A KYP lemma and invariance principle for systems with multiple hysteresis non-linearities

Absolute stability criteria for systems with multiple hysteresis non-linear ities are given in this paper. It is shown that the stability guarantee is achieved with a simple two part test on the linear subsystem. If the linear subsystem satisfies a particular linear matrix inequality and a simple res idue condition, then, as is proven, the non-linear system will be asymptoti cally stable. The main stability theorem is developed using a combination o f passivity, Lyapunov and Popov stability theories to show that the state d escribing the linear system dynamics must converge to an equilibrium positi on of the non-linear closed loop system. The invariant sets that contain al l such possible equilibrium points are described in detail for several comm on types of hystereses. The class of non-linearities covered by the analysi s is very general and includes multiple slope-restricted memoryless non-lin earities as a special case. Simple numerical examples are used to demonstra te the effectiveness of the new analysis in comparison to other recent resu lts, and graphically illustrate state asymptotic stability.