Stability of Nonlinear Feedback Loops

In general, placing a memoryless nonlinearity
in a stable
feedback loop preserves stability provided the gain of the
nonlinearity is less than one, *i.e.*,
. 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
and a noninverting
reflection at its midpoint. A memoryless nonlinearity is a special
case of an arbitrary time-varying gain [#!JOSFP!#]. 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
at the ``other end'' of the ideal string or acoustic pipe. We can
imagine, for example, a terminating dashpot with randomly varying
positive resistance
. The set of all
corresponds to
the set of real reflection coefficients
in the
open interval
. Thus, each instantaneous nonlinearity-gain
corresponds to some instantaneously positive resistance
. The whole system is therefore passive, even as
changes arbitrarily (while remaining positive). (It is perhaps easier
to ponder a charged capacitor
terminated on a randomly varying
resistor
.) 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
for which the gain is bounded by 1 at each frequency, *viz.*,
.

The finite-sampling-rate case can be embedded in a passive infinite-sampling-rate case by replacing each sample with a constant pulse lasting seconds (in the delay line). The continuous-time memoryless nonlinearity is similarly a held version of the discrete-time case . Since the discrete-time case is a simple sampling of the (passive) continuous-time case, we are done.

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