# State space (controls)

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A state space representation shows how the states and outputs of a system change in response to the system inputs. State space models may be described in either continuous or discrete form.

## Continuous formEdit

The continuous form of a state space model shows how the rate vector $\mathbf{\dot x}$ and output vector $\mathbf{y}$ evolve over time $(t)$, dependent upon the system states $\mathbf{x}$ and inputs $\mathbf{u}$

$\mathbf{\dot x}(t) = A \mathbf{x}(t)+ B \mathbf{u}(t)$

$\mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t)$

## Discrete formEdit

The discrete form represents a system that is sampled at discrete intervals. The vector $\mathbf{x}_k$ represents the states at the time of the $k$th sample. Vector $\mathbf{x}_{k+1}$ represents the states at the next sample time.

$\mathbf{x}_{k+1} = A \mathbf{x}_k+ B \mathbf{u}_k$

$\mathbf{y}_k = C \mathbf{x}_k + D \mathbf{u}_k$

## The $A, B, C$ and $D$ matricesEdit

The matrices $A, B, C$ and $D$ matrices are called the state matrix, input matrix, output matrix and transmission matrix respectively.

It should be noted that the matrices have different values depending on whether the model is in continuous or discrete form. Conversion to discrete form from continuous is achieved by discretization.