Remarks on “Robustness analysis of nonlinear feedback systems: an input-output approach”

In our previous paper (1999) Sec. VII contains two examples of adaptive controllers (Examples 8 and 9) for which the nonlinear gap robustness margin is zero and which are destabilized by arbitrarily small perturbations in the gap metric. The proof that the robustness margin is zero makes use of the fact that the parameter estimate Θ(t) tends to infinity as t→∞ when there is a constant input disturbance of size ε>0. The purpose of this present note is: a) to complete the argument given in the above paper to prove the unboundedness of Θ(t) and b) to give an alternative proof that the convergence of the state x(t)→0 is a consequence of the unboundedness of Θ(t) in Example 9