An introduction to the analysis of non-linear control systems with random inputs

Abstract

By way of introduction, the Wiener method of optimization of linear control systems with noisy inputs is briefly set out, and attention is called to the reasons why this method cannot be used directly for systems containing non-linear components. It is then shown that the difficulty with random inputs arises because the probability distribution of the output can be calculated only when that of the error signal is known; but in order to obtain this quantity the distribution of input and output must be combined. Thus it is not, in general, possible to obtain explicit expressions for either the error function or the output. If all the distributions were Gaussian, spectral densities could be used to obtain a solution of the problem, and this is the basis of Burt's approximation (not previously published), which is given in detail. Unfortunately, when the input of a non-linear component is Gaussian, the output will be non-Gaussian, but in some cases it is possible to make the necessary approximation.The mathematical justification for this approximation and for Booton's approximation is given in the Appendix. Booton's approximation consists in dividing the non-linear characteristic into a linear part and what he calls “the distortion factor.” The slope of the linear part is adjusted to give the best fit on a mean-square-error basis, and the distortion factor is then neglected.A method of obtaining an explicit solution to any required degree of accuracy by approximating to the non-linear characteristic by a number of linear domains is given in the Appendix.Finally, the possible use of topological methods to determine the stability of non-linear systems with random inputs is discussed, and it is shown that for useful control systems the phase-plane diagram of error for no input can almost certainly be used in the design of nonlinear systems, provided that the bounds of input and output and their derivatives can be determined.