The regulator equation is the fundamental equation whose solution must be found in order to solve the output regulation problem. It is a system of first-order partial differential equations (PDE) combined with an algebraic equation. The classical approach to its solution is to use the Taylor series with undetermined coefficients. In this contribution, another path is followed: the equation is solved using the finite-element method which is, nevertheless, suitable to solve PDE part only. This paper presents two methods to handle the algebraic condition: the first one is based on iterative minimization of a cost functional defined as the integral of the square of the algebraic expression to be equal to zero. The second method converts the algebraic-differential equation into a singularly perturbed system of partial differential equations only. Both methods are compared and the simulation results are presented including on-line control implementation to some practically motivated laboratory models.
This paper presents a novel error-feedback practical solution for real-time implementation of nonlinear output regulation. Sufficient and necessary conditions for both state- and error-feedback output regulation have been established for linear and nonlinear systems several decades ago. In their most general form, these solutions require solving a set of nonlinear partial differential equations, which may be hard or even impossible to solve analytically. In recent years, a methodology for dynamic calculation of the mappings required for state-feedback regulation has been put forward; following the latter, an error-feedback extension is hereby provided which, when combined with design conditions in the form of linear matrix inequalities, becomes suitable for real-time setups. Real-time results are presented for a nonlinear twin rotor MIMO system. Issues concerning the implementation as well as the solutions adopted, are discussed.