# 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.

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

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

Here, *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.

For more information, see What Are Transfer Function Models?

## Apps

System Identification | Identify models of dynamic systems from measured data |

## Functions

## Topics

### Transfer Function Model Basics

**What Are Transfer Function Models?**

Transfer function models describe the relationship between the inputs and outputs of a system using a ratio of polynomials.**Estimate Transfer Function Models in the System Identification App**

Use the app to set model configuration and estimation options for estimating a transfer function model.**Estimate Transfer Function Models at the Command Line**

General workflow for estimating transfer function models at the command line.**Data Supported by Transfer Function Models**

Characteristics of estimation data for transfer function identification.

### Estimate Transfer Function Models

**Estimate Transfer Function Models by Specifying Number of Poles**

This example shows how to identify a transfer function containing a specified number of poles for given data.**Estimate Transfer Function Models with Transport Delay to Fit Given Frequency-Response Data**

This example shows how to identify a transfer function to fit a given frequency response data (FRD) containing additional phase roll off induced by input delay.**Estimate Transfer Function Models with Prior Knowledge of Model Structure and Constraints**

This example shows how to estimate a transfer function model when the structure of the expected model is known and apply constraints to the numerator and denominator coefficients.**Estimate Transfer Functions with Delays**

This example shows how to estimate transfer function models with I/O delays.**Estimate Transfer Function Models with Unknown Transport Delays**

This example shows how to estimate a transfer function model with unknown transport delays and apply an upper bound on the unknown transport delays.

### Frequency Domain Troubleshooting

**Troubleshoot Frequency-Domain Identification of Transfer Function Models**

Improve frequency-domain model estimation by preprocessing data and applying frequency-dependent weighting filters.

### Model Initialization and Structure Parameters

**Transfer Function Structure Specification**

Specify the values and constraints for the numerator, denominator and transport delays.**Specifying Initial Conditions for Iterative Estimation of Transfer Functions**

Specify how initial conditions are handled during model estimation in the app and at the command line.