# lmireg

Specify LMI regions for pole placement

## Syntax

region = lmireg
region = lmireg(reg1,reg2,...)

## Description

lmireg is an interactive facility to specify the LMI regions involved in multi-objective H synthesis with pole placement constraints (see msfsyn and h2hinfsyn). An LMI region is any convex subset D of the complex plane that can be characterized by an LMI in z and , i.e.,

$D=\left\{z\in C:L+Mz+{M}^{T}\overline{z}<0\right\}$

for some fixed real matrices M and L = LT. This class of regions encompasses half planes, strips, conic sectors, disks, ellipses, and any intersection of the above.

Calling lmireg without argument starts an interactive query/answer session where you can specify the region of your choice. The matrix region = [L, M] is returned upon termination. This matrix description of the LMI region can be passed directly to msfsyn for synthesis purposes.

The function lmireg can also be used to intersect previously defined LMI regions reg1, reg2,.... The output region is then the [L, M] description of the intersection of these regions.

## Examples

collapse all

For LMI controller synthesis with functions like msfsyn and h2hinfsyn, you can restrict the eigenvalues of the closed-loop system to an LMI region. The region is specified as a matrix of the form [L M]. In this example, use lmireg interactively to generate a matrix you can use to restrict the poles of the closed-loop system to Re(z) < –1.

Start the interactive process.

region = lmireg
Select a region among the following:

h)   Half-plane
d)   Disk
c)   Conic sector
e)   Ellipsoid
p)   Parabola
s)   Horizontal strip
m)   Matrix description of the LMI region
q)   Quit
choice:

The software prompts you to select the geometry of the region. For this example, enter h to specify a half-plane region. The software now prompts you to specify left half-plane (Re(z) less than some value) or right half-plane (Re(z) greater than some value).

Orientation (x < x0 -> l , x > x0 -> r):

Enter l to specify a left half-plane. The software prompts you to specify a value for x0.

Specify x0:

Enter -1. You have now completely specified the restriction Re(z) < –1. If you want to specify additional regional constraints on the pole locations, you can select another geometry now and follow the prompts. For this example, enter q to generate the LMI region matrix corresponding to Re(z) < –1.

region =

2.0000 + 1.0000i   1.0000 + 0.0000i

You can now use region with msfsyn or h2hinfsyn.

## Version History

Introduced before R2006a