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 [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 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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.