The Character of an Idempotent-analytic Nonlinear Small Gain Theorem

Abstract

In this paper a general nonlinear input-to-state stability small gain theory is described using idempotent analytic techniques. The theorem is proved within the context of the idempotent semiring $\mathcal{K} \subset$ End₀ ^⊕ $(\mathbb{R}_{\geq 0})$, and may be regarded as an application of theoretical computer science techniques to systems and control theory. We show that particular to power law input-to-state gain functions the deduction of the resulting sufficient condition for input-to-state stability may be performed efficiently, using any suitable dynamic programming algorithm. We indicate, through an example, how an analysis of the (weighted, directed) graph of the system complex gives a computable means to delimit (in an easily understood form) robust input-to-state stability bounds.