Numerical method for the solution of the regulator equation with application to nonlinear tracking

A numerical method to solve the so-called regulator equation is presented here. This equation consists of partial differential equations combined with algebraic ones and arises when solving the output-regulation problem. Solving the regulator equation is becoming difficult especially for the nonminimum phase systems where reducing variables against algebraic part leads to a potentially unsolvable differential part. The proposed numerical method is based on the successive approximation of the differential part of the regulator equation by the finite-element method while trying to minimize a functional expressing the error of its algebraical part. The method is analyzed to obtain theoretical estimates of its convergence and it is tested on an example of the “two-carts with an inverted pendulum” system. Simulations are included to illustrate the suggested approach.