Stochastic problems of absolute stability

Automation and Remote Control

Volumn 67, Issue 11, 2006, Pages 1811-1846

  Pakshin, P ^a   Ugrinovskii, V ^b  

^a Arzamas Polytechnic Institute of RE Alekseev Nizhny Novgorod State Technical University   (Russian Federation)

^b UNIVERSITY OF NEW SOUTH WALES   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

ALGEBRA; NONLINEAR SYSTEMS; PROBLEM SOLVING; RANDOM PROCESSES; THEOREM PROVING;

FREQUENCY THEOREM; STOCHASTIC STABILITY; STOCHASTIC STABILIZATION; STOCHASTIC SYSTEMS;

CONVERGENCE OF NUMERICAL METHODS;

EID: 33751528546     PISSN: 00051179     EISSN: None     Source Type: Journal    
DOI: 10.1134/S0005117906110051     Document Type: Article

References (105)

1. Kozin, F., A Survey of Stability of Stochastic Systems, Automatica, 1969, vol. 5, pp. 95-112.
2. Bertram, J.E. and Sarachik, P.E., Stability of Circuits with Randomly Time-varying Parameters, Trans. IRE, Special Supplement, 1959, vol. PGIT-5, p. 260.
3. Kats, I.Ya. and Krasovskii, N.N., On Stability of Random-parameter Systems, Prikl. Mat. Mekh., 1960, vol. 27, no. 5, pp. 809-823.
4. Khas'minskii, R.Z., Ustoichivost' sistem differentsial'nykh uravnenii pri sluchainykh vozmushcheniyakh ikh parametrov (Stability of Systems of Differential Equations under Random Perturbation of Their Parameters), Moscow: Nauka, 1969.
5. Wong, E. and Zakai, M., On the Relation between Ordinary and Stochastic Differential Equations, Int. J. Engin. Sci., 1965, vol. 3, pp. 213-229.
6. Sussmann, H., On the Gap between Deterministic and Stochastic Ordinary Differential Equations, Ann. Probability, 1978, vol. 60, pp. 19-41.
7. Kushner, H.J., Stochastic Stability and Control, New York: Academic, 1967. Translated under the title Stokhasticheskaya ustoichivost' i upravlenie, Moscow: Mir, 1969.
8. Morozan, T., The Method of V.M. Popov for Control Systems with Random Parameters, J. Math. Anal. Appl., 1966, vol. 16, pp. 201-215.
9. Brusin, V.A. and Tai, M.L., Absolute Stochastic Stability, Izv. Vuzov, Radiofiz., 1967, vol. 10, no. 7.
10. Brusin, V.A., On Absolute Stochastic Stability with Respect to one Class of Perturbations, Izv. Vuzov, Radiofiz., 1968, vol. 11, no. 8, pp. 1003-1018.
11. Gelig, A.Kh., Leonov, G.A., and Yakubovich, V.A., Ustoichivost' nelineinykh sistem s needinstvennym sostoyaniem ravnovesiya (Stability of the Nonlinear Systems with Nonunique Equilibrium State), Moscow: Nauka, 1978
12. Yakubovich, V.A., Frequency Theorem in the Control Theory, Sib. Mat. Zh., 1973, vol. 14, no. 2, pp. 384-420.
13. Yakubovich, V.A., Frequency Theorem for the Case of Hilbert Spaces of States and Controls and Its Use in Some Problems of Design of Optimal Control. I, Sib. Mat. Zh., 1974, vol. 15, no. 3, pp. 639-668.
14. Yakubovich, V.A., Frequency Theorem for the Case of Hilbert Spaces of States and Controls and Its Use in Some Problems of Design of Optimal Control. II, Sib. Mat. Zh., 1975, vol. 16, no. 5, pp. 1083-1102.
15. Yakubovich, V.A., Solution of Some Matrix Inequalities Encountered in the Automatic Control Theory, Dokl. Akad. Nauk SSSR, 1962, vol. 143, no. 6, pp. 1304-1307.
16. Yakubovich, V.A., Solution of Some Matrix Inequalities Encountered in the Nonlinear Control Theory, Dokl. Akad. Nauk SSSR, 1964, vol. 156, no. 2, pp. 278-281.
17. Levit, M.V., Frequency Criterion for Absolute Stochastic Stability of the Nonlinear Systems of Differential Ito Equations, Usp. Mat. Nauk, 1972, vol. 27, no. 4(166), pp. 215, 216.
18. Pakshin, P.V., Stability of a Class of Nonlinear Stochastic Systems, Avtom. Telemekh., 1977, no. 4, pp. 27-36.
19. Mahalanabis, A.K. and Purkayastha, S., Frequency-Domain Criteria for Stability of a Class of Nonlinear Stochastic Systems, IEEE Trans. Automat. Control, 1973, vol. 18, no. 3, pp. 266-270.
20. Shibata, H. and Hata, S., Stochastic Stability for a Multiloop Nonlinear System, IEEE Trans. Automat. Control, 1976, vol. 21, pp. 600, 601.
21. Barsuk, L.O. and Brusin, V.A., Infinite-dimensional Generalization of the Kalman-Yakubovich Lemma, Dinamika Sistem, 1975, no. 8.
22. Likhtarnikov, V.L. and Yakubovich, V.A., Frequency Theorem for the Evolutionary Equations, Sib. Mat. Zh., 1976, vol. 17, no. 5, pp. 1069-1085.
23. Brusin, V.A., Lur'e Equations in the Hilbert Space and Their Solvability, Prikl. Mat. Mekh., 1976, vol. 40, no. 5, pp. 947-956.
24. Ichikawa, A., Stability of Semilinear Stochastic Evolution Equations, J. Math. Anal. Appl., 1982, vol. 90, pp. 12-44.
25. Likhtarnikov, A.L. and Yakubovich, V.A., Frequency Theorem for the Continuous One-parameter Semigroups, Izv. Akad. Nauk SSSR, Mat., 1977, no. 4, pp. 895-911.
26. Lions, J.L., Controle optimal des systèmes gouvernés par des équations aux derivées partielles, Paris: Dunod Gauthier-Villars, 1968. Translated under the title Optimal'noe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi, Moscow: Mir, 1989.
27. Brusin, V.A. and Ugrinovskii, V.A., On Global Asymptotic Stability of the Differential Equations with Random Vector Perturbation, Preprint of NIRFI, Gorkii, 1984, no. 177.
28. Brusin, V.A., Global Stability and Dichotomy of a Class of Nonlinear Random-parameter Systems, Sib. Mat. Zh., 1981, vol. 22, no. 2, pp. 57-73.
29. Brusin, V.A. and Ugrinovskii, V.A., Stochastic Stability of a Class of Nonlinear Differential Equations of Ito Type, Sib. Mat. Zh., 1987, vol. 28, no. 3, pp. 35-50.
30. Ugrinovskii, V.A., Stochastic Counterpart of the Frequency Theorem, Izv. Vuzov, Mat., 1987, no. 10, pp. 37-43.
31. Brusin, V.A. and Ugrinovskii, V.A., Absolute Stability Approach to Stochastic Stability of Infinite-dimensional Nonlinear Systems, Automatica, 1995, vol. 31, no. 10, pp. 1453-1458.
32. Wu, H. and Zhou, X.Y., Stochastic Frequency Characteristics, SIAM J. Control Optim., 2001, vol. 40, no. 2, pp. 557-576.
33. Dokuchaev, N.G., Frequency Criterion for Existence of the Optimal Control of the Ito Equation, Vestn. Leningr. Univ., Ser. Mat. Mekh. Astron., 1983, no. 1, pp. 38-43.
34. Korenevskii, D.G., Algebraic Criterion for Absolute (in Nonlinearity) Stability of the Stochastic Control Systems with Nonlinear Feedback, Ukr. Mat. Zh., 1988, vol. 40, no. 6, pp. 731-736.
35. Korenevskii, D.G., Algebraic Criterion for Absolute (in Nonlinearity) Stability of the Stochastic Discrete Control Systems with Nonlinear Feedback, Ukr. Mat. Zh., 1989, vol. 41, no. 1, pp. 42-48.
36. Korenevskii, D.G., Absolute Mean-square Stability of Continuous and Discrete Nonlinear Stochastic Control Systems. Algebraic Coefficient Criteria, Dokl. Akad. Nauk SSSR, 1989, vol. 306, no. 6, pp. 1316-1319.
37. Korenevskii, D.G., Ustoichivost' dinamicheskikh sistem pri sluchainykh vozmushcheniyakh parametrov. Algebraicheskie kriterii (Stability of the Dynamic Systems under Random Parameter Perturbations. Algebraic Criteria), Kiev: Naukova Dumka, 1989.
38. Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in Control and System Theory, Philadelphia: SIAM, 1994.
39. Ait Rami, M. and El Ghaoui, L., LMI Optimization for Nonstandard Riccati Equation Arising in Stochastic Control, IEEE Trans. Automat. Control, 1996, vol. 41, pp. 1666-1671.
40. Ait Rami, M. and Zhou, X.Y., Linear Matrix Inequalities, Riccati Equations, an Indefinite Stochastic Linear Quadratic Controls, IEEE Trans. Automat. Control, 2000, vol. 45, pp. 1131-1143.
41. Yao, D.D., Zhang, S., and Zhou, X.Y., Stochastic Linear-quadratic Control via Semidefinite Programming, SIAM J. Control Optim., 2001, vol. 40, pp. 801-823.
42. Ugrinovskii, V.A., Exponential Stabilization of Nonlinear Stochastic Systems, Prikl. Mat. Mekh., 1988, vol. 52, no. 2, pp. 16-24.
43. Kazarinov, Yu.F., Stabilization of a Linear Stochastic System Subjected to a Parametric Action of the "White Noise" Type, Prikl. Mat. Mekh., 1977, vol. 41, no. 2, pp. 245-250.
44. Kazarinov, Yu.F., Stabilization of a Linear Stochastic System Subjected to a Parametric Perturbation by a Vector White Noise, Voprosy Mekh. Protsessov Upravl., 1982, no. 5, pp. 64-75.
45. Popkov, Yu.C., Ashimov, A.A., and Asaubaev, K.Sh., Statisticheskaya teoriya avtomaticheskikh sistem s dinamicheskoi chastotno-impul'snoi modulyatsiei (Statistical Theory of Automatic Systems with Dynamic Pulse-frequency Modulation), Moscow: Nauka, 1988.
46. Pakshin, P.V., Exponential Stability of a Class of Nonlinear Stochastic Systems, Avtom. Telemekh., 1980, no. 2, pp. 65-71.
47. Gelig, A.Kh., Stability of the Asynchronous Pulse Systems with Random Parameter Perturbations, Avtom. Telemekh., 1998, no. 5, pp. 181-185.
48. Gelig, A.Kh. and Elkhimova, Yu.V., Stability of Nonlinear Pulse Systems under Random Parameter Perturbations, Avtom. Telemekh., 1995, no. 11, pp. 140-147.
49. Gelig, A.Kh., Elkhimova, Yu.V., and Churilov, A.N., Stability of one Class of Functional Differential Ito Equations, Vestn. S.-Peterburg. Gos. Univ., Ser. 1, 1994, Vol. 2, no. 8, pp. 3-9.
50. Gelig, A.Kh. and Elkhimova, Yu.V., Stability of the Functional Differential Ito Equation with a Monotone Nonlinear Characteristic, Vestn. S.-Peterburg. Gos. Univ., Ser. 1, 1995, vol. 4, pp. 3-7.
51. Gelig, A.Kh. and Churilov, A.N., Kolebaniya i ustoichivost' nelineinykh impul'snykh sistem (Oscillations and Stability of the Nonlinear Pulse Systems), St. Petersburg: S.-Peterburg. Gos. Univ., 1993.
52. Ugrinovskii, V.A. and Petersen, I.R., Absolute Stabilization and Minimax Optimal Control of Uncertain Systems with Stochastic Uncertainty, SIAM J. Control Optim., 1999, vol. 37, pp. 1089-1122.
53. ∞ Methods, New York: Springer, 2000.
54. Yaz, E. and Yildizbayrak, N., Robustness of Feedback-stabilized Systems in the Presence of Nonlinear and Random Perturbation, Int. J. Control, 1985, vol. 41, pp. 345-353.
55. ∞ Control in the Presence of Stochastic Uncertainty, Int. J. Control, 1998, vol. 71, pp. 219-237.
56. ∞ SIAM J. Control Optim., 1998, vol. 36, pp. 1504-1538.
57. Wonham, W.M., Stochastic Differential Equations in Control Theory, in Probabilistic Methods in Applied Mathematics, Bharucha-Reid, A.T., Ed., New York: Academic, 1970, vol. 2, pp. 131-212.
58. Mohler, R.R. and Kolodziej, W.J., An Overview of Bilinear Systems Theory and Applications, IEEE Trans. Syst., Man Cybernet., 1980, vol. 10, pp. 683-688.
59. Mohler, R.R. and Kolodziej, W.J. An Overview of Stochastic Bilinear Control Processes, IEEE Trans. Syst., Man Cybernet., 1980, vol. 10, pp. 913-918.
60. Malyshev, V.V. and Pakshin, P.V., Applied Theory of Stochastic Stability and Optimal Stationary Control (Review). I, Izv. Akad. Nauk SSSR, Tekh. Kibern., 1990, no. 1, pp. 42-66.
61. Malyshev, V.V. and Pakshin, P.V., Applied Theory of Stochastic Stability and Optimal Stationary Control (Review). I, Izv. Akad. Nauk SSSR, Tekh. Kibern., 1990, no. 2, pp. 97-120.
62. Pakshin, P.V., Diskretnye sistemy so sluchainymi parametrami i strukturoi (Discrete Systems with Random Parameters and Structure), Moscow: Fizmatlit, 1994.
63. Barkin, A.I., Zelentsovskii, A.L., and Pakshin, P.V., Absolyutnaya ustoichivost' determinirovannykh i stokhasticheskikh sistem upravleniya (Absolute Stability of the Deterministic and Stochastic Control Systems), Moscow: Mosk. Aviats. Inst., 1992.
64. ∞ Control and Estimation of State-multiplicative Linear Systems Series, New York: Springer, 2005.
65. Brusin, V.A., Some Results of Studying the Nonlinear Discrete Stochastic Control Systems, Avtom. Telemekh., 1970, no. 6, pp. 82-87.
66. Gusev, C.V. and Likhtarnikov, A.L., Kalman-Popov-Yakubovich Lemma and the S-procedure: A Historical Essay, Avtom. Telemekh., 2006, no. 11, pp. 77-121.
67. Meyer, K.R., On the Existence of Liapunov Functions for the Problem of Lur'e, J. SIAM Control, 1966, ser. A, vol. 3, pp. 373-383.
68. Wonham, W.M., A Liapunov Method for the Estimation of Statistical Averages, J. Diff. Equat., 1966, vol. 2, pp. 365-377.
69. Lorenz, J., Czestotliwościowy warunek stochasticznej stabilności jednowymiarowych dyskretnych ukladów regulacji (Frequency Criterion for Stochastic Stability of One-dimensional Discrete Control Systems), Arch. Automat. Telemech., 1969, vol. 14, no. 3, pp. 243-250.
70. Yakubovich, V.A., Frequency Conditions for Absolute Stability of the Control Systems with Multiple Nonlinear or Linear Nonstationary Units, Avtom. Telemekh., 1967, vol. 23, no. 6, pp. 5-30.
71. Tunik, A.A. and Lychak, M.M., Analysis of Stochastic Stability of the Pulse Control Systems in the Frequency Domain, Izv. Akad. Nauk SSSR, Tekh. Kibern., 1973, no. 1, pp. 172-179.
72. Lychak, M.M. and Tunik, A.A., Chastotni umovi stokhastichnnoi stiikosti nelineinykh impul'snikh sistem keruvannya (Frequency Conditions for Stochastic Stability of the Nonlinear Pulse Control Systems), Avtomatika, 1971, no. 4, pp. 81-84.
73. Mishra, M.K.P. and Mahalanabis, A.K., On the Stability of Discrete Non-Linear Feedback Systems with State-dependent Noise, Int. J. Syst. Sci., 1975, vol. 6, no. 5, pp. 479-490.
74. Pakshin, P.V., Stability of the Nonlinear Discrete Random-parameter Systems, Prob. Unpravl. Teor. Inf., 1978, vol. 7, no. 4, pp. 73-81.
75. Levit, M.V. and Churilov, A.N., On the Estimates of Some Functionals of the Solutions of Matrix Inequalities Occurring in the Control Theory, Izv. Vuzov, Mat., 1983, no. 5, pp. 53-59.
76. Churilov, A.N., On the Estimates of the Functional Occurring in the Studies of the Discrete Control Systems, Izv. Vuzov, Mat., 1984, no. 9, pp. 59-65.
77. Mishra, M.K.P. and Mahalanabis, A.K., On the Stability of Non-Linear Feedback Systems with Controldependent Noise, Int. J. Syst. Sci., 1975, vol. 6, no. 10, pp. 945-949.
78. Mishra, M.K.P. and Mahalanabis, A.K., On the Stability of Linear Time-varying Feedback Systems with Random Parameters, Int. J. Syst. Sci., 1976, vol. 7, no. 11, pp. 1315-1322.
79. Socha, L., Application of Yakubovich Criterion for Stability of Nonlinear Stochastic systems, IEEE Trans. Automat. Control, 1980, vol. 25, no. 2, pp. 330-332.
80. Rootenberg, J. and Ghozati, S-A., Cross-coupling and Stability of Nonlinear Stochastic Control Systems, Int. J. Syst. Sci., 1977, vol. 8, no. 5, pp. 539-546.
81. Rootenberg, J. and Ghozati, S-A., Improved Stability for Linear Stochastic Systems, Int. J. Syst. Sci., 1977, vol. 8, no. 4, pp. 413-422.
82. Polushin, I.G., Fradkov, A.L., and Hill, D.J., Passivity and Passification of Nonlinear Systems, Avtom. Telemekh., 2000, no. 3, pp. 3-37.
83. Andrievskii, B.R. and Fradkov, A.L., Method of Passification in the Problems of Adaptive Control, Estimation, and Synchronization, Avtom. Telemekh., 2006, no. 11, pp. 3-37.
84. Brockett, R.W., Lie Algebras and Lie Groups in Control Theory, in Geometric Methods in Systems Theory, Brockett, R. and Mayne D. Eds., Dordrecht: Reidel, 1973, pp. 43-82.
85. Bismut, J.M., Linear Quadratic Optimal Stochastic Control with Random Coefficients, SIAM J. Control Optim., 1976, vol. 14, no. 3, pp. 419-444.
86. Ichikawa, A., Dynamic Programming Approach to Stochastic Evolution Equations, SIAM J. Control Optim., 1979, vol. 17, pp. 152-174.
87. Ichikawa, A., Semilinear Stochastic Evolution Equations: Boundedness, Stability and Invariant Measures, Stochastics, 1984, vol. 12, pp. 1-39.
88. Chen, L., Li, X., and Zhou, X., Stochastic Linear Quadratic Regulators with Indefinite Control Weight Costs, SIAM J. Control Optim., 1998, vol. 36, pp. 1685-1702.
89. Johnson, C.D., The 'Unreachable Poles' Defect in LQR Theory: Analysis and Remedy, Int. J. Control, 1988, vol. 47, pp. 697-709.
90. Pakshin, P.V. and Retinskii, D.M., Robust Stabilization of Random-structure Systems with Feedback Switched According to the Plant Output, Avtom. Telemekh., 2005, no. 7, pp. 144-153.
91. Willems, J.L. and Willems, J.C., Feedback Stabilizability of Stochastic Systems with State and Control Dependent Noise, Automatica, 1976, vol. 12, pp. 277-283.
92. Gelig, A.Kh. and Churilov, A.N., Frequency Methods in the Theory of Pulse-Modulated Control Systems, Avtom. Telemekh., 2006, no. 11, pp. 60-76.
93. Malikov, A.I., Absolute Stability of the Nonlinear Controlled Systems with Random Variaitons of Structure, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 1996, no. 2, pp. 19-30.
94. Willems, J.C., Dissipative Dynamic systems. Part I: General Theory, Arch. Rational Mechan. Anal., 1972, vol. 45, pp. 321-351.
95. Florchinger, P., A Passive System Approach to Feedbakc Stabilization of Nonlinear Control Stochastic Systems, SIAM J. Control Optim., 1999, vol. 37, pp. 1848-1864.
96. Aliyu, M.D.S., Dissipative Analysis and Stability of Nonlinear Stochastic State-delayed Systems, Nonlinear Dynam. Syst. Theory, 2004, vol. 4, pp. 243-256.
97. ∞ Control for Nonlinear Stochastic Systems: The Output-feedback Case, in Prepr. 16th IFAC World Congr., Prague, 2005, pp. 1-6 (CD-ROM).
98. ∞ Control for a Class of Nonlinear Stochastic Systems, SIAM J. Control Optim., 2006, vol. 44, pp. 1973-1991.
99. Hill, D.J. and Moylan, P.J., The Stability of Nonlinear Dissipative Systems, IEEE Trans. Automat. Control, 1976, vol. 21, pp. 708-711.
100. Hill, D.J. and Moylan, P.J., Connection between Finite-gain and Asymptotic Stability, IEEE Trans. Automat. Control, 1980, vol. 25, pp. 931-936.
101. Byrnes, C.I., Isidori, A., and Willems, J.C., Passivity, Feedback Equivalence, and the Global Stabilization of Minimum Phase Nonlinear Systems, IEEE Trans. Automat. Control, 1991, vol. 36, pp. 1228-1240.
102. Oksendal, B., Stochastic Differential Equations. An Introduction with Applications, Berlin: Springer, 2000. Translated under the title Stokhasticheskie differentsial'nye uravneniya. Vvedenie v teoriyu i prilozheniya, Moscow: Mir, 2003.
103. Gikhman, I.I. and Skorokhod, A.V., Vvedenie v teoriyu sluchainykh protsessov (Introduction to the Theory of Random Processes), Moscow: Nauka, 1977.
104. Gikhman, I.I. and Skorokhod, A.V., Stokhasticheskie differentsial'nye uravneniya i ikh prilozheniya (Stochastic Differential Equations and Their Applications), Kiev: Naukova Dumka, 1982.
105. Stratonovich, R.L., A New Notation of the Stochastic Integrals and Equations, Vestn. Mosk. Univ., Ser. 1, 1964, no. 1, pp. 3-12.