## What Are Transfer Function Models?

### Definition of Transfer Function Models

Transfer function models describe the relationship between the inputs and outputs of a system using a ratio of polynomials. The model order is equal to the order of the denominator polynomial. The roots of the denominator polynomial are referred to as the model poles. The roots of the numerator polynomial are referred to as the model zeros.

The parameters of a transfer function model are its poles, zeros and transport delays.

### Continuous-Time Representation

In continuous-time, a transfer function model has the form:

`$Y\left(s\right)=\frac{num\left(s\right)}{den\left(s\right)}U\left(s\right)+E\left(s\right)$`

Where, Y(s), U(s) and E(s) represent the Laplace transforms of the output, input and noise, respectively. num(s) and den(s) represent the numerator and denominator polynomials that define the relationship between the input and the output.

### Discrete-Time Representation

In discrete-time, a transfer function model has the form:

`$\begin{array}{l}y\left(t\right)=\frac{num\left({q}^{-1}\right)}{den\left({q}^{-1}\right)}u\left(t\right)+e\left(t\right)\\ num\left({q}^{-1}\right)={b}_{0}+{b}_{1}{q}^{-1}+{b}_{2}{q}^{-2}+\dots \\ den\left({q}^{-1}\right)=1+{a}_{1}{q}^{-1}+{a}_{2}{q}^{-2}+\dots \end{array}$`

The roots of num(q^-1) and den(q^-1) are expressed in terms of the lag variable q^-1.

If you take the Z-transform, the transfer function has the form:

`$\begin{array}{l}Y\left({z}^{-1}\right)=\frac{num\left({z}^{-1}\right)}{den\left({z}^{-1}\right)}U\left({z}^{-1}\right)+E\left({z}^{-1}\right)\\ num\left({z}^{-1}\right)={b}_{0}+{b}_{1}{z}^{-1}+{b}_{2}{z}^{-2}+\dots \\ den\left({z}^{-1}\right)=1+{a}_{1}{z}^{-1}+{a}_{2}{z}^{-2}+\dots \end{array}$`

Where, Y(z-1), U(z-1) and E(z-1) represent the Z-transforms of the output, input and noise, respectively. z-1 is the Z-transform of the lag operator.

### Delays

In continuous-time, input and transport delays are of the form:

`$Y\left(s\right)=\frac{num\left(s\right)}{den\left(s\right)}{e}^{-s\tau }U\left(s\right)+E\left(s\right)$`

Where τ represents the delay.

In discrete-time:

`$y\left(t\right)=\frac{num}{den}u\left(t-\tau \right)+e\left(t\right)$`

where num and den are polynomials in the lag operator `q^(-1)`.

### Multi-Input Multi-Output Models

A single-input single-output (SISO) continuous transfer function has the form $G\left(s\right)=\frac{num\left(s\right)}{den\left(s\right)}$. The corresponding transfer function model can be represented as:

`$Y\left(s\right)=G\left(s\right)U\left(s\right)+E\left(s\right)$`

A multi-input multi-output (MIMO) transfer function contains a SISO transfer function corresponding to each input-output pair in the system. For example, a continuous-time transfer function model with two inputs and two outputs has the form:

`$\begin{array}{l}{Y}_{1}\left(s\right)={G}_{11}\left(s\right){U}_{1}\left(s\right)+{G}_{12}\left(s\right){U}_{2}\left(s\right)+{E}_{1}\left(s\right)\\ {Y}_{2}\left(s\right)={G}_{21}\left(s\right){U}_{1}\left(s\right)+{G}_{22}\left(s\right){U}_{2}\left(s\right)+{E}_{2}\left(s\right)\end{array}$`

Where, Gij(s) is the SISO transfer function between the ith output and the jth input. E1(s) and E2(s) are the Laplace transforms of the noise corresponding to the two outputs.

The representation of discrete-time MIMO transfer function models is analogous.