## What Are State-Space Models?

### Definition of State-Space 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 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.

### Continuous-Time Representation

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:

`$\begin{array}{l}\stackrel{˙}{x}\left(t\right)=Fx\left(t\right)+Gu\left(t\right)+\stackrel{˜}{K}w\left(t\right)\\ y\left(t\right)=Hx\left(t\right)+Du\left(t\right)+w\left(t\right)\\ x\left(0\right)=x0\end{array}$`

The matrices F, G, H, and D contain elements with physical significance—for example, material constants. x0 specifies the initial states.

Note

$\stackrel{˜}{K}$ = 0 gives the state-space representation of an Output-Error model. For more information, see What Are Polynomial Models?.

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

### Discrete-Time Representation

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

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

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.

Note

K=0 gives the state-space representation of an Output-Error model. For more information about Output-Error models, see What Are Polynomial Models?.

Discrete-time state-space models provide the same type of linear difference relationship between the inputs and outputs as the linear ARMAX model, but are rearranged such that there is only one delay in the expressions.

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

The innovations form uses a single source of noise, e(kT), rather than independent process and measurement noise. If you have prior knowledge about the process and measurement noise, you can use linear grey-box estimation to identify a state-space model with structured independent noise sources. For more information, see Identifying State-Space Models with Separate Process and Measurement Noise Descriptions.

### Relationship Between Continuous-Time and Discrete-Time State Matrices

The relationships between the discrete state-space matrices A, B, C, D, and K and the continuous-time state-space matrices F, G, H, D, and $\stackrel{˜}{K}$ are given for piece-wise-constant input, as follows:

`$\begin{array}{l}A={e}^{FT}\\ B=\underset{0}{\overset{T}{\int }}{e}^{F\tau }Gd\tau \\ C=H\end{array}$`

These relationships assume that the input is piece-wise-constant over time intervals $kT\le t<\left(k+1\right)T$.

The exact relationship between K and $\stackrel{˜}{K}$ is complicated. However, for short sample time T, the following approximation works well:

`$K=\underset{0}{\overset{T}{\int }}{e}^{F\tau }\stackrel{˜}{K}d\tau$`

### State-Space Representation of Transfer Functions

For linear models, the general model description is given by:

`$y=Gu+He$`

G is a transfer function that takes the input u to the output y. H is a transfer function that describes the properties of the additive output noise model.

The relationships between the transfer functions and the discrete-time state-space matrices are given by the following equations:

`$\begin{array}{l}G\left(q\right)=C{\left(}^{q}B+D\\ H\left(q\right)=C{\left(}^{q}K+{I}_{ny}\end{array}$`

Here, Inx is the nx-by-nx identity matrix, and nx is the number of states. Iny is the ny-by-ny identity matrix, and ny is the dimension of y and e.

The state-space representation in the continuous-time case is similar.