Existence Conditions for Regulator Equations

Abstract

The purpose of this Chapter is to show that the existence of solutions for the pair of equations

$$ \begin{array}{*{20}{c}} {\frac{{\partial \pi }}{{\partial w}}s\left( w \right) = f\left( {\pi \left( w \right),c\left( w \right),w} \right)} \\ {0 = h\left( {\pi \left( w \right),w} \right),} \end{array} $$

(3.1)

which, as we have seen in the previous Chapter, determine the existence of solutions of the problem of local output regulation, is intimately related to the properties of the so-calledzero dynamicsof the nonlinear system

$$ \begin{array}{*{20}{c}} {\dot x = f\left( {x,u,w} \right)} \\ {\dot w = s\left( w \right)} \\ {e = h\left( {x,w} \right).} \end{array} $$

(3.2)