Relationships Among Passivity, Positive Realness, and Dissipativity with an Application to Passivity Based Pairing

The notions of passivity and positive realness are fundamental concepts in classical control theory, but the use of the terms has varied. For LTI systems, these two concepts capture the same essential property of dynamical systems, that is, a system with this property does not generate its own energy but only stores and dissipates energy supplied by the environment. This paper summarizes the connection between these two concepts for continuous and discrete time LTI systems. Beyond that, the paper summarizes relationships between classes of strictly passive systems and classes of positive real systems. The more general framework of dissipativity is introduced to connect passivity and positive realness. An application is included to demonstrate how these results can be applied to input-output pairing in MIMO control systems. Two case studies are provided to demonstrate the performance of the proposed methods.