Discrete Transfer Function

Purpose

Model discrete linear time-invariant system as transfer function

Library

Control / Discrete

Description

The Discrete Transfer Function models a discrete linear time-invariant system that is expressed in the \(z\)-domain:

\[\frac{Y(z)}{U(z)}=\frac{n_nz^n + \cdots+n_1z+n_0}{d_nz^n+ \cdots +d_1z+d_0}\]

The transfer function is displayed in the block if it is large enough, otherwise a default text is shown. To resize the block, select it, then drag one of its selection handles.

The Transfer Function block performs scalar expansion: If the input signal is a vector, the transfer function is applied to each element of the input vector individually, and the output signal is a concatenated vector of all system outputs.

Parameters

Numerator coefficients

A vector of the \(z\) term coefficients \([n_n \ldots n_1, n_0]\) for the numerator, written in descending order of powers of z. For example, the numerator \(z^3+2z\) would be entered as [1,0,2,0].

The Transfer Function supports multiple outputs for a single input by entering a matrix for the numerator. Each row of the matrix defines the numerator coefficients of an output.

Denominator coefficients

A vector of the \(z\) term coefficients \([d_n \ldots d_1, d_0]\) for the denominator, written in descending order of powers of \(z\).

Note

The order of the denominator (highest power of \(z\)) must be greater than or equal to the order of the numerator.

Sample time

A scalar specifying the sampling period or a two-element vector specifying the sampling period and offset, in seconds \((\mathrm{s})\). See also the Discrete-Periodic sample time type in section Sample Times.

Initial condition

The initial condition vector of the internal states of the Transfer Function in the form \([x_n \ldots x_1, x_0]\).

The initial condition must be specified for the controller normal form, depicted in Fig. 162 for the transfer function:

\[\frac{Y(z)}{U(z)}=\frac{n_2z^2+n_1z+n_0}{d_2z^2+d_1z+d_0} = b_2\left(a_2 + \frac{a_1z + a_0}{z^2 + b_1z + b_0}\right)\]

Fig. 162 Discrete controller normal form

where

\[\begin{split}\begin{array}{rcll} b_i & = & \frac{\displaystyle d_i}{\displaystyle d_n} & \mbox{for $i<n$ }\\ b_n & = & \frac{\displaystyle 1}{\displaystyle d_n}\\ a_i & = & n_i - \frac{\displaystyle n_n d_i}{\displaystyle d_n} & \mbox{for $i<n$}\\ a_n & = & n_n \end{array}\end{split}\]

For the normalized transfer function (with \(n_n = 0\) and \(d_n = 1\)) this simplifies to \(b_i = d_i\) and \(a_i = n_i\).

Note

The number of internal states is defined by the highest power of \(z\) of the denominator.

Note

A scalar initial condition will be applied to all internal states.

Note

In case of scalar expansion (multiple input signals), the initial condition can also be a matrix, where each row defines the initial condition for the individual inputs.

Probe Signals

Input

The input signal.

Output

The output signal.

State

The internal states of the controller normal form.