On the Existence of Stable, Causal Multipliers for Systems With Slope-Restricted Nonlinearities

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The stability of a feedback interconnection of a linear time invariant (LTI) system and a slope-restricted nonlinearity is revisited. Unlike the normal treatment of this problem, in which multipliers are explicitly chosen and then stability conditions checked, this technical note derives <emphasis emphasistype="italic">existence</emphasis> conditions for a sub-class of these multipliers, namely those which are <formula formulatype="inline"><tex Notation="TeX">L₁</tex></formula> bounded, stable, causal and of order equal to the LTI part of the system. It is proved that for the single-input-single-output case, these existence conditions can be expressed as a set of linear matrix inequalities and thus can be solved efficiently with modern optimization software. Examples illustrate the effectiveness of the results. </para>