Stabilizability of Second Order Bilinear Systems

This note states necessary and sufficient conditions for the existence of a linear state feedback controller such that a second-order bilinear system has a globally asymptotically stable closed loop. A suitable controller is constructed for each system which satisfies the conditions. Comments Copyright 1983 IEEE. Reprinted from IEEE Transactions on Automatic Control, Volume AC-28, Issue 10, October 1983, pages 987-989. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. NOTE: At the time of publication, the author, Daniel Koditschek, was affiliated with Yale University. Currently, he is a faculty member of the School of Engineering at the University of Pennsylvania. This journal article is available at ScholarlyCommons: http://repository.upenn.edu/ese_papers/343 TABLE I11 NUMERICAL RESULT^ CONTAINERSHIP AT 32 KNOTS Solutions of t he ME [ 1 0.1768 x 10-1 -0.1282 X 1 n2 fl.1775 x 10-1 P+ = -0.1281 X l o 2 0.1236 X lo4 0.2285 x l o 2 0.1775 x 10-1 0.2285 x l o2 0.4257 x 101 -0.2828 x 103 -0.5973 x 100 -0.6442 x lo5 -0.8872 x l o 2 -0.8872 x 10’ -0.4497 x 101 3 S t a t e Feedback Gains (Stabi l iz ing Control) Eigenvalues o f t he Closed Loop System, (A’) -0.7115 x 10-1 0.1292 x 102 o.1oao X loo -0.16777 x 10’ -0.10081 x 10’ -0.23708 x Eigenvalues o f P+ PMatrix: 0.37515 x 10’ 0.65655 x l o5 0.8567 x lo1 250000 DWT TANKER AT 15 KNOTS (FULL LOAO) Solutions of t he WE 0.7520 x 10’ -0.3245 x lo3 -0.9354 x [ -0.9354 x 10-2 0.61172 x lo3 0.4469 x l o2 1 [ P’ = -0.3245 x l o3 0.5441 x lo5 0.6872 x lo3 -0.1942 x 102 -0.5705 x-104 -0.1215 x l o2 ] -0.1215 x l o2 -0.2551 x lo4 -a.vol x 102 J P= -0.5705 x lo4 -0.1997 x lo7 -0.2251 X lo4 S t a t e Feedback Gains (Stabi l iz ing Control) : Eigenvalues o f t he Closed Loop System. (A+): -0.2019 x 100 0.4782 x 10‘ 0.3162 x 10’ -0.62846 x 10-1 -0.27199 x 10-1 -0.23213 x IO-‘ Eigenvalues o f P+ PMatrix: 0 .5908~ X la1 o. 21151 3 X 1 o7 0.91736 x 102 The quadratic criterion of (7) is physically well motivated, with the weighting coefficient X being completely defined a priori from the dynamics of the problem. It is possible that many other optimization problems can be successfully posed in this framework and solved in a more truly optimal manner rather than by the classical LR (Q 2 01 formulation where the quadratic weighting coefficients (generally) are iteratively specified by trial-and-error by the system designer on the basis of experimental studies.