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The Method of Averaged Models for Discrete-Time Adaptive Systems

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Abstract

Dynamical processes in nature and technology are usually described by continuous-or discrete-time dynamical models, which have the form of nonlinear stochastic differential or difference equations. Hence, a topical problem is to develop effective methods for a simpler description of dynamical systems. The main requirement to simplification methods is preserving certain properties of a process under study. One group of such methods is represented by the methods of continuous- or discrete-time averagedmodels, which are surveyed in this paper. New results for stochastic networked systems are also introduced. As is shown below, the method of averaged models can be used to reduce the analytical complexity of a closed loop stochastic system. The corresponding upper bounds on the mean square distance between the states of an original stochastic system and its approximate averaged model are obtained.

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References

  1. Bogolyubov, N.N. and Mitropol’skii, Yu.A., Asimptoticheskie metody v teorii nelineinykh kolebanii (Asymptotical Methods in Theory of Nonlinear Oscillations), Moscow: Gostekhizdat, 1955.

    Google Scholar 

  2. Volosov, V.M., Averaging in Systems of Ordinary Differential Equations, Russ. Math. Surveys, 1962, vol. 17, no. 6, pp. 1–126.

    Article  MathSciNet  MATH  Google Scholar 

  3. Mishchenko, E.F. and Rozov, N.Kh., Differentsial’nye uravneniya s malym parametrom i relaksatsionye kolebaniya (Differential Equations with Small Parameter and Relaxation Oscillations, Moscow: Nauka, 1975, vol. 1.

  4. Gerashchenko, E.I. and Gerashchenko, S.M., Metod razdeleniya dvizhenii i optimizatsiya nelineinykh sistem (The Method of Motions Separation and Optimization of Nonlinear Systems), Moscow: Nauka, 1975.

    Google Scholar 

  5. O’Malley, R., Introduction to Singular Perturbations, New York: Academic, 1974.

    MATH  Google Scholar 

  6. Kokotovic, P.V., O’Malley, R., and Sannuti, P., Singular Perturbations and Order Reduction in Control Theory—An Overview, Automatica, 1976, vol. 12, no. 2. P. 123–132.

    Article  MathSciNet  MATH  Google Scholar 

  7. Young, K.-K., Kokotovic, P., and Utkin, V., A Singular Perturbation Analysis of High-Gain Feedback Systems, IEEE Trans. Autom. Control, 1977, vol. 22, no. 6, pp. 931–938.

    Article  MathSciNet  MATH  Google Scholar 

  8. Khas’minskii, R.Z., On Random Processes Defined by Differential Equations with Small Parameter, Teor. Veroyatn. Primen., 1966, vol. 11, no. 2, pp. 240–259.

    Google Scholar 

  9. Khas’minskii, R.Z., On the Averaging Principle for Itô Stochastic Differential Equations, Kibern., 1968, vol. 4, no. 3, pp. 260–279.

    Google Scholar 

  10. Gikhman, I.I. and Skorokhod, A.V., Stokhasticheskie differentsial’nye uravneniya (Stochastic Differential Equations), Kiev: Naukova Dumka, 1968.

    MATH  Google Scholar 

  11. Freidlin, M.I., The Averaging Principle and Theorems on Large Deviations, Russ. Math. Surveys, 1978, vol. 33, no. 5, pp. 117–176.

    Article  MathSciNet  Google Scholar 

  12. Wentzell, A.D. and Freidlin, M.I., Fluktuatsii v dinamicheskikh sistemakh pod deistviem malykh sluchainykh vozmushchenii (Fluctuations in Dynamical Systems Effected by Small Random Perturbations), Moscow: Nauka, 1979.

    Google Scholar 

  13. Derevitskii, D.P. and Fradkov, A.L., Two Models Analyzing the Dynamics of Adaptation Algorithms, Autom. Remote Control, 1974, vol. 35, no. 1, pp. 67–75.

    MathSciNet  Google Scholar 

  14. Derevitskii, D. and Fradkov, A., Investigation of Discrete-Time Adaptive Control Systems Using Continuous-Time Models, Izv. Akad. Nauk SSSR, Tekhn. Kibern., 1975, vol. 5, pp. 93–99.

    Google Scholar 

  15. Ljung, L., Analysis of Recursive Stochastic Algorithms, IEEE Trans. Autom. Control, 1977, vol. 22, no. 4, pp. 551–575.

    Article  MathSciNet  MATH  Google Scholar 

  16. Derevitskii, D.P. and Fradkov, A.L., Prikladnaya teoriya diskretnykh adaptivnykh sistem upravleniya (Applied Theory of Discrete-Time Adaptive Control Systems), Moscow: Nauka, 1981.

    MATH  Google Scholar 

  17. Fradkov, A., Continuous-Time Averaged Models of Discrete-Time Stochastic Systems: Survey and Open Problems, Proc. 2011 50 IEEE Conf. on Decision and Control and Eur. Control Conf. (CDCECC), Orlando, Florida, USA, 2011, pp. 2076–2081.

    Google Scholar 

  18. Fradkov, A.L., Averaged Continuous-Time Models in Identification and Control, Proc. Eur. Control Conf., Strasbourg, 2014, pp. 2822–2826.

    Google Scholar 

  19. Polyak, B.T., Vvedenie v optimizatsiyu, Moscow: Nauka, 1983. Translated under the title Introduction to Optimization, Translations Series in Mathematics and Engineering, New York: Optimization Software, 1987.

    MATH  Google Scholar 

  20. Robbins, H. and Monro, S., A Stochastic Approximation Method, Ann. Math. Stat., 1951, vol. 22, pp. 400–407.

    Article  MathSciNet  MATH  Google Scholar 

  21. Kushner, H.J. and Clark, D.S., Stochastic Approximation Methods for Constrained and Unconstrained Systems, New York: Springer-Verlag, 2012.

    Google Scholar 

  22. Meerkov, S.M., On Simplification of the Description of Slow Markovian Roaming, Autom. Remote Control, 1972, vol. 33, no. 3, pp. 404–414.

    Google Scholar 

  23. Informatsionnye materialy: Kibernetika, tom 68 (Information Materials: Cybernetics, vol. 68), 1973.

  24. Ljung, L., Convergence of Recursive Stochastic Algorithms, Proc. IFAC Sympos. on Stochastic Control, 1974, pp. 551–575.

    Google Scholar 

  25. Basar, E.T., Control Theory: Twenty-Five Seminal Papers (1932–1981), New York: Wiley–IEEE Press, 2001.

    Google Scholar 

  26. Gerencs´er, L., A Representation Theorem for the Error of Recursive Estimators, SIAM J. Control Optim., 2006, vol. 44, no. 6, pp. 2123–2188.

    Article  MathSciNet  Google Scholar 

  27. Polyak, B.T. and Tsypkin, Ya.Z., Pseudogradient Algorithms of Adaptation and Learning, Autom. Remote Control, 1973, vol. 34, no. 3, pp. 45–68.

    MATH  Google Scholar 

  28. Nevel’son, M.B. and Khas’minskii, R.Z., Continuous Procedures of Stochastic Approximation, Probl. Inf. Transm., 1971, vol. 7, no. 2, pp. 139–148.

    MathSciNet  Google Scholar 

  29. Khas’minskii, R.Z., The Averaging Principle for Stochastic Differential Equations, Probl. Inf. Transm., 1968, vol. 4, no. 2, pp. 68–69.

    Google Scholar 

  30. Ibragimov, I.A. and Linnik, Yu.V., Nezavisimye i statsionarno svyazannye velichiny (Independent and Stationary-Bound Variables), Moscow: Nauka, 1965.

    Google Scholar 

  31. Bernstein, S.N., Stokhasticheskie uravneniya v konechnykh raznostyakh i stokhasticheskie differentsial’-nye uravneniya (Stochastic Finite-Difference Equations and Stochastic Differential Equations), Moscow: Akad. Nauk SSSR, 1964, vol. 4, pp. 484–542.

    Google Scholar 

  32. Krasovskii, N., Stability of Motion, Stanford: Stanford Univ. Press, 1963.

    MATH  Google Scholar 

  33. Akhmetkaliev, T., On the Relationship between the Stability of Stochastic Difference and Differential Equations, Differ. Uravn., 1965, no. 8, pp. 1016–1026.

    MathSciNet  Google Scholar 

  34. Polyak, B.T., Convergence and Rate of Convergence of Recursive Stochastic Algorithms. I, Autom. Remote Control, 1976, vol. 37, pp. 537–542.

    Google Scholar 

  35. Nevel’son, M.B. and Khas’minskii, R.Z., Stokhasticheskaya approksimatsiya i rekurrentnoe otsenivanie (Stochastic Approximation and Recursive Estimation), Moscow: Nauka, 1972.

    Google Scholar 

  36. Kushner, H., Convergence of Recursive Adaptive and Identification Procedures via Weak Convergence Theory, IEEE Trans. Autom. Control, 1977, vol. 22, no. 6, pp. 921–930.

    Article  MathSciNet  MATH  Google Scholar 

  37. Derevitskii, D. and Fradkov, A., Application of the Theory of Markov Processes to Analyze the Dynamics of Adaptation Algorithms, Avtom. Vychisl. Tekh., 1974, no. 2, pp. 39–48.

    Google Scholar 

  38. Derevitskii, D. and Fradkov, A., Analysis of the Dynamics of Some Adaptation Algorithms, in Voprosy kibernetiki. Adaptivnye sistemy (Problems of Cybernetics. Adaptive Systems), Moscow: Akad. Nauk SSSR, 1974, pp. 79–84.

    Google Scholar 

  39. Neśić, D., Teel, A.R., and Kokotović, P., Sufficient Conditions for Stabilization of Sampled-Data Nonlinear Systems via Discrete-Time Approximations, Syst. Control Lett., 1999, vol. 38, no. 4, pp. 259–270.

    Article  MathSciNet  MATH  Google Scholar 

  40. Dragan, V. and Khalanai, A., Preserving Exponential Stability In Discrete-Time Control Systems with Adaptive Stabilization, Sib. Math. J., 1990, vol. 31, no. 6, pp. 1046–1050.

    Article  Google Scholar 

  41. Braverman, E. and Pyatnitskii, Yu.S., Passing of Random Signal Through Absolutely Stable Systems, Autom. Remote Control, 1971, vol. 32, no. 2, pp. 202–206.

    Google Scholar 

  42. Derevitskii, D.P., Ripa, K.K., and Fradkov, A.L., Investigation of the Dynamics of Certain Randomsearch Algorithms, Probl. Sluchainogo Poiska, 1975, no. 4, pp. 32–47.

    Google Scholar 

  43. Andrievskii, B.R., Blajkin, A.T., Derevitskii, D.P., and Fradkov, A.L., A Method to Investigate the Dynamics of Digital Adaptive Control Systems for Aerial Vehicles, in Upravlenie v prostranstve (Spatial Control), Moscow: Nauka, 1976, vol. 1, pp. 149–153.

    Google Scholar 

  44. Derevitskii, D.P., Synthesis of a Discrete Stochastic Adaptive Stabilization System Using a Continuous- Time Model, Avtom. Vychisl. Tekh., 1975, no. 6, pp. 50–52.

    MathSciNet  Google Scholar 

  45. Andrievskii, B.R., Blajkin, A.T., Derevitskii, D.P., and Fradkov, A.L., A Synthesis Method of Discrete- Time Adaptive Stabilization Systems for a Stochastic Plant, Referaty dokladov VII Vsesoyuznogo soveshchaniya po problemam upravleniya (Abstracts of papers of the VII All-Union Meeting on Control Problems), 1977, pp. 102–105.

    Google Scholar 

  46. Levitan, V.D., Analysis of the Dynamics of Discrete-Time Adaptation Processes with a Discontinuous Stochastic Model, in Voprosy kibernetiki. Adaptivnye sistemy upravleniya (Problems of Cybernetics. Adaptive Control Systems), Moscow: Akad. Nauk SSSR, 1977, pp. 127–129.

    Google Scholar 

  47. Amelina, N., Fradkov, A., Jiang, Y., and Vergados, D.J., Approximate Consensus in Stochastic Networks with Application to Load Balancing, IEEE Trans. Inform. Theory, 2015, vol. 61, no. 4, pp. 1739–1752.

    Article  MathSciNet  MATH  Google Scholar 

  48. Pezeshki-Esfahani, H. and Heunis, A.J., Strong Diffusion Approximations for Recursive Stochastic Algorithms, IEEE Trans. Inform. Theory, 1997, vol. 43, no. 2, pp. 512–523.

    Article  MathSciNet  MATH  Google Scholar 

  49. Weiss, A. and Mitra, D., Digital Adaptive Filters: Conditions for Convergence, Rates of Convergence, Effects of Noise and Errors Arising from the Implementation, IEEE Trans. Inform. Theory, 1979, vol. 25, no. 6, pp. 637–652.

    Article  MathSciNet  MATH  Google Scholar 

  50. Benveniste, A., Métivier, M., and Priouret, P., Adaptive Algorithms and Stochastic Approximations, Berlin: Springer-Verlag, 2012.

    MATH  Google Scholar 

  51. Amelina, N.O. and Fradkov, A.L., Approximate Consensus in the Dynamic Stochastic Network with Incomplete Information and Measurement Delays, Autom. Remote Control, 2012, vol. 73, no. 11, pp. 1765–1783.

    Article  MathSciNet  MATH  Google Scholar 

  52. Amelina, N. and Fradkov, A., Approximate Consensus in Multi-Agent Nonlinear Stochastic Systems, Proc. Europ. Control Conf. (ECC), 2014, pp. 2833–2838.

    Google Scholar 

  53. Amelina, N., Fradkov, A., and Amelin, K., Approximate Consensus in Multi-Agent Stochastic Systems with Switched Topology and Noise, Proc. IEEE Multiconf. on Systems and Control (MSC’2012), Dubrovnik, Croatia, 2012, pp. 445–450.

    Google Scholar 

  54. Amelin, K., Amelina, N., Granichin, O., and Granichina, O., Multi-Agent Stochastic Systems with Switched Topology and Noise, Proc. 13 ACIS Int. Conf. on Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing (SNPD), Kyoto, Japan, 2012, pp. 438–443.

    Google Scholar 

  55. Amelina, N., Granichin, O., and Kornivetc, A., Local Voting Protocol in Decentralized Load Balancing Problem with Switched Topology, Noise, and Delays, Proc. IEEE 52 Ann. Conf. on Decision and Control (CDC), 2013, pp. 4613–4618.

    Google Scholar 

  56. Gorodeisky, Z., Deterministic Approximation of Best-Response Dynamics for the Matching Pennies Game, Game. Econom. Behav., 2009, vol. 66, no. 1, pp. 191–201.

    Article  MathSciNet  MATH  Google Scholar 

  57. Benveniste, A., Design of Adaptive Algorithms for the Tracking of Time-Varying Systems, Int. J. Adapt. Control Signal Process., 1987, vol. 1, no. 1, pp. 3–29.

    Article  MATH  Google Scholar 

  58. Benveniste, A., Priouret, P., and Metivier, M., Algorithmes adaptatifs et approximations stochastiques: théorie et applications à l’identification, au traitement du signal et à la reconnaissance des formes, Paris: Masson, 1987.

    MATH  Google Scholar 

  59. Benveniste, A. and Ruget, G., A Measure of the Tracking Capability of Recursive Stochastic Algorithms with Constant Gains, IEEE Trans. Autom. Control, 1982, vol. 27, no. 3, pp. 639–649.

    Article  MathSciNet  MATH  Google Scholar 

  60. Dugard, L. and Landau, I., Recursive Output Error Identification Algorithms Theory and Evaluation, Automatica, 1980, vol. 16, no. 5, pp. 443–462.

    Article  MATH  Google Scholar 

  61. Kulchitskii, O., The New Method of Dynamical Stochastic Uncertain Systems Investigation, Preprints of Int. Conf. “Stochastic Optimization,” Part 1, 1984, pp. 127–129.

    Google Scholar 

  62. Kushner, H.J., Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, Cambridge: MIT Press, 1984, vol. 6.

  63. Ljung, L., On Positive Real Transfer Functions and the Convergence of Some Recursive Schemes, IEEE Trans. Autom. Control, 1977, vol. 22, no. 4, pp. 539–551.

    Article  MathSciNet  MATH  Google Scholar 

  64. Ljung, L. and Söderström, T., Theory and Practice of Recursive Identification, Cambridge: MIT Press, 1983.

    MATH  Google Scholar 

  65. Solo, V., The Convergence of AML, IEEE Trans. Autom. Control, 1979, vol. 24, no. 6, pp. 958–962.

    Article  MATH  Google Scholar 

  66. Bozin, A. and Zarrop, M., Self Tuning Optimizer-Convergence and Robustness Properties, Proc. 1st Eur. Control Conf., 1991, pp. 672–677.

    Google Scholar 

  67. Ermol’ev, Yu.M. and Kaniovskii, Yu.M., Asymptotic Properties of Some Stochastic Programming Methods with Constant Step, Zh. Vych. Mat. Mat. Fiz., 1979, vol. 19, no. 2, pp. 356–366.

    MathSciNet  Google Scholar 

  68. Coito, F.J. and Lemos, J.M., Adaptive Optimization with Constraints: Convergence and Oscillatory Behaviour, in Pattern Recogn. Image Anal., New York: Springer, 2005, pp. 19–26.

    Chapter  Google Scholar 

  69. Kulchitskii, O.Yu., The Method of Convergence Analysis of Adaptive Filtration Algorithms, Probl. Inf. Transm., 1985, vol. 21, no. 4, pp. 285–297.

    MathSciNet  Google Scholar 

  70. Kushner, H.J. and Shwartz, A., Weak Convergence and Asymptotic Properties of Adaptive Filters with Constant Gains, IEEE Trans. Inform. Theory, 1984, vol. 30, no. 2, pp. 177–182.

    Article  MathSciNet  MATH  Google Scholar 

  71. Metivier, M. and Priouret, P., Applications of a Kushner and Clark Lemma to General Classes of Stochastic Algorithms, IEEE Trans. Inform. Theory, 1984, vol. 30, no. 2, pp. 140–151.

    Article  MathSciNet  MATH  Google Scholar 

  72. Derevitskii, D.P., The Method of Investigation of the Dynamics of Discrete-Time Adaptive Control Systems of Dynamical Objects, in Voprosy kibernetiki. Adaptivnye sistemy (Problems of Cybernetics. Adaptive Systems), Moscow: Akad. Nauk SSSR, 1976, pp. 64–72.

    Google Scholar 

  73. Andrievsky, B., Blazhkin, A., Derevitsky, D., and Fradkov, A., The Method of Investigation of Dynamics of Digital Adaptive Systems of Flight Control, Proc. 6th IFAC Sympos. on Automatic Control in the Space, Yerevan, 1974.

    Google Scholar 

  74. Derevitskii, D.P. and Fradkov, A.L., The Method of Continuous-Time Models in the Theory of Discrete- Time Adaptive Systems, in Voprosy kibernetiki. Zadachi i metody adaptivnogo upravleniya (Problems of Cybernetics. Problems andMethods of Adaptive Control), Moscow: Akad. Nauk SSSR, 1980, pp. 75–98.

    Google Scholar 

  75. Derevitskii, D. and Fradkov, A., Averaging Method for Discrete-Time Stochastic Systems and Its Application in Adaptive Control, Preprints Int. Conf. “Stochastic Optimization,” vol. 1, pp. 74–76

  76. Fradkov, A.L., Adaptivnoe upravlenie slozhnymi sistemami (Adaptive Control of Complex Systems), Moscow: Nauka, 1990.

    MATH  Google Scholar 

  77. Kulchitskii, O.Y., Algorithms of Stochastic Approximation in Adaptation Loop of Linear Dynamical System. I, II, Autom. Remote Control, 1983, vol. 44, no. 9, pp. 1189–1203; 1984, vol. 45, no. 3, pp. 104–113.

    MathSciNet  Google Scholar 

  78. Mosca, E., Zappa, G., and Lemos, J.M., Robustness of Multipredictor Adaptive Regulators: Musmar, Automatica, 1989, vol. 25, no. 4, pp. 521–529.

    Article  MathSciNet  MATH  Google Scholar 

  79. Mosca, E., Lemos, J.M., Mendonca, T., and Nistri, P., Input Variance Constrained Adaptive Control and Singularly Perturbed ODE’s, Proc. 1st ECC, Grenoble, 1991, pp. 2176–2180.

    Google Scholar 

  80. Tsykunov, A.M., Adaptivnoe upravlenie ob”ektami s posledeistviem (Adaptive Control of Plants with Aftereffect), Moscow: Nauka, 1984.

    MATH  Google Scholar 

  81. Oja, E. and Karhunen, J., On Stochastic Approximation of the Eigenvectors and Eigenvalues of the Expectation of a Random Matrix, J. Math. Anal. Appl., 1985, vol. 106, no. 1, pp. 69–84.

    Article  MathSciNet  MATH  Google Scholar 

  82. Zhdanov, A.I., Recursive Estimation of Minimum Eigenvalues of Information Matrices, Autom. Remote Control, 1987, vol. 48, no. 4, part 1, pp. 443–451.

    MathSciNet  MATH  Google Scholar 

  83. Stankovic, M.S., Johansson, K.H., and Stipanović, D.M., Distributed Seeking of Nash Equilibria in Mobile Sensor Networks, Proc. 49th IEEE Conf. on Decision and Control (CDC), 2010, pp. 5598–5603.

    Chapter  Google Scholar 

  84. Astanovskaya, N.V., Varshavskii, V.I., Revako, V.M., and Fradkov, A.L., The Method of Continuous- Time Models in the Problem of Decentralized Resource Allocation, in Primenenie metodov sluchainogo poiska v SAPR (Application of Random Search in CAD), Tallin: Valgus, 1979.

    Google Scholar 

  85. Borovkov, A.A., Asimptoticheskie metody v teorii massovogo obsluzhivaniya (Asymptotic Methods in Queueing Theory), Moscow: Nauka, 1979.

    Google Scholar 

  86. Kuan, C.-M. and Hornik, K., Convergence of Learning Algorithms with Constant Learning Rates, IEEE Trans. Neural Networks, 1991, vol. 2, no. 5, pp. 484–489.

    Article  Google Scholar 

  87. Choi, J. and Horowitz, R., Learning Coverage Control of Mobile Sensing Agents in One-Dimensional Stochastic Environments, IEEE Trans. Autom. Control, 2010, vol. 55, no. 3, pp. 804–809.

    Article  MathSciNet  MATH  Google Scholar 

  88. Choi, J., Oh, S., and Horowitz, R., Distributed Learning and Cooperative Control for Multi-Agent Systems, Automatica, 2009, vol. 45, no. 12, pp. 2802–2814.

    Article  MathSciNet  MATH  Google Scholar 

  89. Huang, M. and Manton, J., Coordination and Consensus of Networked Agents with Noisy Measurements: Stochastic Algorithms and Asymptotic Behavior, SIAM J. Control Optim., 2009, vol. 48, no. 1, pp. 134–161.

    Article  MathSciNet  MATH  Google Scholar 

  90. Borkar, V.S. and Manjunath, D., Distributed Topology Control of Wireless Networks, Wireless Networks, 2008, vol. 14, no. 5, pp. 671–682.

    Article  Google Scholar 

  91. Granichin, O. and Fomin, V., Adaptive Control Using Test Signals in the Feedback Channel, Autom. Remote Control, 1986, vol. 47, no. 2, part 2, pp. 238–248.

    MathSciNet  MATH  Google Scholar 

  92. Granichin, O., Volkovich, V., and Toledano-Kitai, D., Randomized Algorithms in Automatic Control and Data Mining, Berlin: Springer-Verlag, 2015.

    Book  Google Scholar 

  93. Amelin, K. and Granichin, O., Randomized Control Strategies under Arbitrary External Noise, IEEE Trans. Autom. Control, 2016, vol. 61, no. 5, pp. 1328–1333.

    Article  MathSciNet  MATH  Google Scholar 

  94. Spall, J.C., Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control, vol. 64, Hoboken: Wiley-Interscience, 2003.

  95. Tsypkin, Ya.Z., Adaptatsiya i obuchenie v avtomaticheskikh sistemakh (Adaptation and Learning in Automatic Systems), Moscow: Nauka, 1968.

    Google Scholar 

  96. Yakubovich, V.A., The Method of Recurrent Objective Inequalities in the Theory of Adaptive Systems, in Voprosy kibernetiki. Adaptivnye sistemy upravleniya (Problems of Cybernetics. Adaptive Control Systems), Moscow: Akad. Nauk SSSR, 1976, pp. 32–63.

    Google Scholar 

  97. Andrievskii, B.R. and Fradkov, A.L., Analysis of the Dynamics of an Adaptive Control Algorithm for a Linear Dynamical Plant, in Voprosy kibernetiki. Adaptivnye sistemy upravleniya (Problems of Cybernetics. Adaptive Control Systems), Moscow: Akad. Nauk SSSR, 1976, pp. 99–103.

    Google Scholar 

  98. Fradkov, A.L., Speed-Gradient Scheme and Its Application in Adaptive Control Problems, Autom. Remote Control, 1980, vol. 40, no. 9, pp. 1333–1342.

    MATH  Google Scholar 

  99. Yoshihara, K., Moment Inequalities for Mixing Sequences, Kodai Math. J., 1978, vol. 1, no. 2, pp. 316–328.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported in part by the Russian Foundation for Basic Research, projects nos. 17-08-01728, 19-03-00375. The results on the analysis of continuous-discrete and networked systems in Sections 8-11 were obtained at the Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, under the support of the Russian Science Foundation, project no. 16-19-00057-P.

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  1. St. Petersburg State University, St. Petersburg, Russia

    N. O. Amelina, O. N. Granichin & A. L. Fradkov

  2. Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia

    N. O. Amelina, O. N. Granichin & A. L. Fradkov

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  1. N. O. Amelina
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Russian Text © The Author(s), 2019, published in Avtomatika i Telemekhanika, 2019, No. 10, pp. 3–36.

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Amelina, N.O., Granichin, O.N. & Fradkov, A.L. The Method of Averaged Models for Discrete-Time Adaptive Systems. Autom Remote Control 80, 1755–1782 (2019). https://doi.org/10.1134/S0005117919100011

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