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# Transfer Fcn

Model linear system by transfer function

Continuous

## Description

The Transfer Fcn block models a linear system by a transfer function of the Laplace-domain variable s. The block can model single-input single-output (SISO) and single-input multiple output (SIMO) systems.

### Conditions for Using This Block

The Transfer Fcn block assumes the following conditions:

• The transfer function has the form

where u and y are the system input and outputs, respectively, nn and nd are the number of numerator and denominator coefficients, respectively. num(s) and den(s) contain the coefficients of the numerator and denominator in descending powers of s.

• The order of the denominator must be greater than or equal to the order of the numerator.

• For a multiple-output system, all transfer functions have the same denominator and all numerators have the same order.

### Modeling a Single-Output System

For a single-output system, the input and output of the block are scalar time-domain signals. To model this system:

1. Enter a vector for the numerator coefficients of the transfer function in the Numerator coefficients field.

2. Enter a vector for the denominator coefficients of the transfer function in the Denominator coefficients field.

### Modeling a Multiple-Output System

For a multiple-output system, the block input is a scalar and the output is a vector, where each element is an output of the system. To model this system:

1. Enter a matrix in the Numerator coefficients field.

Each row of this matrix contains the numerator coefficients of a transfer function that determines one of the block outputs.

2. Enter a vector of the denominator coefficients common to all transfer functions of the system in the Denominator coefficients field.

### Specifying Initial Conditions

A transfer function describes the relationship between input and output in Laplace (frequency) domain. Specifically, it is defined as the Laplace transform of the response (output) of a system with zero initial conditions to an impulse input.

Operations like multiplication and division of transfer functions rely on zero initial state. For example, you can decompose a single complicated transfer function into a series of simpler transfer functions. Apply them sequentially to get a response equivalent to that of the original transfer function. This will not be correct if one of the transfer functions assumes a non-zero initial state. Furthermore, a transfer function has infinitely many time domain realizations, most of whose states do not have any physical meaning.

For these reasons, Simulink® presets the initial conditions of the Transfer Fcn block to zero. To specify initial conditions for a given transfer function, convert the transfer function to its controllable, canonical state-space realization using tf2ss . Then, use the State-Space block. The tf2ss utility provides the A, B, C, and D matrices for the system.

For more information, type help tf2ss or see the Control System Toolbox™ documentation.

### Transfer Function Display on the Block

The Transfer Fcn block displays the transfer function depending on how you specify the numerator and denominator parameters.

• If you specify each parameter as an expression or a vector, the block shows the transfer function with the specified coefficients and powers of s. If you specify a variable in parentheses, the block evaluates the variable.

For example, if you specify Numerator coefficients as [3,2,1] and Denominator coefficients as (den), where den is [7,5,3,1], the block looks like this:

• If you specify each parameter as a variable, the block shows the variable name followed by (s).

For example, if you specify Numerator coefficients as num and Denominator coefficients as den, the block looks like this:

## Data Type Support

The Transfer Fcn block accepts and outputs signals of type double.

## Parameters and Dialog Box

### Numerator coefficients

Define the row vector of numerator coefficients.

Default: [1]

#### Tips

• For a single-output system, enter a vector for the numerator coefficients of the transfer function.

• For a multiple-output system, enter a matrix. Each row of this matrix contains the numerator coefficients of a transfer function that determines one of the block outputs.

#### Command-Line Information

 Parameter: Numerator Type: vector or matrix Value: '[1]' Default: '[1]'

### Denominator coefficients

Define the row vector of denominator coefficients.

Default: [1 1]

#### Tips

• For a single-output system, enter a vector for the denominator coefficients of the transfer function.

• For a multiple-output system, enter a vector containing the denominator coefficients common to all transfer functions of the system.

#### Command-Line Information

 Parameter: Denominator Type: vector Value: '[1 1]' Default: '[1 1]'

### Absolute tolerance

Specify the absolute tolerance for computing block states.

#### Settings

Default: auto

• You can enter auto, –1, a positive real scalar or vector.

• If you enter auto or –1, then Simulink uses the absolute tolerance value in the Configuration Parameters dialog box (see Solver Pane) to compute block states.

• If you enter a real scalar, then that value overrides the absolute tolerance in the Configuration Parameters dialog box for computing all block states.

• If you enter a real vector, then the dimension of that vector must match the dimension of the continuous states in the block. These values override the absolute tolerance in the Configuration Parameters dialog box.

#### Command-Line Information

 Parameter: AbsoluteTolerance Type: string, scalar, or vector Value: 'auto' | '-1' | any positive real scalar or vector Default: 'auto'

### State Name (e.g., 'position')

Assign a unique name to each state.

#### Settings

Default: ' '

If this field is blank, no name assignment occurs.

#### Tips

• To assign a name to a single state, enter the name between quotes, for example, 'velocity'.

• To assign names to multiple states, enter a comma-delimited list surrounded by braces, for example, {'a', 'b', 'c'}. Each name must be unique.

• The state names apply only to the selected block.

• The number of states must divide evenly among the number of state names.

• You can specify fewer names than states, but you cannot specify more names than states.

For example, you can specify two names in a system with four states. The first name applies to the first two states and the second name to the last two states.

• To assign state names with a variable in the MATLAB® workspace, enter the variable without quotes. A variable can be a string, cell array, or structure.

#### Command-Line Information

 Parameter: ContinuousStateAttributes Type: string Value: ' ' | user-defined Default: ' '

## Examples

The following Simulink examples show how to use the Transfer Fcn block:

## Characteristics

 Direct Feedthrough Only if the lengths of the Numerator coefficients and Denominator coefficients parameters are equal Sample Time Continuous Scalar Expansion No States Length of Denominator coefficients -1 Dimensionalized Yes, the block expands scalar input into vector output when the transfer function numerator is a matrix. Zero-Crossing Detection No