# impulseest

Nonparametric impulse response estimation

## Description

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sys = impulseest(data) estimates an impulse response model 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 coefficients.

Use nonparametric impulse response estimation to analyze input/output data for feedback effects, delays, and significant time constants.

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sys = impulseest(data,n) estimates an nth-order impulse response model that corresponds to the time range 0:Ts:(n–1)*Ts, where Ts is the data sample time.

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sys = impulseest(data,n,nk) specifies a transport delay of nk samples in the estimated impulse response.

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sys = impulseest(___,opt) specifies estimation options using the options set opt. You can use this syntax with any of the previous input argument combinations.

## Examples

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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.

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 idtf model.

Analyze the impulse response of the identified model from time 0 to 1.

h = impulseplot(sys,1);

Determine the point at which a significant response to the impulse begins. First, display the region that bounds amplitudes that are not significantly different from zero. To do so, right-click the plot and select Characteristics > Confidence Region. For impulse response plots, by default, this selection displays a confidence region with a width of one standard deviation that is centered at zero, instead of one centered at the response values. You can modify these defaults by right-clicking the plot and selecting Properties > Options.

Alternatively, you can use the showConfidence command.

showConfidence(h);

The first response value that is significantly different than zero occurs at 0.24 seconds, or the third sample. This implies that the transport delay is three samples. To generate a model that imposes the three-sample delay, set the transport delay, which is the third argument, to 3. You must also set the second argument, the 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 0.24 seconds.

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

Set n to [] 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 a 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, which 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 1 second. Encapsulate y in an iddata object.

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 zero.

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 ('TC').

sys1 = impulseest(z3);

Estimate a model with no regularization.

opt = impulseestOptions('RegularizationKernel','none');
sys2 = impulseest(z3,opt);

Compare the impulse responses of both models.

h = impulseplot(sys2,sys1,70);
legend('sys2','sys1')

As the plot shows, using regularization makes the response smoother.

Plot the confidence intervals.

showConfidence(h);

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.

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.

impulse(trueSys,m1,m2,50);
legend('trueSys','m1','m2');

The three model responses are similar.

Use the empirical impulse response to measured data to determine whether the data includes feedback effects. Feedback effects can 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.

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 zero-response region at one standard deviation by right-clicking on the plot and selecting Characteristics > Confidence Region. Alternatively, you can use the showConfidence command.

showConfidence(h);

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 can also indicate direct feedback, such as proportional control without some delay so that y(t) partly determines u(t).

Other indications of feedback in the data are the significant response values such as those at -7 seconds and -9 seconds.

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 0.1 seconds. 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 responses.

Estimate the impulse response model sys1 (nonparametric) and state-space model sys2 (parametric) using the estimation data set 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 sys1 and sys2.

step(sys1,'b',sys2,'r');
legend('Impulse response model','State-space model');

## Input Arguments

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Estimation data, specified as an iddata object, an idfrd object, or an frd (Control System Toolbox) object, with at least one input signal and a nonzero sample time.

For time-domain estimation, specify data as an iddata object containing the input and output signal values.

For frequency-domain estimation, specify data as one of the following:

• Frequency response data (idfrd object or frd object)

• iddata object with properties specified as follows:

• InputData — Fourier transform of the input signal

• OutputData — Fourier transform of the output signal

• Domain‘Frequency’

Order of the FIR model, specified as a positive integer, [], or a matrix.

• If 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 positive integer.

• If data contains Nu input channels and Ny output channels, and you want to specify individual model orders for the input/output pairs, specify n as an Ny-by-Nu matrix of positive integers, such that N(i,j) represents the length of the impulse response from input j to output i.

• If you want the function to determine the order automatically, specify n as []. The software uses persistence-of-excitation analysis on the input data to select the order.

Example: sys = impulseest(data,70) estimates an impulse response model of order 70.

Transport delay in the estimated impulse response, specified as a scalar integer, 'negative', or an Ny-by-Nu 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 scalar integers.

• If you do not know the delay value, specify nk as 0. Once you estimate the impulse response, you can determine the true delay from the nonsignificant impulse response values in the beginning portion of the response. For an example of finding a 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 nk as 'negative' to 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, specify nk as 1.

The function stores positive values of nk greater than 1 in the IODelay property of sys (sys.IODelay = max(nk-1,0)), and negative values in the InputDelay property.

Estimation options, specified as an impulseestOptions option set, that specify the following:

• Prefilter order

• Regularization algorithm

• Input and output data offsets

• Advanced options such as structure

Use impulseestOptions to create the options set.

## Output Arguments

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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 the Report property of the model. Report has the following fields.

Report FieldDescription
Status

Summary of the model status, which indicates whether the model was created by construction or obtained by estimation.

Method

Estimation command used.

Fit

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:

FieldDescription
FitPercent

Normalized root mean squared error (NRMSE) measure of how well the response of the model fits the estimation data, expressed as the percentage fitpercent = 100(1-NRMSE).

LossFcn

Value of the loss function when the estimation completes.

MSE

Mean squared error (MSE) measure of how well the response of the model fits the estimation data.

FPE

Final prediction error for the model.

AIC

Raw Akaike Information Criteria (AIC) measure of model quality.

AICc

Small-sample-size corrected AIC.

nAIC

Normalized AIC.

BIC

Bayesian Information Criteria (BIC).

Parameters

Estimated values of model parameters.

OptionsUsed

Option set used for estimation. If no custom options were configured, this field is a set of default options. See impulseestOptions for more information.

RandState

State of the random number stream at the start of estimation. Empty, [], if randomization was not used during estimation. For more information, see rng.

DataUsed

Attributes of the data used for estimation, returned as a structure with the following fields.

FieldDescription
Name

Name of the data set.

Type

Data type.

Length

Number of data samples.

Ts

Sample time.

InterSample

Input intersample behavior, returned as one of the following values:

• 'zoh' — Zero-order hold maintains a piecewise-constant input signal between samples.

• 'foh' — First-order hold maintains a piecewise-linear input signal between samples.

• 'bl' — Band-limited behavior specifies that the continuous-time input signal has zero power above the Nyquist frequency.

InputOffset

Offset removed from time-domain input data during estimation. For nonlinear models, it is [].

OutputOffset

Offset removed from time-domain output data during estimation. For nonlinear models, it is [].

## Tips

• To view the impulse or step response of sys, use either impulseplot or stepplot, respectively.

• A response value that corresponds to a negative time value and that is significantly different from zero in the impulse response of sys indicates the presence of feedback in the data.

• To view the region of responses that are not significantly different from zero (the zero-response region) 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 confidence in significance depends on the number of standard deviations specified in showConfidence or options in the property editor. The default value is 1 standard deviation, which gives 68% confidence. A common choice is 3 standard deviations, which gives 99.7% confidence.

## Algorithms

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:

$y\left(t\right)={\int }_{-\infty }^{t}g\left(\tau \right)u\left(t-\tau \right)d\tau$

In discrete time:

$y\left(t\right)=\sum _{k=1}^{\infty }g\left(k\right)u\left(t-k\right)$

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:

• Regularization — 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 estimate.

This option is essential because the model order n can often be quite large. In cases without regularization, n can be automatically decreased to secure a reasonable variance.

Specify the regularizing kernel using the RegularizationKernel name-value argument of impulseestOptions.

• Prewhitening — 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 algorithm:

1. Defines a filter A of order PW that whitens the input signal u:

1/A = A(u)e, where A is a polynomial and e is white noise.

2. Filters the inputs and outputs with A:

uf = Au, yf = Ay

3. Uses the filtered signals uf and yf for estimation.

Specify prewhitening using the PW name-value pair argument of impulseestOptions.

• Autoregressive Parameters — Complement the basic underlying FIR model by NA autoregressive parameters, making it an ARX model.

$y\left(t\right)=\sum _{k=1}^{n}g\left(k\right)u\left(t-k\right)-\sum _{k=1}^{NA}{a}_{k}y\left(t-k\right)$

This option both gives better results for small n values and allows unbiased estimates when data are generated in closed loop. impulseest sets NA to 5 when t > 0 and sets NA to 0 (no autoregressive component) when t < 0.

• Noncausal effects — Include response to negative lags. Use this option if the estimation data includes output feedback:

$u\left(t\right)=\sum _{k=0}^{\infty }h\left(k\right)y\left(t-k\right)+r\left(t\right)$

where 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, mi = impulseest(data,nk,nb), where nk < 0, returns a model mi with an impulse response

$\left[h\left(-nk\right),h\left(-nk-1\right),...,h\left(0\right),g\left(1\right),g\left(2\right),...,g\left(nb+nk\right)\right]$

that has an alignment that corresponds to the lags $\left[nk,nk+1,..,0,1,2,...,nb+nk\right]$. The algorithm achieves this alignment because the input delay (InputDelay) of model mi is nk.

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 ij element of the matrix g(k) describes the behavior of the ith output after an impulse in the jth input.