Causality of general input–output systems and extended small-gain theorem for their feedback connection

Abstract

For the small-gain theorem derived by Zames in 1966, the later studies after a few decades elaborated on its derivation through defining system causality, which was not assumed by Zames. In connection with the treatment of causality, however, these studies made some unnecessary assumptions on the subsystems in feedback connection and failed to handle general systems described by an input–output relation rather than mapping (which we call input-intolerant/-output-unsolitary systems). On the other hand, although the treatment by Zames can handle such subsystems, it instead turns out to lead to larger values for the induced norms of subsystems compared with the later treatment. This paper is concerned with developing an extended form of the small-gain theorem through the same induced norms as in the later studies while dealing with general input–output causal subsystems. Since causality of subsystems plays a key role in such development, our research direction strongly motivates us to study how causality should be defined for general input–output systems. Thus, much of the arguments in this paper is devoted to such a study, which provides us with profound and thorough understandings on causality of different restricted classes of general input–output systems. Mutual relationships among adequate causality definitions for different classes are also clarified, which should be important in its own right. After deriving an extended form of the small-gain theorem, an example illustrates the importance of dealing with such general subsystems, as well as usefulness of the extension.