Free-running nonlinear servo systems: stability study

Consider the loop system shown in figure 21 , where N (X ) is the equivalent complex gain of a separable nonlinearity and L (p ) is the transfer function of a low-pass filter.

Let's study stability in free (or autonomous) operation, i.e. with zero input.

Let x be the non-linearity input.

The necessary condition for self-oscillation in this loop is cancellation of the denominator of the closed-loop transfer function.

The characteristic equation : $1+N\left(X\right)L\left(\text{jω}\right)=0$