TY - JOUR
T1 - Stabilization via nonsmooth, nonconvex optimization
AU - Burke, James V.
AU - Henrion, Didier
AU - Lewis, Adrian S.
AU - Overton, Michael L.
N1 - Funding Information:
Manuscript received February 28, 2005; revised November 22, 2005 and March 25, 2006. Recommended by Associate Editor A. Astolfi. The work of J. V. Burke was supported in part by National Science Foundation under Grant DMS-0505712. The work of D. Henrion was supported in part by Project 102/05/0011 of the Grant Agency of the Czech Republic and by Project ME 698/2003 of the Ministry of Education of the Czech Republic. The work of A. S. Lewis was supported in part by National Science Foundation under Grant DMS-0504032. The work of M. L. Overton was supported in part by National Science Foundation under Grant DMS-0412049, and in part by Université Paul Sabatier, Toulouse, France.
PY - 2006/11
Y1 - 2006/11
N2 - 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.
AB - 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.
KW - Fixed-order controller design
KW - Nonconvex optimization
KW - Nonsmooth optimization
KW - Polynomials
KW - Stability
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U2 - 10.1109/TAC.2006.884944
DO - 10.1109/TAC.2006.884944
M3 - Article
AN - SCOPUS:36348989500
SN - 0018-9286
VL - 51
SP - 1760
EP - 1769
JO - IEEE Transactions on Automatic Control
JF - IEEE Transactions on Automatic Control
IS - 11
ER -