Output Regulation of Nonlinear Systems

As a preliminary step in our approach to the solution of the problem of local output regulation, for a nonlinear plant modeled by equations of the form 2.1 $$ \begin{array}{*{20}{c}} {\dot x = f\left( {x,u,w} \right)} \\ {e = h\left( {x,w} \right),} \end{array} $$ we establish a set of elementarynecessaryconditions. First of all, we look at the necessary conditions which derive form the existence of a controller fulfilling the requirement of local internal stability in the first approximation. To this end, letA, B, C, P, Q, S, F, G, Hbe matrices defined as follows 2.2 $$ \begin{array}{*{20}{c}} {A = \left[ {\frac{{\partial f}}{{\partial x}}} \right]}{B = {{\left[ {\frac{{\partial f}}{{\partial u}}} \right]}_{\left( {0,0,0} \right)}}}{C = {{\left[ {\frac{{\partial h}}{{\partial x}}} \right]}_{\left( {0,0} \right)}}} \\ {P = {{\left[ {\frac{{\partial f}}{{\partial w}}} \right]}_{\left( {0,0,0} \right)}}}{Q = {{\left[ {\frac{{\partial h}}{{\partial w}}} \right]}_{\left( {0,0} \right)}}}{S = {{\left[ {\frac{{\partial s}}{{\partial w}}} \right]}_{\left( 0 \right)}}} \\ {F = {{\left[ {\frac{{\partial \eta }}{{\partial \xi }}} \right]}_{\left( {0,0} \right)}}}{G = {{\left[ {\frac{{\partial h}}{{\partial e}}} \right]}_{\left( {0,0} \right)}}}{H = {{\left[ {\frac{{\partial \theta }}{{\partial \xi }}} \right]}_{\left( 0 \right)}}.} \end{array} $$