Improved Statistical Linearization for Analysis and Control of Nonlinear Stochastic Systems: Part I: An Extended Statistical Linearization Technique

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.