability and stabilization of invariant sets of nonlinear systems (2000) Syst Control Lett, 39 (5), pp. 327-338; Ingalls, B.P., Sontag, E.D., Wang, Y., Measurement to error stability: a notion of partial detectability for nonlinear systems (2002) Proceedings of the 2002 IEEE Conference on Decision and Control, Las Vegas, Nevada, pp. 3946-3951; Yefimov, D.V., (2005) Robust and Adaptive Control of Nonlinear Oscillations, , Nauka, St. Petersburg; Barbashin, Ye.A., Tabuyeva, V.A., (1969) Dynamical Systems with a Cylindrical Phase Space, , Nauka, Moscow; Oziraner, A.S., A Malkin-Masser theorem (1979) Prikl Mat Mekh, 43 (6), pp. 975-979; Hacker, T., Stability of partially controlled motion of an aircraft (1961) J Aerospace Sci, 28 (1), pp. 15-26; Abramov, S.A., Bronshtein, M., Solution of linear differential and difference systems with respect to a part of the unknowns (2006) Zh Vichisl Mat Mat Fiz, 46 (2), pp. 229-241; Bronstein, M., Computer algebra algorithms for linear ordinary differential and difference equations (2001) Prog Math, 202, pp. 105-119; van der Put, M., Singer, M.F., (2003) Galois Theory of Linear Differential Equations, , Springer, Heidelberg
Correspondence Address Alekseyeva, C.A.; Nizhnii TagilRussian Federation; email: vorot@ntiustu.ru
CODEN JAMMA
Language of Original Document English
Abbreviated Source Title J. Appl. Math. Mech.
Source Scopus