Stabilization via nonsmooth, nonconvex optimization

James V. Burke, Didier Henrion, Adrian S. Lewis, Michael L. Overton

Research output: Contribution to journalArticle

Abstract

Nonsmooth variational analysis and related computational methods are powerful tools that can be effectively applied to identify local minimizers of nonconvex optimization problems arising in fixed-order controller design. We support this claim by applying nonsmooth analysis and methods to a challenging "Belgian chocolate" stabilization problem posed in 1994: find a stable, minimum phase, rational controller that stabilizes a specified second-order plant. Although easily stated, this particular problem remained unsolved until 2002, when a solution was found using an eleventh-order controller. Our computational methods find a stabilizing third-order controller without difficulty, suggesting explicit formulas for the controller and for the closed loop system, which has only one pole with multiplicity 5. Furthermore, our analytical techniques prove that this controller is locally optimal in the sense that there is no nearby controller with the same order for which the closed loop system has all its poles further left in the complex plane. Although the focus of the paper is stabilization, once a stabilizing controller is obtained, the same computational techniques can be used to optimize various measures of the closed loop system, including its complex stability radius or H performance.

Original languageEnglish (US)
Pages (from-to)1760-1769
Number of pages10
JournalIEEE Transactions on Automatic Control
Volume51
Issue number11
DOIs
StatePublished - Nov 2006

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Stabilization
Controllers
Closed loop systems
Computational methods
Poles

Keywords

  • Fixed-order controller design
  • Nonconvex optimization
  • Nonsmooth optimization
  • Polynomials
  • Stability

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Electrical and Electronic Engineering

Cite this

Stabilization via nonsmooth, nonconvex optimization. / Burke, James V.; Henrion, Didier; Lewis, Adrian S.; Overton, Michael L.

In: IEEE Transactions on Automatic Control, Vol. 51, No. 11, 11.2006, p. 1760-1769.

Research output: Contribution to journalArticle

Burke, James V. ; Henrion, Didier ; Lewis, Adrian S. ; Overton, Michael L. / Stabilization via nonsmooth, nonconvex optimization. In: IEEE Transactions on Automatic Control. 2006 ; Vol. 51, No. 11. pp. 1760-1769.
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