IEEE Transactions on Automatic Control, Vol.42, No.7, 980-983, 1997
Robust Tracking, Error Feedback, and 2-Degree-of-Freedom Controllers
In a recent textbook , it is claimed that asymptotic tracking, or the attainment of zero tracking error in the steady state, which is robust with respect to the plant parameters, can be accomplished with a two-degree-of-freedom (2-DOF) compensator without the necessity of error feedback. The purpose of this paper is to establish first, that in the absence of error feedback, asymptotic tracking is a property that is fragile with respect to the controller parameters. This implies that arbitrary small perturbations of the parameters of such a controller will cause asymptotic tracking to fail, Second, we propose a general 2-DOF controller architecture. We prove that in this structure, asymptotic tracking is, in general, fragile with respect to the controller parameters F-r, F-y, and C, while it is robust with respect to the plant parameters as well as the components H and L of the 2-DOF controller, Moreover, the only way to realize a 2-DOF controller in a nonfragile way is to generate the error signal exactly, which amounts to setting F-r = F-y = 1. These results are summarized in Theorem 1, which in essence proves that a maximally robust 2-DOF controller must have the special error structure of Fig. 5; also, it provides robust asymptotic tracking with respect to plant parameters and all controller parameters except M, which must contain an internal model. These results reconcile the requirements of the Internal Model Principle, namely the necessity of error feedback, with the additional design freedom potentially offered by 2-DOF controllers. It specifies the controller architecture necessary for robustness and identifies the blocks whose parameters are available for tuning in the design process.