# uscale

## Description

scales the amount of uncertainty in an uncertain control design block by
`blk_scaled`

= uscale(`blk`

,`factor`

)`factor`

. Typically, `factor`

is a robustness margin
returned by `robstab`

or `robgain`

, or a robust performance returned by `musynperf`

. The
uncertain element `blk_scaled`

is of the same type as
`blk`

, with the amount of uncertainty scaled in normalized units. For
instance, if `factor`

is 0.75, the normalized uncertainty of
`blk_scaled`

is 75% of the normalized uncertainty of
`blk`

.

## Examples

### Find Tolerable Range of Gain and Phase Variations

Consider a feedback loop with the following open-loop gain.

L = tf(3.5,[1 2 3 0]);

Suppose that the system has gain uncertainty of 1.5 (gain can increase or decrease by a factor of 1.5) and phase uncertainty of ±30°.

DGM = getDGM(1.5,30,'tight'); F = umargin('F',DGM)

Uncertain gain/phase "F" with relative gain change in [0.472,1.5] and phase change of ±30 degrees.

Examine the robust stability of the closed-loop system.

T = feedback(L*F,1); SM = robstab(T)

`SM = `*struct with fields:*
LowerBound: 0.8303
UpperBound: 0.8319
CriticalFrequency: 1.4482

`robstab`

shows that the system can only tolerate 0.83 times the modeled uncertainty before going unstable. Scale the `umargin`

block `F`

by this amount to find the largest gain and phase variation that the system can tolerate.

factor = SM.LowerBound; Fsafe = uscale(F,factor)

Uncertain gain/phase "F" with relative gain change in [0.563,1.42] and phase change of ±24.8 degrees.

The scaled uncertainty has smaller ranges of both gain variation and phase variation. Compare these ranges for the original modeled variation and the maximum tolerable variation.

DGM = F.GainChange; DGMsafe = Fsafe.GainChange; diskmarginplot([DGM;DGMsafe]) legend('original','safe')

### Scale All Uncertain Elements in a Model

Consider the uncertain control system of the example "Robust Performance of Closed-Loop System" on the `robgain`

reference page. That example examines the sensitivity of the closed-loop response at the plant output to disturbances at the plant input.

k = ureal('k',10,'Percent',40); delta = ultidyn('delta',[1 1]); G = tf(18,[1 1.8 k]) * (1 + 0.5*delta); C = pid(2.3,3,0.38,0.001); S = feedback(1,G*C)

Uncertain continuous-time state-space model with 1 outputs, 1 inputs, 4 states. The model uncertainty consists of the following blocks: delta: Uncertain 1x1 LTI, peak gain = 1, 1 occurrences k: Uncertain real, nominal = 10, variability = [-40,40]%, 1 occurrences Type "S.NominalValue" to see the nominal value and "S.Uncertainty" to interact with the uncertain elements.

Suppose that you do not want the peak gain of this sensitivity function to exceed 1.5. Use `robgain`

to find out how much of the modeled uncertainty the system can tolerate while the peak gain remains below 1.5.

perfmarg = robgain(S,1.5)

`perfmarg = `*struct with fields:*
LowerBound: 0.7821
UpperBound: 0.7837
CriticalFrequency: 7.8566

With that performance requirement, the system can only tolerate about 78% of the modeled uncertainty. Scale all the uncertain elements in `S`

to create a model of the closed-loop system with the maximum level of uncertainty that meets the performance requirement.

factor = perfmarg.LowerBound; S_scaled = uscale(S,factor)

Uncertain continuous-time state-space model with 1 outputs, 1 inputs, 4 states. The model uncertainty consists of the following blocks: delta: Uncertain 1x1 LTI, peak gain = 0.782, 1 occurrences k: Uncertain real, nominal = 10, variability = [-31.3,31.3]%, 1 occurrences Type "S_scaled.NominalValue" to see the nominal value and "S_scaled.Uncertainty" to interact with the uncertain elements.

The display shows how the uncertain elements in `S_scaled`

have changed: the peak gain of the `ultidyn`

element `delta`

is reduced from 1 to 0.78, and the range of variation of the uncertain real parameter `k`

is reduced from ±40% to ±31.3%.

## Input Arguments

`factor`

— Scaling factor

scalar

Scaling factor, specified as a scalar. This argument is the amount by which
`uscale`

scales the normalized uncertainty of
`blk`

or `M`

. For instance, if
`factor`

= 0.8, then the function reduces the uncertainty to 80%
of its original value, in normalized units. Similarly, if `factor`

=
2, then the function doubles the uncertainty.

Typically, `factor`

is a robustness margin returned by `robstab`

or `robgain`

, or a robust performance returned
by `musynperf`

.
Thus, you can use `uscale`

to find the largest range of modeled
uncertainty in a system for which the system has good robust stability or performance.

## Output Arguments

`M_scaled`

— Scaled uncertain model

`uss`

| `umat`

| `ufrd`

| `genss`

| ...

## Version History

**Introduced in R2020a**

## See Also

`normalized2actual`

| `actual2normalized`

| `musynperf`

| `robstab`

| `robgain`

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