# musynperf

Robust *H*_{∞} performance optimized
by `musyn`

## Syntax

## Description

The robust *H*_{∞} performance of
an uncertain system is the smallest value *γ* such that the I/O gain of
the system stays below *γ* for all modeled uncertainty up to size
1/*γ* (in normalized units). The `musyn`

function synthesizes a robust controller by minimizing this quantity for the closed-loop
system over all possible choices of controller. `musynperf`

computes
this quantity for a specified uncertain model. For a detailed discussion of robust
*H*_{∞} performance and how it is
computed, see Robust Performance Measure for Mu Synthesis.

`[`

calculates the robust `gamma`

,`wcu`

]
= musynperf(`clp`

)*H*_{∞}
performance for an uncertain closed-loop system `clp`

. The robust
*H*_{∞} performance is the smallest
value *γ* for which the peak I/O gain stays below
*γ* for all modeled uncertainty up to 1/*γ*,
in normalized units. For example, a value of *γ* = 1.125 implies
the following:

The I/O gain of

`clp`

remains less than 1.125 as long as the uncertain elements stay within 0.8 normalized units of their nominal values. In other words, for uncertain element values within 0.8 normalized units, the largest possible*H*_{∞}norm is 1.125.For some perturbation of size 0.8 normalized units, the peak I/O gain is 1.125.

The peak I/O gain is the maximum I/O gain over all inputs, which is also the peak
of the largest singular value over all frequencies and uncertainties. In other
words, if *Δ* represents all possible values of the uncertain
parameters in the closed-loop transfer function
*CLP*(*jω*), then

$$\gamma =\underset{\Delta}{\mathrm{max}}\underset{\omega}{\mathrm{max}}{\sigma}_{max}\left(CLP\left(j\omega \right)\right).$$

The output structure `gamma`

contains upper and lower bounds on
the robust *H*_{∞} performance and the
critical frequency at which the I/O gain of `clp`

reaches the
lower bound. The structure `wcu`

contains the uncertain-element
values that drive the peak I/O gain to the lower bound.

`[`

computes the robust `gamma`

,`wcu`

]
= musynperf(`clp`

,`w`

)*H*_{∞} performance
at the frequencies specified by `w`

.

If

`w`

is a cell array of the form`{wmin,wmax}`

, then`musynperf`

restricts the computation to the interval between`wmin`

and`wmax`

.If

`w`

is a vector of frequencies, then`musynperf`

computes the*H*_{∞}performance at the specified frequencies only.

`[`

specifies additional options for the computation. Use `gamma`

,`wcu`

]
= musynperf(___,`opts`

)`robOptions`

to create
`opts`

. You can use this syntax with any of the previous
input-argument combinations.

## Examples

## Input Arguments

## Output Arguments

## Version History

**Introduced in R2019b**

## See Also

`musyn`

| `robgain`

| `robOptions`

| `robstab`

| `normalized2actual`

| `actual2normalized`

| `wcgain`

| `uscale`