## 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:

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

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:

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

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:

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

Where, *Y*(*z ^{-1}*),

*U*(

*z*) and

^{-1}*E*(

*z*) represent the Z-transforms of the output, input and noise, respectively.

^{-1}*z*is the Z-transform of the lag operator.

^{-1}### Delays

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

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

Where *τ* represents the delay.

In discrete-time:

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

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 $$G(s)=\frac{num(s)}{den(s)}$$. The corresponding transfer function model can be represented as:

$$Y(s)=G(s)U(s)+E(s)$$

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:

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

Where, *G _{ij}(s)* is the
SISO transfer function between the

*i*

^{th}output and the

*j*

^{th}input.

*E*and

_{1}(s)*E*are the Laplace transforms of the noise corresponding to the two outputs.

_{2}(s)The representation of discrete-time MIMO transfer function models is analogous.