Main Content

Extract `muinfo`

structure returned by `mussv`

`[VDelta,VSigma,VLmi] = mussvextract(muinfo)`

A structured singular value computation of the form

[bounds,muinfo] = mussv(M,BlockStructure)

returns detailed information in the structure `muinfo`

. `mussvextract`

is
used to extract the compressed information within `muinfo`

into
a readable form.

The most general call to `mussvextract`

extracts
three usable quantities: `VDelta`

, `VSigma`

,
and `VLmi`

. `VDelta`

is used to
verify the lower bound. `VSigma`

is used to verify
the Newlin/Young upper bound and has fields `DLeft`

, `DRight`

, `GLeft`

, `GMiddle`

,
and `GRight`

. `VLmi`

is used to
verify the LMI upper bound and has fields `Dr, Dc, Grc`

,
and `Gcr`

. The relation/interpretation of these quantities
with the numerical results in `bounds`

is described
below.

The upper bound is based on a proof that `det(I - M*Delta)`

is
nonzero for all block-structured matrices `Delta`

with
norm smaller than `1/bounds(1)`

. The Newlin/Young
method consists of finding a scalar β and matrices *D* and *G*,
consistent with `BlockStructure`

, such that

$$\overline{\sigma}\left({\left(I+{G}_{L}^{2}\right)}^{-\text{}\frac{1}{4}}\left(\frac{{D}_{L}M{D}_{R}^{-1}}{\beta}-j{G}_{M}\right){\left(I+{G}_{R}^{2}\right)}^{-\text{}\frac{1}{4}}\right)\le 1$$

Here *D _{L}*,

`DLeft`

, `DRight`

, `GLeft`

, `GMiddle`

,
and `GRight`

fields respectively.Because some uncertainty blocks and `M`

need
not be square, the matrices *D* and *G* have
a few different manifestations. In fact, in the formula above, there
are a left and right *D* and *G*,
as well as a middle *G*. Any such β is an upper
bound of `mussv(M,BlockStructure)`

.

It is true that if `BlockStructure`

consists
only of complex blocks, then all *G* matrices will
be zero, and the expression above simplifies to

$$\overline{\sigma}({D}_{L}M{D}_{R}^{-1})\le \beta .$$

The LMI method consists of finding a scalar β and matrices *D* and *G*,
consistent with `BlockStructure`

, such that

$${M}^{\prime}{D}_{r}M-{\beta}^{2}{D}_{c}+j({G}_{cr}M-{M}^{\prime}{G}_{rc})\le 0$$

is negative semidefinite. Again, *D* and *G* have
a few different manifestations to match the row and column dimensions
of M. Any such β is an upper bound of `mussv(M,BlockStructure)`

.
If `BlockStructure`

consists only of complex blocks,
then all *G* matrices will be zero, and negative
semidefiniteness of *M*´*D _{r} *

The lower bound of `mussv(M,BlockStructure)`

is
based on finding a “small” (hopefully the smallest)
block-structured matrix `VDelta`

that causes ```
det(I
- M*VDelta)
```

to equal 0. Equivalently, the matrix `M*VDelta`

has
an eigenvalue equal to 1. It will always be true that the lower bound `(bounds(2))`

will
be the reciprocal of `norm(VDelta)`

.

Suppose `M`

is a 4-by-4 complex matrix. Take
the block structure to be two 1-by-1 complex blocks and one 2-by-2
complex block.

rng(0,'twister') M = randn(4,4) + sqrt(-1)*randn(4,4); BlockStructure = [1 1;1 1;2 2];

You can calculate bounds on the structured singular value using
the `mussv`

command and extract the scaling matrices
using `mussvextract`

.

[bounds,muinfo] = mussv(M,BlockStructure); [VDelta,VSigma,VLmi] = mussvextract(muinfo);

You can first verify the Newlin/Young upper bound with the information
extracted from `muinfo`

. The corresponding scalings
are `Dl`

and `Dr`

.

Dl = VSigma.DLeft

Dl = 1.0000 0 0 0 0 0.7437 0 0 0 0 1.0393 0 0 0 0 1.0393

Dr = VSigma.DRight

Dr = 1.0000 0 0 0 0 0.7437 0 0 0 0 1.0393 0 0 0 0 1.0393

[norm(Dl*M/Dr) bounds(1)]

ans = 6.2950 6.2950

You can first verify the LMI upper bound with the information
extracted from `muinfo`

. The corresponding scalings
are `Dr`

and `Dc`

.

Dr = VLmi.Dr; Dc = VLmi.Dc; eig(M'*Dr*M - bounds(1)^2*Dc)

ans = -0.0000 - 0.0000i -17.7242 - 0.0000i -33.8550 + 0.0000i -41.2013 - 0.0000i

Note that `VDelta`

matches the structure defined
by `BlockStructure`

, and the norm of `VDelta`

agrees
with the lower bound,

VDelta

VDelta = 0.1301 - 0.0922i 0 0 0 0 -0.0121 - 0.1590i 0 0 0 0 -0.0496 - 0.0708i 0.1272 - 0.0075i 0 0 0.0166 - 0.0163i 0.0076 + 0.0334i

[norm(VDelta) 1/bounds(2)]

ans = 0.1595 0.1595

and that `M*VDelta`

has an eigenvalue exactly
at 1.

eig(M*VDelta)

ans = 1.0000 - 0.0000i -0.2501 - 0.1109i 0.0000 + 0.0000i -0.3022 + 0.2535i

Keep the matrix the same, but change `BlockStructure`

to
be a 2-by-2 repeated, real scalar block and two complex 1-by-1 blocks.
Run `mussv`

with the `'C'`

option
to tighten the upper bound.

BlockStructure2 = [-2 0; 1 0; 1 0]; [bounds2,muinfo2] = mussv(M,BlockStructure2,'C');

You can compare the computed bounds. Note that `bounds2`

should
be smaller than `bounds`

, because the uncertainty
set defined by `BlockStructure2`

is a proper subset
of that defined by `BlockStructure`

.

[bounds; bounds2]

ans = 6.2950 6.2704 5.1840 5.1750

You can extract the *D*, *G* and `Delta`

from `muinfo2`

using `mussvextract`

.

[VDelta2,VSigma2,VLmi2] = mussvextract(muinfo2);

As before, you can first verify the Newlin/Young upper bound
with the information extracted from `muinfo`

. The
corresponding scalings are `Dl, Dr, Gl, Gm and Gr`

.

Dl = VSigma2.DLeft; Dr = VSigma2.DRight; Gl = VSigma2.GLeft; Gm = VSigma2.GMiddle; Gr = VSigma2.GRight; dmd = Dl*M/Dr/bounds2(1) - sqrt(-1)*Gm; SL = (eye(4)+Gl*Gl)^-0.25; SR = (eye(4)+Gr*Gr)^-0.25; norm(SL*dmd*SR)

ans = 1.0000

You can first verify the LMI upper bound with the information
extracted from `muinfo`

. The corresponding scalings
are `Dr`

, `Dc, Grc`

and` Gcr`

.

Dr = VLmi2.Dr; Dc = VLmi2.Dc; Grc = VLmi2.Grc; Gcr = VLmi2.Gcr; eig(M'*Dr*M - bounds(1)^2 *Dc + j*(Gcr*M-M'*Grc))

ans = -69.9757 + 0.0000i -11.2139 - 0.0000i -19.2766 - 0.0000i -40.2869 - 0.0000i

`VDelta2`

matches the structure defined by `BlockStructure`

,
and the norm of `VDelta2`

agrees with the lower bound,

VDelta2

VDelta2 = 0.1932 0 0 0 0 0.1932 0 0 0 0 -0.1781 - 0.0750i 0 0 0 0 0.0941 + 0.1688i

[norm(VDelta2) 1/bounds2(2)]

ans = 0.1932 0.1932

and that `M*VDelta2`

has an eigenvalue exactly
at 1.

eig(M*VDelta2)

ans = 1.0000 + 0.0000i -0.4328 + 0.1586i 0.1220 - 0.2648i -0.3688 - 0.3219i