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# tf

Create transfer function model, convert to transfer function model

## Syntax

```sys = tf(Numerator,Denominator) sys = tf(Numerator,Denominator,Ts) sys = tf(M) sys = tf(Numerator,Denominator,ltisys) tfsys = tf(sys) tfsys = tf(sys, 'measured') tfsys = tf(sys, 'noise') tfsys = tf(sys, 'augmented') ```

## Description

Use `tf` to create real- or complex-valued transfer function models (TF objects) or to convert state-space or zero-pole-gain models to transfer function form. You can also use `tf` to create generalized state-space (`genss`) models or uncertain state-space (`uss`) models.

### Creation of Transfer Functions

`sys = tf(Numerator,Denominator) ` creates a continuous-time transfer function with numerator(s) and denominator(s) specified by `Numerator` and `Denominator`. The output `sys` is:

• A `tf` model object, when `Numerator` and `Denominator` are numeric arrays.

• A generalized state-space model (`genss`) when `Numerator` or `Denominator` include tunable parameters, such as `realp` parameters or generalized matrices (`genmat`).

• An uncertain state-space model (`uss`) when `Numerator` or `Denominator` are uncertain (requires Robust Control Toolbox™ software).

In the SISO case, `Numerator` and `Denominator` are the real- or complex-valued row vectors of numerator and denominator coefficients ordered in descending powers of s. These two vectors need not have equal length and the transfer function need not be proper. For example, `h = tf([1 0],1) ` specifies the pure derivative h(s) = s.

To create MIMO transfer functions, using one of the following approaches:

• Concatenate SISO `tf` models.

• Use the `tf` command with cell array arguments. In this case, `Numerator` and `Denominator` are cell arrays of row vectors with as many rows as outputs and as many columns as inputs. The row vectors `Numerator{i,j}` and `Denominator{i,j}` specify the numerator and denominator of the transfer function from input `j` to output `i`.

For examples of creating MIMO transfer functions, see Examples and MIMO Transfer Functions in the Control System Toolbox™ User Guide.

If all SISO entries of a MIMO transfer function have the same denominator, you can set `denominator` to the row vector representation of this common denominator. See "Examples" for more details.

`sys = tf(Numerator,Denominator,Ts) ` creates a discrete-time transfer function with sample time `Ts` (in seconds). Set `Ts = -1` to leave the sample time unspecified. The input arguments `Numerator` and `Denominator` are as in the continuous-time case and must list the numerator and denominator coefficients in descending powers of z.

`sys = tf(M) ` creates a static gain `M` (scalar or matrix).

`sys = tf(Numerator,Denominator,ltisys) ` creates a transfer function with properties inherited from the dynamic system model `ltisys` (including the sample time).

There are several ways to create arrays of transfer functions. To create arrays of SISO or MIMO TF models, either specify the numerator and denominator of each SISO entry using multidimensional cell arrays, or use a `for` loop to successively assign each TF model in the array. See Model Arrays.

Any of the previous syntaxes can be followed by property name/property value pairs

```'Property',Value ```

Each pair specifies a particular property of the model, for example, the input names or the transfer function variable. For information about the properties of `tf` objects, see Properties. Note that

```sys = tf(Numerator,Denominator,'Property1',Value1,...,'PropertyN',ValueN) ```

is a shortcut for

```sys = tf(Numerator,Denominator) set(sys,'Property1',Value1,...,'PropertyN',ValueN) ```

### Transfer Functions as Rational Expressions in s or z

You can also use real- or complex-valued rational expressions to create a TF model. To do so, first type either:

• `s = tf('s')` to specify a TF model using a rational function in the Laplace variable, `s`.

• `z = `tf`('z',Ts)` to specify a TF model with sample time `Ts` using a rational function in the discrete-time variable, `z`.

Once you specify either of these variables, you can specify TF models directly as rational expressions in the variable `s` or `z` by entering your transfer function as a rational expression in either `s` or `z`.

### Conversion to Transfer Function

```tfsys = tf(sys) ``` converts the dynamic system model `sys` to transfer function form. The output `tfsys` is a `tf` model object representing `sys` expressed as a transfer function.

If `sys` is a model with tunable components, such as a `genss`, `genmat`, `tunableTF`, or `tunableSS` model, the resulting transfer function `tfsys` takes the current values of the tunable components.

### Conversion of Identified Models

An identified model is represented by an input-output equation of the form `y(t) = Gu(t) + He(t)`, where `u(t)` is the set of measured input channels and `e(t)` represents the noise channels. If Λ = LL' represents the covariance of noise `e(t)`, this equation can also be written as: `y(t) = Gu(t) + HLv(t)`, where ```cov(v(t)) = I```.

`tfsys = tf(sys)`, or ```tfsys = tf(sys, 'measured')``` converts the measured component of an identified linear model into the transfer function form. `sys` is a model of type `idss`, `idproc`, `idtf`, `idpoly`, or `idgrey`. `tfsys` represents the relationship between `u` and `y`.

`tfsys = tf(sys, 'noise')` converts the noise component of an identified linear model into the transfer function form. It represents the relationship between the noise input, `v(t)` and output, `y_noise = HL v(t)`. The noise input channels belong to the `InputGroup` `'Noise'`. The names of the noise input channels are `v@yname`, where `yname` is the name of the corresponding output channel. `tfsys` has as many inputs as outputs.

`tfsys = tf(sys, 'augmented')` converts both the measured and noise dynamics into a transfer function. `tfsys` has `ny+nu` inputs such that the first `nu` inputs represent the channels `u(t)` while the remaining by channels represent the noise channels `v(t)`. `tfsys.InputGroup` contains 2 input groups- `'measured'` and `'noise'`. `tfsys.InputGroup.Measured` is set to `1:nu` while `tfsys.InputGroup.Noise` is set to `nu+1:nu+ny`. `tfsys` represents the equation `y(t) = [G HL] [u; v]`.

### Tip

An identified nonlinear model cannot be converted into a transfer function. Use linear approximation functions such as `linearize` and `linapp`.

### Creation of Generalized State-Space Models

You can use the syntax:

`gensys = tf(Numerator,Denominator)`

to create a Generalized state-space (`genss`) model when one or more of the entries `Numerator` and `Denominator` depends on a tunable `realp` or `genmat` model. For more information about Generalized state-space models, see Models with Tunable Coefficients.

## Examples

### Create Transfer Function with One Input and Two Outputs

Create the following transfer function model:

` `

The model has an input current and two outputs, torque and angular velocity.

Specify the numerator and denominator coefficients of the model:

```Numerator = {[1 1] ; 1}; Denominator = {[1 2 2] ; [1 0]};```

Create the transfer function model, specifying the input name, output names, and variable.

```H = tf(Numerator,Denominator,'InputName','current',... 'OutputName',{'torque' 'ang. velocity'},... 'Variable','p')```
```H = From input "current" to output... p + 1 torque: ------------- p^2 + 2 p + 2 1 ang. velocity: - p Continuous-time transfer function. ```

Setting the `Variable` property of the model to `'p'` causes the result to display as a transfer function of the variable p.

### Create Transfer Function Model Using Rational Expression

To use a rational expression to create a SISO transfer function model, first specify `s` as a `tf` object.

`s = tf('s');`

Create a transfer function using `s` in a rational expression.

`H = s/(s^2 + 2*s + 10);`

This method produces the same transfer function as:

`h = tf([1 0],[1 2 10]);`

### Second-Order Transfer Function from Damping and Natural Frequency

Create a `tf` model that represents a second-order system with known natural frequency and damping ratio.

The transfer function of a second-order system, expressed in terms of its damping ratio and natural frequency , is:

` `

Represent this transfer function in MATLAB using the `tf` command. For example, suppose you have a system with = 0.25 and = 3 rad/s.

```zeta = 0.25; w0 = 3; H = tf(w0^2,[1,2*zeta*w0,w0^2])```
```H = 9 --------------- s^2 + 1.5 s + 9 Continuous-time transfer function. ```

Examine the response of this transfer function to a step input.

`stepplot(H)` The plot shows the ringdown expected of a second-order system with a low damping ratio.

### Create MIMO Transfer Function Model

Create a transfer function for the discrete-time, multi-input, multi-output model:

` `

with sample time `Ts = 0.2` seconds.

Specify the numerator coefficients as a 2-by-2 matrix.

`Numerators = {1 [1 0];[-1 2] 3};`

Specify the coefficients of the common denominator as a row vector.

`Denominator = [1 0.3];`

Create the discrete-time transfer function model.

```Ts = 0.2; H = tf(Numerators,Denominator,Ts);```

### Convert State-Space Model to Transfer Function

Compute the transfer function of the following state-space model:

` `

Specify the state-space model.

`sys = ss([-2 -1;1 -2],[1 1;2 -1],[1 0],[0 1]);`

Convert this model to a transfer function.

`tf(sys)`
```ans = From input 1 to output: s - 4.441e-16 ------------- s^2 + 4 s + 5 From input 2 to output: s^2 + 5 s + 8 ------------- s^2 + 4 s + 5 Continuous-time transfer function. ```

### Create Array of Transfer Function Models

You can use a `for` loop to specify an array of SISO transfer function models.

Pre-allocate the array with zero transfer functions.

`H = tf(zeros(1,1,10));`

The first two indices represent the number of outputs and inputs for the models, while the third index is the number of models in the array.

Create the transfer function models.

```s = tf('s'); for k = 1:10 H(:,:,k) = k/(s^2+s+k); end```

### Create Tunable Low-Pass Filter

This example shows how to create a low-pass filter with one tunable parameter a:

` `

You cannot use `tunableTF` to represent `F`, because the numerator and denominator coefficients of a `tunableTF` block are independent. Instead, construct `F` using the tunable real parameter object `realp`.

Create a tunable real parameter with an initial value of `10`.

`a = realp('a',10);`

Use `tf` to create the tunable filter `F`.

`F = tf(a,[1 a]);`

`F` is a genss object which has the tunable parameter `a` in its `Blocks` property. You can connect `F` with other tunable or numeric models to create more complex control system models. For example, see Control System with Tunable Components.

### Extract Transfer Functions from Identified Model

Extract the measured and noise components of an identified polynomial model into two separate transfer functions (requires System Identification Toolbox™). The measured component can serve as a plant model, while the noise component can serve as a disturbance model for control system design.

```load icEngine; z = iddata(y,u,0.04); nb = 2; nf = 2; nc = 1; nd = 3; nk = 3; sys = bj(z, [nb nc nd nf nk]); ```

`sys` is a model of the form: ```y(t) = B/F u(t) + C/D e(t)```, where `B/F` represents the measured component and `C/D` the noise component.

```sysMeas = tf(sys, 'measured') sysNoise = tf(sys, 'noise')```

Alternatively, use can simply use `tf(sys)` to extract the measured component.

## Discrete-Time Conventions

The control and digital signal processing (DSP) communities tend to use different conventions to specify discrete transfer functions. Most control engineers use the z variable and order the numerator and denominator terms in descending powers of z, for example,

`$h\left(z\right)=\frac{{z}^{2}}{{z}^{2}+2z+3}.$`

The polynomials z2 and z2 + 2z + 3 are then specified by the row vectors ```[1 0 0]``` and `[1 2 3]`, respectively. By contrast, DSP engineers prefer to write this transfer function as

`$h\left({z}^{-1}\right)=\frac{1}{1+2{z}^{-1}+3{z}^{-2}}$`

and specify its numerator as `1` (instead of ```[1 0 0]```) and its denominator as `[1 2 3]`.

`tf` switches convention based on your choice of variable (value of the `'Variable'` property).

Variable

Convention

`'z'` (default), `'q'`

Use the row vector `[ak ... a1 a0]` to specify the polynomial ${a}_{k}{z}^{k}+...+{a}_{1}z+{a}_{0}$ (coefficients ordered in descending powers of z or q).

`'z^-1'`

Use the row vector `[b0 b1 ... bk]` to specify the polynomial ${b}_{0}+{b}_{1}{z}^{-1}+...+{b}_{k}{z}^{-k}$ (coefficients in ascending powers of z-1).

For example,

```g = tf([1 1],[1 2 3],0.1); ```

specifies the discrete transfer function

`$g\left(z\right)=\frac{z+1}{{z}^{2}+2z+3}$`

because z is the default variable. In contrast,

```h = tf([1 1],[1 2 3],0.1,'variable','z^-1'); ```

uses the DSP convention and creates

`$h\left({z}^{-1}\right)=\frac{1+{z}^{-1}}{1+2{z}^{-1}+3{z}^{-2}}=zg\left(z\right).$`

See also `filt` for direct specification of discrete transfer functions using the DSP convention.

Note that `tf` stores data so that the numerator and denominator lengths are made equal. Specifically, `tf` stores the values

```Numerator = [0 1 1]; Denominator = [1 2 3]; ```

for `g` (the numerator is padded with zeros on the left) and the values

```Numerator = [1 1 0]; Denominator = [1 2 3]; ```

for `h` (the numerator is padded with zeros on the right).

## Properties

`tf` objects have the following properties:

 `Numerator` Transfer function numerator coefficients. For SISO transfer functions, `Numerator` is a row vector of polynomial coefficients in order of descending power (for `Variable` values `s`, `z`, `p`, or `q`) or in order of ascending power (for `Variable` values `z^-1` or `q^-1`). For MIMO transfer functions with `Ny` outputs and `Nu` inputs, `Numerator` is a `Ny`-by-`Nu` cell array of the numerator coefficients for each input/output pair. `Denominator` Transfer function denominator coefficients. For SISO transfer functions, `Denominator` is a row vector of polynomial coefficients in order of descending power (for `Variable` values `s`, `z`, `p`, or `q`) or in order of ascending power (for `Variable` values `z^-1` or `q^-1`). For MIMO transfer functions with `Ny` outputs and `Nu` inputs, `Denominator` is a `Ny`-by-`Nu` cell array of the denominator coefficients for each input/output pair. `Variable` Transfer function display variable, specified as one of the following: `'s'` — Default for continuous-time models`'z'` — Default for discrete-time models`'p'` — Equivalent to `'s'``'q'` — Equivalent to `'z'``'z^-1'` — Inverse of `'z'``'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. For `Variable = 'z'` or `'q'`, the coefficient vectors are ordered in descending powers of the variable. For `Variable = 'z^-1'` or `'q^-1'`, the coefficient vectors are ordered as ascending powers of the variable. Default: `'s'` `IODelay` Transport delays. `IODelay` is a numeric array specifying a separate transport delay for each input/output pair. For continuous-time systems, specify transport delays in the time unit stored in the `TimeUnit` property. For discrete-time systems, specify transport delays in integer multiples of the sample time, `Ts`. For a MIMO system with `Ny` outputs and `Nu` inputs, set `IODelay` to a `Ny`-by-`Nu` 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. Default: `0` for all input/output pairs `InputDelay` 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, ```InputDelay = 3``` means 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. Default: 0 `OutputDelay` Output delays. `OutputDelay` is a numeric vector specifying a time delay for each output channel. For continuous-time systems, specify output delays in the time unit stored in the `TimeUnit` property. For discrete-time systems, specify output delays in integer multiples of the sample time `Ts`. For example, ```OutputDelay = 3``` means a delay of three sampling periods. For a system with `Ny` outputs, set `OutputDelay` to an `Ny`-by-1 vector, where each entry is a numerical value representing the output delay for the corresponding output channel. You can also set `OutputDelay` to a scalar value to apply the same delay to all channels. Default: 0 for all output channels `Ts` Sample time. For continuous-time models, `Ts = 0`. For discrete-time 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 discrete-time model with unspecified sample time, set ```Ts = -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. Default: `0` (continuous time) `TimeUnit` Units for the time variable, the sample time `Ts`, and any time delays in the model, specified as one of the following values:`'nanoseconds'``'microseconds'``'milliseconds'``'seconds'` `'minutes'``'hours'``'days'``'weeks'``'months'``'years'` 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, specified as one of the following: Character vector — For single-input models, for example, `'controls'`.Cell array of character vectors — For multi-input models. 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)'}`. 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: Identifying channels on model display and plotsExtracting subsystems of MIMO systemsSpecifying connection points when interconnecting models Default: `''` for all input channels `InputUnit` Input channel units, specified as one of the following: Character vector — For single-input models, for example, `'seconds'`.Cell array of character vectors — For multi-input models. Use `InputUnit` to keep track of input signal units. `InputUnit` has no effect on system behavior. Default: `''` 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, specified as one of the following: Character vector — For single-output models. For example, `'measurements'`.Cell array of character vectors — For multi-output models. 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)'}`. 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: Identifying channels on model display and plotsExtracting subsystems of MIMO systemsSpecifying connection points when interconnecting models Default: `''` for all output channels `OutputUnit` Output channel units, specified as one of the following: Character vector — For single-output models. For example, `'seconds'`.Cell array of character vectors — For multi-output models. Use `OutputUnit` to keep track of output signal units. `OutputUnit` has no effect on system behavior. Default: `''` for all output 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 = ; 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, specified as a character vector. For example, `'system_1'`. Default: `''` `Notes` Any text that you want to associate with the system, stored 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.' ``` Default: `[0×1 string]` `UserData` Any type of data you want to associate with system, specified as any MATLAB® data type. Default: `[]` `SamplingGrid` Sampling grid for model arrays, specified as a data structure. For model arrays that are derived by sampling one or more independent variables, this property tracks the variable values associated with each model in the array. 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, suppose you create a 11-by-1 array of linear models, `sysarr`, by taking snapshots of a linear time-varying system at times `t = 0:10`. The following code stores the time samples with the linear models. ` sysarr.SamplingGrid = struct('time',0:10)` Similarly, suppose you create a 6-by-9 model array, `M`, by independently sampling two variables, `zeta` and `w`. The following code attaches the `(zeta,w)` values to `M`. ```[zeta,w] = ndgrid(<6 values of zeta>,<9 values of w>) M.SamplingGrid = struct('zeta',zeta,'w',w)``` When you display `M`, each entry in the array includes the corresponding `zeta` and `w` values. `M` ```M(:,:,1,1) [zeta=0.3, w=5] = 25 -------------- s^2 + 3 s + 25 M(:,:,2,1) [zeta=0.35, w=5] = 25 ---------------- s^2 + 3.5 s + 25 ...``` For model arrays generated by linearizing a Simulink® model at multiple parameter values or operating points, the software populates `SamplingGrid` automatically with the variable values that correspond to each entry in the array. For example, the Simulink Control Design™ commands `linearize` and `slLinearizer` populate `SamplingGrid` in this way. Default: `[]`

## Algorithms

`tf` uses the MATLAB function `poly` to convert zero-pole-gain models, and the functions `zero` and `pole` to convert state-space models.