Parameterization of decoupling control laws for affine nonlinear systems

The problem of parameterizing all the decoupling feedbacks is investigated and applied to the design of a stable noninteracting control for affine nonlinear systems. An algebraic study of controlled invariance is presented. It covers a new criterion for controlled invariance, parameterization and novel properties of the friend set, and computational aspects of the maximal controlled invariant distribution. This analysis is used to derive Falb-Wolovich-like conditions for general decoupling (including disturbance decoupling and noninteracting control) problems. It is shown that to determine all the decoupling feedbacks one usually needs infinitely many (sets of) controlled invariant or controllability distributions that can arise as solutions to the problems. It is also shown that under certain conditions, stable noninteracting design based on first parameterizing the decoupling feedbacks is still a feasible approach in the sense that a feedback can be chosen from the friend set of the maximal solution for the exponentially stable and noninteracting design problem