## Estimate State-Space Time Series Models

### Definition of State-Space Time Series Model

The discrete-time state-space model for a time series is given by the following equations:

`$\begin{array}{l}x\left(kT+T\right)=Ax\left(kT\right)+Ke\left(kT\right)\\ y\left(kT\right)=Cx\left(kT\right)+e\left(kT\right)\end{array}$`

where T is the sample time and y(kT) is the output at time instant kT.

The time series structure corresponds to the general structure with empty B and D matrices.

For information about general discrete-time and continuous-time structures for state-space models, see What Are State-Space Models?.

### Estimate State-Space Models at the Command Line

You can estimate single-output and multiple-output state-space models at the command line for time-domain data (`iddata` object).

The following table provides a brief description of each command. The resulting models are `idss` model objects. You can estimate either continuous-time, or discrete-time models using these commands.

Commands for Estimating State-Space Time Series Models

CommandDescription
`n4sid`

Noniterative subspace method for estimating linear state-space models.

`ssest`

Estimates linear time series models using an iterative estimation method that minimizes the prediction error.