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Performance analysis of stochastic gradient algorithms under weak conditions. (English) Zbl 1145.93050

Summary: By using the stochastic martingale theory, convergence properties of Stochastic Gradient (SG) identification algorithms are studied under weak conditions. The analysis indicates that the parameter estimates by the SG algorithms consistently converge to the true parameters, as long as the information vector is persistently exciting (i.e., the data product moment matrix has a bounded condition number) and that the process noises are zero mean and uncorrelated. These results remove the strict assumptions, made in existing references, that the noise variances and high-order moments exist, and the processes are stationary and ergodic and the strong persistent excitation condition holds. This contribution greatly relaxes the convergence conditions of stochastic gradient algorithms. The simulation results with bounded and unbounded noise variances confirm the convergence conclusions proposed.

MSC:

93E12 Identification in stochastic control theory
93E10 Estimation and detection in stochastic control theory
93E25 Computational methods in stochastic control (MSC2010)
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[1] Ljung L. Consistency of the least-squares identification method. IEEE Trans Automatic Contr, 1976, 21(5): 779–781 · Zbl 0334.93048
[2] Solo V. The convergence of AML. IEEE Trans Automatic Contr, 1979, 24(6): 958–962 · Zbl 0422.62083
[3] Lai T L, Wei C Z. Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Ann Stat, 1982, 10(1): 154–166 · Zbl 0649.62060
[4] Lai T L, Wei C Z. Extended least squares and their applications to adaptive control and prediction in linear systems. IEEE Trans Automatic Contr, 1986, 31(10): 898–906 · Zbl 0603.93060
[5] Wei C Z. Adaptive prediction by least squares prediction in stochastic regression models. Ann Stat, 1987, 15(4): 1667–1682 · Zbl 0643.62058
[6] Lai T L, Ying Z L. Recursive identification and adaptive prediction in linear stochastic systems. SIAM J Contr Optim, 1991, 29(5): 1061–1090 · Zbl 0756.93081
[7] Toussi K, Ren W. On the convergence least squares estimates in white noise. IEEE Trans Automatic Contr, 1994, 39(2): 364–368 · Zbl 0825.93985
[8] Ren W, Kumar P R. Stochastic adaptive prediction and model reference control. IEEE Trans Automatic Contr, 1994, 39(10): 2047–2060 · Zbl 0827.93071
[9] Chen H F, Guo L. Identification and Stochastic Adaptive Control. Boston, MA: Birkhäuser, 1991 · Zbl 0747.93002
[10] Ding F, Yang J B, Ding T. Performance analysis of least mean square algorithm for time-varying systems. Contr Theory Appl, 2001, 18(3): 433–437 · Zbl 0988.93089
[11] Ding F, Ding T, Xiao D Y, et al. Bounded convergence of finite data window least squares identification for time-varying systems. Acta Automat Sin, 2002, 28(5): 754–761
[12] Ding F, Ding T, Yang J B, et al. Convergence of forgetting gradient estimation algorithm for time-varying parameters. Acta Automat Sin, 2002, 28(6): 962–968
[13] Ding F, Chen T. Least squares based self-tuning control of dual-rate systems. Int J Adapt Contr Sig Proc, 2004, 18(8): 697–714 · Zbl 1055.93044
[14] Ding F, Chen T. Combined parameter and output estimation of dual-rate systems using an auxiliary model. Automatica, 2004, 40(10): 1739–1748 · Zbl 1162.93376
[15] Ding F, Chen T. Hierarchical least squares identification methods for multivariable systems. IEEE Trans Automat Contr, 2005, 50(3): 397–402 · Zbl 1365.93551
[16] Ding F, Chen T. Hierarchical gradient-based identification of multivariable discrete-time systems. Automatica, 2005, 41(2): 315–325 · Zbl 1086.93063
[17] Ding F, Chen T. Performance bounds of forgetting factor least squares algorithm for time-varying systems with finite measurement data. IEEE Trans Circ Syst-I: Regular papers, 2005, 52(3): 555–566 · Zbl 1374.93390
[18] Ding F, Chen T. Identification of Hammerstein nonlinear ARMAX systems. Automatica, 2005, 41(9): 1479–1489 · Zbl 1086.93063
[19] Ding F, Chen T. Modeling and identification for multirate systems. Acta Automat Sin, 2005, 31(1): 105–122
[20] Ding F, Chen T. Performance analysis of multi-innovation gradient type identification methods. Automatica, 2007, 43(1): 1–14 · Zbl 1140.93488
[21] Xie X M, Ding F. Adaptive Control Systems. Beijing: Tsinghua University Press, 2002
[22] Goodwin G C, Sin K S. Adaptive Filtering, Prediction and Control. Englewood Cliffs, NJ: Prentice-Hall, 1984 · Zbl 0653.93001
[23] Ding F, Chen T. Parameter estimation of dual-rate stochastic systems by using an output error method. IEEE Trans Automat Contr, 2005, 50(9): 1436–1441 · Zbl 1365.93480
[24] Ding F, Chen T. Adaptive digital control of Hammerstein nonlinear systems with limited output sampling. SIAM J Contr Optim, 2007, 45(6): 2257–2276 · Zbl 1126.93034
[25] Ding F, Shi Y, Chen T. Auxiliary model based least-squares identification methods for Hammerstein output-error systems. Syst Contr Lett, 2007, 5(56): 373–380 · Zbl 1130.93055
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