A practical method of improving the accuracy of the Gaussian statistical linearization technique is presented. The method uses a series expansion of the unknown probability density function which includes up to fourth order terms. It is shown that by the use of the Gram-Charlier expansion a simple generating function can be derived to evaluate analytically the integrals required. It is also shown how simplifying assumptions can be used to substantially reduce the required extra equations, e.g. symmetric or assymetric and single input nonlinearities. It is also shown that the eigenvalues of the statistically linearized system can be used to estimate the stability and speed of response of the nonlinear system. The reduced expansion technique is applied to first and second order nonlinear systems and the predicted mean square response is compared to the Gaussian statistical linearization and the exact solution. The prediction of the time response of the mean of a nonlinear first order system by the use of the statistically linearized eigenvalues is compared to a 300 run Monte Carlo digital solution.
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March 1981
Research Papers
Improved Statistical Linearization for Analysis and Control of Nonlinear Stochastic Systems: Part I: An Extended Statistical Linearization Technique
J. J. Beaman,
J. J. Beaman
Department of Mechanical Engineering, University of Texas at Austin, Austin, Texas 78712
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J. Karl Hedrick
J. Karl Hedrick
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Mass.
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J. J. Beaman
Department of Mechanical Engineering, University of Texas at Austin, Austin, Texas 78712
J. Karl Hedrick
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Mass.
J. Dyn. Sys., Meas., Control. Mar 1981, 103(1): 14-21 (8 pages)
Published Online: March 1, 1981
Article history
Received:
February 25, 1980
Online:
July 21, 2009
Connected Content
A companion article has been published:
Improved Statistical Linearization for Analysis and Control of Nonlinear Stochastic Systems: Part II: Application to Control System Design
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Citation
Beaman, J. J., and Hedrick, J. K. (March 1, 1981). "Improved Statistical Linearization for Analysis and Control of Nonlinear Stochastic Systems: Part I: An Extended Statistical Linearization Technique." ASME. J. Dyn. Sys., Meas., Control. March 1981; 103(1): 14–21. https://doi.org/10.1115/1.3139636
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