It is known that the set of internally stabilizing controller
Cstab​ is non-convex, but it admits convex
characterizations using certain closed-loop maps: a classical result is the
{Youla parameterization}, and two recent notions are the {system-level
parameterization} (SLP) and the {input-output parameterization} (IOP). In this
paper, we address the existence of new convex parameterizations and discuss
potential tradeoffs of each parametrization in different scenarios. Our main
contributions are: 1) We first reveal that only four groups of stable
closed-loop transfer matrices are equivalent to internal stability: one of them
is used in the SLP, another one is used in the IOP, and the other two are new,
leading to two new convex parameterizations of Cstab​. 2)
We then investigate the properties of these parameterizations after imposing
the finite impulse response (FIR) approximation, revealing that the IOP has the
best ability of approximating Cstab​ given FIR
constraints. 3) These four parameterizations require no \emph{a priori}
doubly-coprime factorization of the plant, but impose a set of equality
constraints. However, these equality constraints will never be satisfied
exactly in numerical computation. We prove that the IOP is numerically robust
for open-loop stable plants, in the sense that small mismatches in the equality
constraints do not compromise the closed-loop stability. The SLP is known to
enjoy numerical robustness in the state feedback case; here, we show that
numerical robustness of the four-block SLP controller requires case-by-case
analysis in the general output feedback case.Comment: 20 pages; 5 figures. Added extensions on numerial computation and
robustness of closed-loop parameterization