Nussbaum functions in adaptive control with time-varying unknown control coefficients

Nussbaum functions have been successfully used in adaptive controller design for dealing with unknown control direction since the original work in 1983. However, for time-varying control coefficients of an unknown sign (positive or negative), only a special Nussbaum function can be proved to be effective based on the explicit calculation on the particular function. It remains open whether a general Nussbaum function is sufficient in these scenarios and why. This paper gives a No answer with a counter example. Moreover, it introduces new types of Nussbaum functions and reveals their fundamental characteristics that are sufficient for dealing with time-varying unknown control coefficients in adaptive control. A multivariable version is also introduced.