  We present a conic sector theorem for linear parameter varying (LPV) systems
in which the traditional definition of conicity is violated for certain values
of the parameter. We show that such LPV systems can be defined to be conic in
an average sense if the parameter trajectories are restricted so that the
system operates with such values of the parameter sufficiently rarely. We then
show that such an average definition of conicity is useful in analyzing the
stability of the system when it is connected in feedback with a conic system
with appropriate conic properties. This can be regarded as an extension of the
classical conic sector theorem. Based on this modified conic sector theorem, we
design conic controllers that allow the closed-loop system to operate in
nonconic parameter regions for brief periods of time. Due to this extra degree
of freedom, these controllers lead to less conservative performance than
traditional designs, in which the controller parameters are chosen based on the
largest cone that the plant dynamics are contained in. We demonstrate the
effectiveness of the proposed design in stabilizing a power grid with very high
penetration of renewable energy while minimizing power transmission losses.
