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State-space models with free, canonical, and structured
parameterizations; equivalent ARMAX and OE models

State-space models are models that use state variables to describe a system by a set of
first-order differential or difference equations, rather than by one or more
*n*th-order differential or difference equations. State variables
*x*(*t*) can be reconstructed from the measured
input-output data, but are not themselves measured during an experiment.

The state-space model structure is a good choice for quick estimation because it
requires you to specify only one input, the *model order*,
*n*. The model order is an integer equal to the dimension of
*x*(*t*) and relates to, but is not necessarily equal to,
the number of delayed inputs and outputs used in the corresponding linear difference
equation.

It is often easier to define a parameterized state-space model in continuous time because physical laws are most often described in terms of differential equations. In continuous-time, the state-space description has the following form:

$$\begin{array}{l}\dot{x}(t)=Fx(t)+Gu(t)+\tilde{K}w(t)\\ y(t)=Hx(t)+Du(t)+w(t)\\ x(0)=x0\end{array}$$

The matrices * F*,

You can estimate continuous-time state-space model using both time-domain and frequency-domain data.

The discrete-time state-space model structure is often written in the
*innovations form* that describes noise:

$$\begin{array}{l}x(kT+T)=Ax(kT)+Bu(kT)+Ke(kT)\\ y(kT)=Cx(kT)+Du(kT)+e(kT)\\ x(0)=x0\end{array}$$

where *T* is the sample time,
*u*(*kT*) is the input at time instant
*kT*, and *y*(*kT*) is the output at
time instant *kT*.

You cannot estimate a discrete-time state-space model using continuous-time frequency-domain data.

For more information, see What Are State-Space Models?

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

Estimate State-Space Model | Estimate state-space model using time or frequency data in the Live Editor |

*State-space models* are models that use state variables to describe a
system by a set of first-order differential or difference equations, rather than by one or more
*n*th-order differential or difference equations.

**State-Space Model Estimation Methods**

Choose between noniterative subspace methods, iterative method that uses prediction error minimization algorithm, and noniterative method.

**Estimate State-Space Model With Order Selection**

To estimate a state-space model, you must provide a value of its order, which represents the number of states.

**Canonical State-Space Realizations**

Modal, companion, observable and controllable canonical state-space models.

**Data Supported by State-Space Models**

You can use time-domain and frequency-domain data that is real or complex and has single or multiple outputs.

**Estimate State-Space Models in System Identification App**

Import data into the System Identification app.

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

Perform black-box or structured estimation.

**Estimate State-Space Models with Canonical Parameterization**

*Canonical parameterization* represents a state-space system in a
reduced parameter form where many elements of *A*, *B* and
*C* matrices are fixed to zeros and ones.

**Estimate State-Space Equivalent of ARMAX and OE Models**

This example shows how to estimate ARMAX and OE-form models using the state-space estimation approach.

**Estimate State-Space Models with Free-Parameterization**

The default parameterization of the state-space matrices *A*, *B*, *C*, *D*,
and *K* is free; that is, any elements in the matrices are adjustable by the estimation routines.

**Use State-Space Estimation to Reduce Model Order**

Reduce the order of a Simulink^{®} model by linearizing the model and estimating a lower-order model that retains model dynamics.

**Estimate State-Space Models with Structured Parameterization**

*Structured parameterization* lets you exclude specific parameters from estimation by setting these parameters to specific values.

**Identifying State-Space Models with Separate Process and Measurement Noise Descriptions**

An identified linear model is used to simulate and predict system outputs for given input and noise signals.

**Supported State-Space Parameterizations**

System Identification Toolbox™ software supports the following parameterizations that indicate which parameters are estimated and which remain fixed at specific values:

**Specifying Initial States for Iterative Estimation Algorithms**

When you estimate state-space models, you can specify how the algorithm treats initial states.