Information about variables and term content of LMIs

lmiinfo

`lmiinfo`

provides qualitative
information about the system of LMIs `lmisys`

. This
includes the type and structure of the matrix variables, the number
of diagonal blocks in the inner factors, and the term content of each
block.

`lmiinfo`

is an interactive facility where
the user seeks specific pieces of information. General LMIs are displayed
as

N' * L(x) * N < M' * R(x) * M

where `N,M`

denote the outer factors and `L,R`

the
left and right inner factors. If the outer factors are missing, the
LMI is simply written as

L(x) < R(x)

If its right side is zero, it is displayed as

N' * L(x) * N < 0

Information on the block structure and term content of `L(x)`

and `R(x)`

is
also available. The term content of a block is symbolically displayed
as

C1 + A1*X2*B1 + B1'*X2*A1' + a2*X1 + x3*Q1

with the following conventions:

`X1, X2, x3`

denote the problem variables. Upper-case`X`

indicates matrix variables while lower-case`x`

indicates scalar variables. The labels 1,2,3 refer to the first, second, and third matrix variable in the order of declaration.`Cj`

refers to constant terms. Special cases are`I`

and –`I`

(`I`

= identity matrix).`Aj, Bj`

denote the left and right coefficients of variable terms. Lower-case letters such as`a2`

indicate a scalar coefficient.`Qj`

is used exclusively with scalar variables as in`x3*Q1`

.

The index `j`

in `Aj, Bj, Cj, Qj`

is
a dummy label. Hence `C1`

may appear in several blocks
or several LMIs without implying any connection between the corresponding
constant terms. Exceptions to this rule are the notations `A1*X2*A1'`

and ```
A1*X2*B1
+ B1'*X2'*A1'
```

which indicate symmetric terms and symmetric
pairs in diagonal blocks.

Consider the LMI

$$0\left(\begin{array}{cc}-2X+{A}^{T}YB+{B}^{T}{Y}^{T}A+I& XC\\ {C}^{T}X& -zI\end{array}\right)$$

where the matrix variables are *X* of Type
1, *Y* of Type 2, and *z* scalar.
If this LMI is described in `lmis`

, information about *X* and
the LMI block structure can be obtained as follows:

lmiinfo(lmis) LMI ORACLE ------- This is a system of 1 LMI with 3 variable matrices Do you want information on (v) matrix variables (l) LMIs (q) quit ?> v Which variable matrix (enter its index k between 1 and 3) ? 1 X1 is a 2x2 symmetric block diagonal matrix its (1,1)-block is a full block of size 2 ------- This is a system of 1 LMI with 3 variable matrices Do you want information on (v) matrix variables (l) LMIs (q) quit ?> l Which LMI (enter its number k between 1 and 1) ? 1 This LMI is of the form 0 < R(x) where the inner factor(s) has 2 diagonal block(s) Do you want info on the right inner factor ? (w) whole factor (b) only one block (o) other LMI (t) back to top level ?> w Info about the right inner factor block (1,1) : I + a1*X1 + A2*X2*B2 + B2'*X2'*A2' block (2,1) : A3*X1 block (2,2) : x3*A4 (w) whole factor (b) only one block (o) other LMI (t) back to top level ------- This is a system of 1 LMI with 3 variable matrices Do you want information on (v) matrix variables (l) LMIs (q) quit ?> q It has been a pleasure serving you!

Note that the prompt symbol is ?> and that answers are either
indices or letters. All blocks can be displayed at once with option `(w)`

,
or you can prompt for specific blocks with option `(b)`

.

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