usample
Generate random samples of uncertain or generalized model
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)].
The input A can be any uncertain element, matrix,
or system, such as ureal, umat, uss,
or ufrd. A can
also be any generalized matrix or system, such as genss or genmat,
that contains uncertain blocks and other types of Control
Design Blocks. If A contains non-uncertain
control design blocks, these are unchanged in B.
Thus, for example, usample applied to a genss with
both tunable and uncertain blocks, the result is a genss array
with only tunable blocks.
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 and umargin 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 or umargin elements in
SampleValues will each have magnitude <= BW. If
A is a discrete-time, then sampled GainBounded
ultidyn or umargin 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
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 = 1×3
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 bandwidth limit on the samples of A results in generally slower step responses for those samples.
Compatibility Considerations
See Also
rsampleBlock | ucomplex | ufind | ufrd | ultidyn | umargin | umat | ureal | usample | uss | usubs