Discussion on: 'The Non-Uniform in Time Small-Gain Theorem for a Wide Class of Control Systems with Outputs'

摘要

Small-gain statements play a key role in system analysis. These results state that the interconnection of two stable systems is stable provided that the closed-loop gain is less than unity. This intuitively satisfying statement can be made rigorous in a number of ways depending on the notion of stability and the family of systems under consideration. Such results were originally given in the 1960s by Zames, Sandberg and others who addressed stability of input-output systems by the use of linear gains. When notions of stability are generalized by allowing nonlinear gains, an extended small-gain theorem applies, provided by Mareels and Hill. Further extensions of stability concepts were supplied by Sontag. His notions of input-to-state stability (ISS) and input-to-output stability (IOS) incorporate a bound on both transient and asymptotic behaviour. Jiang, et al. provided a small-gain theorem for ISS and IOS systems in [2]. The current paper by Karafyllis extends the results in [2] by providing a small-gain result in a very general setting. A broad definition of control system is given which encompasses continuous and discrete time systems infinite- and infinite-dimensional state spaces. Systems are allowed to be time-varying and gains are not restricted to be uniform in time. The results thus provide a very satisfactory generalization of existing small-gain statements to a much broader class of systems. The small-gain result in [2] was generalized by a different approach in [1] (Ref. in the paper). The remainder of this note will compare and contrast these two approaches.