Passivity and stability of systems with a state representation†

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Abstract
The concept of passivity is developed for systems with a state representation. This development of passivity is related to network passivity and passivity of operators as used by Zames, Falb, Wu and others. This formulation of passivity naturally leads to general stability theorems for feedback systems. An L 2-input L 2-output stability theorem similar to those arising out of passive (positive) operator theory is derived. Also a theorem for Liapunov stability similar to one of Baker and Bergen is obtained. In addition, this formulation leads to bounded-input bounded-state stability theorems. In the second part of the paper, more specific feedback systems are examined, including a system with a general linear time-varying differential type system in the forward loop and a non-linear amplifier in the return loop. Conditions for the stability of these systems are found.