On the Equivalence of Youla, System-level and Input-output Parameterizations

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.Comment: 8 pages, 3 figure

Robust Stability of Iterative Learning Control Schemes

A notion of robust stability is developed for iterative learning control in the context of disturbance attenuation. The size of the unmodelled dynamics is captured via a gap distance, which in turn is related to the standard H2 gap metric, and the resulting robustness certificate is qualitatively equivalent to that obtained in classical robust H∞ theory. A bound on the robust stability margin for a specific adaptive ILC design is established

System-level, Input-output and New Parameterizations of Stabilizing Controllers, and Their Numerical Computation

It is known that the set of internally stabilizing controller $\mathcal{C}_{\text{stab}}$ 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 $\mathcal{C}_{\text{stab}}$. 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 $\mathcal{C}_{\text{stab}}$ 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