idfrd - Frequency-response data or model

Syntax

Description

h = idfrd(Response,Freq,Ts) constructs an idfrd object that stores the frequency response Response of a linear system at frequency values Freq. Ts is the sampling time interval. For a continuous-time system, set Ts=0.

h = idfrd(Response,Freq,Ts,...
     'CovarianceData',Covariance,'SpectrumData',Spec,...
     'NoiseCovariance',Speccov) also stores the uncertainty of the response Covariance, the spectrum of the additive disturbance (noise) Spec, and the covariance of the noise Speccov.

h = idfrd(Response,Freq,Ts,...
     'P1',V1,'PN',VN) constructs an idfrd object that stores a frequency-response model with properties specified by the idfrd model property-value pairs.

h = idfrd(mod) converts a System Identification Toolbox or Control System Toolbox linear model to frequency-response data at default frequencies, including the output noise spectra and their covariance.

h = idfrd(mod,Freqs) converts a System Identification Toolbox or Control System Toolbox linear model to frequency-response data at frequencies Freqs.

For a model

stores the transfer function estimate , as well as the spectrum of the additive noise (Φ_v) at the output

where λ is the estimated variance of e(t), and T is the sampling interval.

Creating idfrd from Given Responses

Response is a 3-D array of dimension ny-by-nu-by-Nf, with ny being the number of outputs, nu the number of inputs, and Nf the number of frequencies (that is, the length of Freqs). Response(ky,ku,kf) is thus the complex-valued frequency response from input ku to output ky at frequency =Freqs(kf). When defining the response of a SISO system, Response can be given as a vector.

Freqs is a column vector of length Nf containing the frequencies of the response.

Ts is the sampling interval. Ts = 0 means a continuous-time model.

Covariance is a 5-D array containing the covariance of the frequency response. It has dimension ny-by-nu-by-Nf-by-2-by-2. The structure is such that Covariance(ky,ku,kf,:,:) is the 2-by-2 covariance matrix of the response Response(ky,ku,kf). The 1-1 element is the variance of the real part, the 2-2 element is the variance of the imaginary part, and the 1-2 and 2-1 elements are the covariance between the real and imaginary parts. squeeze(Covariance(ky,ku,kf,:,:)) thus gives the covariance matrix of the corresponding response.

The format for spectrum information is as follows:

spec is a 3-D array of dimension ny-by-ny-by-Nf, such that spec(ky1,ky2,kf) is the cross spectrum between the noise at output ky1 and the noise at output ky2, at frequency Freqs(kf). When ky1 = ky2 the (power) spectrum of the noise at output ky1 is thus obtained. For a single-output model, spec can be given as a vector.

speccov is a 3-D array of dimension ny-by-ny-by-Nf, such that speccov(ky1,ky1,kf) is the variance of the corresponding power spectrum.

If only SpectrumData is to be packaged in the idfrd object, set Response = [].

Creating idfrd from a Given Model

idfrd can also be computed from a given linear identified model, mod.

If the frequencies Freqs are not specified, a default choice is made based on the dynamics of the model mod.

The estimated covariances are computed using the Gauss approximation formula from the uncertainty information in mod. For grey-box models (idgrey

), numerical differentiation is applied. The step sizes for the numerical derivatives are determined by nuderst.

Intersample behavior: For discrete-time frequency response data (Ts>0), you can also specify the intersample behavior of the input signal that was in effect when the samples were collected originally from an experiment. To specify the intersample behavior, use:

mf = idfrd(Response,Freq,Ts,'InterSample','zoh');

For multi-input systems, specify the intersample behavior using an Nu-by-1 cell array, where Nu is the number of inputs. The InterSample property is irrelevant for continuous-time data.

Frequency responses for submodels can be obtained by the standard subreferencing, h = idfrd(m(2,3)). h = idfrd(m(:,[])) gives an h that just contains SpectrumData.

The idfrd models can be graphed with bode, spectrum, and nyquist, which all accept mixtures of parametric models, such as idtf and idfrd models as arguments. Note that spa, spafdr, and etfe return their estimation results as idfrd objects.

Constructor

The idfrd represents complex frequency-response data. Before you can create an idfrd object, you must import your data as described in Frequency-Response Data Representation.

Note   The idfrd object can only encapsulate one frequency-response data set. It does not support the iddata equivalent of multiexperiment data.

Use the following syntax to create the data object fr_data:

fr_data = idfrd(response,f,Ts)

Suppose that ny is the number of output channels, nu is the number of input channels, and nf is a vector of frequency values. response is an ny-by-nu-by-nf 3-D array. f is the frequency vector that contains the frequencies of the response.Ts is the sampling time, which is used when measuring or computing the frequency response. If you are working with a continuous-time system, set Ts to 0.

response(ky,ku,kf), where ky, ku, and kf reference the kth output, input, and frequency value, respectively, is interpreted as the complex-valued frequency response from input ku to output ky at frequency f(kf).

You can specify object properties when you create the idfrd object using the constructor syntax:

fr_data = idfrd(response,f,Ts,
               'Property1',Value1,...,'PropertyN',ValueN)

Properties

After creating the object, you can use get or dot notation to access the object property values.

Use set or dot notation to set a property of an existing object.

Property Name	Description
ResponseData	3-D array of the complex-valued frequency response as described above. For SISO systems use Response(1,1,:) to obtain a vector of the response data.
Frequency	Column vector containing the frequencies at which the responses are defined.
CovarianceData	5-D array of the covariance matrices of the response data as described above.
SpectrumData	3-D array containing power spectra and cross spectra of the output disturbances (noise) of the system.
NoiseCovariance	3-D array containing the variances of the power spectra, as explained above.
FrequencyUnit	Unit of the frequency vector. Specify as one of the following: 'rad/TimeUnit', 'cycles/TimeUnit', 'rad/s', 'Hz', 'kHz', 'MHz', 'GHz', or 'rpm'. The units 'rad/TimeUnit' and 'cycles/TimeUnit' are relative to the time units specified in TimeUnit. Changing this property changes the overall system behavior. Use chgFreqUnit to convert between frequency units without modifying system behavior.
Ts	Scalar denoting the sampling interval of the model whose frequency response is stored. 'Ts' = 0 means a continuous-time model.
Name	An optional name for the object.
InputName	String or cell array containing the names of the input channels. It has as many entries as there are input channels.
OutputName	Correspondingly for the output channels.
InputUnit	Units in which the input channels are measured. It has the same format as 'InputName'.
OutputUnit	Correspondingly for the output channels.
InputDelay	Row vector of length equal to the number of input channels. Contains the delays from the input channels. These should thus be appended as phase lags when the response is calculated. This is done automatically by freqresp, bode, and nyquist.
InterSample	Intersample behavior of inputs. Specifies the behavior of the input signals between samples for transformations between discrete-time and continuous-time. This property is meaningful for discrete-time idfrd models only. Specify InterSample as one of the following: 'zoh' — The input signal used for construction/estimation of the frequency response data was subject to a zero-order-hold filter. 'foh' — The input signal was subject to a first-order-hold filter. 'bl' — The input signal has no power above the Nyquist frequency (pi/sys.Ts rad/s). This is typically the case when the input signal is measured experimentally using an anti-aliasing filter and a sampler. Ideally, treat the data as continuous-time. That is, if the signals used for the estimation of the frequency response were subject to anti-aliasing filters, set sys.Ts to zero. For multi-input data, specify InterSample as an Nu-by-1 cell array, where Nu is the number of inputs.
Notes	An arbitrary field to store extra information and notes about the object.
UserData	An arbitrary field for any possible use.
Report	Information about the estimation process that is behind the frequency data. It contains the following fields: Status: Specifies whether the model was created directly, transformed or estimated. Method: The identification routine that created the model. WindowSize: If the model was estimated by spa, spafdr, or etfe, the size of window (input argument M, the resolution parameter) that was used. This is scalar or a vector. DataUsed: Information on the estimation data such as its name, type, sample time and intersample behavior.

To view the properties of the idfrd object, you can use the get command. The following example shows how to create an idfrd object that contains 100 frequency-response values with a sampling time interval of 0.08 s and get its properties:

% Create the idfrd data object
  fr_data = idfrd(response,f,0.08)
% Get property values of data
  get(fr_data)

response and f are variables in the MATLAB Workspace browser, representing the frequency-response data and frequency values, respectively.

To change property values for an existing idfrd object, use the set command or dot notation. For example, to change the name of the idfrd object, type the following command sequence at the prompt:

% Set the name of the f_data object
  set(fr_data,'name','DC_Converter')
% Get fr_data properties and values
  get(fr_data)                    

Property names are not case sensitive. You do not need to type the entire property name if the first few letters uniquely identify the property.

If you import fr_data into the System Identification Tool GUI, this data has the name DC_Converter in the GUI, and not the variable name fr_data.

Subreferencing

The different channels of the idfrd are retrieved by subreferencing.

h(outputs,inputs)

h(2,3) thus contains the response data from input channel 3 to output channel 2, and, if applicable, the output spectrum data for output channel 2. The channels can also be referred to by their names, as in h('power',{'voltage', 'speed'}).

Horizontal Concatenation

Adding input channels,

h = [h1,h2,...,hN]

creates an idfrd model h, with ResponseData containing all the input channels in h1,...,hN. The output channels of hk must be the same, as well as the frequency vectors. SpectrumData is ignored.

Vertical Concatenation

Adding output channels,

h = [h1;h2;... ;hN]

creates an idfrd model h with ResponseData containing all the output channels in h1, h2,...,hN. The input channels of hk must all be the same, as well as the frequency vectors. SpectrumData is also appended for the new outputs. The cross spectrum between output channels of h1, h2,...,hN is then set to zero.

Converting to iddata

You can convert an idfrd object to a frequency-domain iddata object by

Data = iddata(Idfrdmodel)

See iddata.

Examples

Compare the results from spectral analysis and an ARMAX model.

load iddata1 z1;
m = armax(z1,[2 2 2 1]);
g = spa(z1)
g = spafdr(z1,[],{1e-3,10})
bode(g,m)

See Also

bode | etfe | freqresp | nyquist | spa | spafdr | tfest

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