# Documentation

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# 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`.