Spectral characterization of control system nonlinearities

Control system feedback elements are subject to statistical variation from their desired characteristics due to manufacturing tolerances. For most feedback elements, this variation takes the form of a random nonlinear function of the input to the element. Analysis of the effects of such feedback errors is exceedingly difficult because the control system affected must be described by a random nonlinear differential equation. It is proposed that this difficulty be surmounted by computer simulation. For purposes of simulation, a power spectrum is developed for the ensemble of feedback error functions. This power spectrum is then used to synthesize artificial nonlinearities in the simulation of the control system. By simulating groups of components with different statistical properties, production specifications can be generated based on parameters borrowed from noise theory such as power spectral density, noise bandwidth, rms error, etc. This development is followed by two experimental examples which illustrate the application of the method and display the concept of bandwidth as applied to a feedback nonlinearity. It is shown that, in some situations, linearizing approximations permit analytic determination of the effects of random feedback error. In other problems, the utilization of power spectra permits the normalization of computer results.