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# idtf

Transfer function model with identifiable parameters

## Description

An `idtf` model represents a system as a continuous-time or discrete-time transfer function with identifiable (estimable) coefficients. Use `idtf` to create a transfer function model, or to convert Dynamic System Models to transfer function form.

A SISO transfer function is a ratio of polynomials with an exponential term. In continuous time,

`$G\left(s\right)={e}^{-\tau s}\frac{{b}_{n}{s}^{n}+{b}_{n-1}{s}^{n-1}+...+{b}_{0}}{{s}^{m}+{a}_{m-1}{s}^{m-1}+...+{a}_{0}}.$`

In discrete time,

`$G\left({z}^{-1}\right)={z}^{-k}\frac{{b}_{n}{z}^{-n}+{b}_{n-1}{z}^{-n+1}+...+{b}_{0}}{{z}^{-m}+{a}_{m-1}{z}^{-m+1}+...+{a}_{0}}.$`

In discrete time, zk represents a time delay of kTs, where Ts is the sample time.

For `idtf` models, the denominator coefficients a0,...,am–1 and the numerator coefficients b0,...,bn can be estimable parameters. (The leading denominator coefficient is always fixed to 1.) The time delay τ (or k in discrete time) can also be an estimable parameter. The `idtf` model stores the polynomial coefficients a0,...,am–1 and b0,...,bn in the `Denominator` and `Numerator` properties of the model, respectively. The time delay τ or k is stored in the `IODelay` property of the model.

Unlike `idss` and `idpoly`, `idtf` fixes the noise parameter to 1 rather than parameterizing it. So, in $y=Gu+He$, H = 1.

A MIMO transfer function contains a SISO transfer function corresponding to each input-output pair in the system. For `idtf` models, the polynomial coefficients and transport delays of each input-output pair are independently estimable parameters.

## Creation

You can obtain an `idtf` model object in one of three ways.

• Estimate the `idtf` model based on input-output 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 `Numerator`, `Denominator`, and `IODelay` properties of the resulting `idtf` model. When you reference numerator and denominator properties, you can use the shortcuts `num` and `den`. The `Report` property of the resulting model stores information about the estimation, such as handling of initial conditions and options used in estimation. For example, you can use the following commands to estimate and get information about a transfer function.

```sys = tfest(data,nx); num = sys.Numerator; den = sys.den; sys.Report```

For more examples of estimating an `idtf` model, see `tfest`.

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. For example, create an `idtf` model with the numerator and denominator that you specify.

`sys = idtf(num,den)`
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. For examples, see Create Continuous-Time Transfer Function Model and Create Discrete-Time Transfer Function.

• Convert an existing dynamic system model to an `idtf` model using the `idtf` command. For example, convert the state-space model `sys_ss` to a transfer function.

`sys_tf = idtf(sys_ss);`
For a more detailed example, see Convert Identifiable State-Space Model to Identifiable Transfer Function

For information on functions you can use to extract information from or transform `idtf` model objects, see Object Functions.

### Syntax

``sys = idtf(numerator,denominator)``
``sys = idtf(numerator,denominator,Ts)``
``sys = idtf(___,Name,Value)``
``sys = idtf(sys0)``

### Description

#### Create Transfer Function Model

example

````sys = idtf(numerator,denominator)` creates a continuous-time transfer function model with identifiable parameters. `numerator` specifies the current values of the transfer function numerator coefficients. `denominator` specifies the current values of the transfer function denominator coefficients.```

example

````sys = idtf(numerator,denominator,Ts)` creates a discrete-time transfer function model with sample time `Ts`. ```

example

````sys = idtf(___,Name,Value)` creates a transfer function with the properties specified by one or more `Name,Value` pair arguments. Specify name-value pair arguments after any of the input argument combinations in the previous syntaxes.```

#### Convert Dynamic System Model to Transfer Function Model

example

````sys = idtf(sys0)` converts any dynamic system model `sys0` to `idtf` model form.```

### Input Arguments

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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`.

## Properties

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Values of transfer function numerator coefficients, specified as a row vector or a cell array.

For SISO transfer functions, the values of the numerator coefficients are stored as a row vector in the following order:

• Descending powers of s or p (for continuous-time transfer functions)

• Ascending powers of z–1 or q–1 (for discrete-time transfer functions)

Any coefficient whose initial value is not known is stored as `NaN`.

For MIMO transfer functions with `Ny` outputs and `Nu` inputs, `Numerator` is a `Ny`-by-`Nu` cell array of numerator coefficients for each input/output pair. For an example of a MIMO transfer function, see Create MIMO Discrete-Time Transfer Function.

If you create an `idtf` model `sys` using the `idtf` command, `sys.Numerator` contains the initial values of numerator coefficients that you specify with the `numerator` input argument.

If you obtain an `idtf` model by identification using `tfest`, then `sys.Numerator` contains the estimated values of the numerator coefficients.

For an `idtf` model `sys`, the property `sys.Numerator` is an alias for the value of the property `sys.Structure.Numerator.Value`.

Values of transfer function denominator coefficients, specified as a row vector or a cell array.

For SISO transfer functions, the values of the denominator coefficients are stored as a row vector in the following order:

• Descending powers of s or p (for continuous-time transfer functions)

• Ascending powers of z–1 or q–1 (for discrete-time transfer functions)

The leading coefficient in `Denominator` 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, `Denominator` is an Ny-by-Nu cell array of denominator coefficients for each input-output pair. For an example of a MIMO transfer function, see Create MIMO Discrete-Time Transfer Function.

If you create an `idtf` model `sys` using the`idtf` command, `sys.Denominator` contains the initial values of denominator coefficients that you specify with the `denominator` input argument.

If you obtain an `idtf` model `sys` by identification using `tfest`, then `sys.Denominator` contains the estimated values of the denominator coefficients.

For an `idtf` model `sys`, the property `sys.Denominator` is an alias for the value of the property `sys.Structure.Denominator.Value`.

Transfer function display variable, specified as one of the following values:

• `'s'` — Default for continuous-time models

• `'p'` — Equivalent to `'s'`

• `'z^-1'` — Default for discrete-time models

• `'q^-1'` — Equivalent to `'z^-1'`

The value of `Variable` is reflected in the display, and also affects the interpretation of the `Numerator` and `Denominator` coefficient vectors for discrete-time models. When `Variable` is set to `'z^-1'` or `'q^-1'`, the coefficient vectors are ordered as ascending powers of the variable.

For an example of using the `Variable` property, see Specify Transfer Function Display Variable.

Transport delays, specified as a numeric array containing a separate transport delay for each input-output pair.

For continuous-time systems, transport delays are expressed in the time unit stored in the `TimeUnit` property. For discrete-time systems, transport delays are expressed as integers denoting delay of a multiple of the sample time `Ts`.

For a MIMO system with Ny outputs and Nu inputs, set `IODelay` as an Ny-by-Nu 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.

If you create an `idtf` model `sys` using the `idtf` command, then `sys.IODelay` contains the initial values of the transport delay that you specify with a name-value pair argument.

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`.

Property-specific information about the estimable parameters of the `idtf` model, specified as a single structure or an array of structures.

• SISO system — Single structure.

• MIMO system with Ny outputs and Nu inputs — Ny-by-Nu array. The element `Structure(i,j)` contains information corresponding to the transfer function for the `(i,j)` input-output pair.

`Structure.Numerator`, `Structure.Denominator`, and `Structure.IODelay` contain information about the numerator coefficients, denominator coefficients, and transport delay, respectively. Each parameter in `Structure` contains the following fields.

FieldDescriptionExamples
ValueParameter values. Each property is an alias of the corresponding `Value` entry in the `Structure` property of `sys`.`NaN` represents unknown parameter values.`sys.Structure.Numerator.Value` contains the initial or estimated values of SISO numerator coefficients. `sys.Numerator` is an alias of the value of this property. `sys.Numerator{i,j}` is the alias of the MIMO property `sys.Structure(i,j).Numerator.Value`.
MinimumMinimum value that the parameter can assume during estimation. `sys.Structure.IODelay.Minimum = 0.1` constrains the transport delay to values greater than or equal to 0.1. `sys.Structure.IODelay.Minimum` must be greater than or equal to zero.
MaximumMaximum value that the parameter can assume during estimation.`sys.Structure.IODelay.Maximum = 0.5` constrains the transport delay to values less than or equal to 0.5. `sys.Structure.IODelay.Maximum` must be greater than or equal to zero.
FreeBoolean specifying whether the parameter is a free estimation variable. If you want to fix the value of a parameter during estimation, set the corresponding `Free` value to `false`. For denominators, the value of `Free` for the leading coefficient, specified by `sys.Structure.Denominator.Free(1)`, is always `false` (the leading denominator coefficient is always fixed to 1).`sys.Structure.Denominator.Free = false` fixes all of the denominator coefficients in `sys` to the values specified in `sys.Structure.Denominator.Value`.
ScaleScale of the value of the parameter. The estimation algorithm does not use `Scale`.
InfoStructure array that contains the fields `Label` and `Unit` for storing parameter labels and units. Specify parameter labels and units as character vectors.`'Time'`

Variance (covariance matrix) of the model innovations e, specified as a scalar or matrix.

• SISO model — Scalar

• MIMO model with Ny outputs — Ny-by-Ny matrix

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.

This property is read-only.

Summary report that contains information about the estimation options and results for a state-space model obtained using estimation commands, such as `tfest` and `impulseest`. Use `Report` to find estimation information for the identified model, including:

• Estimation method

• Estimation options

• Search termination conditions

• Estimation data fit and other quality metrics

If you create the model by construction, the contents of `Report` are irrelevant.

```sys = idtf([1 4],[1 20 5]); sys.Report.OptionsUsed```
```ans = []```

If you obtain the model using estimation commands, the fields of `Report` contain information on the estimation data, options, and results.

```load iddata2 z2; sys = tfest(z2,3); sys.Report.OptionsUsed```
``` InitializeMethod: 'iv' InitializeOptions: [1×1 struct] InitialCondition: 'auto' Display: 'off' InputOffset: [] OutputOffset: [] EstimateCovariance: 1 Regularization: [1×1 struct] SearchMethod: 'auto' SearchOptions: [1×1 idoptions.search.identsolver] WeightingFilter: [] EnforceStability: 0 OutputWeight: [] Advanced: [1×1 struct] ```

For more information on this property and how to use it, see the Output Arguments section of the corresponding estimation command reference page and Estimation Report.

Input delay for each input channel, specified as a scalar value or numeric vector. For continuous-time systems, specify input delays in the time unit stored in the `TimeUnit` property. For discrete-time systems, specify input delays in integer multiples of the sample time `Ts`. For example, setting `InputDelay` to `3` specifies a delay of three sample times.

For a system with Nu inputs, set `InputDelay` to an Nu-by-1 vector. Each entry of this vector is a numerical value that represents 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 constrains for estimation of time delays, use `sys.Structure.IODelay`.

For identified systems such as `idtf`, `OutputDelay` is fixed to zero.

Sample time, specified as one of the following.

• Continuous-time model — `0`

• Discrete-time model with a specified sampling time — Positive scalar representing the sampling period expressed in the unit specified by the `TimeUnit` property of the model

• Discrete-time model with unspecified sample time — `-1`

Changing this property does not discretize or resample the model. Use `c2d` and `d2c` to convert between continuous- and discrete-time representations. Use `d2d` to change the sample time of a discrete-time system.

Units for the time variable, the sample time `Ts`, and any time delays in the model, specified as a scalar.

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

Input channel names, specified as a character vector or cell array.

• Single-input model — Character vector, for example, `'controls'`

• Multi-input model — Cell array of character vectors

Alternatively, use automatic vector expansion to assign input names for multi-input models. For example, if `sys` is a two-input 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`.

You can use input channel names in several ways, including:

• To identify channels on model display and plots

• To extract subsystems of MIMO systems

• To specify connection points when interconnecting models

Input channel units, specified as a character vector or cell array.

• Single-input model — Character vector

• Multi-input Model — Cell array of character vectors

Use `InputUnit` to keep track of input signal units. `InputUnit` has no effect on system behavior.

Input channel groups, specified as a structure. The `InputGroup` property lets you divide the input channels of MIMO systems into groups so that you can refer to each group by name. In the `InputGroup` structure, set field names to the group names, and field values to the input channels belonging to each group.

For example, create input groups named `controls` and `noise` that include input channels 1, 2 and 3, 5, respectively.

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

You can then extract the subsystem from the `controls` inputs to all outputs using the following syntax:

`sys(:,'controls')`

Output channel names, specified as a character vector or cell array.

• Single-input model — Character vector, for example, `'measurements'`

• Multi-input model — Cell array of character vectors

Alternatively, use automatic vector expansion to assign output names for multi-output models. For example, if `sys` is a two-output model, enter:

`sys.OutputName = 'measurements';`

The output names 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`.

You can use output channel names in several ways, including:

• To identify channels on model display and plots

• To extract subsystems of MIMO systems

• To specify connection points when interconnecting models

Output channel units, specified as a character vector or cell array.

• Single-input model — Character vector, for example, `'seconds'`

• Multi-input Model — Cell array of character vectors

Use `OutputUnit` to keep track of output signal units. `OutputUnit` has no effect on system behavior.

Output channel groups, specified as a structure. The `OutputGroup` property lets you divide the output channels of MIMO systems into groups and refer to each group by name. In the `OutputGroup` structure, set field names to the group names, and field values to the output channels belonging to each group.

For example, create output groups named `temperature` and `measurement` that include output channels 1 and 3, 5, respectively.

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

You can then extract the subsystem from all inputs to the `measurement` outputs using the following syntax:

`sys('measurement',:)`

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

Any text that you want to associate with the system, specified as a string or a cell array of character vectors. The property stores whichever data type you provide. For instance, if `sys1` and `sys2` are dynamic system models, you can set their `Notes` properties as follows.

```sys1.Notes = "sys1 has a string."; sys2.Notes = 'sys2 has a character vector.'; sys1.Notes sys2.Notes```
```ans = "sys1 has a string." ans = 'sys2 has a character vector.' ```

Data to associate with the system, specified as any MATLAB data type.

Sampling grid for model arrays, specified as a structure.

For arrays of identified linear (IDLTI) models that you derive 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 must be numeric and scalar valued, and all arrays of sampled values must match the dimensions of the model array.

For example, suppose that 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)```

Here, `sys` is an array containing three identified models obtained at 1000, 5000, and 10000 rpm, respectively.

For model arrays that you generate by linearizing a Simulink® model at multiple parameter values or operating points, the software populates `SamplingGrid` automatically with the variable values that correspond to each entry in the array.

## Object Functions

In general, any function applicable to Dynamic System Models is applicable to an `idtf` model object. These functions are of four general types.

The following lists contain a representative subset of the functions that you can use with `idtf` models.

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 `translatecov` Translate parameter covariance across model transformation operations `setpar` Set initial parameter values of `idnlgrey` model object `chgTimeUnit` Change time units of dynamic system `d2d` Resample discrete-time model `d2c` Convert model from discrete to continuous time `c2d` Convert model from continuous to discrete time `merge` Merge estimated models

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 `sim` Simulate response of identified model `predict` Predict state and state estimation error covariance at next time step using extended or unscented Kalman filter, or particle filter `compare` Compare identified model output and measured output `impulse` Impulse response plot of dynamic system; impulse response data `step` Step response plot of dynamic system; step response data `bode` Bode plot of frequency response, or magnitude and phase data

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 `tfdata` Access transfer function data `get` Access model property values `getpar` Parameter values and properties of `idnlgrey` model parameters `getcov` Parameter covariance of identified model `advice` Analysis and recommendations for data or estimated linear models

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 `idpoly` Polynomial model with identifiable parameters `idss` State-space model with identifiable parameters `idfrd` Frequency-response data or model

## Examples

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Specify a continuous-time, single-input, single-output (SISO) transfer function with estimable parameters. The initial values of the transfer function are given by the following equation:

`$G\left(s\right)=\frac{s+4}{{s}^{2}+20s+5}$`

```num = [1 4]; den = [1 20 5]; G = idtf(num,den)```
```G = s + 4 -------------- s^2 + 20 s + 5 Continuous-time identified transfer function. Parameterization: Number of poles: 2 Number of zeros: 1 Number of free coefficients: 4 Use "tfdata", "getpvec", "getcov" for parameters and their uncertainties. Status: Created by direct construction or transformation. Not estimated. ```

`G` is an `idtf` model. `num` and `den` specify the initial values of the numerator and denominator polynomial coefficients in descending powers of $s$. The numerator coefficients with initial values 1 and 4 are estimable parameters. The denominator coefficients with initial values 20 and 5 are also estimable parameters. The leading denominator coefficient is always fixed to 1.

You can use `G` to specify an initial parameterization for estimation with `tfest`.

Specify a continuous-time, SISO transfer function with known input delay. The transfer function initial values are given by the following equation:

`$G\left(s\right)={e}^{-5.8s}\frac{5}{s+5}$`

Label the input of the transfer function with the name `'Voltage'` and specify the input units as `volt`.

Use name-value pair arguments to specify the delay, input name, and input unit.

```num = 5; den = [1 5]; input_delay = 5.8; input_name = 'Voltage'; input_unit = 'volt'; G = idtf(num,den,'InputDelay',input_delay,... 'InputName',input_name,'InputUnit',input_unit);```

$G$ is an `idtf` model. You can use `G` to specify an initial parameterization for estimation with `tfest`. If you do so, model properties such as `InputDelay`, `InputName`, and `InputUnit` are applied to the estimated model. The estimation process treats `InputDelay` as a fixed value. If you want to estimate the delay and specify an initial value of 5.8 s, use the `IODelay` property instead.

Specify a discrete-time SISO transfer function with estimable parameters. The initial values of the transfer function are given by the following equation:

`$H\left(z\right)=\frac{z-0.1}{z+0.8}$`

Specify the sample time as 0.2 seconds.

```num = [1 -0.1]; den = [1 0.8]; Ts = 0.2; H = idtf(num,den,Ts);```

`num` and `den` are the initial values of the numerator and denominator polynomial coefficients. For discrete-time systems, specify the coefficients in ascending powers of ${z}^{-1}$.

`Ts` specifies the sample time for the transfer function as 0.2 seconds.

`H` is an `idtf` model. The numerator and denominator coefficients are estimable parameters (except for the leading denominator coefficient, which is fixed to 1).

Specify a discrete-time, two-input, two-output transfer function. The initial values of the MIMO transfer function are given by the following equation:

`$H\left(z\right)=\left[\begin{array}{cc}\frac{1}{z+0.2}& \frac{z}{z+0.7}\\ \frac{-z+2}{z-0.3}& \frac{3}{z+0.3}\end{array}\right]$`

Specify the sample time as 0.2 seconds.

```nums = {1,[1,0];[-1,2],3}; dens = {[1,0.2],[1,0.7];[1,-0.3],[1,0.3]}; Ts = 0.2; H = idtf(nums,dens,Ts);```

`nums` and `dens` specify the initial values of the coefficients in cell arrays. Each entry in the cell array corresponds to the numerator or denominator of the transfer function of one input-output pair. For example, the first row of `nums` is `{1,[1,0]}`. This cell array specifies the numerators across the first row of transfer functions in `H`. Likewise, the first row of `dens`, `{[1,0.2],[1,0.7]}`, specifies the denominators across the first row of `H`.

`Ts` specifies the sample time for the transfer function as 0.2 seconds.

`H` is an `idtf` model. All of the polynomial coefficients are estimable parameters, except for the leading coefficient of each denominator polynomial. These coefficients are always fixed to 1.

Specify the following discrete-time transfer function in terms of `q^-1`:

`$H\left({q}^{-1}\right)=\frac{1+0.4{q}^{-1}}{1+0.1{q}^{-1}-0.3{q}^{-2}}$`

Specify the sample time as 0.1 seconds.

```num = [1 0.4]; den = [1 0.1 -0.3]; Ts = 0.1; convention_variable = 'q^-1'; H = idtf(num,den,Ts,'Variable',convention_variable);```

Use a name-value pair argument to specify the variable `q^-1`.

`num` and `den` are the numerator and denominator polynomial coefficients in ascending powers of ${q}^{-1}$.

`Ts` specifies the sample time for the transfer function as 0.1 seconds.

`H` is an `idtf` model.

Specify a transfer function with estimable coefficients whose initial value is given by the following static gain matrix:

`$H\left(s\right)=\left[\begin{array}{ccc}1& 0& 1\\ 1& 1& 0\\ 3& 0& 2\end{array}\right]$`

```M = [1 0 1; 1 1 0; 3 0 2]; H = idtf(M);```

`H` is an `idtf` model that describes a three input (`Nu = 3`), three output (`Ny = 3`) transfer function. Each input-output channel is an estimable static gain. The initial values of the gains are given by the values in the matrix `M`.

Convert a state-space model with identifiable parameters to a transfer function with identifiable parameters.

Convert the following identifiable state-space model to an identifiable transfer function.

`$\begin{array}{l}\underset{}{\overset{\sim }{x}}\left(t\right)=\left[\begin{array}{cc}-0.2& 0\\ 0& -0.3\end{array}\right]x\left(t\right)+\left[\begin{array}{c}-2\\ 4\end{array}\right]u\left(t\right)+\left[\begin{array}{c}0.1\\ 0.2\end{array}\right]e\left(t\right)\\ y\left(t\right)=\left[\begin{array}{cc}1& 1\end{array}\right]x\left(t\right)\end{array}$`

```A = [-0.2, 0; 0, -0.3]; B = [2;4]; C = [1, 1]; D = 0; K = [0.1; 0.2]; sys0 = idss(A,B,C,D,K,'NoiseVariance',0.1); sys = idtf(sys0);```

`A`, `B`, `C`, `D`, and `K` are matrices that specify `sys0`, an identifiable state-space model with a noise variance of 0.1.

`sys = idtf(sys0)` creates an `idtf` model `sys`.

Load the time-domain system-response data `z1`.

`load iddata1 z1;`

Set the number of poles `np` to `2` and estimate a transfer function.

```np = 2; sys = tfest(z1,np);```

`sys` is an `idtf` model containing the estimated two-pole 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`.

Create an array of transfer function models with identifiable coefficients. Each transfer function in the array is of the form:

`$H\left(s\right)=\frac{a}{s+a}.$`

The initial value of the coefficient $a$ varies across the array, from 0.1 to 1.0, in increments of 0.1.

```H = idtf(zeros(1,1,10)); for k = 1:10 num = k/10; den = [1 k/10]; H(:,:,k) = idtf(num,den); end```

The first command preallocates a one-dimensional, 10-element array, `H`, and fills it with empty `idtf` models.

The first two dimensions of a model array are the output and input dimensions. The remaining dimensions are the array dimensions. `H(:,:,k)` represents the ${k}^{th}$ model in the array. Thus, the `for` loop replaces the ${k}^{th}$ entry in the array with a transfer function whose coefficients are initialized with $a=k/10$.

## See Also

Introduced in R2012a

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