Stabilization via Nonsmooth, Nonconvex Optimization

• Published in: IEEE transactions on automatic control Vol. 51; no. 11; pp. 1760 - 1769
• Main Authors: Burke, J.V., Henrion, D., Lewis, A.S., Overton, M.L.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-11-2006; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Closed loop systems; Computation; Computer programs; Computer science; control theory; systems; Constraint optimization; Control design; Control system synthesis; Control systems; Control theory. Systems; Controllers; Design engineering; Design optimization; Eigenvalues and eigenfunctions; Electrical equipment industry; Exact sciences and technology; Fixed-order controller design; Mathematical analysis; nonconvex optimization; nonsmooth optimization; Optimal control; Optimization; Poles; Polynomials; Stability; Stabilization; Studies; Non convex programming; polynomials; Stabilization; H infinite control; Control synthesis; Fixed-order controller design; nonconvex optimization; nonsmooth optimization; Optimization; Stability radius; Closed feedback; Nonsmooth analysis; Optimal control; Multiplicity; Variational calculus; Minimum phase; stability
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2006.884944

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 infin performance