## Nonlinear system analysis and identification from random dataDescribes procedures to identify and analyze the properties of many types of nonlinear systems from random data measured at the input and output points of physical systems. Improvements are offered in applying older techniques, and problems that traditionally have been difficult to analyze are solved by new, simpler procedures. Formulas are stated for optimum nonlinear system identification in both general models consisting of parallel, linear bilinear and trilinear systems, and special models consisting of parallel linear, finite-memory square-law systems and finite-memory cubic systems. New results, obtained here, show when and how to replace complicated single input/output nonlinear models with simpler alternative multiple input/single output linear models. New error analysis formulas are presented to design experiments and to evaluate estimates obtained from measured data. Includes many illustrative examples. |

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### Contents

LINEAR SYSTEMS RANDOM DATA | 1 |

ZEROMEMORY NONLINEAR SYSTEMS | 15 |

BILINEAR AND TRILINEAR SYSTEMS | 74 |

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### Common terms and phrases

autospectral density function bispectrum Chapter constant-parameter linear system correlation functions cross-spectral density function Cuber cubic system defined delta function derived envelope detector expected value finite-memory nonlinear system frequency response function frequency-domain Gaussian input data Hence identify input x(t input/output linear and nonlinear linear model mean value Gaussian model of Figure model with uncorrelated non-Gaussian nonlinear coherence functions nonlinear wave force normalized random error obtains optimum linear system output autocorrelation function output autospectral density output data output PDF output probability density output spectrum output y(t parallel linear physically realizable Price's theorem probability density function quantities shown in Figure special bispectral spectral density functions square-law system stationary random data Sxxy(f Sybyb(f Sycyc(f system identification system output system with sign Syy(f third-order nonlinear total output transient random data trilinear systems trispectral density functions u)Sxx(f uncorrelated outputs wave force model weighting function yb(t Yc(f yc(t zero mean value zero-memory nonlinear system