Systemlevel, Inputoutput and New Parameterizations of Stabilizing Controllers, and Their Numerical Computation
Abstract
It is known that the set of internally stabilizing controller $\mathcal{C}_{\text{stab}}$ is nonconvex, but it admits convex characterizations using certain closedloop maps: a classical result is the {Youla parameterization}, and two recent notions are the {systemlevel parameterization} (SLP) and the {inputoutput 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 closedloop 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} doublycoprime 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 openloop stable plants, in the sense that small mismatches in the equality constraints do not compromise the closedloop stability. The SLP is known to enjoy numerical robustness in the state feedback case; here, we show that numerical robustness of the fourblock SLP controller requires casebycase analysis in the general output feedback case.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.12346
 Bibcode:
 2019arXiv190912346Z
 Keywords:

 Mathematics  Optimization and Control;
 Electrical Engineering and Systems Science  Systems and Control
 EPrint:
 20 pages