Model linear system by transfer function

Continuous

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.

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.

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

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 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.

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:

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

.

For more information, see Data Types Supported by Simulink in the Simulink documentation.

Define the row vector of numerator coefficients.

**Default:** `[1]`

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.

Parameter: `Numerator` |

Type: vector or matrix |

Value: `'[1]'` |

Default: `'[1]'` |

See the Transfer Fcn block reference page for more information.

Define the row vector of denominator coefficients.

**Default:** `[1 1]`

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.

Parameter: `Denominator` |

Type: vector |

Value: `'[1 1]'` |

Default: `'[1 1]'` |

Specify the absolute tolerance for computing block states.

**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.

Parameter: ` AbsoluteTolerance` |

Type: string, scalar, or
vector |

Value: `'auto'` | `'-1'` |
any positive real scalar or vector |

Default: ` 'auto'` |

Assign a unique name to each state.

**Default:** `' '`

If this field is blank, no name assignment occurs.

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.

Parameter: `ContinuousStateAttributes` |

Type: string |

Value: `' '` |
user-defined |

Default: `' '` |

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

Direct Feedthrough | Only if the lengths of the |

Sample Time | Continuous |

Scalar Expansion | No |

States | Length of |

Dimensionalized | Yes, the block expands scalar input into vector output when the transfer function numerator is a matrix. |

Zero-Crossing Detection | No |

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