Nonparametric impulse response estimation
estimates an impulse response model
sys = impulseest(
sys, also known as a finite
impulse response (FIR) model, using time-domain or frequency-domain data
data. The function uses persistence-of-excitation analysis
on the input data to select the model order (number of nonzero impulse response
Use nonparametric impulse response estimation to analyze input/output data for feedback effects, delays and significant time constants.
Estimate a nonparametric impulse response model using data from a hair dryer. The input is the voltage applied to the heater and the output is the heater temperature. Use the first 500 samples for estimation.
Load the data and use the first 500 samples to estimate the model.
load dry2 ze = dry2(1:500); sys = impulseest(ze);
ze is an
iddata object that contains time-domain data.
sys, the identified nonparametric impulse response model, is an
Analyze the impulse response of the identified model from time 0 to 1.
h = impulseplot(sys,1);
Right-click the plot and select Characteristics > Confidence Region to view the region of statistically zero response. Alternatively, you can use the
The first significantly nonzero response value occurs at 0.24 seconds, or the third lag. This implies that the transport delay is 3 samples. To generate a model that imposes the 3-lag delay, set the transport delay, which is the third argument, to 3. You must also set the second argument, order
n, to its default value of
 as a placeholder.
sys1 = impulseest(ze,,3); h1 = impulseplot(sys1,1); showConfidence(h1);
The response is identically zero until 0.24s.
Load the estimation data
load iddata3 z3;
Estimate a 35th order FIR model.
n = 35; sys = impulseest(z3,n);
You can confirm the model order of
sys by displaying the number of terms.
nsys = size(sys.num)
nsys = 1×2 1 35
 so that the function automatically determines
n. Display the model order.
n = ; sys1 = impulseest(z3,n); nsys1 = size(sys1.Numerator)
nsys1 = 1×2 1 70
The model order is 70. The default value for the order is
, so setting the order to
 is equivalent to omitting the specification.
Estimate an impulse response model with transport delay of 3 samples.
If you know about the presence of delay in the input/output data in advance, use the delay value as a transport delay for impulse response estimation.
Generate data that contains a 3-sample input-to-output lag.
Create a random input signal. Construct an
idpoly model that includes three sample delays that you implement by using three leading zeros in the B polynomial.
u = rand(100,1); A = [1 .1 .4]; B = [0 0 0 4 -2]; C = [1 1 .1]; sys = idpoly(A,B,C);
Simulate the model response
y to the noise signal, using the
AddNoise option and a sample time of 1s. Encapsulate
y in an
opt = simOptions('AddNoise',true); y = sim(sys,u,opt); data = iddata(y,u,1);
Estimate and plot a 20th order model with no transport delay.
n = 20; model1 = impulseest(data,n); impulseplot(model1);
The plot shows that the impulse response includes nonzero samples during the 3-second delay period.
Estimate a model with a 3-sample transport delay.
nk = 3; model2 = impulseest(data,n,nk); impulseplot(model2)
The first three samples are identically 0.
Obtain regularized estimates of impulse response model using the regularizing kernel estimation option.
Estimate a model using regularization.
impulseest performs regularized estimates by default, using the tuned and correlated kernel (
load iddata3 z3; sys1 = impulseest(z3);
Estimate a model with no regularization.
opt = impulseestOptions('RegularizationKernel','none'); sys2 = impulseest(z3,opt);
Compare the impulse response of both models.
h = impulseplot(sys2,sys1,70); legend('sys2','sys1')
As the plot shows, using regularization makes the response smoother.
Plot the confidence interval.
The uncertainty in the computed response is reduced at larger lags for the model using regularization. Regularization decreases variance at the price of some bias. The tuning of the
'TC' regularization is such that the variance error dominates the overall error.
Load the estimation data.
load regularizationExampleData eData;
Recreate the transfer function model that was used for generating the estimation data (true system).
num = [0.02008 0.04017 0.02008]; den = [1 -1.561 0.6414]; Ts = 1; trueSys = idtf(num,den,Ts);
Obtain a regularized impulse response (FIR) model with an order of 70.
opt = impulseestOptions('RegularizationKernel','DC'); m0 = impulseest(eData,70,opt);
Convert the model into a state-space model and reduce the model order.
m1 = idss(m0); m1 = balred(m1,15);
Estimate a second state-space model directly from
eData by using regularized reduction of an ARX model.
m2 = ssregest(eData,15);
Compare the impulse responses of the true system and the estimated models.
The three model responses are similar.
Use the empirical impulse response to measured data to determine whether the data includes feedback effects. Feedback effects may be present when the impulse response includes statistically significant response values for negative time values.
Compute the noncausal impulse response using a fourth-order prewhitening filter and no regularization, automatic order selection, and negative lag.
load iddata3 z3; opt = impulseestOptions('pw',4,'RegularizationKernel','none'); sys = impulseest(z3,,'negative',opt);
sys is a noncausal model containing response values for negative time.
Analyze the impulse response of the identified model.
h = impulseplot(sys);
View the statistically zero-response region by right-clicking on the plot and selecting Characteristics > Confidence Region. Alternatively, you can use the
The large response value at
t=0 (zero lag) suggests that the data comes from a process containing feedthrough. That is, the input affects the output instantaneously. The large response value could also indicate direct feedback, such as proportional control without some delay so that y
(t) partly determines u
Other indications of feedback in the data are the significant response values such as those at -7s and -9s.
Compute an impulse response model for frequency response data.
Load the frequency response data, which contains measured amplitude
AMP and phase
PHA for the frequency vector W
Create the complex frequency response
zfr and encapsulate it in an
idfrd object that has a sample time of 0.1s. Plot the data.
zfr = AMP.*exp(1i*PHA*pi/180); Ts = 0.1; data = idfrd(zfr,W,Ts);
Estimate an impulse response model from
data and plot the response.
sys = impulseest(data); impulseplot(sys)
Identify parametric and nonparametric models for a data set, and compare their step response.
Estimate the impulse response model
sys1 (nonparametric) and state-space model
sys2 (parametric) using the estimation data set
load iddata1 z1; sys1 = impulseest(z1); sys2 = ssest(z1,4);
sys1 is a discrete-time identified transfer function model.
sys2 is a continuous-time identified state-space model.
Compare the step responses for
step(sys1,'b',sys2,'r'); legend('impulse response model','state-space model');
data— Estimation data
For time-domain estimation, specify
data as an
iddata object containing the input and output
For frequency-domain estimation, specify
data as one
of the following:
Frequency response data (
idfrd object or
iddata object with properties specified as
InputData — Fourier transform of the
OutputData — Fourier transform of the
Domain — ‘Frequency’
n— Order of FIR model
(default) | positive integer | matrix
Order of the FIR model, specified as a positive integer,
, or a matrix.
data contains a single input channel
and output channel, or if you want to apply the same model order
to all input/output pairs, specify
n as a
channels and Ny output
channels, and you want to specify individual model orders for
the input/output pairs, specify
n as an
matrix of positive integers, such that
represents the length of impulse response from input
j to output i.
If you want the function to determine the order automatically,
software uses persistence-of-excitation analysis on the input
data to select the
sys = impulseest(data,70) estimates an impulse
response model of order 70.
nk— Transport delay
1| scalar integer | matrix
Transport delay in the estimated impulse response, specified as a scalar
'negative', or an
matrix, where Ny is the number of
outputs and Nu is the number of
inputs. The impulse response (input
j to output
i) coefficients correspond to the time span
nk(i,j)*Ts : Ts : (n(ij)+nk(i,j)-1)*Ts.
If you know the value of the transport delay, specify
nk as a scalar integer or a matrix of
If you do not know the delay value, specify
0. Once you have
estimated the impulse response, you can determine the true delay
from the insignificant impulse response values in the beginning
portion of the response. For an example of finding true delay, see
Identify Nonparametric Impulse Response Model from Data.
To generate the impulse response coefficients for negative time
values, which is useful for feedback analysis, use a negative
integer. If you specify a negative value, the value must be the same
across all output channels. You can also specify
automatically pick negative lags for all input/output channels of
the model. For an example of using negative time values, see Test Measured Data for Feedback Effects.
To create a system whose leading numerator coefficient is zero,
The function stores positive values of
nk greater than
1 in the
IODelay property of
negative values in the
opt— Estimation options
Estimation options, specified as an
impulseestOptions option set,
that specify the following:
Input and output data offsets
Advanced options such as structure
impulseestOptions to create the options
sys— Estimated impulse response model
Estimated impulse response model, returned as an
idtf model that encapsulates
an FIR model.
Information about the estimation results and options used is stored in
Report property of the model.
Report has the following fields:
Summary of the model status, which indicates whether the model was created by construction or obtained by estimation.
Estimation command used.
Quantitative assessment of the estimation, returned as a structure. See Loss Function and Model Quality Metrics for more information on these quality metrics. The structure has the following fields:
Estimated values of model parameters.
Option set used for estimation. If no custom
options were configured, this is a set of default
State of the random number stream at the start of estimation. Empty,
Attributes of the data used for estimation, returned as a structure with the following fields:
For more information on using
Report, see Estimation Report.
A significant value of the impulse response of
negative time values indicates the presence of feedback in the data.
To view the region of insignificant impulse response (statistically zero) in a
plot, right-click on the plot and select Characteristics > Confidence Region. A patch depicting the zero-response region appears on the plot.
The impulse response at any time value is significant only if it lies outside
the zero response region. The level of significance depends on the number of
standard deviations specified in
showConfidence or options in the
property editor. A common choice is 3 standard deviations, which gives 99.7%
Correlation analysis refers to methods that estimate the impulse response of a linear model, without specific assumptions about model orders.
The impulse response, g, is the system output when the input is an impulse signal. The output response to a general input, u(t), is the convolution with the impulse response. In continuous time:
The values of g(k) are the discrete-time impulse response coefficients.
You can estimate the values from observed input/output data in several different ways.
impulseest estimates the first
n coefficients using the
least-squares method to obtain a finite impulse response (FIR) model
of order n.
impulseest provides several important options for the estimation:
Regularize the least-squares estimate. With
regularization, the algorithm forms an estimate of
the prior decay and mutual correlation among
g(k), and then merges this
prior estimate with the current information about
g from the observed data. This
approach results in an estimate that has less
variance but also some bias. You can choose one of
several kernels to encode the prior
This option is essential because the model
n can often be quite
large. In cases where there is no regularization,
n can be automatically
decreased to secure a reasonable variance.
Specify the regularizing kernel using the
pair argument of
Prewhiten the input by applying an input-whitening
filter of order
PW to the data.
Use prewhitening when you are performing
unregularized estimation. Using a prewhitening
filter minimizes the effect of the neglected tail
k > n) of the impulse
response. To achieve prewhitening, the
Defines a filter
PW that whitens the input
1/A = A(u)e, where
A is a polynomial and
e is white noise.
Filters the inputs and outputs with
uf = Au,
Uses the filtered signals
Specify prewhitening using the
PW name-value pair argument of
Parameters — Complement the basic
underlying FIR model by
autoregressive parameters, making it an ARX
This option gives both better results for
n values and allows
unbiased estimates when data are generated in
impulseest uses NA = 5 for t>0 and NA = 0 (no autoregressive component)
Noncausal effects — Include response to negative lags. Use this option if the estimation data includes output feedback:
h(k) is the
impulse response of the regulator and
r is a setpoint or disturbance
term. The algorithm handles the existence and
character of such feedback h,
and estimates h in the same way
as g by simply trading places
between y and
u in the estimation call. Using
impulseest with an indication
of negative delays, returns a model
mi with an impulse
that has an alignment that corresponds to lags . The algorithm achieves this
alignment because the input delay
InputDelay) of model
For a multi-input multi-output system, the impulse response g(k) is an ny-by-nu matrix, where ny is the number of outputs and nu is the number of inputs. The i–j element of the matrix g(k) describes the behavior of the ith output after an impulse in the jth input.