Robustness of the Tuning Functions Adaptive Backstepping Design for Linear Systems

In this paper we study robustness of the recently developed adaptive backstepping design with tuning functions for linear systems. Under assumptions on unmodeled dynamics and disturbances equal to those for certainty equivalence schemes, we address—for the first time—an adaptive scheme not based on the certainty equivalence principle. In the process of redesign for robustness we employ only leakage in the estimator—we do not employ normalization, neither static nor dynamic. A fundamental difference between the tuning functions design and the certainty equivalence designs is that the controller in the former is inherently nonlinear, while in the latter it is nonlinear only in the parameter estimate. As a result, achievable robustness results for the tuning functions scheme are not global but regional, with a region of attraction inversely proportional to the “size” of the unmodeled dynamics. The tracking error is proportional to the size of the uncertainties.