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Generate input signals
The idinput
command generates an
input signal with specified characteristics for your system. You can
use the generated input, and simulate the response of your system
to study system behavior. For example, you can study the system response
to periodic inputs. The system can be an actual physical system or
a model such as a Simulink^{®} model. You can also design optimal
experiments. For example, you can determine which input signals isolate
faults or nonlinearities in your system. You can also use idinput
to
design an input that has sufficient bandwidth to excite the dynamic
range of your system.
u = idinput(N)
u = idinput([N,Nu])
u = idinput([Period,Nu,NumPeriod])
u = idinput(___,Type)
u = idinput(___,Type,Band)
u = idinput(___,Type,Band,Range)
[u,freq]
= idinput(___,'sine',Band,Range,SineData)
[
specifies
the u
,freq
]
= idinput(___,'sine',Band
,Range
,SineData
)Type
as a sum-of-sinusoids signal and specifies
the characteristics of the sine waves used to generate the signal
in SineData
. You can specify characteristics
such as the number of sine waves and their frequency separation. The
frequencies of the sine waves are returned in freq
.
Generate a single-channel random binary input signal with 200 samples.
N = 200; u = idinput(N);
u
is a column vector of length 200. The values in u
are either -1 or 1.
Create an iddata
object from the generated signal. For this example, specify the sample time as 1 second.
u = iddata([],u,1);
To examine the signal, plot it.
plot(u)
The generated signal is a random binary input signal with values -1 or 1. You can use the generated input signal to simulate the output of your system using the sim
command.
Generate a two-channel random binary input signal with 200 samples.
N = 200; u = idinput([N,2]);
u
is a 200-by-2 matrix with values -1 or 1.
Create an iddata
object from the generated signal. For this example, specify the sample time as 1 second.
u = iddata([],u,1);
Plot the signals for the two channels, and examine the signals.
plot(u)
The plot shows the two generated random binary signals with values -1 or 1.
Generate a single-channel periodic random binary input signal with a period of 10 samples and 5 periods in the signal.
NumChannel = 1; Period = 10; NumPeriod = 5; u = idinput([Period,NumChannel,NumPeriod]);
u
is a column vector of length 50 (= Period*NumPeriod). The values in u
are either -1 or 1.
Create an iddata
object from the generated signal. Specify the sample time as 1 second.
u = iddata([],u,1);
Plot the signal.
plot(u)
As specified, the generated single-channel periodic random binary input signal has a period of 10 seconds, and there are 5 whole periods in the signal.
Generate a single-channel periodic random Gaussian input signal with a period of 50 samples and 5 periods in the signal. First generate the signal using the entire frequency range, then specify a passband.
NumChannel = 1;
Period = 50;
NumPeriod = 5;
u = idinput([Period,NumChannel,NumPeriod],'rgs');
u
is a column vector of length 250 (= Period*NumPeriod).
Create an iddata
object from the generated signal, and plot the signal. For this example, specify the sample time as 0.01 seconds.
u = iddata([],u,0.01); plot(u)
The plot shows that u
contains a random segment of 50 samples, repeated 5 times. The signal is a Gaussian white noise signal with zero mean and variance one.
Since the sample time is 0.01 seconds, the generated signal has a period of 0.5 seconds. The frequency content of the signal spans the entire available range (0-50 Hz).
Now specify a passband between 0 and 25 Hz ( = 0.5 times the Nyquist frequency).
Band = [0 0.5];
u2 = idinput([Period,NumChannel,NumPeriod],'rgs',Band);
Create an iddata
object, and plot the signal.
u2 = iddata([],u2,0.01); plot(u2)
The frequency content of the generated signal u2
is limited to 0-25 Hz.
A pseudorandom binary input signal (PRBS) is a deterministic signal whose frequency properties mimic white noise. A PRBS is inherently periodic with a maximum period length of , where integer n is the order of the PRBS. For more information, see Pseudorandom Binary Signals.
Specify that the single-channel PRBS value switches between -2 and 2.
Range = [-2,2];
Specify the clock period of the signal as 1 sample. That is, the signal value can change at each time step. For PRBS signals, the clock period is specified in Band
= [0 B], where B
is the inverse of the required clock period.
Band = [0 1];
Generate a nonperiodic PRBS of length 100 samples.
u = idinput(100,'prbs',Band,Range);
Warning: The PRBS signal delivered is the 100 first values of a full sequence of length 127.
A PRBS is inherently periodic. To generate a nonperiodic signal, the software generates a maximum length PRBS of length 127 that has a period greater than the required number of samples, 100. The software returns the first 100 samples of the generated PRBS. This action ensures that the generated signal is not periodic, as indicated in the generated warning.
Create an iddata
object from the generated signal. For this example, specify the sample time as 1 second.
u = iddata([],u,1);
Plot, and examine the generated signal.
plot(u);
title('Non-Periodic Signal')
The generated signal is a nonperiodic PRBS of length 100 that switches between -2 and 2.
Specify that the pseudorandom binary input signal (PRBS) switches between -2 and 2.
Range = [-2,2];
Specify the clock period of the signal as 1 sample. That is, the signal value can change at each time step. For PRBS signals, the clock period is specified in Band
= [0 B], where B
is the inverse of the required clock period.
Band = [0 1];
Generate a single-channel, periodic PRBS with a period of 100 samples and 3 periods in the signal.
u1 = idinput([100,1,3],'prbs',Band,Range);
Warning: The period of the PRBS signal was changed to 63. Accordingly, the length of the generated signal will be 189.
A PRBS is inherently periodic with a maximum period length of
, where integer n is the order of the PRBS. If the period you specify is not equal to a maximum length PRBS, the software adjusts the period of the generated signal to obtain an integer number of maximum length PRBS, and issues a warning. For more information about maximum length PRBS, see Pseudorandom Binary Signals. In this example, the desired period, 100, is not equal to a maximum length PRBS, thus the software instead generates a maximum length PRBS of order n = floor(log2(Period)) = 6
. Thus, the period of the PRBS signal is 63 ( =
), and the length of the generated signal is 189 (= NumPeriod
*63). This result is indicated in the generated warning.
Create an iddata
object from the generated signal, and plot the signal. Specify the period of the signal as 63 samples.
u1 = iddata([],u1,1,'Period',63); plot(u1) title('Periodic Signal')
The generated signal is a periodic PRBS with three periods.
Generate periodic and nonperiodic pseudorandom binary input signals (PRBS) with specified clock period.
Generate a single-channel PRBS that switches between -2 and 2. Specify the clock period of the signal as 4 samples. That is, the signal has to stay constant for at least 4 consecutive samples before it can change. For PRBS signals, the clock period is specified in Band
= [0 B], where B
is the inverse of the required clock period.
Range = [-2,2]; Band = [0 1/4];
First generate a nonperiodic signal of length 100.
u1 = idinput(100,'prbs',Band,Range);
Warning: The PRBS signal delivered is the 100 first values of a full sequence of length 124.
To understand the generated warning, first note that the code is equivalent to generating a single-channel PRBS with a 100-sample period and 1 period.
u1 = idinput([100,1,1],'prbs',Band,Range);
The generated PRBS signal has to remain constant for at least 4 samples before the value can change. To satisfy this requirement, the software first computes the order of the smallest possible maximum length PRBS as n = floor(log2(Period*B)) = 4
and period
. For information about maximum length PRBS, see Pseudorandom Binary Signals. The software then stretches this PRBS such that the period of the stretched signal is
.
However, since this period is less than the specified length, 100, the software computes instead a maximum length PRBS of order m = n+1 = 5
. The software then stretches this PRBS such that the period is now
. The software returns the first 100 samples of this signal as u1
. This result ensures that the generated signal is not periodic but is constant for every 4 samples.
Create an iddata
object from the generated signal. For this example, specify the sample time as 1 second.
u1 = iddata([],u1,1);
Plot, and examine the signal.
plot(u1);
title('Nonperiodic Signal')
The generated signal is a nonperiodic PRBS of length 100. The signal remains constant for at least 4 samples before each change in value. Thus, the signal satisfies the clock period specified in Band
.
Now generate a periodic signal with a 100-sample period and 3 periods.
u2 = idinput([100,1,3],'prbs',Band,Range);
Warning: The period of the PRBS signal was changed to 60. Accordingly, the length of the generated signal will be 180.
To generate a periodic signal with specified clock period, the software generates u2
as 3 repetitions of the original stretched signal of period P = 60
. Thus, the length of u2
is P*NumPeriod = 60*3 = 180
. This change in period and length of the generated signal is indicated in the generated warning.
Create an iddata
object from the generated signal, and plot the signal. Specify the period of the signal as 60 seconds.
u2 = iddata([],u2,1,'Period',60); plot(u2) title('Periodic Signal')
The generated signal is a periodic PRBS with a 60-second period and 3 periods. The signal remains constant for at least 4 samples before each change in value. Thus, the signal satisfies the specified clock period.
You can generate a sum-of-sinusoids signal using default characteristics for the sine waves. Alternatively, you configure the number of sine waves, and the frequencies and phases of the sine waves. This example shows both approaches.
Specify that the signal has 50 samples in each period and 3 periods. Also specify that the signal amplitude range is between -1 and 1.
Period = 50; NumPeriod = 3; Range = [-1 1];
Specify the frequency range of the signal. For a sum-of-sinusoids signal, you specify the lower and upper frequencies of the passband in fractions of the Nyquist frequency. In this example, use the entire frequency range between 0 and Nyquist frequency.
Band = [0 1];
First generate the signal using default characteristics for the sine waves. By default, the software uses 10 sine waves to generate the signal. The software assigns a random phase to each sinusoid, and then changes these phases 10 times to get the smallest signal spread. The signal spread is the difference between the minimum and the maximum value of the signal over all samples.
[u,freq] = idinput([Period 1 NumPeriod],'sine',Band,Range);
The software returns the sum-of-sinusoids signal in u
and the frequencies of the sinusoids in freq
. The values in freq
are scaled assuming that the sample time is 1 time unit. Suppose that the sample time is 0.01 hours. To retrieve the actual frequencies in rad/hours, divide the values by the sample time.
Ts = 0.01; % Sample time in hours
freq = freq/Ts;
freq(1)
ans = 12.5664
freq(1)
is the frequency of the first sine wave. To see how the software chooses the frequencies, see the SineData
argument description on the idinput
reference page.
To verify that 10 sine waves were used to generate the signal, you can view the frequency content of the signal. Perform a Fourier transform of the signal, and plot the single-sided amplitude spectrum of the signal.
ufft = fft(u); Fs = 2*pi/Ts; % Sampling frequency in rad/hour L = length(u); w = (0:L-1)*Fs/L; stem(w(1:L/2),abs(ufft(1:L/2))) % Plot until Nyquist frequency title('Single-Sided Amplitude Spectrum of u(t)') xlabel('Frequency (rad/hour)') ylabel('Amplitude')
The generated plot shows the frequencies of the 10 sine waves used to generate the signal. For example, the plot shows that the first sine wave has a frequency of 12.57 rad/hour, the same as freq(1)
.
Convert the generated signal into an iddata
object, and plot the signal. Specify the sample time as 0.01 hours.
u = iddata([],u,Ts,'TimeUnit','hours'); plot(u)
The signal u
is generated using 10 sinusoids and has a period of 0.5 hours and 3 periods.
Now modify the number, frequency, and phase of the sinusoids that are used to generate the sum-of-sinusoids signal. Use 12 sinusoids and try 15 different sets of phases. To set the frequencies of the sinusoids, specify GridSkip
= 2. The software selects the frequencies of the sinusoids from the intersection of the frequency grid 2*pi*[1:GridSkip:fix(Period/2)]/Period
and the passband pi*Band
.
NumSinusoids = 12;
NumTrials = 15;
GridSkip = 2;
SineData = [NumSinusoids,NumTrials,GridSkip];
u2 = idinput([Period 1 NumPeriod],'sine',Band,Range,SineData);
Convert the generated signal into an iddata
object, and plot the signal.
u2 = iddata([],u2,Ts,'TimeUnit','hours'); plot(u2)
The signal u2
is generated using 12 sinusoids and has a period of 0.5 hours and 3 periods.
N
— Number of generated input data samplesNu
— Number of input channels1
(default) | real positive integerNumber of input channels in generated signal, specified as a real positive integer.
Period
— Number of samples in each periodNumber of samples in each period of generated signal, specified
as a real positive integer. Use this input to specify a periodic signal.
Also specify the number of periods in NumPeriod
.
Each generated input channel signal has NumPeriod*Period
samples.
NumPeriod
— Number of periods in generated signal1
(default) | real positive integerNumber of periods in generated signal, specified as a real positive
integer. Use this input to specify a periodic signal. Also specify
the signal Period
. Each generated input channel
signal has NumPeriod*Period
samples.
Type
— Type of generated signal'rbs'
(default) | 'rgs'
| 'prbs'
| 'sine'
Type of generated signal, specified as one of the following values:
'rbs'
— Generates a random
binary signal. A random binary signal is a random process that assumes
only two values. You can specify these values using Range
.
To generate a band-limited signal, specify the passband in Band
.
To generate a periodic signal, specify Period
and NumPeriod
.
'rgs'
— Generates a random
Gaussian signal. The generated Gaussian signal has mean μ and
standard deviation σ such that [μ-σ, μ+σ]
equals Range
. To generate a band-limited Gaussian
signal, specify the passband in Band
. To generate
a periodic Gaussian signal with an n
samples period
that repeats itself m
times, specify Period
as n
and NumPeriod
as m
.
'prbs'
— Generates a pseudorandom
binary signal (PRBS). A PRBS is a periodic, deterministic signal with
white-noise-like properties that shifts between two values. You can
specify these two values using Range
. You can
also specify the clock period, the minimum number of sampling intervals
for which the value of the signal does not change. You specify the
inverse of the clock period in Band
.
The length of the generated signal is not always the same as what you specify. The length depends on whether you require a periodic or nonperiodic signal and also on the clock period you specify. For more information, see Pseudorandom Binary Signals.
'sine'
— Generates a signal
that is a sum-of-sinusoids. The software selects the frequencies of
the sinusoids to be equally spread over a chosen grid and assigns
each sinusoid a random phase. The software then tries several random
phases for each sinusoid and selects the phases that give the smallest
signal spread. The signal spread is the difference between the minimum
and the maximum value of the signal over all samples. The amplitude
of the generated sum-of-sinusoids signal is scaled to satisfy the Range
you
specify.
You can specify the characteristics of the sine waves used to
generate the signal, such as the number of sine waves and their frequency
separation, in the SineData
argument.
Band
— Frequency range of generated signal[0 1]
(default) | 1-by-2 row vectorFrequency range of generated signal, specified as a 1-by-2 row vector containing minimum and maximum frequency values.
If Type
is 'rgs'
, 'rbs'
,
or 'sine'
— Specify Band
as
a passband [wlow whigh]
. Where, wlow
and whigh
are
the lower and upper frequencies of the passband, expressed in fractions
of the Nyquist frequency. For example, to generate an input with white
noise characteristics, use Band = [0 1]
.
The software achieves the frequency contents for a random Guassian
signal ('rgs'
) using idfilt
with
an eighth-order Butterworth, noncausal filter. For generating a random
binary signal ('rbs'
), the software uses the same
filter and then makes the signal binary. Thus, the frequency content
in the generated random binary signal may not match the specified
passband.
For 'sine'
signals, the frequencies of the
sinusoids are selected to be equally spread over a chosen grid in
the specified passband. For more information, see the SineData
argument
description.
If Type
is 'prbs'
—
Specify Band
as [0 B]
, where B
is
the inverse of the clock period of the signal. The clock period is
the minimum number of sampling intervals for which the value of the
signal does not change. Thus, the generated signal is constant over
intervals of length 1/B
samples. If 1/B
is
not an integer, the software uses floor(1/B)
as
the clock period.
Range
— Generated input signal range[-1,1]
(default) | two-element row vectorGenerated input signal range, specified as a two-element row
vector of the form [umin,umax]
.
If Type
is 'rbs'
or 'prbs'
—
The generated signal u
has values umin
or umax
.
If Type
is 'sine'
—
The generated signal u
has values between umin
and umax
.
If Type
is 'rgs'
—
The generated Gaussian signal has mean μ and standard deviation
σ such that umin
and umax
are
equal to μ-σ and μ+σ, respectively. For example, Range
= [-1,1]
returns a Gaussian white noise signal with zero
mean and variance one.
SineData
— Characterization of sinusoids[10,10,1]
(default) | three-element row vector [NumSinusoids,NumTrials,GridSkip]
Characterization of sinusoids used to generate a sum-of-sinusoids
signal, specified as a three-element row vector [NumSinusoids,NumTrials,GridSkip]
.
Where,
NumSinusoids
is the number of sinusoids
used to generate the signal. The default value is 10
.
NumTrials
is the number of different
random relative phases of the sinusoids that the software tries to
find the lowest signal spread. The signal spread is the difference
between the minimum and the maximum value of the signal over all samples.
The maximum amplitude of the sum-of-sinusoids signal depends
on the relative phases of the different sinusoids. To find the phases
that give the smallest signal spread, the software tries NumTrials
different
random choices of phases to find the best phase values. For example,
suppose that NumSinusoids
is 20
and NumTrials
is 5
.
The software tries 5 different sets of relative phases for the 20
sinusoids, and selects the phases that give the smallest signal spread.
The default value for NumTrials
is 10
.
GridSkip
is used to characterize
the frequency of the sinusoids. The software selects the frequency
of the sinusoids from the intersection of the frequency grid 2*pi*[1:GridSkip:fix(Period/2)]/Period
and
the pass band pi*[Band(1) Band(2)]
. For multichannel
input signals, the software uses different frequencies from this frequency
grid to generate the different input channels. You can use GridSkip
for
controlling odd and even frequency multiples, for example, to detect
nonlinearities of different kinds.
To extract the frequencies freq
that are
selected by the software to generate the signal, use the following
syntax.
[u,freq] = idinput(__)
u
— Generated input signalGenerated input signal, returned as a column vector of length N
for
a single-channel input or an N
-by-Nu
matrix
for an Nu
-channel signal. You use the generated
signal to simulate the response of your system using sim
.
You can create an iddata
object
from u
by specifying output data as []
.
u = iddata([],u);
In the iddata
object, you can also specify
the properties of the signal such as sample time, input names, and
periodicity.
freq
— Frequencies of sine wavesFrequencies of sine waves used for sum-of-sinusoids signal,
returned as a column vector of length equal to the number of sinusoids, NumSinusoids
.
You specify NumSinusoids
in the SineData
argument.
The frequency values are scaled assuming the sample time is 1 time
unit. To retrieve the actual frequencies, divide the values by the
sample time. For an example, see Generate a Sum-of-Sinusoids Signal.
For multichannel input signals, freq
is
an Nu
-by-NumSinusoids
matrix
where the kth row contains the frequencies corresponding
to the kth channel. For information about how the
software selects the frequencies, see the SineData
argument
description.
A pseudorandom binary signal (PRBS) is a periodic, deterministic signal with white-noise-like properties that shifts between two values.
A PRBS is generated as:
$$u(t)=rem\left({a}_{1}u(t-1)+\dots +{a}_{n}u(t-n),2\right)$$
Here, $$u(t-1),\dots u(t-n)$$ is
the vector of past inputs, n is the PRBS order,
and rem
denotes the remainder when $$\left({a}_{1}u(t-1)+\dots {a}_{n}u(t-n)\right)$$ is
divided by 2. Thus, a PRBS can only take the values 0 and 1. The software
scales these values according to the Range
you
specify. In addition, the vector of past inputs $$u(t-1),\dots u(t-n)$$ can
only take 2^{n}
values.
Out of these values, the state with all zeros is ignored because it
will result in future signals equal to zero. Thus, a PRBS is an inherently
periodic signal with a maximum period length of 2^{n}-1
.
The following table lists the maximum length possible for different
orders n of the PRBS.
Order n | Maximum length PRBS (2^{n}-1 ) |
---|---|
2 | 3 |
3 | 7 |
4 | 15 |
5 | 31 |
6 | 63 |
7 | 127 |
⋮ | ⋮ |
18 | 262143 |
Note: The software does not generate signals with period greater than 262143 (n = 18). |
Since PRBS are inherently periodic, the length and period of
the generated signal depends on the clock period that you specify
and whether you require a periodic or nonperiodic signal. The clock
period is the minimum number of sampling intervals for which the value
of the signal does not change. You specify the clock period in Band
.
Clock period = 1 sample (Band = [0 B] = [0
1]
):
To generate a nonperiodic signal of
length N
, (NumPeriod
= 1),
the software first computes a maximum length PRBS with a period greater
than N
. The software then returns the first N
samples
of the PRBS as u
. This action ensures that u
is
not periodic. For example, if N
is 100, the software
creates a maximum length PRBS of period 127 (order 7), and returns
the first 100 samples as u
.
For an example, see Generate a Nonperiodic Pseudorandom Binary Input Signal.
To generate a periodic signal (NumPeriod
>
1), the software adjusts the period of the signal to obtain an integer
number of maximum length PRBS. To do so, the software computes a PRBS
of order n = floor(log2(Period))
and period P
= 2^{n}-1
. The signal u
is
then generated as NumPeriod
repetitions of this
PRBS signal of period P
. Thus, the length of u
is P*NumPeriod
.
For an example, see Generate a Periodic Pseudorandom Binary Input Signal.
In the multiple-input channel case, the signals are maximally
shifted. That is, the overlap between the different inputs is minimized.
This means Period/NumPeriod
is an upper bound for
the model orders that you can estimate using such a signal.
Clock period > 1 sample (Band = [0 B],
where B<1
):
The generated signal has to remain constant for at least 1/B
samples.
To satisfy this requirement, the software first computes the order
of the smallest possible maximum length PRBS as n = floor(log2(Period*B))
and
period 2^{n}-1
. The software
then stretches the PRBS such that period of the stretched signal is P
= B^{-1}(2^{n}-1)
.
To generate a nonperiodic signal of
length N
, if the period P
of
the stretched signal is greater than or equal to N
,
the software returns the first N
samples of the
stretched signal as u
. This ensures that u
is
nonperiodic but constant for every 1/B
samples.
Note that for a nonperiodic signal, Period
is
equal to N
.
If the period P
is less than N
,
the software computes instead a maximum length PRBS of order n2
= n+1
. The software then stretches this PRBS such that the
period is now P_{2} = B^{-1}(2^{n2}-1)
.
The software then returns the first N
samples
of this signal as u
.
To generate a periodic signal,
the software generates u
as NumPeriod
repetitions
of the stretched signal of period P
. Thus, the
length of u
is P*NumPeriod
.
For an example, see Generate Pseudorandom Binary Input Signal with Specified Clock Period.
[1] Söderström, T. and P. Stoica., Chapter C5.3 in System Identification, Prentice Hall, 1989.
[2] Ljung, L., Section 13.3 in System Identification: Theory for the User, Prentice Hall PTR, 1999.
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