Transfer Fcn

Model linear system as transfer function

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Libraries:
Simulink / 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

  $$H(s)=\frac{y(s)}{u(s)}=\frac{num(s)}{den(s)}=\frac{num(1)s^{nn-1}+num(2)s^{nn-2}+\dots +num(nn)}{den(1)s^{nd-1}+den(2)s^{nd-2}+\dots +den(nd)},$$

  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 using the specified coefficients for the powers of s. If you specify a variable in parentheses, the block evaluates the variable.

  For example, if you specify the Numerator coefficients parameter value as [3 2 1] and the Denominator coefficients parameter value as (den), where den is a workspace variable with a value of [7 5 3 1], the block displays the equation using the specified values.

  

  Tip

  When the block size is too small to accommodate the full numerator or denominator, the block icon displays the numerator as num(s) and the denominator as den(s).

  

  If you want the block to show the equation for the transfer function it implements, resize the block by dragging a corner.

  

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

  For example, if you specify the Numerator coefficients parameter as num and the Denominator coefficients parameter as den, the block icon shows the numerator of the transfer function as num(s) and the denominator as den(s).

  

Examples

Aircraft Longitudinal Flight Control

Model flight control for the longitudinal motion of an aircraft using first-order linear approximations.

• Open Model

Model an Anti-Lock Braking System

Model a simple Anti-Lock Braking System (ABS). The model simulates the dynamic behavior of a vehicle under hard braking conditions. The model represents a single wheel, which may be replicated a number of times to create a model for a multi-wheel vehicle.

• Open Model

Inverted Pendulum with Animation

Use Simulink® to model and animate an inverted pendulum system. An inverted pendulum has its center of mass above its pivot point. To stably maintain this position, the system implements control logic to move the pivot point below the center-of mass as the pendulum starts to fall. The inverted pendulum is a classic dynamics problem used to test control strategies.

• Open Model

Ports

Input

Port_1 — Input signal

scalar

Input signal, specified as a scalar with data type double.

This port has direct feedthrough only when the numerator of the transfer function is 0.

Data Types: double

Output

Port_1 — Output signal

scalar | vector

Output signal, provided as a scalar or vector with data type double.

• For a single-output system, the input and output of the block are scalar time-domain signals.

• For a multiple-output system, the input is a scalar, and the output is a vector, where each element is an output of the system.

Data Types: double

Parameters

Numerator coefficients — Vector or matrix of numerator coefficients

[1] (default) | vector | matrix

Define the numerator coefficients of the transfer function.

• 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.

Programmatic Use

Block Parameter: Numerator

Type: character vector, string

Values: vector | matrix

Default: '[1]'

Denominator coefficients — Row vector of denominator coefficients

[1 1] (default) | vector

Define the row vector of denominator coefficients.

• 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.

Programmatic Use

Block Parameter: Denominator

Type: character vector | string

Values: vector

Default: '[1 1]'

Parameter tunability — Tunable representation of block parameters in code

Auto (default) | Optimized | Unconstrained

Tunability level of the numerator and denominator coefficients for Accelerated simulation modes and deployed simulations using Simulink Compiler™. Set this parameter to Auto to allow Simulink to choose the appropriate level of parameter tunability.

Set this parameter to Optimized to generate a representation of numerator and denominator coefficients in generated code for accelerated and deployed simulations that is optimized for better simulation performance.

Set this parameter to Unconstrained to generate a fully tunable (between simulations) representation of the numerator and denominator coefficients in the generated code for accelerated and deployed simulations. To let Simulink determine the appropriate tunability level, select Auto.

Programmatic Use

Block Parameter: ParameterTunability

Type: character vector, string

Values: 'Auto' | 'Optimized' | 'Unconstrained'

Default: 'Auto'

Absolute tolerance — Absolute tolerance for computing block states

auto (default) | scalar | vector

Variable-step solvers use absolute and relative tolerances when choosing the step size to determine whether the error in state calculations is acceptable.

To inherit the absolute tolerance from the Absolute tolerance configuration parameter, specify this parameter value as auto or -1.

To specify an absolute tolerance for this block that overrides the value specified for the Absolute tolerance configuration parameter:

• Enter a real, positive scalar value to use to compute all block states.

• Enter a real vector with dimensions that match the dimensions of the continuous states for the block.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	AbsoluteTolerance
Values:	'auto' (default) | '-1' | positive real scalar number | vector of positive real scalar numbers
Data Types:	char | string

Example: set_param("MyModel/Descriptor State-Space",AbsoluteTolerance="-1")

State Name (e.g., 'position') — Option to assign unique names to states

'' (default) | character vector | cell array of character vectors | MATLAB^® variable | ...

Use this parameter to optionally assign names to the states of this block. The names you assign apply only to the states of this block.

• To use default state names, leave this field blank ('').

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

• To assign names to multiple states, specify this parameter value as a cell array of character vectors. Each name in the cell array must be unique. For example, to assign the names a, b, and c, enter {'a','b','c'}.

• To specify the names using a MATLAB variable, enter the name of the variable without quotes. For example, to use the variable names to specify the state names, enter names.

You can specify a number of names that is less than the number of states in the block. In this case, the state names are used for multiple states, and the number of states must divide evenly into the number of state names. For example, when you specify two names for a block that has four states, the first name is used for the first two states, and the second name is used for the last two states.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

To get the block parameter value programmatically, use the get_param function.

Parameter:	ContinuousStateAttributes
Values:	'' (default) | valid MATLAB variable name
Data Types:	char | string | cell

Example: set_param("MyModel/Descriptor State-Space",ContinuousStateAttributes={'position','velocity'})

Block Characteristics

Data Types	double
Direct Feedthrough	noa
Multidimensional Signals	no
Variable-Size Signals	no
Zero-Crossing Detection	no
a Direct feedthrough characteristics for this block depend on block parameter values.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using Simulink® Coder™.

Not recommended for production-quality code. Relates to resource limits and restrictions on speed and memory often found in embedded systems. The code generated can contain dynamic allocation and freeing of memory, recursion, additional memory overhead, and widely-varying execution times. While the code is functionally valid and generally acceptable in resource-rich environments, smaller embedded targets often cannot support such code.

In general, consider using the Simulink Model Discretizer to map continuous blocks into discrete equivalents that support production code generation. To start the Model Discretizer, in the Simulink Editor, on the Apps tab, under Apps, under Control Systems, click Model Discretizer. One exception is the Second-Order Integrator block because, for this block, the Model Discretizer produces an approximate discretization.

Version History

Introduced before R2006a