Regulation of a controlled Burgers´ equation: Tracking and disturbance rejection for general time dependent signals

In this paper we describe a significant extension of our earlier work on design of feedback laws for regulation of nonlinear distributed parameter systems. In our previous work we presented a technique that was primarily applicable for set-point control. This method was based on the geometric theory of output regulation. In this paper we show that a generalization, motivated by the same theory, can be derived to design feedback laws for solving regulation problems for very general time dependent reference and disturbance signals. In this work the usual assumptions used in the geometric theory do not strictly apply, but nevertheless, we show that some analogs of the regulator equations from the geometric theory can be derived and used to obtain accurate approximations of the control inputs. We note that this methodology is not a simple extension of the set point tracking problem. In particular the infinite dimensional controller involves the solution of a singular DAE. An important part of this work is the development of an iterative regularization scheme that is used to obtain a sequence of time dependent control laws which provide an approximate solution of the regulation problem. Rather than attempt to address the general abstract theory, we demonstrate the method for a multi-input multi-output regulation problem which involves a tracking/disturbance rejection problem for a nonlinear distributed parameter system governed by a one dimensional viscous Burgers´ equation. This work represents a significant advance over our earlier work, allowing tracking and disturbance rejection for very general time dependent signals.