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frd

Frequency-response data model

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Description

Usefrd创建实或复值-频率response data models, or to convertdynamic system modelsto frequency-response data model form.

Frequency-response data models store complex frequency response data with corresponding frequency points. For example, a frequency-response data modelH(jwi), stores the frequency response at each input frequencywi, wherei= 1,…,n. Thefrdmodel object can represent SISO or MIMO frequency-response data models in continuous time or discrete time. For more information, seeFrequency Response Data (FRD) Models.

You can also usefrdto create generalized frequency-response data (genfrd) models.

Creation

You can obtainfrdmodels in one of the following ways.

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

example

sys= frd(response,frequency)creates a continuous-time frequency-response data (frd) model, setting theResponseDataandFrequencyproperties.frequencycan contain both negative and positive frequencies.

example

sys= frd(response,frequency,ts)creates a discrete-timefrdmodel with the sample timets. To leave the sample time unspecified, settsto –1.

example

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

example

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.

example

sys= frd(ltiSys,frequency)converts the dynamic system modelltiSysto a frequency-response data model.frdcomputes the frequency response at frequencies specified byfrequency.sysinherits its frequency units rad/TimeUnitfromltiSys.TimeUnit.

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

Input Arguments

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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 infrequency.

  • For MIMO systems withNuinputs andNyoutputs, specify aNy-by-Nu-by-Nfarray, whereNfis the number of frequency points.

  • For anS1-...-by-Snarray of models withNuinputs andNyoutputs, specify a multidimensional array of size [NyNuNfS1…Sn].

    For instance, aresponseof size [Ny,Nu,Nf,3,4] represents the response data for a 3-by-4 array of models. Each model hasNyoutputs,Nuinputs, andNffrequency points.

This input sets theResponseDataproperty.

Frequency points corresponding toresponse, specified as a vector that containsNfpoints.frequencycan contain both positive and negative frequencies.

This input sets theFrequencyproperty.

Sample time, specified as a scalar.

The input sets theTsproperty.

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

  • Continuous-time or discrete-time numeric LTI models, such astf,zpk,ss, orpidmodels.

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

    The resultingfrdmodel assumes:

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

    • Nominal model values for uncertain control design blocks

  • Identified LTI models, such asidtf(System Identification Toolbox),idss(System Identification Toolbox),idproc(System Identification Toolbox),idpoly(System Identification Toolbox), andidgrey(System Identification Toolbox)models.(Using identified models requires System Identification Toolbox™ software.)

Properties

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

  • For SISO systems,ResponseDatais a1-by-1-by-Nfarray of frequency response values at theNffrequency points specified in theFrequencyproperty.

  • For MIMO systems withNuinputs andNyoutputs,ResponseDatais anNy-by-Nu-by-Nfarray, whereNfis the number of frequency points.

    For instance,ResponseData(肯塔基州,ku, k中f)represents the frequency response from the inputkuto the outputkyat the frequencyFrequency(kf).

  • For anS1-...-by-Snarray of models withNuinputs andNyoutputs,ResponseDatais a multidimensional array of size [NyNuNfS1…Sn].

    For instance, aResponseDataof size [Ny,Nu,Nf,3,4] represents the response data for a 3-by-4 array of models. Each model hasNyoutputs,Nuinputs, andNffrequency points.

Frequency points corresponding toResponseData, specified as a vector that containsNfpoints in the units specified byFrequencyUnit.

Units of the frequency vector in theFrequencyproperty, 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 theTimeUnitproperty.

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

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-Nuarray — Specify separate transport delays for each input/output pair of a MIMO system. Here,Nyis the number of outputs andNuis the number of inputs.

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

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.

  • Nu1的向量— Specify separate input delays for input of a multi-input system, whereNuis the number of inputs.

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

For more information, seeTime Delays in Linear Systems.

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.

  • Ny1的向量— Specify separate output delays for output of a multi-output system, whereNyis the number of outputs.

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

For more information, seeTime Delays in Linear Systems.

Sample time, specified as:

  • 0for continuous-time systems.

  • A positive scalar representing the sampling period of a discrete-time system. SpecifyTsin the time unit specified by theTimeUnitproperty.

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

Note

ChangingTsdoes not discretize or resample the model.

Time variable units, specified as one of the following:

  • 'nanoseconds'

  • 'microseconds'

  • 'milliseconds'

  • 'seconds'

  • 'minutes'

  • 'hours'

  • 'days'

  • 'weeks'

  • 'months'

  • 'years'

ChangingTimeUnithas no effect on other properties, but changes the overall system behavior. UsechgTimeUnit时间单位,无需修改sy之间转换stem behavior.

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, ifsysis 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 notationuto refer to theInputNameproperty. For example,sys.u相当于sys.InputName.

UseInputNameto:

  • Identify channels on model display and plots.

  • Extract subsystems of MIMO systems.

  • Specify connection points when interconnecting models.

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.

UseInputUnitto specify input signal units.InputUnithas no effect on system behavior.

Input channel groups, specified as a structure. UseInputGroupto assign the input channels of MIMO systems into groups and refer to each group by name. The field names ofInputGroupare the group names and the field values are the input channels of each group. For example, enter the following to create input groups namedcontrolsandnoisethat include input channels1and2, and3and5, respectively.

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

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

sys(:,'controls')

By default,InputGroupis a structure with no fields.

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, ifsysis 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 notationyto refer to theOutputNameproperty. For example,sys.y相当于sys.OutputName.

UseOutputNameto:

  • Identify channels on model display and plots.

  • Extract subsystems of MIMO systems.

  • Specify connection points when interconnecting models.

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.

UseOutputUnitto specify output signal units.OutputUnithas no effect on system behavior.

Output channel groups, specified as a structure. UseOutputGroupto assign the output channels of MIMO systems into groups and refer to each group by name. The field names ofOutputGroupare the group names and the field values are the output channels of each group. For example, create output groups namedtemperatureandmeasurementthat include output channels1, and3and5, respectively.

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

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

sys('measurement',:)

By default,OutputGroupis a structure with no fields.

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

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'.

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

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

UseSamplingGridto 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 timest = 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,zetaandw. The following code maps the(zeta,w)values toM.

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

When you displayM, each entry in the array includes the correspondingzetaandwvalues.

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 populatesSamplingGridautomatically with the variable values that correspond to each entry in the array. For instance, theSimulink Control Designcommandslinearize(Simulink Control Design)andslLinearizer(Simulink Control Design)populateSamplingGridautomatically.

By default,SamplingGridis a structure with no fields.

Object Functions

The following lists contain a representative subset of the functions you can use withfrdmodels. In general, many functions applicable toDynamic System Modelsare also applicable to afrdobject.frdmodels do not work with any time-domain analysis functions.

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bode Bode plot of frequency response, or magnitude and phase data
sigma Singular value plot of dynamic system
nyquist Nyquist plot of frequency response
nichols Nichols chart of frequency response
bandwidth Frequency response bandwidth
freqresp Frequency response over grid
margin Gain margin, phase margin, and crossover frequencies
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 anfrdmodel 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
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
pidtune PID tuning algorithm for linear plant model

Examples

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Open Live Script

Create anfrdobject from frequency response data.

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

loadwtankData.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.

Plotsys.

bode(sys)

图包含2轴对象。坐标轴对象1续ains an object of type line. This object represents sys. Axes object 2 contains an object of type line. This object represents sys.

Open Live Script

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 timeTsto 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 ---------------- -------- -------- 0.0100 0.5377 + 0.3192i 1.8339 + 0.3129i 0.0464 -0.4336 + 1.0933i 0.3426 + 1.1093i 0.2154 0.7254 - 0.0068i -0.0631 + 1.5326i 1.0000 1.4090 - 1.0891i 1.4172 + 0.0326i 4.6416 0.4889 - 1.4916i 1.0347 - 0.7423i 21.5443 0.8884 - 0.1924i -1.1471 + 0.8886i 100.0000 0.3252 - 0.1774i -0.7549 - 0.1961i From input 1 to: Frequency(rad/s) output 3 ---------------- -------- 0.0100 -2.2588 - 0.8649i 0.0464 3.5784 - 0.8637i 0.2154 0.7147 - 0.7697i 1.0000 0.6715 + 0.5525i 4.6416 0.7269 - 1.0616i 21.5443 -1.0689 - 0.7648i 100.0000 1.3703 + 1.4193i From input 2 to: Frequency(rad/s) output 1 output 2 ---------------- -------- -------- 0.0100 0.8622 - 0.0301i 0.3188 - 0.1649i 0.0464 2.7694 + 0.0774i -1.3499 - 1.2141i 0.2154 -0.2050 + 0.3714i -0.1241 - 0.2256i 1.0000 -1.2075 + 1.1006i 0.7172 + 1.5442i 4.6416 -0.3034 + 2.3505i 0.2939 - 0.6156i 21.5443 -0.8095 - 1.4023i -2.9443 - 1.4224i 100.0000 -1.7115 + 0.2916i -0.1022 + 0.1978i From input 2 to: Frequency(rad/s) output 3 ---------------- -------- 0.0100 -1.3077 + 0.6277i 0.0464 3.0349 - 1.1135i 0.2154 1.4897 + 1.1174i 1.0000 1.6302 + 0.0859i 4.6416 -0.7873 + 0.7481i 21.5443 1.4384 + 0.4882i 100.0000 -0.2414 + 1.5877i Sample time: 5 seconds Discrete-time frequency response.

The specified data results in a two-input, three-outputfrdmodel.

Open Live Script

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

Create a transfer functionsys1with theTimeUnitproperty set to'minutes'andInputDelayproperty 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.
propValues1 = {sys1.TimeUnit,sys1.InputDelay}
propValues1=1×2 cell array{'minutes'} {[3]}

Create anfrdmodel with properties inherited fromsys1.

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.
propValues2 = {sys2.TimeUnit,sys2.InputDelay}
propValues2=1×2 cell array{'minutes'} {[3]}

Observe that thefrdmodelsys2has that same properties assys1.

Open Live Script

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

loadwtankData.mat

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

Create anfrdmodel, specifying the input and output names.

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

Plot the frequency response.

bode(sys)

图包含2轴对象。Axes object 1 with title From: Voltage To: Height contains an object of type line. This object represents sys. Axes object 2 contains an object of type line. This object represents 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.

Open Live Script

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

A = [ - 2 - 1 1 - 2 ] , B = [ 1 1 2 - 1 ] , C = [ 1 0 ] , D = [ 0 1 ]

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 modelltiSysto afrdmodel 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--')

Figure contains 4 axes objects. Axes object 1 with title From: In(1) contains 2 objects of type line. These objects represent ltiSys, sys. Axes object 2 contains 2 objects of type line. These objects represent ltiSys, sys. Axes object 3 with title From: In(2) contains 2 objects of type line. These objects represent ltiSys, sys. Axes object 4 contains 2 objects of type line. These objects represent ltiSys, sys.

The responses are identical.

Open Live Script

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

For instance, when you specify the response data as a numeric array of size [NYNUNFS1...Sn], the function returns aS1-by-...-by-Snarray offrdmodels. Each of these models hasNYoutputs,NUinputs, andNFfrequency 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.
Open Live Script

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--')

图包含2轴对象。坐标轴对象1续ains 2 objects of type line. These objects represent Gc, sys. Axes object 2 contains 2 objects of type line. These objects represent Gc, sys.

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

See Also

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