Transfer function model

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## Description

Use `tf`

to create real-valued or complex-valued transfer function models, or to convert dynamic system models to transfer function form.

Transfer functions are a frequency-domain representation of linear time-invariant systems. For instance, consider a continuous-time SISO dynamic system represented by the transfer function `sys(s) = N(s)/D(s)`

, where `s = jw`

and `N(s)`

and `D(s)`

are called the numerator and denominator polynomials, respectively. The `tf`

model object can represent SISO or MIMO transfer functions in continuous time or discrete time.

You can create a transfer function model object either by specifying its coefficients directly, or by converting a model of another type (such as a state-space model `ss`

) to transfer-function form. For more information, see Transfer Functions.

You can also use `tf`

to create generalized state-space (genss) models or uncertain state-space (uss (Robust Control Toolbox)) models.

## Creation

### Syntax

`sys = tf(numerator,denominator)`

`sys = tf(numerator,denominator,ts)`

`sys = tf(numerator,denominator,ltiSys)`

`sys = tf(m)`

`sys = tf(___,Name,Value)`

`sys = tf(ltiSys)`

`sys = tf(ltiSys,component)`

`s = tf('s')`

`z = tf('z',ts)`

### Description

example

`sys = tf(numerator,denominator)`

creates a continuous-time transfer function model, setting the `Numerator`

and `Denominator`

properties. For instance, consider a continuous-time SISO dynamic system represented by the transfer function `sys(s) = N(s)/D(s)`

, the input arguments `numerator`

and `denominator`

are the coefficients of `N(s)`

and `D(s)`

, respectively.

example

`sys = tf(numerator,denominator,ts)`

creates a discrete-time transfer function model, setting the `Numerator`

, `Denominator`

, and `Ts`

properties. For instance, consider a discrete-time SISO dynamic system represented by the transfer function `sys(z) = N(z)/D(z)`

, the input arguments `numerator`

and `denominator`

are the coefficients of `N(z)`

and `D(z)`

, respectively. To leave the sample time unspecified, set `ts`

input argument to `-1`

.

example

`sys = tf(numerator,denominator,ltiSys)`

creates a transfer function model with properties inherited from the dynamic system model `ltiSys`

, including the sample time.

example

`sys = tf(m)`

creates a transfer function model that represents the static gain, `m`

.

example

`sys = tf(___,`

sets properties of the transfer function model using one or more `Name,Value`

)`Name,Value`

pair arguments for any of the previous input-argument combinations.

example

`sys = tf(ltiSys)`

converts the dynamic system model `ltiSys`

to a transfer function model.

example

`sys = tf(ltiSys,component)`

converts the specified `component`

of `ltiSys`

to transfer function form. Use this syntax only when `ltiSys`

is an identified linear time-invariant (LTI) model.

example

`s = tf('s')`

creates special variable `s`

that you can use in a rational expression to create a continuous-time transfer function model. Using a rational expression can sometimes be easier and more intuitive than specifying polynomial coefficients.

example

`z = tf('z',ts)`

creates special variable `z`

that you can use in a rational expression to create a discrete-time transfer function model. To leave the sample time unspecified, set `ts`

input argument to `-1`

.

### Input Arguments

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`numerator`

— Numerator coefficients of the transfer function

row vector | `Ny`

-by-`Nu`

cell array of row vectors

Numerator coefficients of the transfer function, specified as:

A row vector of polynomial coefficients.

An

`Ny`

-by-`Nu`

cell array of row vectors to specify a MIMO transfer function, where`Ny`

is the number of outputs, and`Nu`

is the number of inputs.

When you create the transfer function, specify the numerator coefficients in order of descending power. For instance, if the transfer function numerator is `3s^2-4s+5`

, then specify `numerator`

as `[3 -4 5]`

. For a discrete-time transfer function with numerator `2z-1`

, set `numerator`

to `[2 -1]`

.

Also a property of the `tf`

object. For more information, see Numerator.

`denominator`

— Denominator coefficients of the transfer function

row vector | `Ny`

-by-`Nu`

cell array of row vectors

Denominator coefficients, specified as:

A row vector of polynomial coefficients.

An

`Ny`

-by-`Nu`

cell array of row vectors to specify a MIMO transfer function, where`Ny`

is the number of outputs and`Nu`

is the number of inputs.

When you create the transfer function, specify the denominator coefficients in order of descending power. For instance, if the transfer function denominator is `7s^2+8s-9`

, then specify `denominator`

as `[7 8 -9]`

. For a discrete-time transfer function with denominator `2z^2+1`

, set `denominator`

to `[2 0 1]`

.

Also a property of the `tf`

object. For more information, see Denominator.

`ts`

— Sample time

scalar

Sample time, specified as a scalar. Also a property of the `tf`

object. For more information, see Ts.

`ltiSys`

— Dynamic system

dynamic system model | model array

Dynamic system, specified as a SISO or MIMO dynamic system model or array of dynamic system models. Dynamic systems that you can use include:

Continuous-time or discrete-time numeric LTI models, such as

`tf`

, zpk, ss, or pid models.Generalized or uncertain LTI models such as genss or uss (Robust Control Toolbox) models. (Using uncertain models requires Robust Control Toolbox™ software.)

The resulting transfer function assumes

current values of the tunable components for tunable control design blocks.

nominal model values for uncertain control design blocks.

Identified LTI models, such as idtf (System Identification Toolbox), idss (System Identification Toolbox), idproc (System Identification Toolbox), idpoly (System Identification Toolbox), and idgrey (System Identification Toolbox) models. To select the component of the identified model to convert, specify

`component`

. If you do not specify`component`

,`tf`

converts the measured component of the identified model by default. (Using identified models requires System Identification Toolbox™ software.)

`m`

— Static gain

scalar | matrix

Static gain, specified as a scalar or matrix. Static gain or steady state gain of a system represents the ratio of the output to the input under steady state condition.

`component`

— Component of identified model

`'measured'`

(default) | `'noise'`

| `'augmented'`

Component of identified model to convert, specified as one of the following:

`'measured'`

— Convert the measured component of`sys`

.`'noise'`

— Convert the noise component of`sys`

`'augmented'`

— Convert both the measured and noise components of`sys`

.

`component`

only applies when `sys`

is an identified LTI model.

For more information on identified LTI models and their measured and noise components, see Identified LTI Models.

### Output Arguments

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`sys`

— Output system model

`tf`

model object | `genss`

model object | `uss`

model object

Output system model, returned as:

A transfer function (

`tf`

) model object, when`numerator`

and`denominator`

input arguments are numeric arrays.A generalized state-space model (

`genss`

) object, when the`numerator`

or`denominator`

input arguments includes tunable parameters, such as realp parameters or generalized matrices (genmat). For an example, see Tunable Low-Pass Filter.An uncertain state-space model (

`uss`

) object, when the`numerator`

or`denominator`

input arguments includes uncertain parameters. Using uncertain models requires Robust Control Toolbox software. For an example, see Transfer Function with Uncertain Coefficients (Robust Control Toolbox).

## Properties

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`Numerator`

— Numerator coefficients

row vector | `Ny`

-by-`Nu`

cell array of row vectors

Numerator coefficients, specified as:

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'`

).An

`Ny`

-by-`Nu`

cell array of row vectors to specify a MIMO transfer function, where`Ny`

is the number of outputs and`Nu`

is the number of inputs. Each element of the cell array specifies the numerator coefficients for a given input/output pair. If you specify both`Numerator`

and`Denominator`

as cell arrays, they must have the same dimensions.

The coefficients of `Numerator`

can be either real-valued or complex-valued.

`Denominator`

— Denominator coefficients

row vector | `Ny`

-by-`Nu`

cell array of row vectors

Denominator coefficients, specified as:

A row vector of polynomial coefficients in order of descending power (for values

`Variable`

values`'s'`

,`'z'`

,`'p'`

, or`'q'`

) or in order of ascending power (for`Variable`

values`'z^-1'`

or`'q^-1'`

).An

`Ny`

-by-`Nu`

cell array of row vectors to specify a MIMO transfer function, where`Ny`

is the number of outputs and`Nu`

is the number of inputs. Each element of the cell array specifies the numerator coefficients for a given input/ output pair. If you specify both`Numerator`

and`Denominator`

as cell arrays, they must have the same dimensions.

If all SISO entries of a MIMO transfer function have the same denominator, you can specify `Denominator`

as the row vector while specifying `Numerator`

as a cell array.

The coefficients of `Denominator`

can be either real-valued or complex-valued.

`Variable`

— Transfer function display variable

`'s'`

(default) | `'z'`

| `'p'`

| `'q'`

| `'z^-1'`

| `'q^-1'`

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`

values`'s'`

,`'z'`

,`'p'`

, or`'q'`

, the coefficients are ordered in descending powers of the variable. For example, consider the row vector`[ak ... a1 a0]`

. The polynomial order is specified as $${a}_{k}{z}^{k}+\mathrm{...}+{a}_{1}z+{a}_{0}$$.For

`Variable`

values`'z^-1'`

or`'q^-1'`

, the coefficients are ordered in ascending powers of the variable. For example, consider the row vector`[b0 b1 ... bk]`

. The polynomial order is specified as $${b}_{0}+{b}_{1}{z}^{-1}+\mathrm{...}+{b}_{k}{z}^{-k}$$.

For examples, see Specify Polynomial Ordering in Discrete-Time Transfer Function, Transfer Function Model Using Rational Expression, and Discrete-Time Transfer Function Model Using Rational Expression.

`IODelay`

— Transport delay

`0`

(default) | scalar | `Ny`

-by-`Nu`

array

Transport delay, specified as one of the following:

Scalar — Specify the transport delay for a SISO system or the same transport delay for all input/output pairs of a MIMO system.

`Ny`

-by-`Nu`

array — Specify separate transport delays for each input/output pair of a MIMO system. Here,`Ny`

is the number of outputs and`Nu`

is the number of inputs.

For continuous-time systems, specify transport delays in the time unit specified by the `TimeUnit`

property. For discrete-time systems, specify transport delays in integer multiples of the sample time, `Ts`

.

`InputDelay`

— Input delay

`0`

(default) | scalar | `Nu`

-by-1 vector

Input delay for each input channel, specified as one of the following:

Scalar — Specify the input delay for a SISO system or the same delay for all inputs of a multi-input system.

`Nu`

-by-1 vector — Specify separate input delays for input of a multi-input system, where`Nu`

is the number of inputs.

For continuous-time systems, specify input delays in the time unit specified by the `TimeUnit`

property. For discrete-time systems, specify input delays in integer multiples of the sample time, `Ts`

.

For more information, see Time Delays in Linear Systems.

`OutputDelay`

— Output delay

`0`

(default) | scalar | `Ny`

-by-1 vector

Output delay for each output channel, specified as one of the following:

Scalar — Specify the output delay for a SISO system or the same delay for all outputs of a multi-output system.

`Ny`

-by-1 vector — Specify separate output delays for output of a multi-output system, where`Ny`

is the number of outputs.

For continuous-time systems, specify output delays in the time unit specified by the `TimeUnit`

property. For discrete-time systems, specify output delays in integer multiples of the sample time, `Ts`

.

For more information, see Time Delays in Linear Systems.

`Ts`

— Sample time

`0`

(default) | positive scalar | `-1`

Sample time, specified as:

`0`

for continuous-time systems.A positive scalar representing the sampling period of a discrete-time system. Specify

`Ts`

in the time unit specified by the`TimeUnit`

property.`-1`

for a discrete-time system with an unspecified sample time.

**Note**

Changing `Ts`

does not discretize or resample the model. To convert between continuous-time and discrete-time representations, use c2d and d2c. To change the sample time of a discrete-time system, use d2d.

`TimeUnit`

— Time variable units

`'seconds'`

(default) | `'nanoseconds'`

| `'microseconds'`

| `'milliseconds'`

| `'minutes'`

| `'hours'`

| `'days'`

| `'weeks'`

| `'months'`

| `'years'`

| ...

Time variable units, specified as one of the following:

`'nanoseconds'`

`'microseconds'`

`'milliseconds'`

`'seconds'`

`'minutes'`

`'hours'`

`'days'`

`'weeks'`

`'months'`

`'years'`

Changing `TimeUnit`

has no effect on other properties, but changes the overall system behavior. Use chgTimeUnit to convert between time units without modifying system behavior.

`InputName`

— Input channel names

`''`

(default) | character vector | cell array of character vectors

Input channel names, specified as one of the following:

A character vector, for single-input models.

A cell array of character vectors, for multi-input models.

`''`

, no names specified, for any input channels.

Alternatively, you can assign input names for multi-input models using automatic vector expansion. For example, if `sys`

is a two-input model, enter the following:

`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`

.

Use `InputName`

to:

Identify channels on model display and plots.

Extract subsystems of MIMO systems.

Specify connection points when interconnecting models.

`InputUnit`

— Input channel units

`''`

(default) | character vector | cell array of character vectors

Input channel units, specified as one of the following:

A character vector, for single-input models.

A cell array of character vectors, for multi-input models.

`''`

, no units specified, for any input channels.

Use `InputUnit`

to specify input signal units. `InputUnit`

has no effect on system behavior.

`InputGroup`

— Input channel groups

structure

Input channel groups, specified as a structure. Use `InputGroup`

to assign the input channels of MIMO systems into groups and refer to each group by name. The field names of `InputGroup`

are the group names and the field values are the input channels of each group. For example, enter the following to create input groups named `controls`

and `noise`

that include input channels `1`

and `2`

, and `3`

and `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.

`sys(:,'controls')`

By default, `InputGroup`

is a structure with no fields.

`OutputName`

— Output channel names

`''`

(default) | character vector | cell array of character vectors

Output channel names, specified as one of the following:

A character vector, for single-output models.

A cell array of character vectors, for multi-output models.

`''`

, no names specified, for any output channels.

Alternatively, you can assign output names for multi-output models using automatic vector expansion. For example, if `sys`

is a two-output model, enter the following.

`sys.OutputName = 'measurements';`

The output names automatically expand to `{'measurements(1)';'measurements(2)'}`

.

You can also use the shorthand notation `y`

to refer to the `OutputName`

property. For example, `sys.y`

is equivalent to `sys.OutputName`

.

Use `OutputName`

to:

Identify channels on model display and plots.

Extract subsystems of MIMO systems.

Specify connection points when interconnecting models.

`OutputUnit`

— Output channel units

`''`

(default) | character vector | cell array of character vectors

Output channel units, specified as one of the following:

A character vector, for single-output models.

A cell array of character vectors, for multi-output models.

`''`

, no units specified, for any output channels.

Use `OutputUnit`

to specify output signal units. `OutputUnit`

has no effect on system behavior.

`OutputGroup`

— Output channel groups

structure

Output channel groups, specified as a structure. Use `OutputGroup`

to assign the output channels of MIMO systems into groups and refer to each group by name. The field names of `OutputGroup`

are the group names and the field values are the output channels of each group. For example, create output groups named `temperature`

and `measurement`

that include output channels `1`

, and `3`

and `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.

`sys('measurement',:)`

By default, `OutputGroup`

is a structure with no fields.

`Name`

— System name

`''`

(default) | character vector

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

.

`Notes`

— User-specified text

`{}`

(default) | character vector | cell array of character vectors

User-specified text that you want to associate with the system, specified as a character vector or cell array of character vectors. For example, `'System is MIMO'`

.

`UserData`

— User-specified data

`[]`

(default) | any MATLAB^{®} data type

User-specified data that you want to associate with the system, specified as any MATLAB data type.

`SamplingGrid`

— Sampling grid for model arrays

structure array

Sampling grid for model arrays, specified as a structure array.

Use `SamplingGrid`

to track the variable values associated with each model in a model array, including identified linear time-invariant (IDLTI) model arrays.

Set the field names of the 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 scalars, and all arrays of sampled values must match the dimensions of the model array.

For example, you can create an 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, you can create a 6-by-9 model array, `M`

, by independently sampling two variables, `zeta`

and `w`

. The following code maps 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 instance, the Simulink Control Design™ commands linearize (Simulink Control Design) and slLinearizer (Simulink Control Design) populate `SamplingGrid`

automatically.

By default, `SamplingGrid`

is a structure with no fields.

## Object Functions

The following lists contain a representative subset of the functions you can use with `tf`

models. In general, any function applicable to Dynamic System Models is applicable to a `tf`

object.

expand all

### Linear Analysis

step | Step response of dynamic system |

impulse | Impulse response plot of dynamic system; impulse response data |

lsim | Plot simulated time response of dynamic system to arbitrary inputs; simulated response data |

bode | Bode plot of frequency response, or magnitude and phasedata |

nyquist | Nyquist plot of frequency response |

nichols | Nichols chart of frequency response |

bandwidth | Frequency response bandwidth |

### Stability Analysis

pole | Poles of dynamic system |

zero | Zeros and gain of SISO dynamic system |

pzplot | Pole-zero plot of dynamic system model with additional plot customization options |

margin | Gain margin, phase margin, and crossover frequencies |

### Model Transformation

zpk | Zero-pole-gain model |

ss | State-space model |

c2d | Convert model from continuous to discrete time |

d2c | Convert model from discrete to continuous time |

d2d | Resample discrete-timemodel |

### Model Interconnection

feedback | Feedback connection of multiple models |

connect | Block diagram interconnections of dynamic systems |

series | Series connection of two models |

parallel | Parallel connection of two models |

### Controller Design

pidtune | PID tuning algorithm for linear plant model |

rlocus | Root locus plot of dynamic system |

lqr | Linear-Quadratic Regulator (LQR) design |

lqg | Linear-Quadratic-Gaussian (LQG) design |

lqi | Linear-Quadratic-Integral control |

kalman | Design Kalman filter for state estimation |

## Examples

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### SISO Transfer Function Model

Open Live Script

For this example, consider the following SISO transfer function model:

$$sys(s)=\frac{1}{2{s}^{2}+3s+4}.$$

Specify the numerator and denominator coefficients ordered in descending powers of `s`

, and create the transfer function model.

numerator = 1;denominator = [2,3,4];sys = tf(numerator,denominator)

sys = 1 --------------- 2 s^2 + 3 s + 4 Continuous-time transfer function.

### Discrete-Time SISO Transfer Function Model

Open Live Script

For this example, consider the following discrete-time SISO transfer function model:

$$sys(z)=\frac{2z}{4{z}^{3}+3z-1}.$$

Specify the numerator and denominator coefficients ordered in descending powers of `z`

and the sample time of 0.1 seconds. Create the discrete-time transfer function model.

numerator = [2,0];denominator = [4,0,3,-1];ts = 0.1;sys = tf(numerator,denominator,ts)

sys = 2 z --------------- 4 z^3 + 3 z - 1 Sample time: 0.1 secondsDiscrete-time transfer function.

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

Open Live Script

For this example, consider a transfer function 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 $$\zeta $$ and natural frequency $${\omega}_{0}$$, is:

$$sys\left(s\right)=\frac{{\omega}_{0}^{2}}{{s}^{2}+2\zeta {\omega}_{0}s+{\omega}_{0}^{2}}.$$

Assuming a damping ratio, $$\zeta $$ = 0.25 and natural frequency, $${\omega}_{0}$$ = 3 rad/s, create the second order transfer function.

zeta = 0.25;w0 = 3;numerator = w0^2;denominator = [1,2*zeta*w0,w0^2];sys = tf(numerator,denominator)

sys = 9 --------------- s^2 + 1.5 s + 9 Continuous-time transfer function.

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

stepplot(sys)

The plot shows the ringdown expected of a second-order system with a low damping ratio.

### Discrete-Time MIMO Transfer Function Model

Open Live Script

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

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

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 MIMO transfer function model.

ts = 0.2;sys = tf(numerators,denominator,ts)

sys = From input 1 to output... 1 1: ------- z + 0.3 -z + 2 2: ------- z + 0.3 From input 2 to output... z 1: ------- z + 0.3 3 2: ------- z + 0.3 Sample time: 0.2 secondsDiscrete-time transfer function.

For more information on creating MIMO transfer functions, see MIMO Transfer Functions.

### Concatenate SISO Transfer Functions into MIMO Transfer Function Model

Open Live Script

In this example, you create a MIMO transfer function model by concatenating SISO transfer function models. Consider the following single-input, two-output transfer function:

$$sys(s)=\left[\begin{array}{c}\frac{s-1}{s+1}\\ \frac{s+2}{{s}^{2}+4s+5}\end{array}\right].$$

Specify the MIMO transfer function model by concatenating the SISO entries.

sys1 = tf([1 -1],[1 1]);sys2 = tf([1 2],[1 4 5]);sys = [sys1;sys2]

sys = From input to output... s - 1 1: ----- s + 1 s + 2 2: ------------- s^2 + 4 s + 5 Continuous-time transfer function.

For more information on creating MIMO transfer functions, see MIMO Transfer Functions.

### Transfer Function Model Using Rational Expression

Open Live Script

For this example, create a continuous-time transfer function model using rational expressions. Using a rational expression can sometimes be easier and more intuitive than specifying polynomial coefficients of the numerator and denominator.

Consider the following system:

$$sys(s)=\frac{s}{{s}^{2}+2s+10}.$$

To create the transfer function model, first specify `s`

as a `tf`

object.

`s = tf('s')`

s = s Continuous-time transfer function.

Create the transfer function model using s in the rational expression.

sys = s/(s^2 + 2*s + 10)

sys = s -------------- s^2 + 2 s + 10 Continuous-time transfer function.

### Discrete-Time Transfer Function Model Using Rational Expression

Open Live Script

For this example, create a discrete-time transfer function model using a rational expression. Using a rational expression can sometimes be easier and more intuitive than specifying polynomial coefficients.

Consider the following system:

$$sys(z)=\frac{z-1}{{z}^{2}-1.85z+0.9}.$$

To create the transfer function model, first specify `z`

as a `tf`

object and the sample time `Ts`

.

`ts = 0.1;z = tf('z',ts)`

z = z Sample time: 0.1 secondsDiscrete-time transfer function.

Create the transfer function model using `z`

in the rational expression.

sys = (z - 1) / (z^2 - 1.85*z + 0.9)

sys = z - 1 ------------------ z^2 - 1.85 z + 0.9 Sample time: 0.1 secondsDiscrete-time transfer function.

### Transfer Function Model with Inherited Properties

Open Live Script

For this example, create a transfer function model with properties inherited from another transfer function model. Consider the following two transfer functions:

$$sys1(s)=\frac{2s}{{s}^{2}+8s}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}and\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}sys2(s)=\frac{s-1}{7{s}^{4}+2{s}^{3}+9}.$$

For this example, create `sys1`

with the `TimeUnit`

and `InputDelay`

property set to '`minutes`

'.

numerator1 = [2,0];denominator1 = [1,8,0];sys1 = tf(numerator1,denominator1,'TimeUnit','minutes','InputUnit','minutes')

sys1 = 2 s --------- s^2 + 8 s Continuous-time transfer function.

propValues1 = [sys1.TimeUnit,sys1.InputUnit]

`propValues1 = `*1x2 cell* {'minutes'} {'minutes'}

Create the second transfer function model with properties inherited from `sys1`

.

numerator2 = [1,-1];denominator2 = [7,2,0,0,9];sys2 = tf(numerator2,denominator2,sys1)

sys2 = s - 1 ----------------- 7 s^4 + 2 s^3 + 9 Continuous-time transfer function.

propValues2 = [sys2.TimeUnit,sys2.InputUnit]

`propValues2 = `*1x2 cell* {'minutes'} {'minutes'}

Observe that the transfer function model `sys2`

has that same properties as `sys1`

.

### Array of Transfer Function Models

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You can use a `for`

loop to specify an array of transfer function models.

First, pre-allocate the transfer function array with zeros.

sys = tf(zeros(1,1,3));

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 model array using a rational expression in the `for`

loop.

s = tf('s'); for k = 1:3 sys(:,:,k) = k/(s^2+s+k); endsys

sys(:,:,1,1) = 1 ----------- s^2 + s + 1 sys(:,:,2,1) = 2 ----------- s^2 + s + 2 sys(:,:,3,1) = 3 ----------- s^2 + s + 3 3x1 array of continuous-time transfer functions.

### Convert State-Space Model to Transfer Function

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For this example, compute the transfer function of the following state-space model:

$$A=\left[\begin{array}{cc}-2& -1\\ 1& -2\end{array}\right],\phantom{\rule{0ex}{0ex}}B=\left[\begin{array}{cc}1& 1\\ 2& -1\end{array}\right],\phantom{\rule{0ex}{0ex}}C=\left[\begin{array}{cc}1& 0\end{array}\right],\phantom{\rule{0ex}{0ex}}D=[\phantom{\rule{0ex}{0ex}}\begin{array}{cc}0& 1\end{array}].$$

Create the state-space model using the state-space matrices.

A = [-2 -1;1 -2];B = [1 1;2 -1];C = [1 0];D = [0 1];ltiSys = ss(A,B,C,D);

Convert the state-space model `ltiSys`

to a transfer function.

sys = tf(ltiSys)

sys = From input 1 to output: s ------------- s^2 + 4 s + 5 From input 2 to output: s^2 + 5 s + 8 ------------- s^2 + 4 s + 5 Continuous-time transfer function.

### Extract Transfer Functions from Identified Model

This example uses:

- Control System ToolboxControl System Toolbox
- System Identification ToolboxSystem Identification Toolbox

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For this example, extract the measured and noise components of an identified polynomial model into two separate transfer functions.

Load the Box-Jenkins polynomial model `ltiSys`

in `identifiedModel.mat`

.

load('identifiedModel.mat','ltiSys');

`ltiSys`

is an identified discrete-time model of the form: $$y(t)=\frac{B}{F}u(t)+\frac{C}{D}e(t)$$, where $$\frac{B}{F}$$ represents the measured component and $$\frac{C}{D}$$ the noise component.

Extract the measured and noise components as transfer functions.

`sysMeas = tf(ltiSys,'measured') `

sysMeas = From input "u1" to output "y1": -0.1426 z^-1 + 0.1958 z^-2 z^(-2) * ---------------------------- 1 - 1.575 z^-1 + 0.6115 z^-2 Sample time: 0.04 secondsDiscrete-time transfer function.

`sysNoise = tf(ltiSys,'noise')`

sysNoise = From input "v@y1" to output "y1": 0.04556 + 0.03301 z^-1 ---------------------------------------- 1 - 1.026 z^-1 + 0.26 z^-2 - 0.1949 z^-3 Input groups: Name Channels Noise 1 Sample time: 0.04 secondsDiscrete-time transfer function.

The measured component can serve as a plant model, while the noise component can be used as a disturbance model for control system design.

### Specify Input and Output Names for MIMO Transfer Function Model

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Transfer function model objects include model data that helps you keep track of what the model represents. For instance, you can assign names to the inputs and outputs of your model.

Consider the following continuous-time MIMO transfer function model:

$$sys(s)=\left[\begin{array}{c}\frac{s+1}{{s}^{2}+2s+2}\\ \frac{1}{s}\end{array}\right]$$

The model has one input $$-$$ Current, and two outputs $$-$$ Torque and Angular velocity.

First, specify the numerator and denominator coefficients of the model.

numerators = {[1 1] ; 1};denominators = {[1 2 2] ; [1 0]};

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

sys = tf(numerators,denominators,'InputName','Current',... 'OutputName',{'Torque' 'Angular Velocity'})

sys = From input "Current" to output... s + 1 Torque: ------------- s^2 + 2 s + 2 1 Angular Velocity: - s Continuous-time transfer function.

### Specify Polynomial Ordering in Discrete-Time Transfer Function

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For this example, specify polynomial ordering in discrete-time transfer function models using the '`Variable`

' property.

Consider the following discrete-time transfer functions with sample time 0.1 seconds:

$$sys1(z)=\frac{{z}^{2}}{{z}^{2}+2z+3}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}sys2({z}^{-1})=\frac{1}{1+2{z}^{-1}+3{z}^{-2}}.$$

Create the first discrete-time transfer function by specifying the `z`

coefficients.

numerator = [1,0,0];denominator = [1,2,3];ts = 0.1;sys1 = tf(numerator,denominator,ts)

sys1 = z^2 ------------- z^2 + 2 z + 3 Sample time: 0.1 secondsDiscrete-time transfer function.

The coefficients of `sys1`

are ordered in descending powers of `z`

.

`tf`

switches convention based on the value of the '`Variable`

' property. Since `sys2`

is the inverse transfer function model of `sys1`

, specify '`Variable`

' as '`z^-1`

' and use the same numerator and denominator coefficients.

sys2 = tf(numerator,denominator,ts,'Variable','z^-1')

sys2 = 1 ------------------- 1 + 2 z^-1 + 3 z^-2 Sample time: 0.1 secondsDiscrete-time transfer function.

The coefficients of `sys2`

are now ordered in ascending powers of `z^-1`

.

Based on different conventions, you can specify polynomial ordering in transfer function models using the '`Variable`

' property.

### Tunable Low-Pass Filter

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In this example, you will create a low-pass filter with one tunable parameter *a*:

$$F=\frac{a}{s+a}$$

Since the numerator and denominator coefficients of a `tunableTF`

block are independent, you cannot use `tunableTF`

to represent `F`

. Instead, construct `F`

using the tunable real parameter object `realp`

.

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

.

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

a = Name: 'a' Value: 10 Minimum: -Inf Maximum: Inf Free: 1Real scalar parameter.

Use `tf`

to create the tunable low-pass filter `F`

.

numerator = a;denominator = [1,a];F = tf(numerator,denominator)

Generalized continuous-time state-space model with 1 outputs, 1 inputs, 1 states, and the following blocks: a: Scalar parameter, 2 occurrences.Type "ss(F)" to see the current value and "F.Blocks" to interact with the blocks.

`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 an example, see Control System with Tunable Components.

### Static Gain MIMO Transfer Function Model

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In this example, you will create a static gain MIMO transfer function model.

Consider the following two-input, two-output static gain matrix `m`

:

$$m=\left[\begin{array}{cc}2& 4\\ 3& 5\end{array}\right]$$

Specify the gain matrix and create the static gain transfer function model.

`m = [2,4;... 3,5];sys1 = tf(m)`

sys1 = From input 1 to output... 1: 2 2: 3 From input 2 to output... 1: 4 2: 5 Static gain.

You can use static gain transfer function model `sys1`

obtained above to cascade it with another transfer function model.

For this example, create another two-input, two-output discrete transfer function model and use the `series`

function to connect the two models.

numerators = {1,[1,0];[-1,2],3};denominator = [1,0.3];ts = 0.2;sys2 = tf(numerators,denominator,ts)

sys2 = From input 1 to output... 1 1: ------- z + 0.3 -z + 2 2: ------- z + 0.3 From input 2 to output... z 1: ------- z + 0.3 3 2: ------- z + 0.3 Sample time: 0.2 secondsDiscrete-time transfer function.

sys = series(sys1,sys2)

sys = From input 1 to output... 3 z^2 + 2.9 z + 0.6 1: ------------------- z^2 + 0.6 z + 0.09 -2 z^2 + 12.4 z + 3.9 2: --------------------- z^2 + 0.6 z + 0.09 From input 2 to output... 5 z^2 + 5.5 z + 1.2 1: ------------------- z^2 + 0.6 z + 0.09 -4 z^2 + 21.8 z + 6.9 2: --------------------- z^2 + 0.6 z + 0.09 Sample time: 0.2 secondsDiscrete-time transfer function.

## Limitations

Transfer function models are ill-suited for numerical computations. Once created, convert them to state-space form before combining them with other models or performing model transformations. You can then convert the resulting models back to transfer function form for inspection purposes

An identified nonlinear model cannot be directly converted into a transfer function model using

`tf`

. To obtain a transfer function model:Convert the nonlinear identified model to an identified LTI model using linapp (System Identification Toolbox), idnlarx/linearize (System Identification Toolbox), or idnlhw/linearize (System Identification Toolbox).

Then, convert the resulting model to a transfer function model using

`tf`

.

## Version History

**Introduced before R2006a**

## See Also

filt | frd | get | set | ss | tfdata | zpk | genss | realp | genmat | tunableTF

### Topics

- What Are Model Objects?
- Transfer Functions
- Discrete-Time Numeric Models
- MIMO Transfer Functions
- Using the Right Model Representation
- Using FEEDBACK to Close Feedback Loops

### External Websites

- Transfer Function Analysis of Dynamic Systems (MathWorks Teaching Resources)

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