On the existence of stable, causal multipliers for systems with slope-restricted nonlinearities

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 paper derives existence conditions for a sub-class of these multipliers, namely those which are L₁ bounded, stable, causal and of order equal to the LTI part of the system. It is proved that for the single-input-single-output (SISO) case, these existence conditions can be expressed as a set of linear matrix inequalities (LMIs) and thus can be solved efficiently with modern optimisation software. Examples illustrate the effectiveness of the results.