State Observers with Random Sampling Times and Convergence Analysis of Double-Indexed and Randomly Weighted Sums of Mixing Processes

Algorithms for system identification, estimation, and adaptive control in stochastic systems rely mostly on different types of signal averaging to achieve uncertainty reduction, convergence, stability, and performance enhancement. The core of such algorithms is various types of laws of large numbers that reduce the effect of noises when they are averaged. Many of the noise sequences encountered are often correlated and nonwhite. In the case of state estimation using quantized information such as in networked systems, convergence must be analyzed on double-indexed and randomly weighted sums of mixing-type stochastic processes, which are correlated with the remote past and distant future being asymptotically independent. This paper presents new results on convergence analysis of such processes. Strong laws of large numbers and convergence rates for such problems are established. These results resolve some fundamental issues in state observer designs with random sampling times, quantized information processing, and other applications.

[1]  Magda Peligrad,et al.  Almost-Sure Results for a Class of Dependent Random Variables , 1999 .

[2]  R. C. Bradley.  Basic properties of strong mixing conditions. A survey and some open questions , 2005, math/0511078.

[3]  An Jun,et al.  Complete convergence of weighted sums for ρ∗ -mixing sequence of random variables , 2008 .

[4]  K. Åström,et al.  Comparison of Periodic and Event Based Sampling for First-Order Stochastic Systems , 1999 .

[5]  Wlodzimierz Bryc,et al.  Moment conditions for almost sure convergence of weakly correlated random variables , 1993 .

[6]  G. Yin,et al.  System Identification with Quantized Observations , 2010 .

[7]  M. Loève.  On Almost Sure Convergence , 1951 .

[8]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[9]  Le Yi Wang,et al.  Non-commercial Research and Educational Use including without Limitation Use in Instruction at Your Institution, Sending It to Specific Colleagues That You Know, and Providing a Copy to Your Institution's Administrator. All Other Uses, Reproduction and Distribution, including without Limitation Comm , 2022 .

[10]  Magda Peligrad,et al.  Maximal Inequalities and an Invariance Principle for a Class of Weakly Dependent Random Variables , 2003 .

[11]  P. N. Paraskevopoulos,et al.  Modern Control Engineering , 2001 .

[12]  H. Kushner,et al.  Stochastic Approximation and Recursive Algorithms and Applications , 2003 .

[13]  Richard C. Bradley,et al.  On the spectral density and asymptotic normality of weakly dependent random fields , 1992 .

[14]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .

[15]  R. C. Bradley.  Every “lower psi-mixing” Markov chain is “interlaced rho-mixing” , 1997 .

[16]  Joono Sur,et al.  State Observer for Linear Time-Invariant Systems With Quantized Output , 1998 .

[17]  Paul Erdös,et al.  On a Theorem of Hsu and Robbins , 1949 .

[18]  Lei Guo,et al.  Estimation of nonstationary ARMAX models based on the Hannan-Rissanen method , 1990 .

[19]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[20]  Le Yi Wang,et al.  System identification using binary sensors , 2003, IEEE Trans. Autom. Control..

[21]  Antonio Vicino,et al.  Information-Based Complexity and Nonparametric Worst-Case System Identification , 1993, J. Complex..

[22]  L. Baum,et al.  Convergence rates in the law of large numbers , 1963 .

[23]  R. C. Bradley.  Basic Properties of Strong Mixing Conditions , 1985 .

[24]  R. C. Bradley.  A Stationary Rho-Mixing Markov Chain Which Is Not “Interlaced” Rho-Mixing , 2001 .

[25]  Han-Fu Chen,et al.  Identification and Stochastic Adaptive Control , 1991 .

[26]  A. Kolmogorov,et al.  On Strong Mixing Conditions for Stationary Gaussian Processes , 1960 .

[27]  H Robbins,et al.  Complete Convergence and the Law of Large Numbers. , 1947, Proceedings of the National Academy of Sciences of the United States of America.

[28]  Robert L. Taylor.  Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces , 1978 .

[29]  Fredrik Gustafsson,et al.  Event based sampling with application to vibration analysis in pneumatic tires , 2001, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221).