Documentation 
Transfer function model with identifiable parameters
sys = idtf(num,den) creates a continuoustime transfer function with identifiable parameters (an idtf model). num specifies the current values of the transfer function numerator coefficients. den specifies the current values of the transfer function denominator coefficients.
sys = idtf(num,den,Ts) creates a discretetime transfer function with identifiable parameters. Ts is the sampling time.
sys = idtf(___,Name,Value) creates a transfer function with properties specified by one or more Name,Value pair arguments.
sys = idtf(sys0) converts any dynamic system model, sys0, to idtf model form.
An idtf model represents a system as a continuoustime or discretetime transfer function with identifiable (estimable) coefficients.
A SISO transfer function is a ratio of polynomials with an exponential term. In continuous time,
$$G\left(s\right)={e}^{\tau s}\frac{{b}_{n}{s}^{n}+{b}_{n1}{s}^{n1}+\mathrm{...}+{b}_{0}}{{s}^{m}+{a}_{m1}{s}^{m1}+\mathrm{...}+{a}_{0}}.$$
In discrete time,
$$G\left({z}^{1}\right)={z}^{k}\frac{{b}_{n}{z}^{n}+{b}_{n1}{z}^{n+1}+\mathrm{...}+{b}_{0}}{{z}^{m}+{a}_{m1}{z}^{m+1}+\mathrm{...}+{a}_{0}}.$$
In discrete time, z^{–k} represents a time delay of kT_{s}, where T_{s} is the sampling time.
For idtf models, the denominator coefficients a_{0},...,a_{m–1} and the numerator coefficients b_{0},...,b_{n} can be estimable parameters. (The leading denominator coefficient is always fixed to 1.) The time delay τ (or kin discrete time) can also be an estimable parameter. The idtf model stores the polynomial coefficients a_{0},...,a_{m–1} and b_{0},...,b_{n} in the den and num properties of the model, respectively. The time delay τ or k is stored in the ioDelay property of the model.
A MIMO transfer function contains a SISO transfer function corresponding to each inputoutput pair in the system. For idtf models, the polynomial coefficients and transport delays of each inputoutput pair are independently estimable parameters.
There are three ways to obtain an idtf model.
Estimate the idtf model based on inputoutput measurements of a system, using tfest. The tfest command estimates the values of the transfer function coefficients and transport delays. The estimated values are stored in the num, den, and ioDelay properties of the resulting idtf model. The Report property of the resulting model stores information about the estimation, such as handling of initial conditions and options used in estimation.
When you obtain an idtf model by estimation, you can extract estimated coefficients and their uncertainties from the model. To do so, use commands such as tfdata, getpar, or getcov.
Create an idtf model using the idtf command.
You can create an idtf model to configure an initial parameterization for estimation of a transfer function to fit measured response data. When you do so, you can specify constraints on such values as the numerator and denominator coefficients and transport delays. For example, you can fix the values of some parameters, or specify minimum or maximum values for the free parameters. You can then use the configured model as an input argument to tfest to estimate parameter values with those constraints.
Convert an existing dynamic system model to an idtf model using the idtf command.
num 
Initial values of transfer function numerator coefficients. For SISO transfer functions, specify the initial values of the numerator coefficients num as a row vector. Specify the coefficients in order of:
Use NaN for any coefficient whose initial value is not known. For MIMO transfer functions with Ny outputs and Nu inputs, num is a NybyNu cell array of numerator coefficients for each input/output pair. 
den 
Initial values of transfer function denominator coefficients. For SISO transfer functions, specify the initial values of the denominator coefficients den as a row vector. Specify the coefficients in order of:
The leading coefficient in den must be 1. Use NaN for any coefficient whose initial value is not known. For MIMO transfer functions with Ny outputs and Nu inputs, den is a NybyNu cell array of denominator coefficients for each input/output pair. 
Ts 
Sampling time. For continuoustime models, Ts = 0. For discretetime models, Ts is a positive scalar representing the sampling period. This value is expressed in the unit specified by the TimeUnit property of the model. To denote a discretetime model with unspecified sampling time, set Ts = 1. Changing this property does not discretize or resample the model. Use c2d and d2c to convert between continuous and discretetime representations. Use d2d to change the sampling time of a discretetime system. Default: 0 (continuous time) 
sys0 
Dynamic system. Any dynamic system to convert to an idtf model. When sys0 is an identified model, its estimated parameter covariance is lost during conversion. If you want to translate the estimated parameter covariance during the conversion, use translatecov. 
Specify optional commaseparated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.
Use Name,Value arguments to specify additional properties of idtf models during model creation. For example, idtf(num,den,'InputName','Voltage') creates an idtf model with the InputName property set to Voltage.
idtf object properties include:
num 
Values of transfer function numerator coefficients. If you create an idtf model sys using the idtf command, sys.num contains the initial values of numerator coefficients that you specify with the num input argument. If you obtain an idtf model by identification using tfest, then sys.num contains the estimated values of the numerator coefficients. For an idtf model sys, the property sys.num is an alias for the value of the property sys.Structure.num.Value. For SISO transfer functions, the values of the numerator coefficients are stored as a row vector in order of:
Any coefficient whose initial value is not known is stored as NaN. For MIMO transfer functions with Ny outputs and Nu inputs, num is a NybyNu cell array of numerator coefficients for each input/output pair. 
den 
Values of transfer function denominator coefficients. If you create an idtf model sys using the idtf command, sys.den contains the initial values of denominator coefficients that you specify with the den input argument. If you obtain an idtf model sys by identification using tfest, then sys.den contains the estimated values of the denominator coefficients. For an idtf model sys, the property sys.den is an alias for the value of the property sys.Structure.den.Value. For SISO transfer functions, the values of the denominator coefficients are stored as a row vector in order of:
The leading coefficient in den is fixed to 1. Any coefficient whose initial value is not known is stored as NaN. For MIMO transfer functions with Ny outputs and Nu inputs, den is a NybyNu cell array of denominator coefficients for each input/output pair. 
Variable 
String specifying the transfer function display variable. Variable requires one of the following values:
The value of Variable is reflected in the display, and also affects the interpretation of the num and den coefficient vectors for discretetime models. For Variable = 'z^1' or 'q^1', the coefficient vectors are ordered as ascending powers of the variable. 
ioDelay 
Transport delays. ioDelay is a numeric array specifying a separate transport delay for each input/output pair. If you create an idtf model sys using the idtf command, sys.ioDelay contains the initial values of the transport delay that you specify with a Name,Value argument pair. If you obtain an idtf model sys by identification using tfest, then sys.ioDelay contains the estimated values of the transport delay. For an idtf model sys, the property sys.ioDelay is an alias for the value of the property sys.Structure.ioDelay.Value. For continuoustime systems, transport delays are expressed in the time unit stored in the TimeUnit property. For discretetime systems, specify transport are expressed as integers denoting delay of a multiple of the sampling period Ts. For a MIMO system with Ny outputs and Nu inputs, set ioDelay as a NybyNu array. Each entry of this array is a numerical value representing the transport delay for the corresponding input/output pair. You can set ioDelay to a scalar value to apply the same delay to all input/output pairs. Default: 0 for all input/output pairs 
Structure 
Information about the estimable parameters of the idtf model. Structure.num, Structure.den, and Structure.ioDelay contain information about the numerator coefficients, denominator coefficients, and transport delay, respectively. Each contains the following fields:
For a MIMO model with Ny outputs and Nu input, Structure is an NybyNu array. The element Structure(i,j) contains information corresponding to the transfer function for the (i,j) inputoutput pair. 
NoiseVariance 
The variance (covariance matrix) of the model innovations e. An identified model includes a white, Gaussian noise component e(t). NoiseVariance is the variance of this noise component. Typically, the model estimation function (such as tfest) determines this variance. For SISO models, NoiseVariance is a scalar. For MIMO models, NoiseVariance is a N_{y}byN_{y} matrix, where N_{y} is the number of outputs in the system. 
Report 
Information about the estimation process. Report contains the following fields:

InputDelay 
Input delays. InputDelay is a numeric vector specifying a time delay for each input channel. For continuoustime systems, specify input delays in the time unit stored in the TimeUnit property. For discretetime systems, specify input delays in integer multiples of the sampling period Ts. For example, InputDelay = 3 means a delay of three sampling periods. For a system with Nu inputs, set InputDelay to an Nuby1 vector. Each entry of this vector is a numerical value representing the input delay for the corresponding input channel. You can also set InputDelay to a scalar value to apply the same delay to all channels. Estimation treats InputDelay as a fixed constant of the model. Estimation uses the ioDelay property for estimating time delays. To specify initial values and constraints for estimation of time delays, use sys.Structure.ioDelay. Default: 0 for all input channels 
OutputDelay 
Output delays. For identified systems, like idtf, OutputDelay is fixed to zero. 
Ts 
Sampling time. For continuoustime models, Ts = 0. For discretetime models, Ts is a positive scalar representing the sampling period. This value is expressed in the unit specified by the TimeUnit property of the model. To denote a discretetime model with unspecified sampling time, set Ts = 1. Changing this property does not discretize or resample the model. Use c2d and d2c to convert between continuous and discretetime representations. Use d2d to change the sampling time of a discretetime system. Default: 0 (continuous time) 
TimeUnit 
String representing the unit of the time variable. This property specifies the units for the time variable, the sampling time Ts, and any time delays in the model. Use any of the following values:
Changing this property has no effect on other properties, and therefore changes the overall system behavior. Use chgTimeUnit to convert between time units without modifying system behavior. Default: 'seconds' 
InputName 
Input channel names. Set InputName to a string for singleinput model. For a multiinput model, set InputName to a cell array of strings. Alternatively, use automatic vector expansion to assign input names for multiinput models. For example, if sys is a twoinput model, enter: sys.InputName = 'controls'; The input names automatically expand to {'controls(1)';'controls(2)'}. When you estimate a model using an iddata object, data, the software automatically sets InputName to data.InputName. You can use the shorthand notation u to refer to the InputName property. For example, sys.u is equivalent to sys.InputName. Input channel names have several uses, including:
Default: Empty string '' for all input channels 
InputUnit 
Input channel units. Use InputUnit to keep track of input signal units. For a singleinput model, set InputUnit to a string. For a multiinput model, set InputUnit to a cell array of strings. InputUnit has no effect on system behavior. Default: Empty string '' for all input channels 
InputGroup 
Input channel groups. The InputGroup property lets you assign the input channels of MIMO systems into groups and refer to each group by name. Specify input groups as a structure. In this structure, field names are the group names, and field values are the input channels belonging to each group. For example: sys.InputGroup.controls = [1 2]; sys.InputGroup.noise = [3 5]; creates input groups named controls and noise that include input channels 1, 2 and 3, 5, respectively. You can then extract the subsystem from the controls inputs to all outputs using: sys(:,'controls') Default: Struct with no fields 
OutputName 
Output channel names. Set OutputName to a string for singleoutput model. For a multioutput model, set OutputName to a cell array of strings. Alternatively, use automatic vector expansion to assign output names for multioutput models. For example, if sys is a twooutput model, enter: sys.OutputName = 'measurements'; The output names to automatically expand to {'measurements(1)';'measurements(2)'}. When you estimate a model using an iddata object, data, the software automatically sets OutputName to data.OutputName. You can use the shorthand notation y to refer to the OutputName property. For example, sys.y is equivalent to sys.OutputName. Output channel names have several uses, including:
Default: Empty string '' for all input channels 
OutputUnit 
Output channel units. Use OutputUnit to keep track of output signal units. For a singleoutput model, set OutputUnit to a string. For a multioutput model, set OutputUnit to a cell array of strings. OutputUnit has no effect on system behavior. Default: Empty string '' for all input channels 
OutputGroup 
Output channel groups. The OutputGroup property lets you assign the output channels of MIMO systems into groups and refer to each group by name. Specify output groups as a structure. In this structure, field names are the group names, and field values are the output channels belonging to each group. For example: sys.OutputGroup.temperature = [1]; sys.InputGroup.measurement = [3 5]; creates output groups named temperature and measurement that include output channels 1, and 3, 5, respectively. You can then extract the subsystem from all inputs to the measurement outputs using: sys('measurement',:) Default: Struct with no fields 
Name 
System name. Set Name to a string to label the system. Default: '' 
Notes 
Any text that you want to associate with the system. Set Notes to a string or a cell array of strings. Default: {} 
UserData 
Any type of data you wish to associate with system. Set UserData to any MATLAB^{®} data type. Default: [] 
SamplingGrid 
Sampling grid for model arrays, specified as a data structure. For arrays of identified linear (IDLTI) models that are derived by sampling one or more independent variables, this property tracks the variable values associated with each model. This information appears when you display or plot the model array. Use this information to trace results back to the independent variables. Set the field names of the data 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 should be numeric and scalar valued, and all arrays of sampled values should match the dimensions of the model array. For example, if you collect data at various operating points of a system, you can identify a model for each operating point separately and then stack the results together into a single system array. You can tag the individual models in the array with information regarding the operating point: nominal_engine_rpm = [1000 5000 10000];
sys.SamplingGrid = struct('rpm', nominal_engine_rpm)
where sys is an array containing three identified models obtained at rpms 1000, 5000 and 10000, respectively. Default: [] 