frd

Frequency-response data model

Description

Use frd to create real-valued or complex-valued frequency-response data models, or to convert dynamic system models to frequency-response data model form.

Frequency-response data models store complex frequency response data with corresponding frequency points. For example, a frequency-response data model H(jw_i), stores the frequency response at each input frequency w_i, where i = 1,…,n. The frd model object can represent SISO or MIMO frequency-response data models in continuous time or discrete time. For more information, see Frequency Response Data (FRD) Models.

You can also use frd to create generalized frequency-response data (genfrd) models.

Creation

You can obtain frd models in one of the following ways.

• Create the model from frequency response data using the frd command. For example, you can create an frd model with frequency response data taken at specific frequencies.

  For an example, see SISO Frequency-Response Data Model.

• Convert a linear model such as an ss model into an frd model by computing the frequency response of the model at specified frequencies.

  For an example, see Convert State-Space Model to Frequency-Response Data Model.

• Estimate the model using offline frequency response estimation workflows. These workflows require Simulink^® Control Design™ software.

  For more information, see Estimate Frequency Response at the Command Line (Simulink Control Design) and Estimate Frequency Response Using Model Linearizer (Simulink Control Design).

Syntax

sys = frd(response,frequency)

sys = frd(response,frequency,ts)

sys = frd(response,frequency,ltiSys)

sys = frd(___,Name,Value)

sys = frd(ltiSys,frequency)

sys = frd(ltiSys,frequency,FrequencyUnits)

Description

sys = frd(response,frequency) creates a continuous-time frequency-response data (frd) model, setting the ResponseData and Frequency properties. frequency can contain both negative and positive frequencies.

sys = frd(response,frequency,ts) creates a discrete-time frd model with the sample time ts. To leave the sample time unspecified, set ts to –1.

sys = frd(response,frequency,ltiSys) creates a frequency-response data model with properties inherited from the dynamic system model ltiSys, including the sample time.

sys = frd(___,Name,Value) sets properties of the frequency-response data model using one or more name-value arguments for any of the previous input-argument combinations.

sys = frd(ltiSys,frequency) converts the dynamic system model ltiSys to a frequency-response data model. frd computes the frequency response at frequencies specified by frequency. sys inherits its frequency units rad/TimeUnit from ltiSys.TimeUnit.

sys = frd(ltiSys,frequency,FrequencyUnits) interprets frequencies in the units specified by FrequencyUnit.

Input Arguments

response — Frequency response data

vector | multidimensional array

Frequency response data, specified as a vector or a multidimensional array of complex numbers.

• For SISO systems, specify a vector of frequency response values at the frequency points specified in frequency.

• For MIMO systems with Nu inputs and Ny outputs, specify a Ny-by-Nu-by-Nf array, where Nf is the number of frequency points.

• For an S1-...-by-Sn array of models with Nu inputs and Ny outputs, specify a multidimensional array of size [Ny Nu Nf S1 … Sn].

  For instance, a response of size [Ny,Nu,Nf,3,4] represents the response data for a 3-by-4 array of models. Each model has Ny outputs, Nu inputs, and Nf frequency points.

This input sets the ResponseData property.

frequency — Frequency points

vector

Frequency points corresponding to response, specified as a vector that contains Nf points. frequency can contain both positive and negative frequencies.

This input sets the Frequency property.

ts — Sample time

scalar

Sample time, specified as a scalar.

The input sets the Ts property.

ltiSys — Dynamic system

dynamic system model | model array

Dynamic system, specified as a SISO or MIMO dynamic system model or an array of dynamic system models. Dynamic systems that you can use include:

• Continuous-time or discrete-time numeric LTI models, such as tf, zpk, ss, or pid models.

• Generalized or uncertain LTI models such as genss or uss (Robust Control Toolbox) models. (Using uncertain models requires Robust Control Toolbox™ software.)

  The resulting frd model assumes:

  • Current values of the tunable components for tunable control design blocks

  • Nominal model values for uncertain control design blocks

• Identified LTI models, such as idtf (System Identification Toolbox), idss (System Identification Toolbox), idproc (System Identification Toolbox), idpoly (System Identification Toolbox), and idgrey (System Identification Toolbox) models. (Using identified models requires System Identification Toolbox™ software.)

Properties

ResponseData — Frequency response data

multidimensional array of complex numbers

Frequency response data, specified as a multidimensional array of complex numbers.

• For SISO systems, ResponseData is a 1-by-1-by-Nf array of frequency response values at the Nf frequency points specified in the Frequency property.

• For MIMO systems with Nu inputs and Ny outputs, ResponseData is an Ny-by-Nu-by-Nf array, where Nf is the number of frequency points.

  For instance, ResponseData(ky,ku,kf) represents the frequency response from the input ku to the output ky at the frequency Frequency(kf).

• For an S1-...-by-Sn array of models with Nu inputs and Ny outputs, ResponseData is a multidimensional array of size [Ny Nu Nf S1 … Sn].

  For instance, a ResponseData of size [Ny,Nu,Nf,3,4] represents the response data for a 3-by-4 array of models. Each model has Ny outputs, Nu inputs, and Nf frequency points.

Frequency — Frequency points

vector

Frequency points corresponding to ResponseData, specified as a vector that contains Nf points in the units specified by FrequencyUnit.

FrequencyUnit — Units for frequency vector

'rad/TimeUnit' (default) | 'cycles/TimeUnit' | 'rad/s' | 'Hz' | 'kHz' | 'MHz' | 'GHz' | 'rpm'

Units of the frequency vector in the Frequency property, specified as one of the following values:

• 'rad/TimeUnit'

• 'cycles/TimeUnit'

• 'rad/s'

• 'Hz'

• 'kHz'

• 'MHz'

• 'GHz'

• 'rpm'

The units 'rad/TimeUnit' and 'cycles/TimeUnit' are relative to the time units specified in the TimeUnit property.

Changing this property does not resample or convert the data. Modifying the property changes only the interpretation of the existing data. Use chgFreqUnit to convert the data to different frequency units.

IODelay — Transport delay

0 (default) | scalar | Ny-by-Nu array

Transport delay, specified as one of the following:

• Scalar — Specify the transport delay for a SISO system or the same transport delay for all input/output pairs of a MIMO system.

• Ny-by-Nu array — Specify separate transport delays for each input/output pair of a MIMO system. Here, Ny is the number of outputs and Nu is the number of inputs.

For continuous-time systems, specify transport delays in the time unit specified by the TimeUnit property. For discrete-time systems, specify transport delays in integer multiples of the sample time, Ts.

InputDelay — Input delay

0 (default) | scalar | Nu-by-1 vector

Input delay for each input channel, specified as one of the following:

• Scalar — Specify the input delay for a SISO system or the same delay for all inputs of a multi-input system.

• Nu-by-1 vector — Specify separate input delays for input of a multi-input system, where Nu is the number of inputs.

For continuous-time systems, specify input delays in the time unit specified by the TimeUnit property. For discrete-time systems, specify input delays in integer multiples of the sample time, Ts.

For more information, see Time Delays in Linear Systems.

OutputDelay — Output delay

0 (default) | scalar | Ny-by-1 vector

Output delay for each output channel, specified as one of the following:

• Scalar — Specify the output delay for a SISO system or the same delay for all outputs of a multi-output system.

• Ny-by-1 vector — Specify separate output delays for output of a multi-output system, where Ny is the number of outputs.

For continuous-time systems, specify output delays in the time unit specified by the TimeUnit property. For discrete-time systems, specify output delays in integer multiples of the sample time, Ts.

For more information, see Time Delays in Linear Systems.

Ts — Sample time

0 (default) | positive scalar | -1

Sample time, specified as:

• 0 for continuous-time systems.

• A positive scalar representing the sampling period of a discrete-time system. Specify Ts in the time unit specified by the TimeUnit property.

• -1 for a discrete-time system with an unspecified sample time.

Note

Changing Ts does not discretize or resample the model.

TimeUnit — Time variable units

'seconds' (default) | 'nanoseconds' | 'microseconds' | 'milliseconds' | 'minutes' | 'hours' | 'days' | 'weeks' | 'months' | 'years' | ...

Time variable units, specified as one of the following:

• 'nanoseconds'

• 'microseconds'

• 'milliseconds'

• 'seconds'

• 'minutes'

• 'hours'

• 'days'

• 'weeks'

• 'months'

• 'years'

Changing TimeUnit has no effect on other properties, but changes the overall system behavior. Use chgTimeUnit to convert between time units without modifying system behavior.

InputName — Input channel names

'' (default) | character vector | cell array of character vectors

Input channel names, specified as one of the following:

• A character vector, for single-input models.

• A cell array of character vectors, for multi-input models.

• '', no names specified, for any input channels.

Alternatively, you can assign input names for multi-input models using automatic vector expansion. For example, if sys is a two-input model, enter the following.

sys.InputName = 'controls';

The input names automatically expand to {'controls(1)';'controls(2)'}.

You can use the shorthand notation u to refer to the InputName property. For example, sys.u is equivalent to sys.InputName.

Use InputName to:

• Identify channels on model display and plots.

• Extract subsystems of MIMO systems.

• Specify connection points when interconnecting models.

InputUnit — Input channel units

'' (default) | character vector | cell array of character vectors

Input channel units, specified as one of the following:

• A character vector, for single-input models.

• A cell array of character vectors, for multi-input models.

• '', no units specified, for any input channels.

Use InputUnit to specify input signal units. InputUnit has no effect on system behavior.

InputGroup — Input channel groups

structure

Input channel groups, specified as a structure. Use InputGroup to assign the input channels of MIMO systems into groups and refer to each group by name. The field names of InputGroup are the group names and the field values are the input channels of each group. For example, enter the following to create input groups named controls and noise that include input channels 1 and 2, and 3 and 5, respectively.

sys.InputGroup.controls = [1 2];
sys.InputGroup.noise = [3 5];

You can then extract the subsystem from the controls inputs to all outputs using the following.

sys(:,'controls')

By default, InputGroup is a structure with no fields.

OutputName — Output channel names

'' (default) | character vector | cell array of character vectors

Output channel names, specified as one of the following:

• A character vector, for single-output models.

• A cell array of character vectors, for multi-output models.

• '', no names specified, for any output channels.

Alternatively, you can assign output names for multi-output models using automatic vector expansion. For example, if sys is a two-output model, enter the following.

sys.OutputName = 'measurements';

The output names automatically expand to {'measurements(1)';'measurements(2)'}.

You can also use the shorthand notation y to refer to the OutputName property. For example, sys.y is equivalent to sys.OutputName.

Use OutputName to:

• Identify channels on model display and plots.

• Extract subsystems of MIMO systems.

• Specify connection points when interconnecting models.

OutputUnit — Output channel units

'' (default) | character vector | cell array of character vectors

Output channel units, specified as one of the following:

• A character vector, for single-output models.

• A cell array of character vectors, for multi-output models.

• '', no units specified, for any output channels.

Use OutputUnit to specify output signal units. OutputUnit has no effect on system behavior.

OutputGroup — Output channel groups

structure

Output channel groups, specified as a structure. Use OutputGroup to assign the output channels of MIMO systems into groups and refer to each group by name. The field names of OutputGroup are the group names and the field values are the output channels of each group. For example, create output groups named temperature and measurement that include output channels 1, and 3 and 5, respectively.

sys.OutputGroup.temperature = [1];
sys.OutputGroup.measurement = [3 5];

You can then extract the subsystem from all inputs to the measurement outputs using the following.

sys('measurement',:)

By default, OutputGroup is a structure with no fields.

Name — System name

'' (default) | character vector

System name, specified as a character vector. For example, 'system_1'.

Notes — User-specified text

{} (default) | character vector | cell array of character vectors

User-specified text that you want to associate with the system, specified as a character vector or cell array of character vectors. For example, 'System is MIMO'.

UserData — User-specified data

[] (default) | any MATLAB^® data type

User-specified data that you want to associate with the system, specified as any MATLAB data type.

SamplingGrid — Sampling grid for model arrays

structure array

Sampling grid for model arrays, specified as a structure array.

Use SamplingGrid to track the variable values associated with each model in a model array, including identified linear time-invariant (IDLTI) model arrays.

Set the field names of the structure to the names of the sampling variables. Set the field values to the sampled variable values associated with each model in the array. All sampling variables must be numeric scalars, and all arrays of sampled values must match the dimensions of the model array.

For example, you can create an 11-by-1 array of linear models, sysarr, by taking snapshots of a linear time-varying system at times t = 0:10. The following code stores the time samples with the linear models.

 sysarr.SamplingGrid = struct('time',0:10)

Similarly, you can create a 6-by-9 model array, M, by independently sampling two variables, zeta and w. The following code maps the (zeta,w) values to M.

[zeta,w] = ndgrid(<6 values of zeta>,<9 values of w>)
M.SamplingGrid = struct('zeta',zeta,'w',w)

When you display M, each entry in the array includes the corresponding zeta and w values.

M

M(:,:,1,1) [zeta=0.3, w=5] =
 
        25
  --------------
  s^2 + 3 s + 25
 

M(:,:,2,1) [zeta=0.35, w=5] =
 
         25
  ----------------
  s^2 + 3.5 s + 25
 
...

For model arrays generated by linearizing a Simulink model at multiple parameter values or operating points, the software populates SamplingGrid automatically with the variable values that correspond to each entry in the array. For instance, the Simulink Control Design commands linearize (Simulink Control Design) and slLinearizer (Simulink Control Design) populate SamplingGrid automatically.

By default, SamplingGrid is a structure with no fields.

Object Functions

The following lists contain a representative subset of the functions you can use with frd models. In general, many functions applicable to Dynamic System Models are also applicable to a frd object. frd models do not work with any time-domain analysis functions.

Frequency Response Analysis

bode	Bode frequency response of dynamic system
sigma	Singular values of frequency response of dynamic system
nyquist	Nyquist response of dynamic system
nichols	Nichols response of dynamic system
bandwidth	Frequency response bandwidth
freqresp	Evaluate system response over a grid of frequencies
margin	Gain margin, phase margin, and crossover frequencies

Model Transformation

chgFreqUnit	Change frequency units of frequency-response data model
chgTimeUnit	Change time units of dynamic system
frdfun	Apply a function to the frequency response value at each frequency of an frd model object
fselect	Select frequency points or range in FRD model
interp	Interpolate FRD model
fcat	Concatenate FRD models along frequency dimension
fnorm	Pointwise peak gain of FRD model

Model Interconnection

feedback	Feedback connection of multiple models
connect	Block diagram interconnections of dynamic systems
series	Series connection of two models
parallel	Parallel connection of two models

Controller Design

pidtune

PID tuning algorithm for linear plant model

Examples

SISO Frequency-Response Data Model

Create an frd object from frequency response data.

For this example, load the frequency response data collected for a water tank model.

load wtankData.mat

This data contains the frequency response data collected for the frequency range ${10}^{-3}$ rad/s to ${10}^2$ rad/s.

Create the model.

sys = frd(response,frequency)

sys =
 
    Frequency(rad/s)          Response     
    ----------------          --------     
          0.0010        1.562e+01 - 1.9904i
          0.0018        1.560e+01 - 2.0947i
          0.0034        1.513e+01 - 3.3670i
          0.0062        1.373e+01 - 5.4306i
          0.0113        1.047e+01 - 7.5227i
          0.0207        5.829e+00 - 7.6529i
          0.0379        2.340e+00 - 5.6271i
          0.0695        7.765e-01 - 3.4188i
          0.1274        2.394e-01 - 1.9295i
          0.2336        7.216e-02 - 1.0648i
          0.4281        2.157e-02 - 0.5834i
          0.7848        6.433e-03 - 0.3188i
          1.4384        1.916e-03 - 0.1740i
          2.6367        5.705e-04 - 0.0950i
          4.8329        1.698e-04 - 0.0518i
          8.8587        5.055e-05 - 0.0283i
         16.2378        1.505e-05 - 0.0154i
         29.7635        4.478e-06 - 0.0084i
         54.5559        1.333e-06 - 0.0046i
        100.0000        3.967e-07 - 0.0025i
 
Continuous-time frequency response.
Model Properties

Plot sys.

bode(sys)

Discrete-Time MIMO Frequency-Response Data Model

For this example, consider randomly generated response data and frequencies.

Generate a 3-by-2-by-7 complex array and a frequency vector with seven points between 0.01 and 100 rad/s. Set the sample time Ts to 5 seconds.

rng(0)
r = randn(3,2,7)+1i*randn(3,2,7);
w = logspace(-2,2,7);
Ts = 5;

Create the model.

sys = frd(r,w,Ts)

sys =

  From input 1 to:

    Frequency(rad/s)         output 1           output 2           output 3    
    ----------------         --------           --------           --------    
          0.0100         0.5377 + 0.3192i   1.8339 + 0.3129i  -2.2588 - 0.8649i
          0.0464        -0.4336 + 1.0933i   0.3426 + 1.1093i   3.5784 - 0.8637i
          0.2154         0.7254 - 0.0068i  -0.0631 + 1.5326i   0.7147 - 0.7697i
          1.0000         1.4090 - 1.0891i   1.4172 + 0.0326i   0.6715 + 0.5525i
          4.6416         0.4889 - 1.4916i   1.0347 - 0.7423i   0.7269 - 1.0616i
         21.5443         0.8884 - 0.1924i  -1.1471 + 0.8886i  -1.0689 - 0.7648i
        100.0000         0.3252 - 0.1774i  -0.7549 - 0.1961i   1.3703 + 1.4193i

  From input 2 to:

    Frequency(rad/s)         output 1           output 2           output 3    
    ----------------         --------           --------           --------    
          0.0100         0.8622 - 0.0301i   0.3188 - 0.1649i  -1.3077 + 0.6277i
          0.0464         2.7694 + 0.0774i  -1.3499 - 1.2141i   3.0349 - 1.1135i
          0.2154        -0.2050 + 0.3714i  -0.1241 - 0.2256i   1.4897 + 1.1174i
          1.0000        -1.2075 + 1.1006i   0.7172 + 1.5442i   1.6302 + 0.0859i
          4.6416        -0.3034 + 2.3505i   0.2939 - 0.6156i  -0.7873 + 0.7481i
         21.5443        -0.8095 - 1.4023i  -2.9443 - 1.4224i   1.4384 + 0.4882i
        100.0000        -1.7115 + 0.2916i  -0.1022 + 0.1978i  -0.2414 + 1.5877i
 
Sample time: 5 seconds
Discrete-time frequency response.
Model Properties

The specified data results in a two-input, three-output frd model.

Frequency-Response Data Model with Inherited Properties

For this example, create a frequency-response data model with properties inherited from a transfer function model.

Create a transfer function sys1 with the TimeUnit property set to 'minutes' and InputDelay property set to 3.

numerator1 = [2,0];
denominator1 = [1,8,0];
sys1 = tf(numerator1,denominator1,'TimeUnit','minutes','InputDelay',3)

sys1 =
 
                 2 s
  exp(-3*s) * ---------
              s^2 + 8 s
 
Continuous-time transfer function.
Model Properties

propValues1 = {sys1.TimeUnit,sys1.InputDelay}

propValues1=1×2 cell array
    {'minutes'}    {[3]}

Create an frd model with properties inherited from sys1.

rng(0)
response = randn(1,1,7)+1i*randn(1,1,7);
w = logspace(-2,2,7);
sys2 = frd(response,w,sys1)

sys2 =
 
    Frequency(rad/minute)         Response    
    ---------------------         --------    
             0.0100           0.5377 + 0.3426i
             0.0464           1.8339 + 3.5784i
             0.2154          -2.2588 + 2.7694i
             1.0000           0.8622 - 1.3499i
             4.6416           0.3188 + 3.0349i
            21.5443          -1.3077 + 0.7254i
           100.0000          -0.4336 - 0.0631i
 
  Input delays (minutes): 3 
 
Continuous-time frequency response.
Model Properties

propValues2 = {sys2.TimeUnit,sys2.InputDelay}

propValues2=1×2 cell array
    {'minutes'}    {[3]}

Observe that the frd model sys2 has that same properties as sys1.

Specify State and Input Names for Frequency-Response Data Model

For this example, load the frequency response data collected for a water tank model.

load wtankData.mat

The model has one input, Voltage, and one output, Water height.

Create an frd model, specifying the input and output names.

sys = frd(response,frequency,'InputName','Voltage','OutputName','Height');

Plot the frequency response.

bode(sys)

The input and output names appear on the Bode plot. Naming the inputs and outputs can be useful when dealing with response plots for MIMO systems.

Convert State-Space Model to Frequency-Response Data Model

For this example, compute the frd model of the following state-space model:

$$\mathit{A}=\left[\begin{array}{cc}-2& -1\\ 1& -2\end{array}\right],\mathit{B}=\left[\begin{array}{cc}1& 1\\ 2& -1\end{array}\right],\mathit{C}=\left[\begin{array}{cc}1& 0\end{array}\right],\mathit{D}=\left[\begin{array}{cc}0& 1\end{array}\right]$$

Create a state-space model using the state-space matrices.

A = [-2 -1;1 -2];
B = [1 1;2 -1];
C = [1 0];
D = [0 1];
ltiSys = ss(A,B,C,D);

Convert the state-space model ltiSys to a frd model for frequencies between 0.01 and 100 rad/s.

w = logspace(-2,2,50);
sys = frd(ltiSys,w);

Compare the frequency responses.

bode(ltiSys,'b',sys,'r--')

The responses are identical.

Array of Frequency-Response Data Models

To create arrays of frd models, you can specify a multidimensional array of frequency response data.

For instance, when you specify the response data as a numeric array of size [NY NU NF S1 ... Sn], the function returns a S1-by-...-by-Sn array of frd models. Each of these models has NY outputs, NU inputs, and NF frequency points.

Generate a 2-by-3 array of random response data with one-output, two-input models at 10 frequency points between 0.1 and 10 rad/s.

w = logspace(-1,1,10);
r = randn(1,2,10,2,3)+1i*randn(1,2,10,2,3);
sys = frd(r,w);

Extract the model at the index (2,1) from the model array.

sys21 = sys(:,:,2,1)

sys21 =

  From input 1 to:

    Frequency(rad/s)         output 1    
    ----------------         --------    
          0.1000         0.6715 + 0.0229i
          0.1668         0.7172 - 1.7502i
          0.2783         0.4889 - 0.8314i
          0.4642         0.7269 - 1.1564i
          0.7743         0.2939 - 2.0026i
          1.2915         0.8884 + 0.5201i
          2.1544        -1.0689 - 0.0348i
          3.5938        -2.9443 + 1.0187i
          5.9948         0.3252 - 0.7145i
         10.0000         1.3703 - 0.2248i

  From input 2 to:

    Frequency(rad/s)         output 1    
    ----------------         --------    
          0.1000        -1.2075 - 0.2620i
          0.1668         1.6302 - 0.2857i
          0.2783         1.0347 - 0.9792i
          0.4642        -0.3034 - 0.5336i
          0.7743        -0.7873 + 0.9642i
          1.2915        -1.1471 - 0.0200i
          2.1544        -0.8095 - 0.7982i
          3.5938         1.4384 - 0.1332i
          5.9948        -0.7549 + 1.3514i
         10.0000        -1.7115 - 0.5890i
 
Continuous-time frequency response.
Model Properties

Frequency-Response Data Model with Negative Frequencies

You can specify negative frequency values in an frd object. This capability is useful when you want to capture the frequency response data of models with complex coefficients.

Create a frequency vector with both positive and negative values.

w0 = sort([-logspace(-2,2,50) 0 logspace(-2,2,50)]);

Create a state-space model with complex coefficients.

A = [-3.50,-1.25-0.25i;2,0];
B = [1;0];
C = [-0.75-0.5i,0.625-0.125i];
D = 0.5;
Gc = ss(A,B,C,D);

Convert the model to an frd model at the specified frequencies.

sys = frd(Gc,w0);

Plot the frequency response of the models.

bode(Gc,'b',sys,'r--')

The plot responses match closely. The plot shows two branches for models with complex coefficients, one for positive frequencies, with a right-pointing arrow, and one for negative frequencies, with a left-pointing arrow. In both branches, the arrows indicate the direction of increasing frequencies.

Version History

Introduced before R2006a