LMI demos: largest ellipse inside other poliedra/ellipses (2), largest area (geomean)

Antonio Sala, UPV

Difficulty: ***** ,       Relevance: ,      Duration: 16:18

Summary:

This video discusses how to obtain, using linear matrix inequalities (LMI), the largest ellipse within a polygon (well, it also supports curved sides that are an ellipse segment). It is a continuation of [lmielin1EN] where the problem was posed and some possibilities were discussed with an ellipse in the form $x^{T}Px\le 1$, $P$ being a decision variable.

In this second video, $Q$ is considered a decision variable that defines the ellipse $x^T{Q}^{-1}x\le 1$. The ’inverse’ form is used for convenience to express the optimization problem as a convex one. Indeed, maximizing the determinant of $Q$ (well, actually its square root, geometric mean of the eigenvalues $\sqrt{{\lambda }_1{\lambda }_2}$) can be done with the geomean operator of Yalmip.

Conveniently, the area of the ellipse is proportional to said geometric mean, so that we can solve the requested problem of maximum area. Furthermore, translating the restrictions to inverse form will require using ’congruence’ and ’Schur’s complement’ so that the video will serve to illustrate the application of these results, which are very useful in control theory developments with LMIs.

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.