  This paper reformulates and streamlines the core tools of robust stability
and performance for LTI systems using now-standard methods in convex
optimization. In particular, robustness analysis can be formulated directly as
a primal convex (semidefinite program or SDP) optimization problem using sets
of gramians whose closure is a semidefinite cone. This allows various
constraints such as structured uncertainty to be included directly, and
worst-case disturbances and perturbations constructed directly from the primal
variables. Well known results such as the KYP lemma and various scaled small
gain tests can also be obtained directly through standard SDP duality. To
readers familiar with robustness and SDPs, the framework should appear obvious,
if only in retrospect. But this is also part of its appeal and should enhance
pedagogy, and we hope suggest new research. There is a key lemma proving
closure of a grammian that is also obvious but our current proof appears
unnecessarily cumbersome, and a final aim of this paper is to enlist the help
of experts in robust control and convex optimization in finding simpler
alternatives.
