# iv4

ARX model estimation using four-stage instrumental variable method

## Syntax

## Description

### Estimate ARX Polynomial Model

estimates an ARX polynomial model `sys`

= iv4(`tt`

,```
[na
nb nk]
```

)`sys`

using the time-domain data in
the timetable `tt`

. `[na nb nk]`

specifies the ARX
structure orders of the *A* and *B* polynomials and the
input-to-output delay.

The software estimates `sys`

using the four-stage instrumental
variable method. The estimation algorithm is insensitive to the color of the noise
term.

`sys`

is an ARX model, which has the following form.

$$A(q)y(t)=B(q)u(t-nk)+v(t)$$

For more details on the ARX model structure, see `arx`

.

uses the time-domain input and output signals in the comma-separated matrices
`sys`

= iv4(`u`

,`y`

,`[na nb nk]`

)`u`

,`y`

. The software assumes that the data sample
time is one second. To change the sample time, set `Ts`

using
name-value syntax.

uses the time-domain or frequency-domain data in the data object
`sys`

= iv4(`data`

,`[na nb nk]`

)`data`

. Use this syntax especially when you want to estimate a model
using frequency-domain or frequency-response data, or when you want to take advantage of
the additional information, such as data sample time or experiment labeling, that data
objects provide.

### Specify Additional Options

specifies additional options using one or more name-value arguments.`sys`

= iv4(___,`Name,Value`

)

You can use this syntax with any of the previous input-argument combinations.

### Return Estimated Initial Conditions

`[`

returns the estimated initial conditions as an `sys`

,`ic`

] = iv4(___)`initialCondition`

object. For more information on `ic`

, see the `ic`

argument description.

Use this syntax if you plan to simulate or predict the model response using the same estimation input data and then compare the response with the same estimation output data. Incorporating the initial conditions yields a better match during the first part of the simulation.

## Examples

## Input Arguments

## Output Arguments

## Algorithms

Estimation is performed in 4 stages. The first stage uses the `arx`

function. The resulting model generates the instruments for a second-stage
IV estimate. The residuals obtained from this model are modeled as a high-order AR model. At
the fourth stage, the input-output data is filtered through this AR model and then subjected
to the IV function with the same instrument filters as in the second stage.

For the multiple-output case, optimal instruments are obtained only if the noise sources at the different outputs have the same color. The estimates obtained with the routine are reasonably accurate, however, even in other cases.

## References

[1] Ljung, Lennart. System Identification: Theory for the User, equations (15.21) through (15.26). 2nd ed. Prentice Hall Information and System Sciences Series. Upper Saddle River, NJ: Prentice Hall PTR, 1999.

## Version History

**Introduced before R2006a**