RKH Space Methods for Low Level Monitoring and Control of Nonlinear Systems II. A Vector-Case Example: The Lorenz System

By using techniques from the theory of reproducing kernel Hilbert (RKH) spaces, we continue the exploration of the stochastic linearization method for possibly unknown and/or noise corrupted nonlinear systems. The aim of this paper is twofold: (a) the stochastic linearization formalism is explicitly extended to the vector case; and (b) as an illustration, the performance of the stochastic linearization for monitoring and control is assessed in the case of the Lorenz system for which the dynamic behavior is known independently.

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Low level monitoring and control of nonlinear systems

Proceedings of 1994 33rd IEEE Conference on Decision and Control ◽

10.1109/cdc.1994.411348 ◽

2002 ◽

Author(s):

A. Cover ◽

S. Lenhart ◽

V. Protopopescu ◽

J.A. Reneke

Keyword(s):

Nonlinear Systems ◽

Monitoring And Control ◽

Low Level ◽

And Control ◽

Level Monitoring

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RKH SPACE METHODS FOR LOW LEVEL MONITORING AND CONTROL OF NONLINEAR SYSTEMS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202596000067 ◽

1996 ◽

Vol 06 (01) ◽

pp. 77-96 ◽

Cited By ~ 4

Author(s):

ALAN COVER ◽

JAMES RENEKE ◽

MICHAEL FRYER ◽

SUZANNE LENHART ◽

VLADIMIR PROTOPOPESCU

Keyword(s):

Reproducing Kernel ◽

Control Function ◽

Reproducing Kernel Hilbert Space ◽

Smart Devices ◽

Monitoring And Control ◽

Low Level ◽

Stochastic Linearization ◽

Control Devices ◽

Computational Resources ◽

And Control

A monitor or controller is smart provided that the device is equipped with local computational resources for analyzing data, detecting changes, and making decisions. The problem for the monitoring function is to design algorithms to flag model shifts for dynamic systems in a context requiring many interacting system components and system reconfigurations. The problem for the control function is to improve system performance by updating control feedbacks after the model shift has been detected. Therefore it is desirable that smart monitors and controllers be adaptive and update with minimal intervention from a central director. We present here an approach to designing smart monitoring and control devices based on a stochastic linearization of the system whose dynamics is noisy and unknown. This linearization is obtained by factoring the discrete system covariance matrix, estimated from observations, and applying reproducing kernel Hilbert space techniques. The method is nonparametric which allows the smart devices to operate with only a low level logic.

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On an extension of the stochastic linearization

Vietnam Journal of Mechanics ◽

10.15625/0866-7136/10213 ◽

1993 ◽

Vol 15 (4) ◽

pp. 1-6

Author(s):

Di Paola Mario ◽

Nguyen Dong Anh

Keyword(s):

Nonlinear Systems ◽

Nonlinear Equation ◽

Random Excitation ◽

Linearization Method ◽

Mean Square ◽

Fundamental Idea ◽

Stochastic Linearization ◽

The Mean ◽

The Difference

Stochastic linearization method is one of the most useful tools for analysis of nonlinear systems under random excitation. The fundamental idea of the classical stochastic linearization consists in replacing the original nonlinear equation by a linear one in such a way that the difference between two equations is minimized in the mean square value. In this paper a new version of the stochastic linearization is proposed. It is shown that for two nonlinear systems considered the new version gives good results for both the weak and strong nonlinearities.

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The True Linearization Coefficients for Nonlinear Systems Under Parametric White Noise Excitations

Journal of Applied Mechanics ◽

10.1115/1.1940665 ◽

2004 ◽

Vol 74 (1) ◽

pp. 161-163 ◽

Cited By ~ 2

Author(s):

Giovanni Falsone

Keyword(s):

Parameter Estimation ◽

Nonlinear Systems ◽

Nonlinear Response ◽

Original System ◽

Linearization Method ◽

Maximum Likelihood Estimates ◽

Stationary Case ◽

Stochastic Linearization ◽

Linearization Coefficients ◽

White Noises

In this paper some properties of the stochastic linearization method applied to nonlinear systems excited by parametric Gaussian white noises are discussed. In particular, it is shown that the linearized quantities, obtained by the author in another paper by linearizing the coefficients of the Ito differential rule related to the original system, show the same properties found by Kozin with reference to nonlinear system excited by external white noises. The first property is that these coefficients are the true linearized quantities, in the sense that their exact values are able to give the first two statistical moments of the true response. The second property is that, in the stationary case and in the field of the parameter estimation theory, they represent the maximum likelihood estimates of the linear model quantities fitting the original nonlinear response.

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