Stabilization via Nonsmooth, Nonconvex Optimization

James V. Burke, Didier Henrion, Adrian S. Lewis, Michael L. Overton
2006 IEEE Transactions on Automatic Control  
Nonsmooth variational analysis and computational methods are powerful tools that can be effectively applied to find 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
more » ... ned unsolved until 2002, when a solution was found using an 11th order controller. Our computational methods find a stabilizing 3rd 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 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.
doi:10.1109/tac.2006.884944 fatcat:akxutzgwmbcv5edb2o5y3paaau