IUPP.sq

Interactive pole placement for a system with ellipsoidal uncertainties

The goal of the SQ file IUPP.sq is to show how the ellipsoidal uncertainties of a linear system are mapped to the plane of closed-loop poles, not to provide an efficient design method. In its current version, the model corresponds to the flexible arm of the Laboratoire d'Automatique de Grenoble (ENSIEG, France) which was proposed as a benchmark in 1995.

The uncertainty described by the three models of the flexible arm is overbounded by the least conservative ellipsoidal uncertainty, which is flat in the space of the coefficients of the model transfer function and in the space of the coefficients of the characteristic polynomial in all but two directions. The locus of the closed-loop poles is displayed. A point can be clicked and moved; it is considered as a pole of the model closest to the nominal model, and moved by pole placement. The whole locus is dragged with it, more or less.

As it is implemented in IUPP.sq, this synthesis method should not be used to design real controllers. The roots alone do not give a very good idea of the performances. Enclosing uncertainties may be overly conservative. The complex plane is sampled and unstable poles may be missed. But it is mostly the design method which may be difficult to use; indeed, moving the closed-loop poles of the model nearest to the nominal model may be difficult, especially where the locus is narrow. In addition to the beauty of the locus, the goal of IUPP.sq is mainly to give ideas for further research in the field of linear systems with structured uncertainties. It demonstrates the command erlocus.