Stable Process Approach to Analysis of Systems Under Heavy-Tailed Noise: Modeling and Stochastic Linearization

The Wiener process has provided a lot of practically useful mathematical tools to model stochastic noise in many applications. However, this framework is not enough for modeling extremal events, since many statistical properties of dynamical systems driven by the Wiener process are inevitably Gaussian. The goal of this work is to develop a framework that can represent a heavy-tailed distribution without losing the advantages of the Wiener process. To this end, we investigate models based on stable processes (this term “stable” has nothing to do with “dynamical stability”) and clarify their fundamental properties. In addition, we propose a method for stochastic linearization, which enables us to approximately linearize static nonlinearities in feedback systems under heavy-tailed noise, and analyze the resulting error theoretically. The proposed method is applied to assessing wind power fluctuation to show the practical usefulness.

[1]  João Pedro Hespanha,et al.  State Estimation of Multiagent Systems Under Impulsive Noise and Disturbances , 2015, IEEE Transactions on Control Systems Technology.

[2]  Vladimir I. Bogachev,et al.  ON UNIQUENESS OF INVARIANT MEASURES FOR FINITE- AND INFINITE-DIMENSIONAL DIFFUSIONS , 1999 .

[3]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

[4]  R. Bhattacharya.  Criteria for Recurrence and Existence of Invariant Measures for Multidimensional Diffusions , 1978 .

[5]  B. Øksendal.  Stochastic differential equations : an introduction with applications , 1987 .

[6]  M. Fragoso,et al.  Continuous-Time Markov Jump Linear Systems , 2012 .

[7]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[8]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[9]  Ken-iti Sato.  Lévy Processes and Infinitely Divisible Distributions , 1999 .

[10]  Pierre Bernard.  Stochastic linearization: what is available and what is not , 1998 .

[11]  Stephen P. Boyd,et al.  Smoothed state estimates under abrupt changes using sum-of-norms regularization , 2012, Autom..

[12]  Stephen H. Crandall.  A half-century of stochastic equivalent linearization , 2006 .

[13]  J. J. Westman,et al.  The LQGP problem: a manufacturing application , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[14]  Kenji Kashima,et al.  On Weak Approximation of Stochastic Differential Equations through Hard Bounds by Mathematical Programming , 2013, SIAM J. Sci. Comput..

[15]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[16]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[17]  Reiichiro Kawai,et al.  Sample Path Generation of Lévy-Driven Continuous-Time Autoregressive Moving Average Processes , 2017 .

[18]  P. Rousseeuw,et al.  Wiley Series in Probability and Mathematical Statistics , 2005 .

[19]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[20]  James C. Spall.  The Kantorovich inequality for error analysis of the Kalman filter with unknown noise distributions , 1995, Autom..

[21]  Kenji Kashima,et al.  Modeling and linearization of systems under heavy-tailed stochastic noise with application to renewable energy assessment , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[22]  Daniel Liberzon,et al.  Nonlinear feedback systems perturbed by noise: steady-state probability distributions and optimal control , 2000, IEEE Trans. Autom. Control..

[23]  J. L. Maryak,et al.  Use of the Kalman filter for inference in state-space models with unknown noise distributions , 1996, IEEE Transactions on Automatic Control.