On the application of Nussbaum function to the control of constrained nonlinear systems

Nussbaum functions have been widely employed in control design for dynamic systems with unknown control direction, and this approach has been extended to handle input constraints for nonlinear systems. While several studies have adopted this formulation to address system constraints in generic dynamic models, we demonstrate that the existing theoretical framework requires a critical, previously unstated assumption: the boundedness of the commanded control signal. This revelation has significant implications for the method's practical applicability. Thus, it is shown that a well-known alternative to this method, i.e., the Modified Tracking Error (MTE), which necessitates the same theoretical assumptions as that of the Nussbaum function method to tackle input constraints, can be a more effective and simpler option to deal with this issue, particularly in the case of uncertain dynamic systems. We extend this analysis to systems with dead-zone nonlinearities, where the MTE method demonstrates superior performance due to its inherent compensation mechanism through the auxiliary state variable. Through extensive comparisons between these control schemes, we illustrate the limitations and capabilities of each method in the case of constrained dynamic systems. The results highlight that despite being more complicated, the Nussbaum function method does not provide advantages over the MTE approach, while the latter enables effective identification of model uncertainties even in the presence of input constraints and dead-zone nonlinearities.