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 nth-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.
Defining a parameterized state-space model in continuous time is often easier than in discrete time because physical laws are most often described in terms of differential equations. In continuous time, the state-space description has the following form:
The matrices F, G, H, and D contain elements with physical significance—for example, material constants. K contains the disturbance matrix. x0 specifies the initial states.
You can estimate a 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, which describes noise:
Here, T is the sample time, u(kT) is the input at the time instant kT, and y(kT) is the output at the 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|
Live Editor Tasks
|Estimate State-Space Model||Estimate state-space model using time or frequency data in the Live Editor|
Create State-Space Model
|State-space model with identifiable parameters|
|Estimate state-space model using time-domain or frequency-domain data|
|Estimate state-space model by reduction of regularized ARX model|
|Estimate state-space model using subspace method with time-domain or frequency-domain data|
|Prediction error minimization for refining linear and nonlinear models|
Model Initialization and Structure Parameters
Extract or Set Model Parameters
State-Space Model Basics
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 nth-order differential or difference equations.
Choose between noniterative subspace methods, iterative methods that use prediction error minimization algorithm, and noniterative methods.
Select a model order for a state-space model structure in the app and at the command line.
Modal, companion, observable and controllable canonical 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
Use the app to specify model configuration options and estimation options for model estimation.
Perform black-box or structured estimation.
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.
This example shows how to estimate ARMAX and OE-form models using the state-space estimation approach.
Free Parameterization is the default; the estimation routines adjust all the parameters of the state-space matrices.
Reduce the order of a Simulink® model by linearizing the model and estimating a lower order model that retains model dynamics.
Structured Estimation, Innovations Form
Structured parameterization lets you exclude specific parameters from estimation by setting these parameters to specific values.
An identified linear model is used to simulate and predict system outputs for given input and noise signals.
Set State-Space model Options
System Identification Toolbox™ software supports various parameterization combinations that determine which parameters are estimated and which parameters remain fixed to specific values.
When you estimate state-space models, you can specify how the algorithm treats initial states.