Rejection of persistent bounded disturbances : Nonlinear controllers

This paper considers nonlinear time-varying (NLTV) compensation for linear time-invariant (LTI) plants subject to persistent bounded disturbances. It is shown using two different approaches that using NLTV compensation instead of LTI compensation does not improve the optimal rejection of persistent bounded disturbances. The first approach is to derive a bound on the achievable performance over all stabilizing NLTV controllers. Using results from t' Loptimal control, it follows that in some special cases this bound can be achieved by LTI compensation. This approach involves the introduction of an operator analogous to the Hankel operator in ~-optimal control and is of independent interest. The second approach is to assume the NLTV controller is sufficiently smooth to admit a time-varying linearization. This time-varying linearization is then used to construct an LTI controller which achieves the same performance as the original NLTV controller. These results extend previous work by the authors regarding linear time-varying compensation. Notation LTI .'= linear time-invariant. LTV := linear time-varying. NLTV := nonlinear time-varying.