tfest
Estimate transfer function model
Syntax
Description
Estimate Transfer Function Model
estimates the continuoustime transfer function sys
= tfest(tt
,np
)sys
with
np
poles, using all the input and output signals in the timetable
tt
. The number of zeros in sys is max(np
1,0).
You can use this syntax for SISO and MISO systems. The function assumes that the last
variable in the timetable is the single output signal.
You cannot use tfest
to estimate timeseries models, which are
models that contain no inputs. Use ar
, arx
, or armax
for timeseries models instead.
estimates a continuoustime transfer function using the timedomain input signals and
output signals in the matrices sys
= tfest(u
,y
,np
)u
,y
. The software
assumes that the data sample time is 1 second. You cannot change this assumed sample time.
If you want to estimate a model from data with a sample time other than 1 second, you have
two alternatives:
Estimate a discretetime system instead by setting the sample time using the
'Ts'
namevalue argument. For example,sys = tfest(u,y,np,'Ts',0.1)
sets the sample time to0.1
. You can use this syntax with SISO, MISO, and MIMO systems.Convert your matrix data to a
timetable
oriddata
object prior to estimating a continuoustime system. These formats allow you to incorporate sampletime knowledge into the data. For more information, seeu
,y
.
Estimating continuoustime models from matrixbased data is not recommended.
uses the timedomain or frequencydomain data in sys
= tfest(data
,np
)data
. Use this
syntax especially when you want to estimate a transfer function using frequencydomain or
frequency response data, or when you want to take advantage of the additional information,
such as intersample behavior, data sample time, or experiment labeling, that data objects
provide.
uses additional model options specified by one or more namevalue pair arguments. For
example, specify a discretetime system from matrix data that has a sample time of 0.1
using sys
= tfest(___,Name,Value
)sys = tfest(um,ym,np,'Ts',0.1)
. Specify input and output signal
variable names that correspond with the variables to use for MIMO timetable data using
sys =
tfest(data,np,nz,'InputNames',["u1","u2"],'OutputNames',["y1","y3"])
.
Configure Initial Parameters
Specify Additional Estimation Options
Return Estimated Initial Conditions
[
returns the estimated initial conditions as an sys
,ic
] = tfest(___)initialCondition
object. Use this syntax if you plan to simulate or predict the model response using the
same estimation input data and then compare the response with the same estimation output
data. Incorporating the initial conditions yields a better match between measured and
simulated or predicted data during the early stage of the simulation.
Examples
Estimate Transfer Function Model by Specifying Number of Poles
Load the timedomain systemresponse data in timetable tt1
.
load sdata1.mat tt1;
Set the number of poles np
to 2
and estimate a transfer function.
np = 2; sys = tfest(tt1,np);
sys
is an idtf
model containing the estimated twopole transfer function.
View the numerator and denominator coefficients of the resulting estimated model sys
.
sys.Numerator
ans = 1×2
2.4554 176.9856
sys.Denominator
ans = 1×3
1.0000 3.1625 23.1631
To view the uncertainty in the estimates of the numerator and denominator and other information, use tfdata
.
Specify Number of Poles and Zeros in Estimated Transfer Function
Load timedomain system response data z2
and use it to estimate a transfer function that contains two poles and one zero.
load iddata2 z2; np = 2; nz = 1; sys = tfest(z2,np,nz);
sys
is an idtf
model containing the estimated transfer function.
Estimate Transfer Function Containing Known Transport Delay
Load the data z2
, which is an iddata
object that contains timedomain system response data.
load iddata2 z2;
Estimate a transfer function model sys
that contains two poles and one zero, and which includes a known transport delay iodelay
.
np = 2; nz = 1; iodelay = 0.2; sys = tfest(z2,np,nz,iodelay);
sys
is an idtf
model containing the estimated transfer function, with the IODelay
property set to 0.2 seconds.
Estimate Transfer Function Containing Unknown Transport Delay
Load timedomain system response data z2
and use it to estimate a twopole onezero transfer function for the system. Specify an unknown transport delay for the transfer function by setting the value of iodelay
to NaN
.
load iddata2 z2; np = 2; nz = 1; iodelay = NaN; sys = tfest(z2,np,nz,iodelay);
sys
is an idtf
model containing the estimated transfer function, whose IODelay
property is estimated using the data.
Estimate DiscreteTime Transfer Function
Load timedomain system response data, which is contained in input and output matrices umat2
and ymat2
.
load sdata2.mat umat2 ymat2
Estimate a discretetime transfer function with two poles and one zero. Specify the sample time Ts
as 0.1 seconds and the transport delay iodelay
as 2 seconds.
np = 2;
nz = 1;
iodelay = 2;
Ts = 0.1;
sysd = tfest(umat2,ymat2,np,nz,iodelay,'Ts',Ts)
sysd = From input "u1" to output "y1": 1.8 z^1 z^(2) *  1  1.418 z^1 + 0.6613 z^2 Sample time: 0.1 seconds Discretetime identified transfer function. Parameterization: Number of poles: 2 Number of zeros: 1 Number of free coefficients: 3 Use "tfdata", "getpvec", "getcov" for parameters and their uncertainties. Status: Estimated using TFEST on time domain data. Fit to estimation data: 80.26% FPE: 2.095, MSE: 2.063
By default, the model has no feedthrough, and the numerator polynomial of the estimated transfer function has a zero leading coefficient b0
. To estimate b0
, specify the Feedthrough
property during estimation.
Estimate DiscreteTime Transfer Function with Feedthrough
Load the estimation data z5
.
load iddata5 z5
First, estimate a discretetime transfer function model with two poles, one zero, and no feedthrough. Get the sample time from the Ts
property of z5
.
np = 2;
nz = 1;
sys = tfest(z5,np,nz,'Ts',z5.Ts);
The estimated transfer function has the following form:
$$H({z}^{1})=\frac{b1{z}^{1}+b2{z}^{2}}{1+a1{z}^{1}+a2{z}^{2}}$$
By default, the model has no feedthrough, and the numerator polynomial of the estimated transfer function has a zero leading coefficient b0
. To estimate b0
, specify the Feedthrough
property during estimation.
sys = tfest(z5,np,nz,'Ts',z5.Ts,'Feedthrough',true);
The numerator polynomial of the estimated transfer function now has a nonzero leading coefficient:
$$H({z}^{1})=\frac{b0+b1{z}^{1}+b2{z}^{2}}{1+a1{z}^{1}+a2{z}^{2}}$$
Analyze Origin of Delay in Measured Data
Compare two discretetime models with and without feedthrough and transport delay.
If there is a delay from the measured input to output, it can be attributed either to a lack of feedthrough or to an actual transport delay. For discretetime models, absence of feedthrough corresponds to a lag of one sample between the input and output. Estimating a model using Feedthrough = false
and iodelay = 0
thus produces a discretetime system that is equivalent to a system estimated using Feedthrough = true
and iodelay = 1
. Both systems show the same time and frequencydomain responses, for example, on step and Bode plots. However, you get different results if you reduce these models using balred
or convert them to their continuoustime representations. Therefore, a best practice is to check if the observed delay can be attributed to a transport delay or to a lack of feedthrough.
Estimate a discretetime model with no feedthrough.
load iddata1 z1 np = 2; nz = 2; sys1 = tfest(z1,np,nz,'Ts',z1.Ts);
Because sys1
has no feedthrough and therefore has a numerator polynomial that begins with $${z}^{1}$$, sys1
has a lag of one sample. The IODelay
property is 0.
Estimate another discretetime model with feedthrough and with a reduction from two zeros to one, incurring a onesample inputoutput delay.
sys2 = tfest(z1,np,nz1,1,'Ts',z1.Ts,'Feedthrough',true);
Compare the Bode responses of the models.
bode(sys1,sys2);
The discrete equations that underlie sys1
and sys2
are equivalent, and so are the Bode responses.
Convert the models to continuous time and compare the Bode responses for these models.
sys1c = d2c(sys1); sys2c = d2c(sys2); bode(sys1c,sys2c); legend
As the plot shows, the Bode responses of the two models do not match when you convert them to continuous time. When there is no feedthrough, as with sys1c
, there must be some lag. When there is feedthrough, as with sys2c
, there can be no lag. Continuoustime feedthrough maps to discretetime feedthrough. Continuoustime lag maps to discretetime delays.
Estimate MISO DiscreteTime Transfer Function with Feedthrough and Delay Specifications for Individual Channels
Estimate a twoinput, oneoutput discretetime transfer function with a delay of two samples on the first input and zero samples on the second input. Both inputs have no feedthrough.
Load the data and split the data into estimation and validation data sets.
load iddata7 z7 ze = z7(1:300); zv = z7(200:400);
Estimate a twoinput, oneoutput transfer function with two poles and one zero for each inputtooutput transfer function.
Lag = [2;0]; Ft = [false,false]; model = tfest(ze,2,1,'Ts',z7.Ts,'Feedthrough',Ft,'InputDelay',Lag);
The Feedthrough
value you choose dictates whether the leading numerator coefficient is zero (no feedthrough) or not (nonzero feedthrough). Delays are generally expressed separately using the InputDelay
or IODelay
property. This example uses InputDelay
only to express the delays.
Validate the estimated model. Exclude the data outliers for validation.
I = 1:201;
I(114:118) = [];
opt = compareOptions('Samples',I);
compare(zv,model,opt)
Estimate Transfer Function Model Using Regularized Impulse Response Model
Identify a 15th order transfer function model by using regularized impulse response estimation.
Load the data.
load regularizationExampleData m0simdata;
Obtain a regularized impulse response (FIR) model.
opt = impulseestOptions('RegularizationKernel','DC'); m0 = impulseest(m0simdata,70,opt);
Convert the model into a transfer function model after reducing the order to 15.
m = idtf(balred(idss(m0),15));
Compare the model output with the data.
compare(m0simdata,m);
Estimate Transfer Function Using Estimation Option Set
Create an option set for tfest
that specifies the initialization and search methods. Also set the display option, which specifies that the lossfunction values for each iteration be shown.
opt = tfestOptions('InitializeMethod','n4sid','Display','on','SearchMethod','lsqnonlin');
Load timedomain system response data z2
and use it to estimate a transfer function with two poles and one zero. Specify opt
for the estimation options.
load iddata2 z2; np = 2; nz = 1; iodelay = 0.2; sys = tfest(z2,np,nz,iodelay,opt);
sys
is an idtf
model containing the estimated transfer function.
Specify Model Properties of Estimated Transfer Function
Load the timedomain system response data z2
, and use it to estimate a twopole, onezero transfer function. Specify an input delay.
load iddata2 z2; np = 2; nz = 1; input_delay = 0.2; sys = tfest(z2,np,nz,'InputDelay',input_delay);
sys
is an idtf
model containing the estimated transfer function with an input delay of 0.2 seconds.
Convert FrequencyResponse Data into Transfer Function
Use bode
to obtain the magnitude and phase response for the following system:
$$H(s)=\frac{s+0.2}{{s}^{3}+2{s}^{2}+s+1}$$
Use 100 frequency points, ranging from 0.1 rad/s to 10 rad/s, to obtain the frequencyresponse data. Use frd
to create a frequencyresponse data object.
freq = logspace(1,1,100); [mag,phase] = bode(tf([1 0.2],[1 2 1 1]),freq); data = frd(mag.*exp(1j*phase*pi/180),freq);
Estimate a threepole, onezero transfer function using data
.
np = 3; nz = 1; sys = tfest(data,np,nz);
sys
is an idtf
model containing the estimated transfer function.
Estimate Transfer Function with Known Transport Delays for Multiple Inputs
Load the timedomain system response data co2data
, which contains the data from two experiments, each with two inputs and one output. Convert the data from the first experiment into an iddata
object data
with a sample time of 0.5 seconds.
load co2data;
Ts = 0.5;
data = iddata(Output_exp1,Input_exp1,Ts);
Specify estimation options for the search method and the input and output offsets. Also specify the maximum number of search iterations.
opt = tfestOptions('SearchMethod','gna'); opt.InputOffset = [170;50]; opt.OutputOffset = mean(data.y(1:75)); opt.SearchOptions.MaxIterations = 50;
Estimate a transfer function using the measured data and the estimation option set opt
. Specify the transport delays from the inputs to the output.
np = 3; nz = 1; iodelay = [2 5]; sys = tfest(data,np,nz,iodelay,opt);
iodelay
specifies the inputtooutput delay from the first and second inputs to the output as 2 seconds and 5 seconds, respectively.
sys
is an idtf
model containing the estimated transfer function.
Estimate Transfer Function with Known and Unknown Transport Delays
Load timedomain system response data and use it to estimate a transfer function for the system. Specify the known and unknown transport delays.
load co2data;
Ts = 0.5;
data = iddata(Output_exp1,Input_exp1,Ts);
data
is an iddata
object with two input channels and one output channels, and which has a sample rate of 0.5 seconds.
Create an option set opt
. Specify estimation options for the search method and the input and output offsets. Also specify the maximum number of search iterations.
opt = tfestOptions('Display','on','SearchMethod','gna'); opt.InputOffset = [170; 50]; opt.OutputOffset = mean(data.y(1:75)); opt.SearchOptions.MaxIterations = 50;
Specify the unknown and known transport delays in iodelay
, using 2
for a known delay of 2 seconds and nan
for the unknown delay. Estimate the transfer function using iodelay
and opt
.
np = 3; nz = 1; iodelay = [2 nan]; sys = tfest(data,np,nz,iodelay,opt);
sys
is an idtf
model containing the estimated transfer function.
Estimate Transfer Function with Unknown, Constrained Transport Delays
Create a transfer function model with the expected numerator and denominator structure and delay constraints.
In this example, the experiment data consists of two inputs and one output. Both transport delays are unknown and have an identical upper bound. Additionally, the transfer functions from both inputs to the output are identical in structure.
init_sys = idtf(NaN(1,2),[1,NaN(1,3)],'IODelay',NaN);
init_sys.Structure(1).IODelay.Free = true;
init_sys.Structure(1).IODelay.Maximum = 7;
init_sys
is an idtf
model describing the structure of the transfer function from one input to the output. The transfer function consists of one zero, three poles, and a transport delay. The use of NaN
indicates unknown coefficients.
init_sys.Structure(1).IODelay.Free = true
indicates that the transport delay is not fixed.
init_sys.Structure(1).IODelay.Maximum = 7
sets the upper bound for the transport delay to 7 seconds.
Specify the transfer function from both inputs to the output.
init_sys = [init_sys,init_sys];
Load timedomain system response data and use it to estimate a transfer function. Specify options in the tfestOptions
option set opt
.
load co2data; Ts = 0.5; data = iddata(Output_exp1,Input_exp1,Ts); opt = tfestOptions('Display','on','SearchMethod','gna'); opt.InputOffset = [170;50]; opt.OutputOffset = mean(data.y(1:75)); opt.SearchOptions.MaxIterations = 50; sys = tfest(data,init_sys,opt);
sys
is an idtf
model containing the estimated transfer function.
Analyze the estimation result by comparison. Create a compareOptions
option set opt2
and specify input and output offsets, and then use compare
.
opt2 = compareOptions; opt2.InputOffset = opt.InputOffset; opt2.OutputOffset = opt.OutputOffset; compare(data,sys,opt2)
Estimate Transfer Function Containing Different Numbers of Poles for InputOutput Pairs
Estimate a multipleinput, singleoutput transfer function containing different numbers of poles for inputoutput pairs for given data.
Obtain frequencyresponse data.
For example, use frd
to create a frequencyresponse data model for the following system:
$$G=\left[\begin{array}{c}{e}^{4s}\frac{s+2}{{s}^{3}+2{s}^{2}+4s+5}\\ {e}^{0.6s}\frac{5}{{s}^{4}+2{s}^{3}+{s}^{2}+s}\end{array}\right]$$
Use 100 frequency points, ranging from 0.01 rad/s to 100 rad/s, to obtain the frequencyresponse data.
G = tf({[1 2],[5]},{[1 2 4 5],[1 2 1 1 0]},0,'IODelay',[4 0.6]);
data = frd(G,logspace(2,2,100));
data
is an frd
object containing the continuoustime frequency response for G
.
Estimate a transfer function for data
.
np = [3 4]; nz = [1 0]; iodelay = [4 0.6]; sys = tfest(data,np,nz,iodelay);
np
specifies the number of poles in the estimated transfer function. The first element of np
indicates that the transfer function from the first input to the output contains three poles. Similarly, the second element of np
indicates that the transfer function from the second input to the output contains four poles.
nz
specifies the number of zeros in the estimated transfer function. The first element of nz
indicates that the transfer function from the first input to the output contains one zero. Similarly, the second element of np
indicates that the transfer function from the second input to the output does not contain any zeros.
iodelay
specifies the transport delay from the first input to the output as 4 seconds. The transport delay from the second input to the output is specified as 0.6 seconds.
sys
is an idtf
model containing the estimated transfer function.
Estimate Transfer Function for Unstable System
Estimate a transfer function describing an unstable system using frequencyresponse data.
Use idtf
to construct a transfer function model G
of the following system:
$$G=\left[\begin{array}{c}\frac{s+2}{{s}^{3}+2{s}^{2}+4s+5}\\ \frac{5}{{s}^{4}+2{s}^{3}+{s}^{2}+s+1}\end{array}\right]$$
G = idtf({[1 2], 5},{[1 2 4 5],[1 2 1 1 1]});
Use idfrd
to obtain a frequencyresponse data model data
for G
. Specify 100 frequency points ranging from 0.01 rad/s to 100 rad/s.
data = idfrd(G,logspace(2,2,100));
data
is an idfrd
object.
Estimate a transfer function for data
.
np = [3 4]; nz = [1 0]; sys = tfest(data,np,nz);
np
specifies the number of poles in the estimated transfer function. The first element of np
indicates that the transfer function from the first input to the output contains three poles. Similarly, the second element of np
indicates that the transfer function from the second input to the output contains four poles.
nz
specifies the number of zeros in the estimated transfer function. The first element of nz
indicates that the transfer function from the first input to the output contains one zero. Similarly, the second element of nz
indicates that the transfer function from the second input to the output does not contain any zeros.
sys
is an idtf
model containing the estimated transfer function.
pole(sys)
ans = 7×1 complex
1.5260 + 0.0000i
0.2370 + 1.7946i
0.2370  1.7946i
1.4656 + 0.0000i
1.0000 + 0.0000i
0.2328 + 0.7926i
0.2328  0.7926i
sys
is an unstable system, as the pole display indicates.
Estimate Transfer Function using High Modal Density Frequency Response Data
Load the highdensity frequencyresponse measurement data. The data corresponds to an unstable process maintained at equilibrium using feedback control.
load HighModalDensityData FRF f
Package the data as an idfrd
object for identification and find the Bode magnitude response.
G = idfrd(permute(FRF,[2 3 1]),f,0,'FrequencyUnit','Hz'); bodemag(G)
Estimate a transfer function with 32 poles and 32 zeros, and compare the Bode magnitude response.
sys = tfest(G,32,32); bodemag(G, sys) xlim([0.01,2e3]) legend
Obtain and Apply Estimated Initial Conditions
Load and plot the data.
load iddata1ic z1i plot(z1i)
Examine the initial value of the output data y(1)
.
ystart = z1i.y(1)
ystart = 3.0491
The measured output does not start at 0.
Estimate a secondorder transfer function sys
and return the estimated initial condition ic
.
[sys,ic] = tfest(z1i,2,1); ic
ic = initialCondition with properties: A: [2x2 double] X0: [2x1 double] C: [0.2957 5.2441] Ts: 0
ic
is an initialCondition
object that encapsulates the free response of sys
, in statespace form, to the initial state vector in X0
.
Simulate sys
using the estimation data but without incorporating the initial conditions. Plot the simulated output with the measured output.
y_no_ic = sim(sys,z1i); plot(y_no_ic,z1i) legend('Model Response','Output Data')
The measured and simulated outputs do not agree at the beginning of the simulation.
Incorporate the initial condition into the simOptions
option set.
opt = simOptions('InitialCondition',ic); y_ic = sim(sys,z1i,opt); plot(y_ic,z1i) legend('Model Response','Output Data')
The simulation combines the model response to the input signal with the free response to the initial condition. The measured and simulated outputs now have better agreement at the beginning of the simulation. This initial condition is valid only for the estimation data z1i
.
Input Arguments
tt
— Timetablebased estimation data
timetable
 cell array of timetables
Estimation data, specified as a uniformly sampled timetable
that contains both input and output signal variables
or, for multiexperiment data, a cell array of
timetables.
Use Entire Timetable
If you want to use all the input and output variables in tt
, and the
variables are organized so that the set of input
variables is followed by the set of output
variables, then:
For SISO systems, specify
tt
as an N_{s}by2timetable
, where N_{s} is the number of samples and the twotimetable
variables represent the measured input signal and output signal respectively.For MIMO systems, specify
tt
as an N_{s}by(N_{u}+N_{y})timetable
, where N_{u} is the number of inputs and N_{y} is the number of outputs. The first N_{u} variables must contain the input signals and the remaining N_{y} variables must contain the output signals.When you are estimating state space or transfer function models, you must also explicitly specify the input and output channels, as the following section describes.
For multiexperiment data, specify data as an N_{e}by1 cell array of timetables, where N_{e} is the number of experiments. The sample times of all the experiments must match.
Use Selected Variables from Timetable
If you want to use a subset of variables from the timetable
, or if the input
and output variables are intermixed, use the
'InputName'
and
'OutputName'
namevalue
arguments to specify which variables to
use.
For example, suppose that tt
contains six variables:
"u1"
, "u2"
, "u3"
, and
"y1"
, "y2"
, "y3"
. For estimation,
you want to use the variables "u1"
and "u2"
as the
inputs and the variables "y1"
and "y3"
as the outputs.
Use the following command to perform the estimation:
sys = tfest(tt,__,'InputName',["u1" "u2"],'OutputName',["y1"
"y3"])
For more information about working with estimation data types, see Data Domains and Data Types in System Identification Toolbox.
u
, y
— Matrixbased estimation data
matrices  cell array of matrices
Estimation data, specified for SISO systems as a pair of N_{s}by1 realvalued matrices that contain uniformly sampled input and output timedomain signal values. Here, N_{s} is the number of samples.
For MIMO systems, specify u
,y
as an
input/output matrix pair with the following dimensions:
u
— N_{s}byN_{u}, where N_{u} is the number of inputs.y
— N_{s}byN_{y}, where N_{y} is the number of outputs.
For multiexperiment data, specify u
,y
as a
pair of 1byN_{e} cell arrays, where
N_{e} is the number of experiments. The
sample times of all the experiments must match.
Limitations
Matrixbased data does not support estimation from frequencydomain data. You must use a data object such as an
iddata
object oridfrd
object (seedata
).Using matrices for estimation data is not recommended for continuoustime estimation since the data does not provide the sample time. The software assumes that the data is sampled at 1 Hz. For continuoustime estimation, it is recommended that you convert the input and output matrix pair into a single
timetable
. For example, to convert the singlecolumn matricesum
andym
to atimetable
tt
with a sample time of 0.5 minutes, use the following command.For a more detailed example of converting matrixbased SISO data to a timetable, see Convert SISO Matrix Data to Timetable. For an example of converting a MIMO matrix pair to a timetable, see Convert MIMO Matrix Data to Timetable for ContinuousTime Model Estimation.tt = timetable(um,ym,'rowtimes',minutes(0.5*(1:size(u,1))))
For more information about working with estimation data types, see Data Domains and Data Types in System Identification Toolbox.
data
— Estimation data object
iddata
object  frd
object  idfrd
object
Estimation data object, specified as an iddata
object, an
frd
object, or an idfrd
object that contains
uniformly sampled input and output values. By default, the software sets the sample time
of the model to the sample time of the estimation data.
For multiexperiment data, the sample times and intersample behavior of all the experiments must match.
For timedomain estimation, data
must be an iddata
object containing the input and output signal values.
For frequencydomain estimation, data
can be one of the
following:
Limitations
You cannot estimate continuoustime models using discretetime frequencydomain data.
np
— Number of poles
nonnegative integer  matrix
Number of poles in the estimated transfer function, specified as a nonnegative integer or a matrix.
For systems that have multiple inputs and/or multiple outputs, you can apply either
a global value or individual values of np
to the input/output
pairs, as follows:
Same number of poles for every pair — Specify
np
as a scalar.Individual number of poles for each pair — Specify
np
as an N_{y}byN_{u} matrix. N_{y} is the number of outputs and N_{u} is the number of inputs.
For an example, see Estimate Transfer Function Model by Specifying Number of Poles.
nz
— Number of zeros
nonnegative integer  matrix
Number of zeros in the estimated transfer function, specified as a nonnegative integer or a matrix.
For systems that have multiple inputs, multiple outputs, or both, you can apply
either a global value or individual values of nz
to the
input/output pairs, as follows:
Same number of poles for every pair — Specify
nz
as a scalar.Individual number of poles for each pair — Specify
nz
as an N_{y}byN_{u} matrix. N_{y} is the number of outputs and N_{u} is the number of inputs.
For a continuoustime model estimated using discretetime data, set
nz
<= np
.
For discretetime model estimation, specify nz
as the number of
zeros of the numerator polynomial of the transfer function. For example,
tfest(tt,2,1,'Ts',data.Ts)
estimates a transfer function of the
form $${b}_{1}{z}^{1}/(1+{a}_{1}{z}^{1}+{b}_{2}{z}^{2})$$, while tfest(tt,2,2,'Ts',data.Ts)
estimates $$({b}_{1}{z}^{1}+{b}_{2}{z}^{2})/(1+{a}_{1}{z}^{1}+{b}_{2}{z}^{2})$$. Here, z^{1} is the
Ztransform lag variable. For more information about discretetime transfer functions,
see DiscreteTime Representation. For an example, see Estimate DiscreteTime Transfer Function.
iodelay
— Transport delay
[]
(default)  nonnegative integer  matrix
Transport delay, specified as a nonnegative integer, a NaN
scalar, or a matrix.
For continuoustime systems, specify transport delays in the time unit stored in the
TimeUnit
property of data
. For discretetime
systems, specify transport delays as integers denoting delays of a multiple of the
sample time Ts
.
For a MIMO system with N_{y} outputs and
N_{u} inputs, set
iodelay
to an
N_{y}byN_{u}
array. Each entry of this array is a numerical value that represents the transport delay
for the corresponding input/output pair. You can also set iodelay
to a scalar value to apply the same delay to all input/output pairs.
The specified values are treated as fixed delays. To denote unknown transport
delays, specify NaN
in the iodelay
matrix.
Use []
or 0
to indicate no transport
delay.
For an example, see Estimate Transfer Function Containing Known Transport Delay.
opt
— Estimation options
tfestOptions
option set
Estimation options, specified as a tfestOptions
option set. Options specified by opt
include:
Estimation objective
Handling of initial conditions
Numerical search method to be used in estimation
Intersample behavior
For an example, see Estimate Transfer Function Using Estimation Option Set.
init_sys
— Linear system that configures initial parameterization of sys
idtf
model  linear model  structure
Linear system that configures the initial parameterization of
sys
, specified as an idtf
model, a linear model , or a
structure. You obtain init_sys
either by performing an estimation
using measured data or by direct construction.
If init_sys
is an idtf
model,
tfest
uses the parameter values of init_sys
as the initial guess for estimating sys
.
Use the Structure
property of init_sys
to
configure initial parameter values and constraints for the numerator, denominator, and
transport lag. For instance:
To specify an initial guess for the A matrix of
init_sys
, setinit_sys.Structure.Numerator.Value
to the initial guess.To specify constraints for the B matrix of
init_sys
:Set
init_sys.Structure.Numerator.Minimum
to the minimum numerator coefficient values.Set
init_sys.Structure.Numerator.Maximum
to the maximum numerator coefficient values.Set
init_sys.Structure.Numerator.Free
to indicate which numerator coefficients are free for estimation.
For an example, see Estimate Transfer Function with Unknown, Constrained Transport Delays.
If
init_sys
is not an idtf
model, the software
first converts init_sys
to a transfer function.
tfest
uses the parameters of the resulting model as the initial
guess for estimation.
If you do not specify opt
, and init_sys
was obtained by estimation rather than construction, then the software uses estimation
options from init_sys.Report.OptionsUsed
.
NameValue Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Namevalue arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: sys = tfest(data,np,nz,'Ts',0.1)
InputName
— Input channel names
string  character vector  string array  cell array of character vectors
Input channel names, specified as a string, character vector, string array, or cell array of character vectors.
If you are using a timetable for the data source, the names in
InputName
must be a subset of the timetable variables.
Example: sys = tfest(tt,__,'InputName',["u1" "u2"])
selects
the variables u1
and u2
as the input channels from
the timetable tt
to use for the estimation.
OutputName
— Output channel names
string  character vector  string array  cell array of character vectors
Output channel names, specified as a string, character vector, string array, or cell array of character vectors.
If you are using a timetable for the data source, the names in
OutputName
must be a subset of the timetable variables.
Example: sys = tfest(tt,__,'OutputName',["y1" "y3"])
selects
the variables y1
and y3
as the output channels
from the timetable tt
to use for the estimation.
Ts
— Sample time of estimated model
0
(continuous time) (default)  positive scalar
Sample time of the estimated model, specified as either 0
or a
positive scalar.
For continuoustime models, specify
'Ts'
as0
.For discretetime models, specify
'Ts'
as the data sample time in units defined by the following:For timetablebased data — The timetable
Time
columnFor matrixbased data — Seconds
For data objects, such as
iddata
objects — Thedata.TimeUnit
property
In the discrete case,
np
andnz
refer to the number of roots of z^{1} for the numerator and denominator polynomials.To obtain the data sample time for a timetable
tt
, use the timetable propertytt.Properties.Timestep
.
For an example, see Estimate DiscreteTime Transfer Function.
InputDelay
— Input delays
0
(default)  scalar  vector
Input delay for each input channel, specified as a scalar or a numeric vector.
For continuoustime models, specify
'InputDelay'
in the time units stored in theTimeUnit
property.For discretetime models, specify
'InputDelay'
in integer multiples of the sample timeTs
. For example, setting'InputDelay'
to3
specifies a delay of three sampling periods.
For a system with N_{u} inputs, set
InputDelay
to an
N_{u}by1 vector. Each entry of this vector is
a numerical value that represents the input delay for the corresponding input
channel.
To apply the same delay to all channels, specify InputDelay
as a scalar.
For an example, see Specify Model Properties of Estimated Transfer Function.
Feedthrough
— Feedthrough for discretetime transfer function
0
(default)  1
 logical matrix
Feedthrough for discretetime transfer functions, specified as a logical scalar or
an
N_{y}byN_{u}
logical matrix. N_{y} is the number of outputs
and N_{u} is the number of inputs. To use the
same feedthrough for all input/output channels, specify Feedthrough
as a scalar.
Consider a discretetime model with two poles and three zeros:
$$H({z}^{1})=\frac{b0+b1{z}^{1}+b2{z}^{2}+b3{z}^{3}}{1+a1{z}^{1}+a2{z}^{2}}$$
When the model has direct feedthrough, b0 is a free parameter whose value is estimated along with the rest of the model parameters b1, b2, b3, a1, and a2. When the model has no feedthrough, b0 is fixed to zero. For an example, see Estimate DiscreteTime Transfer Function with Feedthrough.
Output Arguments
sys
— Identified transfer function
idtf
model
Identified transfer function, returned as an idtf
model. This model is created using the specified model orders, delays,
and estimation options.
Information about the estimation results and options used is stored in the
Report
property of the model. Report
has the
following fields.
Report Field  Description 

Status  Summary of the model status, which indicates whether the model was created by construction or obtained by estimation 
Method  Estimation command used 
InitializeMethod  Algorithm used to initialize the numerator and denominator for estimation of continuoustime transfer functions using timedomain data, returned as one of the following values:
This field is especially useful to view the algorithm used
when the 
N4Weight  Weighting matrices used in the singularvalue decomposition step when
This field is especially useful to view the weighting
matrices used when the 
N4Horizon  Forward and backward prediction horizons used when

InitialCondition  Handling of initial conditions during model estimation, returned as one of the following values:
This field is especially useful to view
how the initial conditions were handled when the 
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 these fields.

Parameters  Estimated values of model parameters 
OptionsUsed  Option set used for estimation. If no custom options were configured,
this is a set of default options. See 
RandState  State of the random number stream at the start of estimation. Empty,

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

Termination  Termination conditions for the iterative search used for prediction error minimization, returned as a structure with these fields.
For estimation methods that do not require numerical search
optimization, the 
For more information on using Report
, see Estimation Report.
ic
— Initial conditions
initialCondition
object  object array of initialCondition
values
Estimated initial conditions, returned as an initialCondition
object or an object array of
initialCondition
values.
For a singleexperiment data set,
ic
represents, in statespace form, the free response of the transfer function model (A and C matrices) to the estimated initial states (x_{0}).For a multipleexperiment data set with N_{e} experiments,
ic
is an object array of length N_{e} that contains one set ofinitialCondition
values for each experiment.
If tfest
returns ic
values of
0
and you know that you have nonzero initial conditions, set the
'InitialCondition'
option in tfestOptions
to 'estimate'
and pass the updated option
set to tfest
. For
example:
opt = tfestOptions('InitialCondition','estimate') [sys,ic] = tfest(data,np,nz,opt)
'auto'
setting of 'InitialCondition'
uses
the 'zero'
method when the initial conditions have a negligible
effect on the overall estimationerror minimization process. Specifying
'estimate'
ensures that the software estimates values for
ic
.
For more information, see initialCondition
. For an example of using this argument, see Obtain and Apply Estimated Initial Conditions.
Algorithms
The details of the estimation algorithms used by tfest
vary depending
on various factors, including the sampling of the estimated model and the estimation
data.
ContinuousTime Transfer Function Estimation Using TimeDomain Data
The estimation algorithm initializes the estimable parameters using the method
specified by the InitializeMethod
estimation option. The default method
is the Instrument Variable (IV) method.
The StateVariable Filters (SVF) approach and the Generalized Poisson Moment Functions
(GPMF) approach to continuoustime parameter estimation use prefiltered data [1]
[2]. The constant $$\frac{1}{\lambda}$$ in [1] and [2] corresponds to the initialization option (InitializeOptions
) field
FilterTimeConstant
. IV is the simplified refined IV method and is
called SRIVC in [3]. This method has a prefilter that is the denominator of the current model, initialized
with SVF. This prefilter is iterated up to MaxIterations
times, until
the model change is less than Tolerance
.
MaxIterations
and Tolerance
are options that you
can specify using the InitializeOptions
structure. The
'n4sid'
initialization option estimates a discretetime model, using
the N4SID estimation algorithm, that it transforms to continuoustime using d2c
.
Use tfestOptions
to create the option set used
to estimate a transfer function.
The initialized parameters are updated using a nonlinear leastsquares search method,
specified by the SearchMethod
estimation option. The objective of the
search method is to minimize the weighted prediction error norm.
DiscreteTime Transfer Function Estimation Using TimeDomain Data
ContinuousTime Transfer Function Estimation Using ContinuousTime FrequencyDomain Data
The estimation algorithm performs the following tasks:
Perform a bilinear mapping to transform the domain (frequency grid) of the transfer function. For continuoustime models, the imaginary axis is transformed to the unit disk. For discretetime models, the original domain unit disk is transformed to another unit disk.
Perform SK iterations [4] to solve a nonlinear leastsquares problem — Consider a multiinput singleoutput system. The nonlinear leastsquares problem is to minimize the following loss function:
$$\underset{D,{N}_{i}}{\text{minimize}}{\displaystyle \sum _{k=1}^{{n}_{f}}{\leftW({\omega}_{k})\left(y({\omega}_{k}){\displaystyle \sum _{i=1}^{{n}_{u}}\frac{{N}_{i}({\omega}_{k})}{D({\omega}_{k})}{u}_{i}({\omega}_{k})}\right)\right}^{2}}$$
Here, W is a frequencydependent weight that you specify. D is the denominator of the transfer function model that is to be estimated, and N_{i} is the numerator corresponding to the ith input. y and u are the measured output and input data, respectively. n_{f} and n_{u} are the number of frequencies and inputs, and w is the frequency. Rearranging the terms gives
$$\underset{D,{N}_{i}}{\text{minimize}}{\displaystyle \sum _{k=1}^{{n}_{f}}{\left\frac{W({\omega}_{k})}{D({\omega}_{k})}\left(D({\omega}_{k})y({\omega}_{k}){\displaystyle \sum _{i=1}^{{n}_{u}}{N}_{i}({\omega}_{k}){u}_{i}({\omega}_{k})}\right)\right}^{2}}$$
To perform the SK iterations, the algorithm iteratively solves
$$\underset{{D}_{m},{N}_{i,m}}{\text{minimize}}{\displaystyle \sum _{k=1}^{{n}_{f}}{\left\frac{W({\omega}_{k})}{{D}_{m1}({\omega}_{k})}\left({D}_{m}({\omega}_{k})y({\omega}_{k}){\displaystyle \sum _{i=1}^{{n}_{u}}{N}_{i,m}({\omega}_{k}){u}_{i}({\omega}_{k})}\right)\right}^{2}}$$
Here, m is the current iteration, and D_{m1}(ω) is the denominator response identified at the previous iteration. Now each step of the iteration is a linear leastsquares problem, where the identified parameters capture the responses D_{m}(ω) and N_{i,m}(ω) for i = 1,2,...n_{u}. The iteration is initialized by choosing D_{0}(ω) = 1.
The first iteration of the algorithm identifies D_{1}(ω). The D_{1}(ω) and N_{i,1}(ω) polynomials are expressed in monomial basis.
The second and following iterations express the polynomials D_{m}(ω) and N_{i,m}(ω) in terms of orthogonal rational basis functions on the unit disk. These basis functions have the form
$${B}_{j,m}(\omega )=\left(\frac{\sqrt{1{\left{\lambda}_{j,m1}\right}^{2}}}{q{\lambda}_{j,m1}}\right){\displaystyle \prod _{r=0}^{j1}\frac{1{({\lambda}_{j,m1})}^{*}q(\omega )}{q(\omega ){\lambda}_{r,m1}}}$$
Here, λ_{j,m1} is the jth pole that is identified at the previous step m1 of the iteration. λ_{j,m1}^{*} is the complex conjugate of λ_{j,m1}, and q is the frequencydomain variable on the unit disk.
The algorithm runs for a maximum of 20 iterations. The iterations are terminated early if the relative change in the value of the loss function is less than 0.001 in the last three iterations.
If you specify bounds on transfer function coefficients, these bounds correspond to affine constraints on the identified parameters. If you have only equality constraints (fixed transfer function coefficients), the corresponding equality constrained leastsquares problem is solved algebraically. To do so, the software computes an orthogonal basis for the null space of the equality constraint matrix, and then solves the leastsquares problem within this null space. If you have upper or lower bounds on transfer function coefficients, the corresponding inequality constrained leastsquares problem is solved using interiorpoint methods.
Perform linear refinements — The SK iterations, even when they converge, do not always yield a locally optimal solution. To find a critical point of the optimization problem that can yield a locally optimal solution, a second set of iterations are performed. The critical points are solutions to a set of nonlinear equations. The algorithm searches for a critical point by successively constructing a linear approximation to the nonlinear equations and solving the resulting linear equations in the leastsquares sense. The equations follow.
Equation for the jth denominator parameter:
$$0=2{\displaystyle \sum _{k=1}^{{n}_{f}}\mathrm{Re}\left\{\frac{{\leftW({\omega}_{k})\right}^{2}{B}_{j}^{*}({\omega}_{k}){\displaystyle \sum _{i=1}^{{n}_{u}}{N}_{i,m1}^{*}({\omega}_{k}){u}_{i}^{*}({\omega}_{k})}}{{D}_{m1}^{*}({\omega}_{k}){\left{D}_{m1}({\omega}_{k})\right}^{2}}\left({D}_{m}({\omega}_{k})y({\omega}_{k}){\displaystyle \sum _{i=1}^{{n}_{u}}{N}_{i,m}({\omega}_{k}){u}_{i}({\omega}_{k})}\right)\right\}}$$
Equation for the jth numerator parameter that corresponds to input l:
$$0=2{\displaystyle \sum _{k=1}^{{n}_{f}}\mathrm{Re}\left\{\frac{{\leftW({\omega}_{k})\right}^{2}{B}_{j}^{*}({\omega}_{k}){u}_{l}^{*}({\omega}_{k})}{{\left{D}_{m1}({\omega}_{k})\right}^{2}}\left({D}_{m}({\omega}_{k})y({\omega}_{k}){\displaystyle \sum _{i=1}^{{n}_{u}}{N}_{i,m}({\omega}_{k}){u}_{i}({\omega}_{k})}\right)\right\}}$$
The first iteration is started with the best solution found for the numerators N_{i} and denominator D parameters during SK iterations. Unlike SK iterations, the basis functions B_{j}(ω) are not changed at each iteration; the iterations are performed with the basis functions that yielded the best solution in the SK iterations. As before, the algorithm runs for a maximum of 20 iterations. The iterations are terminated early if the relative change in the value of the loss function is less than 0.001 in the last three iterations.
If you specify bounds on transfer function coefficients, these bounds are incorporated into the necessary optimality conditions using generalized Lagrange multipliers. The resulting constrained linear leastsquares problems are solved using the same methods explained in the SK iterations step.
Return the transfer function parameters corresponding to the optimal solution — Both the SK and linear refinement iteration steps do not guarantee an improvement in the loss function value. The algorithm tracks the best parameter value observed during these steps, and returns these values.
Invert the bilinear mapping performed in step 1.
Perform an iterative refinement of the transfer function parameters using the nonlinear leastsquares search method specified in the
SearchMethod
estimation option. This step is implemented in the following situations:When you specify the
EnforceStability
estimation option astrue
(stability is requested), and the result of step 5 of this algorithm is an unstable model. The unstable poles are reflected inside the stability boundary and the resulting parameters are iteratively refined. For information about estimation options, seetfestOptions
.When you add a regularization penalty to the loss function using the
Regularization
estimation option. For an example about regularization, see Regularized Identification of Dynamic Systems.You estimate a continuoustime model using discretetime data (see DiscreteTime Transfer Function Estimation Using DiscreteTime FrequencyDomain Data).
You use frequency domain input/output data to identify a multiinput model.
If you are using the estimation algorithm from R2016a or earlier (see tfest Estimation Algorithm Update) for estimating a continuoustime model using continuoustime frequencydomain data, then for continuoustime data and fixed delays, the OutputError algorithm is used for model estimation. For continuoustime data and free delays, the statespace estimation algorithm is used. In this algorithm, the model coefficients are initialized using the N4SID estimation method. This initialization is followed by nonlinear leastsquares searchbased updates to minimize a weighted prediction error norm.
DiscreteTime Transfer Function Estimation Using DiscreteTime FrequencyDomain Data
The estimation algorithm is the same as for continuoustime transfer function estimation using continuoustime frequencydomain data, except discretetime data is used.
If you are using the estimation algorithm from R2016a or earlier (see tfest Estimation Algorithm Update), the algorithm is the same as the algorithm for discretetime transfer function estimation using timedomain data.
Note
The software does not support estimation of a discretetime transfer function using continuoustime frequencydomain data.
ContinuousTime Transfer Function Estimation Using DiscreteTime FrequencyDomain Data
The tfest
command first estimates a discretetime model from the
discretetime data. The estimated model is then converted to a continuoustime model using
the d2c
command. The frequency response of the
resulting continuoustime model is then computed over the frequency grid of the estimation
data. A continuoustime model of the desired (userspecified) structure is then fit to this
frequency response. The estimation algorithm for using the frequencyresponse data to obtain
the continuoustime model is the same as the algorithm for continuoustime transfer function
estimation using continuoustime data.
If you are using the estimation algorithm from R2016a or earlier (see tfest Estimation Algorithm Update), the statespace estimation algorithm is used for estimating continuoustime models from discretetime data. In this algorithm, the model coefficients are initialized using the N4SID estimation method. This initialization is followed by nonlinear leastsquares searchbased updates to minimize a weighted prediction error norm.
Delay Estimation
When delay values are specified as
NaN
, the software usesdelayest
to estimate them separately from the model numerator and denominator coefficients.tfest
then treats these delay values as fixed during the iterative update of the rest of the model. Therefore, the delay values are not iteratively updated.By default, for discretetime data (
Ts
>0),delayest
limits the search for delays to a range of 0–30 samples. For continuoustime models, this range translates to 0–30Ts
time units. For continuoustime data (Ts
= 0),delayest
limits the search range to 0–10 time units. You can change these limits by first creating a template modelinit_sys
usingidtf
and then, setting the values ofinit_sys.Structure.IODelay.Minimum
andinit_sys.Structure(i,j).IODelay.Maximum
.For an initial model,
init_sys
, with:init_sys.Structure.IODelay.Value
specified as finite valuesinit_sys.Structure.IODelay.Free
specified astrue
the initial delay values are left unchanged.
Estimation of delays is often a difficult problem. A best practice is to assess the
presence and the value of a delay. To do so, use physical insight of the process being
modeled and functions such as arxstruc
, delayest
, and impulseest
. For an example of determining
input delay, see Model Structure Selection: Determining Model Order and Input Delay.
References
[1] Garnier, H., M. Mensler, and A. Richard. “ContinuousTime Model Identification from Sampled Data: Implementation Issues and Performance Evaluation.” International Journal of Control 76, no. 13 (January 2003): 1337–57. https://doi.org/10.1080/0020717031000149636.
[2] Ljung, Lennart. “Experiments with Identification of Continuous Time Models.” IFAC Proceedings Volumes 42, no. 10 (2009): 1175–80. https://doi.org/10.3182/200907063FR2004.00195.
[3] Young, Peter, and Anthony Jakeman. “Refined Instrumental Variable Methods of Recursive TimeSeries Analysis Part III. Extensions.” International Journal of Control 31, no. 4 (April 1980): 741–64. https://doi.org/10.1080/00207178008961080.
[4] Drmač, Z., S. Gugercin, and C. Beattie. “QuadratureBased Vector Fitting for Discretized H_{2} Approximation.” SIAM Journal on Scientific Computing 37, no. 2 (January 2015): A625–52. https://doi.org/10.1137/140961511.
[5] Ozdemir, Ahmet Arda, and Suat Gumussoy. “Transfer Function Estimation in System Identification Toolbox via Vector Fitting.” IFACPapersOnLine 50, no. 1 (July 2017): 6232–37. https://doi.org/10.1016/j.ifacol.2017.08.1026.
Version History
Introduced in R2012aR2022b: Timedomain estimation data is accepted in the form of timetables and matrices
Most estimation, validation, analysis, and utility functions now accept timedomain
input/output data in the form of a single timetable that contains both input and output data
or a pair of matrices that contain the input and output data separately. These functions
continue to accept iddata
objects as a data source as well, for
both timedomain and frequencydomain data.
R2018a: Advanced Options are deprecated for SearchOptions
when SearchMethod
is 'lsqnonlin'
Specification of lsqnonlin
 related advanced options are deprecated,
including the option to invoke parallel processing when estimating using the
lsqnonlin
search method, or solver, in Optimization Toolbox™.
R2016b: tfest
Estimation Algorithm Update
Starting in R2016b, a new algorithm is used for performing transfer function estimation
from frequencydomain data. You are likely to see faster and more accurate results with the
new algorithm, particularly for data with dynamics over a large range of frequencies and
amplitudes. However, the estimation results might not match results from previous releases.
To perform estimation using the previous estimation algorithm, append
'R2016a'
to the syntax.
For example, suppose that you are estimating a transfer function model with
np
poles using the frequencydomain data
data
.
sys = tfest(data,np)
To use the previous estimation algorithm, use the following syntax.
sys = tfest(data,np,'R2016a')
See Also
tfestOptions
 idtf
 timetable
 ssest
 procest
 ar
 arx
 oe
 bj
 polyest
 greyest
Topics
 Estimate Transfer Function Models at the Command Line
 Estimate Transfer Function Models with Transport Delay to Fit Given FrequencyResponse Data
 Estimate Transfer Function Models with Prior Knowledge of Model Structure and Constraints
 Apply Initial Conditions When Simulating Identified Linear Models
 Troubleshoot FrequencyDomain Identification of Transfer Function Models
 What are Transfer Function Models?
 Regularized Estimates of Model Parameters
 Estimating Models Using FrequencyDomain Data
MATLABBefehl
Sie haben auf einen Link geklickt, der diesem MATLABBefehl entspricht:
Führen Sie den Befehl durch Eingabe in das MATLABBefehlsfenster aus. Webbrowser unterstützen keine MATLABBefehle.
Select a Web Site
Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select: .
You can also select a web site from the following list:
How to Get Best Site Performance
Select the China site (in Chinese or English) for best site performance. Other MathWorks country sites are not optimized for visits from your location.
Americas
 América Latina (Español)
 Canada (English)
 United States (English)
Europe
 Belgium (English)
 Denmark (English)
 Deutschland (Deutsch)
 España (Español)
 Finland (English)
 France (Français)
 Ireland (English)
 Italia (Italiano)
 Luxembourg (English)
 Netherlands (English)
 Norway (English)
 Österreich (Deutsch)
 Portugal (English)
 Sweden (English)
 Switzerland
 United Kingdom (English)