## Generate Samples of Uncertain Systems

Use the `usample`

function to randomly sample an
uncertain model, returning one or more non-uncertain instances of the uncertain model.

### Generating One Sample

If `A`

is an uncertain object, then `usample(A)`

generates a single sample of `A`

.

For example, a sample of a `ureal`

is a scalar
`double`

.

A = ureal('A',6); B = usample(A) B = 5.7298

Create a 1-by-3 `umat`

with `A`

and an
uncertain complex parameter `C`

. A single sample of this `umat`

is a 1-by-3 double.

C = ucomplex('C',2+6j); M = [A C A*A]; usample(M) ans = 5.9785 1.4375 + 6.0290i 35.7428

### Generating Many Samples

If `A`

is an uncertain object, then `usample(A,N)`

generates `N`

samples of `A`

.

For example, 20 samples of a `ureal`

gives a 1-by-1-20
`double`

array.

B = usample(A,20); size(B) ans = 1 1 20

Similarly, 30 samples of the 1-by-3 `umat`

`M`

yields a 1-by-3-by-30 array.

size(usample(M,30)) ans = 1 3 30

See Sample Uncertain Elements to Create Arrays for more information on sampling uncertain objects.

### Sampling Uncertain LTI Dynamics

When sampling an `ultidyn`

element or an uncertain object that contains a `ultidyn`

element, the result is always a state-space (`ss`

) object. The property `SampleStateDimension`

of the `ultidyn`

class determines the state dimension of the samples. The same is true when sampling `umargin`

objects, since these are a type of dynamic uncertainty.

Create a 1-by-1, gain bounded `ultidyn`

object with gain bound 4. Verify that the default state dimension for samples is 3.

del = ultidyn('del',[1 1],'Bound',4); del.SampleStateDimension

ans = 3

Sample the uncertain element at 30 points. Verify that this creates a 30-by-1 `ss`

array of 1-input, 1-output, 1-state systems.

```
rng(0) % for reproducibility
delS = usample(del,30);
size(delS)
```

30x1 array of state-space models. Each model has 1 outputs, 1 inputs, and 3 states.

Plot the Nyquist plot of these samples and add a disk of radius 4, the gain bound of `del`

.

nyquist(delS) hold on; theta = linspace(-pi,pi); plot(del.Bound*exp(sqrt(-1)*theta),'r'); hold off;

Change `SampleStateDimension`

to 1, and repeat entire procedure. The Nyquist plots again satisfy the gain bound, but the Nyquist plots are all circles, indicative of 1st order systems.

del.SampleStateDimension = 1; delS = usample(del,30); nyquist(delS) hold on; theta = linspace(-pi,pi); plot(del.Bound*exp(sqrt(-1)*theta),'r'); hold off;

With `SampleStateDimension `

= 1, all Nyquist plots touch the gain boundary at either (–1,0) or (1,0) (frequency = 0 or `Inf`

). Higher sampling dimension yields a Nyquist curve that reaches the gain bound at more frequencies, yielding more thorough coverage.

Create a `umargin`

object using the default `SampleStateDimension`

. The `umargin`

block models uncertain gain and phase. The modeled variations are within bounded ranges. For this example use a `umargin`

block that captures relative gain variations of a factor of two in either direction, and phase variations of ±30°.

DGM = getDGM(2,30,'tight'); F = umargin('F')

F = Uncertain gain/phase "F" with relative gain change in [0.5,2] and phase change of ±36.9 degrees.

The samples of a `umargin`

block are also state-space models.

Fs = usample(F,30); size(Fs)

30x1 array of state-space models. Each model has 1 outputs, 1 inputs, and 3 states.

Plot the samples on the Nyquist plane.

nyquist(Fs)

The Nyquist plot of any sample of `F`

stays within the disk of uncertainty modeled by `F`

. To confirm this bound, use `plot`

to examine the uncertainty disk. Compare the Nyquist plot above with the right side of the following plot.

plot(F)

For further details about the gain and phase uncertainty model, see `umargin`

.