On the Use of Stable Distributions in Design Problems

The role of stable distributions in nonlinear design problems is pointed out. Stable distributions are controlled invariant distributions for which the drift dynamics restricted to the integral manifold through an equilibrium is exponentially stable. These distributions can be calculated by prolongating a certain stable manifold through the equilibrium. The nonlinear analogue Δs*of Vs*, the maximal stabilizable controlled invariant subspace in the kernel of the output mapping, is such a stable distribution. It is proved that (under some extra conditions) the Disturbance Decoupling Problem with Stability is (locally) solvable if and only if the disturbance vectorfields are contained in Δs*. An important role in the Noninteracting Control Problem with Stability is played by the radical. It is shown that generically the radical is a stable distribution.