Theory of nonlinear systems

Abstract

This paper gives a general review of the Theory of Nonlinear Systems. In 1960, the author presented a paper “Theory of Nonlinear Control” at the First IFAC Congress at Moscow. Professor Norbert Wiener, who attended this Congress, drew attention to his work on the synthesis and analysis of nonlinear systems in terms of Hermitian polynomials in the Laguerre coefficients of the past of the input. Wiener's original idea was to use white noise as a probe on any nonlinear system. Applying this input to a Laguerre network gives u 1, u 2,…, u s , and then to a Hermite polynomial generator gives V(α)'s. Applying the same input to the actual nonlinear system gives output c( t). Putting c( t) and V(α)'s through a product averaging device, we get c(t)V(α) = A α 2л s 2 , where the upper bar denotes time average and A α's can be considered as characteristic coefficients of the nonlinear system. A desired output z(itt) may replace c(itt) to get a new set of A α's. The Volterra functional method suggested by Wiener in 1942 has been greatlydeveloped from 1955 to the present. The method involves a multi-dimensional convolution integral with multi- dimensional kernels. The associated multi-dimensional transforms are given by Y.H. Ku and A.A. Wolf ( J. Franklin Inst., Vol. 281, pp. 9–26, 1966). Wiener extended the Volterra functionals by forming an orthogonal set of functionals known as G-functionals, using Gaussian white noise as input. Volterra kernels and Wiener kernels can be correlated and form the characteristic functions of nonlinear systems. From an extension of the linear system to the nonlinear system, the input-output crosscorrelation φ xy can be shown to be equal to the convolution of system impulse response h 1 with the autocorrelation φ xx . Using the white noise as input, where its power density spectrum is a constant, say, A, the crosscorrelation is given by φ xy ( σ) = Ah 1( σ), while the autocorrelation is φ xx ( τ) = Au( τ). This extension forms the basis of an optimum method for nonlinear system identification. Measurement of kernels can be made through proper circuitry. Parallel to the Volterra series and the Wiener series, another series based on Taylor-Cauchy transforms developed since 1959 are given for comparison. The Taylor-Cauchy transform method can be applied in the analysis of simultaneous nonlinear systems. It is noted that the Volterra functional method and the Taylor-Cauchy transform method give identical final results. A selected Bibliography is appended not only to include other aspects of nonlinear system theory but also to show the wide application of nonlinear system characterization and identification to problems in biology, ecology, physiology, cybernetics, control theory, socio- economic systems, etc.