A geometrical formulation to unify construction of Lyapunov functions for interconnected iISS systems

In recent years, the ability to accommodate various nonlinearities has become even more important to support systems design and analysis in a broad area of engineering and science. In this line of research, this paper discusses usefulness of the notion of integral input-to-state stability (iISS) in assessing and establishing system properties through interconnection of component systems. The focus is to construct Lyapunov functions which explain mechanism and provide estimate of stability and robustness of interconnected systems. Unique issues arising in dealing with iISS systems are reviewed in comparison with interconnections of input-to-state stable (ISS) systems. The max-separable Lyapunov function and the sum-separable Lyapunov function which are popular for ISS and iISS, respectively, are revisited. The max-separable function cannot be qualified as a Lyapunov function when component systems are not ISS. Level sets of the max-separable function are rectangles, and the rectangles cannot be expanded to encompass the entire state space in the presence of non-ISS components. The sum-separable function covers iISS components which are not ISS. However, it has practical limitations when stability margins are small. To overcome the limitations, this paper brings in a new idea emerged recently in the literature, and proposes a new type of construction looking at level sets of a Lyapunov function. It is shown how an implicit function allows us to draw chamfered rectangles based on fictitious gain functions of component systems so that they provide reasonable estimates of forward invariant sets producing a Lyapunov function applicable to both iISS and ISS systems equally.