A Generalized Passivity Theory Over Abstract Time Domains
Alessio Moreschini, Michelangelo Bin, Alessandro Astolfi, Thomas Parisini
Abstract
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, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varrho$</tex-math></inline-formula> -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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varrho$</tex-math></inline-formula> -passivity is preserved under a class of interconnections, thereby extending the existing passivity results. Moreover, we explore the relationship between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varrho$</tex-math></inline-formula> -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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varrho$</tex-math></inline-formula> -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.