# mussv

Compute bounds on structured singular value (µ)

## Syntax

## Description

calculates upper and lower bounds on the structured singular value, or
`bounds`

= mussv(`M`

,`BlockStructure`

)*µ*, of the system described by `M`

and
a given block-diagonal uncertainty `Delta`

configured as
follows.

`BlockStructure`

is a matrix encoding the block-diagonal
structure of `Delta`

. *μ* is a generalization
of the singular value for uncertain systems. `M`

is a numeric
array, a state-space (`ss`

) model, or a frequency response
(`frd`

) model.

In practice, *μ* is difficult to compute exactly, so the
software instead computes lower and upper bounds, $$\underset{\xaf}{\mu}$$ and $$\overline{\mu}$$, returned in `bounds`

. The upper bound $$\overline{\mu}$$ can be used as a measure of the system's robust performance.
(See Robust Performance Measure for Mu Synthesis.)

`[`

returns a structure containing more detailed information. To extract the
information in `bounds`

,`muinfo`

] = mussv(`M`

,`BlockStructure`

)`muinfo`

, use `mussvextract`

.

`[`

specifies additional computation options. `bounds`

,`muinfo`

]
= mussv(`M`

,`BlockStructure`

,`opt`

)

## Examples

## Input Arguments

## Output Arguments

## Algorithms

`mussv`

computes the lower bound using a power method of [7] and [4], and the upper
bound using the balanced/AMI technique of [6] for computing the
upper bound from [2]. In the upper
bound computation, the matrix is first balanced using either a variation of Osborne's
method ([3]) generalized to
handle *repeated scalar* and *full* blocks, or a
Perron approach. This computation generates the standard upper bound for the associated
complex *µ* problem. The Perron eigenvector method is based on an idea
of Safonov ([5]). It gives the
exact computation of *µ* for positive matrices with scalar blocks, but
is comparable to Osborne on general matrices. Both the Perron and Osborne methods have
been modified to handle *repeated scalar* and
*full* blocks. Perron is faster for small matrices but has a
growth rate of *n*^{3}, compared with less than
*n*^{2} for Osborne. This is partly due to
the MATLAB^{®} implementation, which greatly favors Perron. The default is to use Perron
for simple block structures and Osborne for more complicated block structures. A
sequence of improvements to the upper bound is then made based on various equivalent
forms of the upper bound. A number of descent techniques are used that exploit the
structure of the problem, concluding with general purpose LMI optimization ([1]) to obtain the
final answer.

Peter Young and Matt Newlin wrote the original version of
`mussv`

.

## References

[1] Boyd, S. and L. El Ghaoui,
“Methods of centers for minimizing generalized eigenvalues,”
*Linear Algebra and Its Applications*, Vol. 188–189, 1993, pp.
63–111.

[2] Fan, M., A. Tits, and J.
Doyle, “Robustness in the presence of mixed parametric uncertainty and unmodeled
dynamics,” *IEEE Transactions on Automatic Control*, Vol.
AC–36, 1991, pp. 25–38.

[3] Osborne, E., “On
preconditioning of matrices,” *Journal of Associated Computer
Machines*, Vol. 7, 1960, pp. 338–345.

[4] Packard, A.K., M. Fan and J.
Doyle, “A power method for the structured singular value,”
*Proc. of 1988 IEEE Conference on Control and Decision*,
December 1988, pp. 2132–2137.

[5] Safonov, M., “Stability
margins for diagonally perturbed multivariable feedback systems,” *IEEE
Proc.*, Vol. 129, Part D, 1992, pp. 251–256.

[6] Young, P. and J. Doyle,
“Computation of with real and complex uncertainties,”
*Proceedings of the 29th IEEE Conference on Decision and
Control*, 1990, pp. 1230–1235.

[7] Young, P., M. Newlin, and J.
Doyle, “Practical computation of the mixed problem,”
*Proceedings of the American Control Conference*, 1992, pp.
2190–2194.

## Version History

**Introduced before R2006a**

## See Also

`mussvextract`

| `robstab`

| `robgain`

| `wcgain`

| `wcdiskmargin`