Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

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

`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

`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

`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

`mussv(M,BlockStructure)`

.
If `BlockStructure`

consists only of complex blocks,
then all 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*,

`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

Was this topic helpful?