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A Case of $\mu$-Synthesis as a Quadratic Semidefinite Program

Lookup NU author(s): Professor Jim Agler, Dr Zinaida LykovaORCiD, Professor Nicholas Young

Abstract

We analyse a special case of the robust stabilization problem under structured uncertainty. We obtain a new criterion for the solvability of the spectral Nevanlinna-Pick problem, which is a special case of the $\mu$-synthesis problem of $H^\infty$ control in which $\mu$ is the spectral radius. Given $n$ distinct points $\la_1,\dots,\la_n$ in the unit disc and $2\times 2$ nonscalar complex matrices $W_1,\dots,W_n$, the problem is to determine whether there is an analytic $2\times 2$ matrix function $F$ on the disc such that $F(\la_j)=W_j$ for each $j$ and the supremum of the spectral radius of $F(\la)$ is less than 1 for $\la$ in the disc. The condition is that the minimum of a quadratic function of pairs of positive $3n$-square matrices subject to certain linear matrix inequalities in the data be attained and be zero.

Author(s): Agler J, Lykova ZA, Young NJ

Publication type: Article

Publication status: Published

Journal: SIAM Journal on Control and Optimization

Year: 2013

Volume: 51

Issue: 3

Pages: 2472-2508

Date deposited: 21/03/2013

ISSN (print): 0363-0129

ISSN (electronic): 1095-7138

Publisher: SIAM Publications

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Funding

Funder referenceFunder name
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
EP/K50340X/1UK Engineering and Physical Sciences Research Council
DMS 1068830National Science Foundation grant on Extending Hilbert Space Operators
EP/J004545/1UK Engineering and Physical Sciences Research Council