A convex parameterization of internally stabilizing controllers is fundamental for many controller synthesis procedures. The celebrated Youla parameterization relies on a doubly-coprime factorization of the system, while the recent system-level and input-output characterizations require no doubly-coprime factorization but a set of equality constraints for achievable closed-loop responses. In this paper, we present explicit affine mappings among Youla, system-level and input-output parameterizations. Two direct implications of the affine mappings are 1) any convex problem in Youla, system level, or input-output parameters can be equivalently and convexly formulated in any other one of these frameworks, including the convex system-level synthesis (SLS); 2) the condition of quadratic invariance (QI) is sufficient and necessary for the classical distributed control problem to admit an equivalent convex reformulation in terms of Youla, system-level, or input-output parameters.
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