A Generalized Passivity Theory over Abstract Time Domains

Passivity is a well-established concept for continuous-time systems. Yet, its application to discrete-time, delay, or other classes of systems is somewhat limited, leading to inconsistencies and disparities. In this article, we study a new notion, <tex-math notation="LaTeX"></tex-math> -passivity, that reduces to standard passivity in the continuous-time case but addresses some of the aforementioned limitations when applied to other classes of systems. In particular, in an abstract input-output setting, we show that <tex-math notation="LaTeX"></tex-math> -passivity is preserved under a class of interconnections, thereby extending the existing passivity results. Moreover, we explore the relationship between <tex-math notation="LaTeX"></tex-math> -passivity and stability, and we derive sufficient conditions for high-gain, low-gain, and causal stabilizability by static output feedback. Finally, in contrast to the standard passivity notion, we prove that <tex-math notation="LaTeX"></tex-math> -passivity is preserved under sampling for a class of nonlinear systems and discretization methods. Overall, the results of this article constitute the first step towards a unifying passivity theory embracing all the different domains and systems classes relevant to systems and control theory.