LMI demos: largest ellipse inside other poliedra/ellipses (1)

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 16:49

Summary:

This video discusses how to obtain using linear matrix inequalities (LMI) the largest ellipse within a polygon (well, it also supports sides that are an ellipse segment).

The presentation is divided into two videos.

In this first video the problem is posed, and we consider how to obtain the matrix $P$ such that $x^{T}Px\le 1$ is within the predetermined intersection set of polygons/ellipses.

The restrictions are coded as LMIs, and two possible cost indices are proposed:

• Minimize maximum eigenvalue of $P$, which is equivalent to maximizing semi-minor axis, that is, finding the circle with the largest radius contained in the search region.

• Minimize the trace of $P$... minimize the sum of the inverse squares of the lengths of the semi-axes... This gives a ‘large’ ellipse as a result of the optimization but its meaning is not entirely clear . Indeed, searching for the ellipse with the largest area will be the objective of the video [lmielin2EN], a continuation of this one.

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