$\mathcal{H}_\infty $ Controller Synthesis byJ-Lossless Coprime Factorization

Abstract

This paper develops a coprime factorization approach to the synthesis of internally stabilizing controllers for a given system such that the $\mathcal{H}_\infty $ norm of the closed loop is strictly less than a given bound.By the use of coprime factorizations, it is shown that the $\mathcal{H}_\infty $ control problem is fundamentally related to the so-called analytic systems considered by Helton et al. [Regional Conference Series in Mathematics 68, 1987]. Such problems admit a solution if and only if a certain J-lossless factorization exists. Interpreted in the $\mathcal{H}_\infty $ control context, this means that for a controller of the requisite type to exist the plant must admit a certain J-lossless coprime factorization. The full $\mathcal{H}_\infty $ synthesis problem requires that two nested J-lossless factorizations exist. The results are independent of whether discrete time or continuous time systems are being considered.It is then shown that J-lossless factorization is equivalent to the existen...