Model linear system by transfer function
Simulink / Continuous
The Transfer Fcn block models a linear system by a transfer function of the Laplacedomain
variable s
. The block can model singleinput singleoutput
(SISO) and singleinput multipleoutput (SIMO) systems.
The Transfer Fcn block assumes the following conditions:
The transfer function has the form
$$H(s)=\frac{y(s)}{u(s)}=\frac{num(s)}{den(s)}=\frac{num(1){s}^{nn1}+num(2){s}^{nn2}+\dots +num(nn)}{den(1){s}^{nd1}+den(2){s}^{nd2}+\dots +den(nd)},$$
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 multipleoutput system, all transfer functions have the same denominator and all numerators have the same order.
For a singleoutput system, the input and output of the block are scalar timedomain signals. To model this system:
Enter a vector for the numerator coefficients of the transfer function in the Numerator coefficients field.
Enter a vector for the denominator coefficients of the transfer function in the Denominator coefficients field.
For a multipleoutput 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:
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.
Enter a vector of the denominator coefficients common to all transfer functions of the system in the Denominator coefficients field.
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 nonzero 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 statespace realization using
tf2ss
. Then, use the StateSpace 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.
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 Types 

Direct Feedthrough 

Multidimensional Signals 

VariableSize Signals 

ZeroCrossing Detection 
