Overcoming limitations in stability theorems based on multiple Nussbaum functions

Among system uncertainties, unknown control direction is a rather essential one, whose compensation should entail the so-called Nussbaum function. Recently, strategies based on multiple Nussbaum functions (MNFs) have been proposed to make possible continuous adaptive control for the systems with nonidentical unknown control directions, such as large-scale systems and multi-agent systems, which cannot be addressed by a single Nussbaum function. But the existing stability theorems required: MNFs have explicit expressions; MNFs are all odd or all even. This paper aims to overcome the limitations of the theorems. We first delineate MNFs by two aspects of basic properties, replacing their explicit expressions as in the literature. Specifically, MNFs have the bivariate-product form, where one variate delineates frequent changes of signs and persistent intensity of MNFs, and the other delineates the rapid growth of MNFs. Such a delineation can capture the essence of MNFs and cover as many MNFs as possible. Based on the delineated MNFs, we present two stability theorems overcoming the aforementioned limitations. Notably, the two theorems do not require MNFs to have explicit expressions. Specifically, Theorem 1 allows nonmonotonic dynamic variables of MNFs while requiring MNFs to be odd, avoiding too large gains and excessive control cost. Theorem 4 does not require all MNFs to be odd while requiring dynamic variables of MNFs to be monotonic, giving users more freedom in selecting MNFs. The two stability theorems are applied to a large-scale system and a nonlinear system with parameterized uncertainties, both with nonidentical unknown control directions, to avoid monotonicity of dynamic gains and overparameterization, respectively, in control design.