## Estimating Models Using Frequency-Domain Data

The System Identification Toolbox™ software lets you use frequency-domain data to identify linear models at the command line and in the System Identification app. You can estimate both continuous-time and discrete-time linear models using frequency-domain data. This topic presents an overview of model estimation in the toolbox using frequency-domain data. For an example of model estimation using frequency-domain data, see Frequency Domain Identification: Estimating Models Using Frequency Domain Data.

Frequency-domain data can be of two types:

• Frequency domain input-output data — You obtain the data by computing Fourier transforms of time-domain input, u(t), and output, y(t), signals. The data is the set of input, U(ω), and output, Y(ω), signals in frequency domain. In the toolbox, frequency-domain input-output data is represented using iddata objects. For more information, see Representing Frequency-Domain Data in the Toolbox.

• Frequency-response data — Also called frequency function or frequency-response function (FRF), the data consists of transfer function measurements, G(), of a system at a discrete set of frequencies ω. Frequency-response data at a frequency ω tells you how a linear system responds to a sinusoidal input of the same frequency. In the toolbox, frequency-response data is represented using idfrd objects. For more information, see Representing Frequency-Domain Data in the Toolbox. You can obtain frequency-response data in the following ways:

• Measure the frequency-response data values directly, such as by using a spectrum analyzer.

• Perform spectral analysis of time-domain or frequency-domain input-output data (iddata objects) using commands such as spa and spafdr.

• Compute the frequency-response of an identified linear model using commands such as freqresp, bode, and idfrd.

The workflow for model estimation using frequency-domain data is the same as that for estimation using time-domain data. If needed, you first prepare the data for model identification by removing outliers and filtering the data. You then estimate a linear parametric model from the data, and validate the estimation.

### Advantages of Using Frequency-Domain Data

Using frequency-domain data has the following advantages:

• Data compression — You can compress long records of data when you convert time-domain data to frequency domain. For example, you can use logarithmically spaced frequencies.

• Non uniformity — Frequency-domain data does not have to be uniformly spaced. Your data can have frequency-dependent resolution so that more data points are used in the frequency regions of interest. For example, the frequencies of interest could be the bandwidth range of a system, or near the resonances of a system.

• Prefiltering — Prefiltering of data in the frequency-domain becomes simple. It corresponds to assigning different weights to different frequencies of the data.

• Continuous-time signal - You can represent continuous-time signals using frequency-domain data and use the data for estimation.

### Representing Frequency-Domain Data in the Toolbox

Before performing model estimation, you specify the frequency-domain data as objects in the toolbox. You can specify both continuous-time and discrete-time frequency-domain data.

• Frequency domain input-output data — Specify as an iddata object. In the object, you store U(ω), Y(ω), and frequency vector ω. The Domain property of the object is 'Frequency', to specify that the object contains frequency-domain signals. If U(ω), Y(ω) are discrete-time Fourier transforms of discrete-time signals, sampled with sampling interval Ts, denote the sampling interval in the iddata object. If U(ω), Y(ω) are Fourier transforms of continuous-time signals, specify Ts as 0 in the iddata object.

To plot the data at the command line, use the plot command.

For example, you can plot the phase and magnitude of frequency-domain input-output data.

Load time-domain input-output data.

The time-domain inputs u and outputs y are stored in z1, an iddata object whose Domain property is set to 'Time'.

Fourier-transform the data to obtain frequency-domain input-output data.

zf = fft(z1);

The Domain property of zf is set to 'Frequency', indicating that it is frequency-domain data.

Plot the magnitude and phase of the frequency-domain input-output data.

plot(zf)

• Frequency-response data — Specify as an idfrd object. If you have Control System Toolbox™ software, you can also specify the data as an frd (Control System Toolbox) object.

To plot the data at the command line, use the bode command.

For example, you can plot the frequency-response of a transfer function model.

Create a transfer function model of your system.

sys = tf([1 0.2],[1 2 1 1]);

Calculate the frequency-response of the transfer function model, sys, at 100 frequency points. Specify the range of the frequencies as 0.1 rad/s to 10 rad/s.

freq = logspace(-1,1,100);
frdModel = idfrd(sys,freq);

Plot the frequency-response of the model.

bode(frdModel)

For more information about the frequency-domain data types and how to specify them, see Frequency-Domain Data Representation.

You can also transform between frequency-domain and time-domain data types using the following commands.

Original Data FormatTo Time-Domain Data
(iddata object)
To Frequency-Domain Data
(iddata object)
To Frequency-Response Data
(idfrd object)
Time-Domain Data
(iddata object)
N/AUse fft
• Use etfe, spa, or spafdr.

• Estimate a linear parametric model from the iddata object, and use idfrd to compute frequency-response data.

Frequency-Domain Data
(iddata object)
Use ifft (works only for evenly spaced frequency-domain data).N/A
• Use etfe, spa, or spafdr.

• Estimate a linear parametric model from the iddata object, and use idfrd to compute frequency-response data.

Frequency-Response Data
(idfrd object)
Not supportedUse iddata. The software creates a frequency-domain iddata object that has the same ratio between output and input as the original idfrd object frequency-response data.
• Use spafdr. The software calculates frequency-response data with a different resolution (number and spacing of frequencies) than the original data.

For more information about transforming between data types in the app or at the command line, see the Transform Data category page.

#### Continuous-Time and Discrete-Time Frequency-Domain Data

Unlike time-domain data, the sample time Ts of frequency-domain data can be zero. Frequency-domain data with zero Ts is called continuous-time data. Frequency-domain data with Ts greater than zero is called discrete-time data.

You can obtain continuous-time frequency-domain data (Ts = 0) in the following ways:

• Generate the data from known continuous-time analytical expressions.

For example, suppose that you know the frequency-response of your system is $G\left(\omega \right)=1/\left(b+j\omega \right)$, where b is a constant. Also assume that the time-domain inputs to your system are, $u\left(t\right)={e}^{-at}\mathrm{sin}{w}_{0}t$, where a is a constant greater than zero, and u(t) is zero for all times t less than zero. You can compute the Fourier transform of u(t) to obtain

$U\left(\omega \right)={\omega }_{0}/\left[{\left(a+j\omega \right)}^{2}+{w}_{0}{}^{2}\right]$

Using U(ω) and G(ω) you can then get the frequency-domain expression for the outputs:

$Y\left(\omega \right)=G\left(\omega \right)U\left(\omega \right)$

You can now evaluate the analytical expressions for Y(ω) and U(ω) over a grid of frequency values $\left({\omega }_{grid}={\omega }_{1},{\omega }_{2},...,{\omega }_{n}\right)$, and get a vector of frequency-domain input-output data values $\left({Y}_{grid},{U}_{grid}\right)$. You can package the input-output data as a continuous-time iddata object by specifying a zero sample time, Ts.

Ts = 0;
zf = iddata(Ygrid,Ugrid,Ts,'Frequency',wgrid)
• Compute the frequency response of a continuous-time linear system at a grid of frequencies.

For example, in the following code, you generate continuous-time frequency-response data, FRDc, from a continuous-time transfer function model, sys for a grid of frequencies, freq.

sys = idtf(1,[1 2 2]);
freq = logspace(-2,2,100);
FRDc = idfrd(sys,freq);
• Measure amplitudes and phases from a sinusoidal experiment, where the measurement system uses anti-aliasing filters. You measure the response of the system to sinusoidal inputs at different frequencies, and package the data as an idfrd object. For example, the frequency-response data measured with a spectrum analyzer is continuous-time.

You can also conduct an experiment by using periodic, continuous-time signals (multiple sine waves) as inputs to your system and measuring the response of your system. Then you can package the input and output data as an iddata object.

You can obtain discrete-time frequency-domain data (Ts >0) in the following ways:

• Transform the measured time-domain values using a discrete Fourier transform.

For example, in the following code, you compute the discrete Fourier transform of time-domain data, y, that is measured at discrete time-points with sample time 0.01 seconds.

t = 0:0.01:10;
y = iddata(sin(2*pi*10*t),[],0.01);
Y = fft(y);
• Compute the frequency response of a discrete-time linear system.

For example, in the following code, you generate discrete-time frequency-response data, FRDd, from a discrete-time transfer function model, sys. You specify a non-zero sample time for creating the discrete-time model.

Ts = 1;
sys = idtf(1,[1 0.2 2.1],Ts);
FRDd = idfrd(sys,logspace(-2,2,100));

You can use continuous-time frequency-domain data to identify only continuous-time models. You can use discrete-time frequency-domain data to identify both discrete-time and continuous-time models. However, identifying continuous-time models from discrete-time data requires knowledge of the intersample behavior of the data. For more information, see Estimating Continuous-Time and Discrete-Time Models.

Note

For discrete-time data, the software ignores frequency-domain data above the Nyquist frequency during estimation.

### Preprocessing Frequency-Domain Data for Model Estimation

After you have represented your frequency-domain data using iddata or idfrd objects, you can prepare the data for estimation by removing spurious data and by filtering the data.

To view the spurious data, plot the data in the app, or use the plot (for iddata objects) or bode (for idfrd objects) commands. After identifying the spurious data in the plot, you can remove them. For example, if you want to remove data points 20–30 from zf, a frequency-domain iddata object, use the following syntax:

zf(20:30) = [];

Since frequency-domain data does not have to be specified with a uniform spacing, you do not need to replace the outliers.

You can also prefilter high-frequency noise in your data. You can prefilter frequency-domain data in the app, or use idfilt at the command line. Prefiltering data can also help remove drifts that are low-frequency disturbances. In addition to minimizing noise, prefiltering lets you focus your model on specific frequency bands. The frequency range of interest often corresponds to a passband over the breakpoints on a Bode plot. For example, if you are modeling a plant for control-design applications, you can prefilter the data to enhance frequencies around the desired closed-loop bandwidth.

### Estimating Linear Parametric Models

After you have preprocessed the frequency-domain data, you can use it to estimate continuous-time and discrete-time models.

#### Supported Model Types

You can estimate the following linear parametric models using frequency-domain data. The noise component of the models is not estimated, except for ARX models.

Model TypeAdditional InformationEstimation CommandsEstimation in the App
Transfer Function Models See Estimate Transfer Function Models in the System Identification App.
State-Space ModelsEstimated K matrix of the state-space model is zero.
See Estimate State-Space Models in System Identification App.
Process ModelsDisturbance model is not estimated.See Estimate Process Models Using the App.
Input-Output Polynomial ModelsYou can estimate only output-error and ARX models.
See Estimate Polynomial Models in the App.
Linear Grey-Box ModelsModel parameters that are only related to the noise matrix K are not estimated.Grey-box model estimation is not available in the app.
Correlation Models
(Impulse-response models)
See Estimate Impulse-Response Models Using System Identification App.
Frequency-Response Models
(Estimated as idfrd objects)

See Estimate Frequency-Response Models in the App.

Before performing the estimation, you can specify estimation options, such as how the software treats initial conditions of the estimation data. To do so at the command line, use the estimation option set corresponding to the estimation command. For example, suppose that you want to estimate a transfer function model from frequency-domain data, zf, and you also want to estimate the initial conditions of the data. Use the tfestOptions option set to specify the estimation options, and then estimate the model.

opt = tfestOptions('InitialCondition','estimate');
sys = tfest(zf,opt);

sys is the estimated transfer function model. For information about extracting estimated parameter values from the model, see Extracting Numerical Model Data. After performing the estimation, you can validate the estimated model.

Note

A zero initial condition for time-domain data does not imply a zero initial condition for the corresponding frequency-domain data. For time-domain data, zero initial conditions mean that the system is assumed to be in a state of rest before the start of data collection. In the frequency-domain, initial conditions can be ignored only if the data collected is periodic in nature. Thus, if you have time-domain data collected with zero initial conditions, and you convert it to frequency-domain data to estimate a model, you have to estimate the initial conditions as well. You cannot specify them as zero.

You cannot perform the following estimations using frequency-domain data:

• Estimation of the noise component of a linear model, except for ARX models.

• Estimation of time series models using spectrum data only. Spectrum data is the power spectrum of a signal, commonly stored in the SpectrumData property of an idfrd object.

#### Estimating Continuous-Time and Discrete-Time Models

You can estimate all the supported linear models as discrete-time models, except for process models. Process models are defined in continuous-time only. For the estimation of discrete-time models, you must use discrete-time data.

You can estimate all the supported linear models as continuous-time models, except for correlation models (see impulseest). You can estimate continuous-time models using both continuous-time and discrete-time data. For information about continuous-time and discrete-time data, see Continuous-Time and Discrete-Time Frequency-Domain Data.

If you are estimating a continuous-time model using discrete-time data, you must specify the intersample behavior of the data. The specification of intersample behavior depends on the type of frequency-domain data.

• Discrete-time frequency-domain input-output data (iddata object) — Specify the intersample behavior of the time-domain input signal u(t) that you Fourier transformed to obtain the frequency-domain input signal U(ω).

• Discrete-time frequency-response data (idfrd object) — The data is generated by computing the frequency-response of a discrete-time model. Specify the intersample behavior as the discretization method assumed to compute the discrete-time model from an underlying continuous-time model. For an example, see Specify Intersample Behavior for Discrete-Time Frequency-Response Data.

You can specify the intersample behavior to be piecewise constant (zero-order hold), linearly interpolated between the samples (first-order hold), or band-limited. If you specify the discrete-time data from your system as band-limited (that is no power above the Nyquist frequency), the software treats the data as continuous-time by setting the sample time to zero. The software then estimates a continuous-time model from the data. For more information, see Effect of Input Intersample Behavior on Continuous-Time Models.

#### Specify Intersample Behavior for Discrete-Time Frequency-Response Data

This example shows the effect of intersample behavior on the estimation of continuous-time models using discrete-time frequency-response data.

Generate discrete-time frequency-response data. To do so, first construct a continuous-time transfer function model, sys. Then convert it to a discrete-time model, sysd, using the c2d command and first-order hold (FOH) method. Use the discrete-time model sysd to generate frequency-response data at specified frequencies, freq.

sys = idtf([1 0.2],[1 2 1 1]);
sysd = c2d(sys,1,c2dOptions('Method','foh'));
freq = logspace(-1,0,10);
FRdata = idfrd(sysd,freq);

FRdata is discrete-time data. The software sets the InterSample property of FRdata to 'foh', which is the discretization method that was used to obtain sysd from sys.

Estimate a third-order continuous-time transfer function from the discrete-time data.

model1 = tfest(FRdata,3,1)
model1 =

s + 0.2
-------------------
s^3 + 2 s^2 + s + 1

Continuous-time identified transfer function.

Parameterization:
Number of poles: 3   Number of zeros: 1
Number of free coefficients: 5
Use "tfdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:
Estimated using TFEST on frequency response data "FRdata".
Fit to estimation data: 100%
FPE: 4.265e-31, MSE: 4.175e-32

model1 is a continuous-time model, estimated using discrete-time frequency-response data. The underlying continuous-time dynamics of the original third-order model sys are retrieved in model1 because the correct intersample behavior is specified in FRdata.

Now, specify the intersample behavior as zero-order hold (ZOH), and estimate a third-order transfer function model.

FRdata.InterSample = 'zoh';
model2 = tfest(FRdata,3,1)
model2 =

-15.49 s - 3.264
---------------------------------
s^3 - 30.03 s^2 - 6.872 s - 17.02

Continuous-time identified transfer function.

Parameterization:
Number of poles: 3   Number of zeros: 1
Number of free coefficients: 5
Use "tfdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:
Estimated using TFEST on frequency response data "FRdata".
Fit to estimation data: 94.82%
FPE: 0.004674, MSE: 0.001557

model2 does not capture the dynamics of the original model sys. Thus, sampling related errors are introduced in the model estimation when the intersample behavior is not correctly specified.

#### Convert Frequency-Response Data Model to a Transfer Function

This example shows how to convert a frequency-response data (FRD) model to a transfer function model. You treat the FRD model as estimation data and then estimate the transfer function.

Obtain an FRD model.

For example, use bode to obtain the magnitude and phase response data for the following fifth-order system:

$G\left(s\right)=\frac{s+0.2}{{s}^{5}+{s}^{4}+0.8{s}^{3}+0.4{s}^{2}+0.12s+0.04}$

Use 100 frequency points between 0.1 rad/s to 10 rad/s to obtain the FRD model. Use frd to create a frequency-response model object.

freq = logspace(-1,1,100);
sys0 = tf([1 0.2],[1 1  0.8  0.4  0.12  0.04]);
[mag,phase] = bode(sys0,freq);
frdModel = frd(mag.*exp(1j*phase*pi/180),freq);

Obtain the best third-order approximation to the system dynamics by estimating a transfer function with 3 zeros and 3 poles.

np = 3;
nz = 3;
sys = tfest(frdModel,np,nz);

sys is the estimated transfer function.

Compare the response of the FRD Model and the estimated transfer function model.

bode(frdModel,sys,freq(1:50));

The FRD model is generated from the fifth-order system sys0. While sys, a third-order approximation, does not capture the entire response of sys0, it captures the response well until approximately 0.6 rad/s.

### Validating Estimated Model

After estimating a model for your system, you can validate whether it reproduces the system behavior within acceptable bounds. It is recommended that you use separate data sets for estimating and validating your model. You can use time-domain or frequency-domain data to validate a model estimated using frequency-domain data. If you are using input-output validation data to validate the estimated model, you can compare the simulated model response to the measured validation data output. If your validation data is frequency-response data, you can compare it to the frequency response of the model. For example, to compare the output of an estimated model sys to measured validation data zv, use the following syntax:

compare(zv,sys);

You can also perform a residual analysis. For more information see, Validating Models After Estimation.

#### Troubleshooting Frequency-Domain Identification

When you estimate a model using frequency-domain data, the estimation algorithm minimizes a loss (cost) function. For example, if you estimate a SISO linear model from frequency-response data f, the estimation algorithm minimizes the following least-squares loss function:

$\underset{G\left(\omega \right)}{\text{minimize}}\sum _{k=1}^{{N}_{f}}{|W\left({\omega }_{k}\right)\left(G\left({\omega }_{k}\right)-f\left({\omega }_{k}\right)\right)|}^{2}$

Here W is a frequency-dependent weight that you specify, G is the linear model that is to be estimated, ω is the frequency, and Nf is the number of frequencies at which the data is available. The quantity $\left(G\left({\omega }_{k}\right)-f\left({\omega }_{k}\right)\right)$ is the frequency-response error. For frequency-domain input-output data, the algorithm minimizes the weighted norm of the output error instead of the frequency-response error. For more information, see Loss Function and Model Quality Metrics. During estimation, spurious or uncaptured dynamics in your data can effect the loss function and result in unsatisfactory model estimation.

• Unexpected, spurious dynamics — Typically observed when the high magnitude regions of data have low signal-to-noise ratio. The fitting error around these portions of data has a large contribution to the loss function. As a result the estimation algorithm may overfit and assign unexpected dynamics to noise in these regions. To troubleshoot this issue:

• Improve signal-to-noise ratio — You can gather more than one set of data, and average them. If you have frequency-domain input-output data, you can combine multiple data sets by using the merge command. Use this data for estimation to obtain an improved result. Alternatively, you can filter the dataset, and use it for estimation. For example, use a moving-average filter over the data to smooth the measured response. Apply the smoothing filter only in regions of data where you are confident that the unsmoothness is due to noise, and not due to system dynamics.

• Reduce the impact of certain portions of data on the loss function — You can specify a frequency-dependent weight. For example, if you are estimating a transfer function model, specify the weight in the WeightingFilter option of the estimation option set tfestOptions. Specify a small weight in frequency regions where the spurious dynamics exist. Alternatively, use fewer data points around this frequency region.

• Uncaptured dynamics — Typically observed when the dynamics you want to capture have a low magnitude relative to the rest of data. Since a poor fit to low magnitude data contributes less to the loss function, the algorithm may ignore these dynamics to reduce errors at other frequencies. To troubleshoot this issue:

• Specify a frequency-dependent weight — Specify a large weight for the frequency region where you would like to capture dynamics.

• Use more data points around this region.

For an example of these troubleshooting techniques, see Troubleshoot Frequency-Domain Identification of Transfer Function Models.

If you do not achieve a satisfactory model using these troubleshooting techniques, try a different model structure or estimation algorithm.

### Next Steps After Identifying a Model

After estimating a model, you can perform model transformations, extract model parameters, and simulate and predict the model response. Some of the tasks you can perform are:

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