Geometric characterization on the solvability of regulator equations

Abstract

The solvability of the regulator equation for a general nonlinear system is discussed in this paper by using geometric method. The 'feedback' part of the regulator equation, that is, the feasible controllers for the regulator equation, is studied thoroughly. The concepts of minimal output zeroing control invariant submanifold and left invertibility are introduced to find all the possible controllers for the regulator equation under the condition of left invertibility. Useful results, such as a necessary condition for the output regulation problem and some properties of friend sets of controlled invariant manifolds, are also obtained. ᭧

Geometric Steady States of Nonlinear Systems

“…Remark 2: The hypothesis A1'') guarantees the controlled invariance of some manifold. By the analytic method in [5] or the geometric approach for the solvability of regulator equation and the parametrization of friend sets in [10], one can simplify the computation of the feasible controllers (i.e., friends) which ensure the controlled invariance. Note that such feasible controllers may not be unique, therefore the results in [10] will be helpful to find all the possible friends.…”

“…By the analytic method in [5] or the geometric approach for the solvability of regulator equation and the parametrization of friend sets in [10], one can simplify the computation of the feasible controllers (i.e., friends) which ensure the controlled invariance. Note that such feasible controllers may not be unique, therefore the results in [10] will be helpful to find all the possible friends. Now one can try to find, among all the possible friends, a controller u which satisfies furthermore hypotheses A2'') and A3'').…”

“…); ! ), which can be simplified by our recent approach for the solvability of regulator equation and the parametrization of friend sets (see [10]). Therefore the main theorems of this technical note focusses on the conditions that ensure the attractiveness.…”