مرکز منطقه ای اطلاع رساني علوم و فناوري

Abstract :

The paper attempts to combine the Wiener-Bose method for characterizing and synthesizing nonlinear systems with the Ku-Wolf method for analyzing nonlinear systems with random inputs. A simple partition theory is first presented. It is shown that a general nonlinear system can be partitioned into two portions: one linear portion with memory or storage, and one nonlinear portion which may also include linear elements. The partition method, the Taylor-Cauchy transform method, and the transform-ensemble method are developed, and illustrated by an example. It is shown that the output of a nonlinear system to a random input can be expressed as the summation of $a_{n}q_{n}(t)$ , for $n = 0, 1, 2,$ and so on, where $q_{n}(t)$ depends upon the form of the functional representation of the modified forcing function or the actuating signal, and a. denotes a set of random variables which are related to the statistics of the random input. Wiener\´s theory of nonlinear systems is then reviewed. The Wiener-Bose method is outlined as follows. Let the output of a shot-noise generator be the standard probe for the study of non-linear systems. The standard random input is fed to a Laguerre network giving Laguerre coefficients $u_{1},u_{2},\\cdots .$ The output of the over-all system is then expressed as Hermite function expansions of the Laguerre coefficients. By the ergodic hypothesis it is then possible to express the output as the summation of $A_{\\alpha } V(\\alpha ) e^{-u^{2}/2}$ By taking the time average of $c(t) V(\\alpha )$ , where $c(t)$ represents either the actual output or the desired output, we get the coefficients $A_{\\alpha }$ , which characterize the actual system or the system to be designed. Knowing $A_{\\alpha }$ , the synthesis procedure is obtained from the summation of $A_{\\alpha } V(\\alpha )$ . By combining the output $c(t)$ , obtained from the Ku-Wolf analysis, with the output $V(\\alpha )$ from the Laguerre network and Hermite function generator, we can get the-characterizing coefficients $A_{\\alpha }$ . It is suggested that the correlation of $a_n$ , a set of random variables related to the random input, and $A_{\\alpha }$ , the characterizing coefficients, may sh- ed light on a unified approach for the analysis and synthesis of nonlinear systems with random inputs.