FEM-BASED SOLUTION OF THE REGULATOR EQUATION ARISING FROM THE NONLINEAR OUTPUT-REGULATION PROBLEM

REHAK, B.; ČELIKOVSKÝ, S.

Abstract

The nonlinear output regulation problem consists in finding a feedback compensator guaranteeing asymptotical tracking of a given reference with simultaneous rejection of the unknown and undesired disturbance. The specifics of output regulation are that both references and disturbances are generated by an autonoumous system (the exosystem). The problem is equivalent to driving asymptotically to zero an output of a certain extended system, what justifies adopted terminology. This topic has been studied both for linear and nonlinear systems. Many variations of the output regulation problem have been addressed. The the resulting control consists of the sum of this feedforward and a stabilizing feedback. The crucial point in is the solution of so-called regulator equation. It consists of a system of partial differential equations of first order combined with an algebraic condition. Another nonstandard point is the fact that this equation is solved on the whole space Rn (n is the order of the exosystem). The regulator equation used to be solved by decomposition of the solution as well as the other involved functions into the Taylor series and then comparing the coefficients. Here, the regulator equation is solved directly via the finite-element. In order to achieve solvability of the regulator equation the system was stabilized first. Then the regulator equation was built for this stabilized system. Its solution was divided into two parts that were repeated iteratively. First the feedforward control was chosen and the part of the regulator equation consisting of the partial differential equations was solved. After that the error (expressed as the error in the algebraic condition measured in the L2 norm) was evaluated. Finally the change of the proposed feedforward was carried out so that this error decreases. The results are demonstrated on an example (inverted pendulum with an added chart). It was described earlier and has already been used for evaluating of the output regulation results.

Coresponding author e-mail: rehakb[at]control[dot]felk[dot]cvut[dot]cz

Session: Linear and Non-linear Control System Design