Abstract :
The paper considers the behaviour of a simple saturating control system subjected to a random input signal, evaluating the variation of error with forward loop gain. Booton´s linearization technique is the foundation of the analysis, and it is shown that this method predicts that the error will decrease monotonically to a constant value as the forward gain is increased indefinitely. However, this approximate method neglects the additional error introduced by the distorting effect of the non-linear characteristic. As the forward gain is increased so that the non-linear operation becomes more pronounced, distortion, and the error due to this distortion, increase. Therefore, to determine optimum operating conditions it is necessary to show how the total error, including that due to distortion, varies with forward loop gain. For the particular system considered, it is found that, as the forward loop gain is increased, the reduction in error of the linear system always exceeds the increase in error owing to greater distortion. As a result, the mean-square value of the error is a minimum for infinite forward loop gain. Distortion has an important influence on the error spectrum at low frequencies, and fundamentally limits performance at frequencies much less than the natural frequency of the system. Zero error at these very low frequencies is possible only when the forward loop gain is so small that saturation has a negligible effect. Thus the optimum forward loop gain depends on the particular application of the control system. To minimize the mean-square magnitude of the error signal, the forward loop gain should be as high as possible. When the criterion of performance is based on the error at very low frequencies, then a lower forward loop gain is preferable. Experimental results are presented for the particular system analysed.
