## Modify a System of LMIs

Once specified, a system of LMIs can be modified in several ways with the functions
`dellmi`

, `delmvar`

, and `setmvar`

.

### Deleting an LMI

The first possibility is to remove an entire LMI from the system with `dellmi`

. For instance, suppose that
the LMI system of Specify LMI System is described in
`LMISYS`

and that we want to remove the positivity constraint
on *X*. This is done by

NEWSYS = dellmi(LMISYS,2)

where the second argument specifies deletion of the second LMI. The resulting
system of two LMIs is returned in `NEWSYS`

.

The LMI identifiers (*initial* ranking of the LMI in the LMI
system) are not altered by deletions. As a result, the last LMI

*S* > *I*

remains known as the third LMI even though it now ranks second in the modified
system. To avoid confusion, it is safer to refer to LMIs via the identifiers
returned by `newlmi`

. If `BRL`

,
`Xpos`

, and `Slmi`

are the identifiers
attached to the three LMIs described in Specify LMI System, `Slmi`

keeps pointing to
*S* > *I* even after deleting the
second LMI by

NEWSYS = dellmi(LMISYS,Xpos)

### Deleting a Matrix Variable

Another way of modifying an LMI system is to delete a matrix variable, that is, to
remove all variable terms involving this matrix variable. This operation is
performed by `delmvar`

. For instance, consider
the LMI

*A*^{T}*X +
XA + BW +
W*^{T}*B** ^{T}*
+

*I*< 0

with variables *X* =
*X** ^{T}* ∊

*R*

^{4}×

^{4}and

*W*∊

*R*

^{2}×

^{4}. This LMI is defined by

setlmis([]) X = lmivar(1,[4 1]) % X W = lmivar(2,[2 4]) % W lmiterm([1 1 1 X],1,A,'s') lmiterm([1 1 1 W],B,1,'s') lmiterm([1 1 1 0],1) LMISYS = getlmis

To delete the variable `W`

, type the command

NEWSYS = delmvar(LMISYS,W)

The resulting `NEWSYS`

now describes the Lyapunov
inequality

*A*^{T}*X*
+ *XA* + *I* < 0

Note that `delmvar`

automatically removes all
LMIs that depended only on the deleted matrix variable.

The matrix variable identifiers are not affected by deletions and continue to
point to the same matrix variable. For subsequent manipulations, it is therefore
advisable to refer to the remaining variables through their identifier. Finally,
note that deleting a matrix variable is equivalent to setting it to the zero matrix
of the same dimensions with `setmvar`

.

### Instantiating a Matrix Variable

The function `setmvar`

is used to set a matrix
variable to some given value. As a result, this variable is removed from the problem
and all terms involving it become constant terms. This is useful, for instance, to
fix`setmvar`

some variables and
optimize with respect to the remaining ones.

Consider again Specify LMI System and suppose we
want to know if the peak gain of *G* itself is less than one,
that is, if

∥*G*∥^{∞} < 1

This amounts to setting the scaling matrix *D* (or
equivalently, *S* = *D*^{T}*D*) to a multiple of the
identity matrix. Keeping in mind the constraint *S* >
*I*, a legitimate choice is *S* =
2-βψ-*I*. To set *S* to this value,
enter

NEWSYS = setmvar(LMISYS,S,2)

The second argument is the variable identifier `S`

, and the third
argument is the value to which *S* should be set. Here the value
2 is shorthand for 2-by-*I*. The resulting system
`NEWSYS`

reads

$$\begin{array}{c}\left(\begin{array}{cc}{A}^{T}X+XA+2{C}^{TC}& XB\\ {B}^{T}X& -2I\end{array}\right)<0\\ X>0\\ 2I>I.\end{array}$$

Note that the last LMI is now free of variable and trivially satisfied. It could, therefore, be deleted by

NEWSYS = dellmi(NEWSYS,3)

or

NEWSYS = dellmi(NEWSYS,Slmi)

if `Slmi`

is the identifier returned by `newlmi`

.