uss/usample

Generate random samples of uncertain object

Syntax

B = usample(A);
B = usample(A,N)
[B,SampleValues] = usample(A,N)
[B,SampleValues] = usample(A,Names,N)
[B,SampleValues] = usample(A,Names1,N1,Names2,N2,...)
[B,SampleValues] = usample(A,N,Wmax)
[B,SampleValues] = usample(A,Names,N,Wmax)

Description

B = usample(A) substitutes a random sample of the uncertain objects in A, returning a certain (i.e., not uncertain) array of size [size(A)].

B = usample(A,N) substitutes N random samples of the uncertain objects in A, returning a certain (i.e., not uncertain) array of size [size(A) N].

[B,SampleValues] = usample(A,N) additionally returns the specific sampled values (as a Struct whose field names are the names of A's uncertain elements) of the uncertain elements. Hence, B is the same as usubs(A,SampleValues).

[B,SampleValues] = usample(A,Names,N) samples only the uncertain elements listed in the Names variable (cell, or char array). If Names does not include all the uncertain objects in A, then B will be an uncertain object. Any entries of Names that are not elements of A are simply ignored. Note that usample(A,fieldnames(A.Uncertainty),N) is the same as usample(A,N).

[B,SampleValues] = usample(A,Names1,N1,Names2,N2,...) takes N1 samples of the uncertain elements listed in Names1, and N2 samples of the uncertain elements listed in Names2, and so on. size(B) will equal [size(A) N1 N2 ...].

The scalar parameter Wmax in

[B,SampleValues] = usample(A,N,Wmax)
[B,SampleValues] = usample(A,Names,N,Wmax) 
[B,SampleValues] = usample(A,Names,N,Wmax)

affects how ultidyn elements within A are sampled, restricting the poles of the samples. If A is a continuous-time uss or ufrd, then the poles of sampled GainBounded ultidyn elements in SampleValues will each have magnitude <= BW. If A is a discrete-time, then sampled GainBounded ultidyn elements are obtained by Tustin transformation, using BW/(2*TS) as the (continuous) pole magnitude bound. In this case, BW should be < 1. If the ultidyn type is PositiveReal, then the samples are obtained by bilinearly transforming (see Normalizing Functions for Uncertain Elements) the GainBounded elements described above.

Examples

expand all

Sample Real Parameter

Create a real uncertain parameter, sample it, and plot a histogram of the sampled values.

A = ureal('A',5);
Asample = usample(A,500);

Examine the size of the parameter and the sample array.

size(A)
Uncertain real scalar.
size(Asample)
ans =

     1     1   500

A is a scalar parameter. The dimensions of Asample reflect that A is a 1-by-1 parameter. Examine the data type of Asample.

class(Asample)
ans =

double

The samples of the scalar parameter are numerical values.

Plot the histogram of sampled values.

hist(Asample(:))

Sample Responses of Uncertain Control System Model

This example illustrates how to sample the open and closed-loop response of an uncertain plant model for Monte Carlo analysis.

Create two uncertain real parameters and an uncertain plant.

gamma = ureal('gamma',4);
tau = ureal('tau',.5,'Percentage',30);
P = tf(gamma,[tau 1]);

Create an integral controller based on the nominal values of plant uncertainties.

KI = 1/(2*tau.Nominal*gamma.Nominal);
C = tf(KI,[1 0]);

Now create an uncertain closed-loop system.

CLP = feedback(P*C,1);

Sample the plant at 20 values, distributed uniformly about the tau and gamma parameter cube.

[Psample1D,Values1D] = usample(P,20);
size(Psample1D)
20x1 array of state-space models.
Each model has 1 outputs, 1 inputs, and 1 states.

This sampling returns an array of 20 state-space models, each representing the closed-loop system within the uncertainty.

Now sample the plant at 10 values of tau and 15 values of gamma.

[Psample2D,Values2D] = usample(P,'tau',10,'gamma',15);
size(Psample2D)
10x15 array of state-space models.
Each model has 1 outputs, 1 inputs, and 1 states.

Plot the step responses of the 1-D sampled plant.

subplot(2,1,1);
step(Psample1D)

Evaluate the uncertain closed-loop model at the same values using usubs, and plot the step response.

subplot(2,1,2);
step(usubs(CLP,Values1D))

Restrict Pole Locations in Sampled Uncertain Dynamics

To see the effect of limiting the bandwidth of sampled models with Wmax, create two ultidyn objects.

A = ultidyn('A',[1 1]);
B = ultidyn('B',[1 1]);

Sample 10 instances of each, using a bandwidth limit of 1 rad/sec on A, and 20 rad/sec on B.

Npts = 10;
As = usample(A,Npts,1);
Bs = usample(B,Npts,20);

Plot 10-second step responses, for the two sample sets.

step(As,'r',Bs,'b--',10)

The lower bandwith limit on the samples of A results in generally slower step responses for those samples.

See Also

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