Stabilization in Spite of Matched Unmodelled Dynamics and an Equivalent Deenition of Input-to-state Stability

We consider nonlinear systems with input-to-output stable (IOS) unmodelled dynamics which are in the \range" of the input. Assuming the nominal system is globally asymptotically stabilizable and a nonlinear small gain condition is satissed, we propose a rst control law such that all solutions of the perturbed system are bounded and the state of the nominal system is captured by an arbitrarily small neighborhood of the origin. The design of this controller is based on a gain assignment result which allows us to prove our statement via a Small-Gain Theorem JTP, Theorem 2.1]. However, this control law exhibits a high gain feature for all values. Since this may be undesirable, in a second stage we propose another controller with diierent characteristics in this respect. This controller requires more a priori knowledge on the unmodelled dynamics, as it is dynamic and incorporates a signal bounding the unmodelled eeects. However this is only possible by restraining the IOS property into the exp-IOS property. Nevertheless we show that, in the case of input-to-state stability (ISS) | the output is the state itself |, ISS and exp-ISS are in fact equivalent properties.