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Stability of Nonlinear Feedback Loops

In general, placing a memoryless nonlinearity $ f(x)$ in a stable feedback loop preserves stability provided the gain of the nonlinearity is less than one, i.e., $ \vert f(x)\vert\le \vert x\vert$. A simple proof for the case of a loop consisting of a continuous-time delay-line and memoryless-nonlinearity is as follows.


The delay line can be interpreted as a waveguide model of an ideal string or acoustic pipe having wave impedance $ R>0$ and a noninverting reflection at its midpoint. A memoryless nonlinearity is a special case of an arbitrary time-varying gain [449]. By hypothesis, this gain has magnitude less than one. By routing the output of the delay line back to its input, the gain plays the role of a reflectance $ g$ at the ``other end'' of the ideal string or acoustic pipe. We can imagine, for example, a terminating dashpot with randomly varying positive resistance $ \mu(t)>0$. The set of all $ \mu>0$ corresponds to the set of real reflection coefficients $ g=(\mu-R)/(\mu+R)$ in the open interval $ (-1,1)$. Thus, each instantaneous nonlinearity-gain $ -1<g<1$ corresponds to some instantaneously positive resistance $ \mu>0$. The whole system is therefore passive, even as $ \mu(t)$ changes arbitrarily (while remaining positive). (It is perhaps easier to ponder a charged capacitor $ C$ terminated on a randomly varying resistor $ R(t)$.) This proof method immediately extends to nonlinear feedback around any transfer function that can be interpreted as the reflectance of a passive physical system, i.e., any transfer function $ H(s)$ for which the gain is bounded by 1 at each frequency, viz., $ \vert H(j\omega)\vert\le 1$. The finite-sampling-rate case can be embedded in a passive infinite-sampling-rate case by replacing each sample with a constant pulse lasting $ T$ seconds (in the delay line). The continuous-time memoryless nonlinearity $ g[x(t)]$ is similarly a held version of the discrete-time case $ g[x(nT)]$. Since the discrete-time case is a simple sampling of the (passive) continuous-time case, we are done.
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