Generalized Cumulants Representing a Transient Random Process in a Linear System

[+] Author and Article Information
Arthur Mayer

The Perkin-Elmer Corporation, Optical Technology Division, Danbury, Conn.

J. Dyn. Sys., Meas., Control 99(2), 103-108 (Jun 01, 1977) (6 pages) doi:10.1115/1.3427081 History: Received April 20, 1977; Online July 13, 2010


The state vector of a linear system responding to gaussian noise satisfies a Langevin equation, and the moment-generating function of the probability distribution of the state vector satisfies a partial differential equation. The logarithm of the moment-generating function is expanded in a power series, whose coefficients are organized into a sequence of symmetric tensors. These are the generalized cumulants of the time-dependent distribution of the state vector. They separately satisfy an infinite sequence of uncoupled ordinary differential tensor equations. The normal modes of each of the generalized cumulants are given by an easy formula. This specifies transient response and proves that all cumulants of a stable system are stable. Also, all cumulants of an unstable system are unstable. As an example, a particular non-gaussian initial distribution is assumed for the state vector of a second-order tracking system, and the transient fourth cumulant is calculated.

Copyright © 1977 by ASME
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