Dissipativity, stability, and connections: Progress in omplexity

Energy-related concepts, such as passivity, dissipation, and energy storage, are appealing and widely appreciated in the study of physical systems. The theory of dissipativity (see 'Summary') enables such properties in abstract system descriptions and can lead to significant progress in the study of dynamic properties and stabilization. The key early result is the celebrated Kalman-Yakubovich-Popov (KYP) lemma or positive real lemma (PRL). This article includes a brief historical review (see 'Some History') that emphasizes a key feature of this result, namely, that it connects input-output (IO) and state-space (SS) system descriptions via the concept of passivity. The later development of nonlinear system versions and the subject of dissipative systems were initially motivated with reference to IO models but were actually expressed in a purely SS framework. A general theory by Hill and Moylan will be reviewed to show how it restored the basic IO-SS connection and includes a precise generalization of the PRL for abstract systems.