setmvar

Instantiate matrix variable and evaluate all LMI terms involving this matrix variable

Syntax

```mnewsys = setmvar(lmisys,X,Xval)
```

Description

`setmvar` sets the matrix variable X with identifier `X` to the value `Xval`. All terms involving X are evaluated, the constant terms are updated accordingly, and X is removed from the list of matrix variables. A description of the resulting LMI system is returned in `newsys`.

The integer `X` is the identifier returned by `lmivar` when X is declared. Instantiating X with `setmvar` does not alter the identifiers of the remaining matrix variables.

The function `setmvar` is useful to freeze certain matrix variables and optimize with respect to the remaining ones. It saves time by avoiding partial or complete redefinition of the set of LMI constraints.

Examples

Consider the system

= Ax + Bu

and the problem of finding a stabilizing state-feedback law u = Kx where K is an unknown matrix.

By the Lyapunov Theorem, this is equivalent to finding P > 0 and K such that

(A + BK)P + P(A + BKT) + I < 0.

With the change of variable Y := KP, this condition reduces to the LMI

AP + PAT + BY + YTBT + I < 0.

This LMI is entered by the commands

```n = size(A,1) % number of states ncon = size(B,2) % number of inputs setlmis([]) P = lmivar(1,[n 1]) % P full symmetric Y = lmivar(2,[ncon n]) % Y rectangular lmiterm([1 1 1 P],A,1,'s') % AP+PA' lmiterm([1 1 1 Y],B,1,'s') % BY+Y'B' lmiterm([1 1 1 0],1) % I lmis = getlmis ```

To find out whether this problem has a solution K for the particular Lyapunov matrix P = I, set P to I by typing

```news = setmvar(lmis,P,1) ```

The resulting LMI system `news` has only one variable Y = K. Its feasibility is assessed by calling `feasp`:

```[tmin,xfeas] = feasp(news) Y = dec2mat(news,xfeas,Y) ```

The computed Y is feasible whenever `tmin` < 0.