Stability analysis of control systems having two non-linear elements with calculations for saturation and backlash

Control systems are considered in which two non-linear relationships are present. No difficulty arises if these are adjacent in the control sequence in the sense that the two may be considered as a single nonlinear element. The more difficult case is when the non-linear relationships are separated, so that the input of the second depends upon the output of the first in a manner which, although linear, involves time. The paper shows how this case may be analysed, and presents a method of determining values of the various system parameters that will inhibit sustained oscillations. To illustrate the procedure, the case is considered of systems in which there are elements having, respectively, torque limitation and backlash, other elements being linear. Calculations are given for two variants of a second-order control system. In the first case stabilization is by feedback from a point in the control sequence after the second non-linear element. In the second, the system is stabilized by feeding back a signal from a point between the two elements. The results are compared with measurements taken from an analogue computer, and are shown to be generally in good agreement, but the presence of sudden changes in amplitude and frequency at particular values of relevant parameter ratios (`jump¿ phenomena) are not revealed by the calculations using describing functions.