A functional analysis approach to L∞stability and its applications to systems with hysteresis

The object of this paper is to obtain a bounded-input, bounded-output type of stability criterion for systems with non-linearities that do not satisfy the sector condition. A criterion is obtained that is based on the nature of the operation of the elements of the system on the derivatives of the input and output functions. This criterion has the following form when applied to a feedback system made up of a time-invariant linear element in cascade with a nonlinear element. If the slope of the input-output characteristic of the nonlinear element is bounded, and if the linear element satisfies a circle or Popov condition for these bounds, then the system maps input functions whose derivatives belong to the space
\{x(t)/ \ni \sigma > 0 \ni e^{\sigma t} x(t) \in L_{2}[0, \infty)\}
into output functions of the same class. This class of functions is a subspace of the space
L_{\infty} [0, \infty)
, and as a consequence, a bounded-input, bounded-output type of stability criterion is established. Experiments were performed with feedback systems containing hysteresis-type nonlinear elements, and the feedback limit gain for stability so determined was compared with that predicted by the stability criterion. For different experimental systems and for different assumptions, the theoretically predicted limit gain ranged from about 1/3 to about 9/10 the limit gain observed experimentally.