Generalized Controlled Invariance for Nonlinear Systems

Abstract

Abstract. A general setting is developed which describes controlled invariance for nonlinear control systems and which incorporates the previous approaches dealing with controlled invariant (co -) distributions. A special class of controlled invariant subspaces, called controllability cospaces, is introduced. These geometric notions are shown to be useful for deriving a (geometric) solution to the dynamic disturbance decoupling problem and for characterizing the so-called fixed dynamics for noninteracting control. These fixed dynamics are a central issue in studying noninteracting control with stability. The class of quasi-static state feedbacks is used.

Disturbance Decoupling for Nonlinear Time‐delay Systems

“…Though a full geometric theory is not available for nonlinear time-delay systems, basic concepts can be introduced that generalize results valid for linear or nonlinear systems (without delays) [8]. They are shown in Section 3 to be instrumental for solving the DDP for the class of nonlinear time-delay systems.…”

Solvability Conditions of Disturbance Rejection by Measurement Feedback for Mimo Nolinear Systems

“…The controlled invariant subspace of X * was defined by (Huijberts and Andiarti, 1997) with nonexact one-forms and it is a dual-notion of the controlled invariant distribution in the differential geometry approach ( (Isidori, 11995).…”

“…In the remainder of this paper, it is assumed that the system (1) is observable from the measurement of y and its DDP is solvable. Furthermore, we introduce a controlled invariant subspace Ω * of X * , which is equivalent to that of 4.3 in (Huijberts and Andiarti, 1997) or (Xia and Moog, 1999) and constructed by applying an Ω * -Algorithm, such that…”

Generalized controlled invariance for discrete-time nonlinear systems with an application to the dynamic disturbance decoupling problem

“…Despite the remarkable success of the geometric approach in tackling synthesis problems involving static-state feedback, the same cannot be said for synthesis problems involving dynamic feedback. Recently, for continuous-time systems a generalized notion of controlled invariance has been introduced under the enlarged class of quasi-static-state feedback transformations [11], [12], and is shown to be useful to derive a geometric solution to the DDDP. The proposed geometric solution to the DDDP is completely parallel to the solution of the static disturbance decoupling problem: the only difference in the solvability conditions is that the classical controlled invariant codistribution is replaced by the generalized controlled invariant subspace.…”