Editing State-space representation (section)

=== Feedback ===
[[Image:Typical State Space model with feedback.svg|framed|Typical state-space model with feedback]]

A common method for feedback is to multiply the output by a matrix ''K'' and setting this as the input to the system: <math>\mathbf{u}(t) = K \mathbf{y}(t)</math>.
Since the values of ''K'' are unrestricted the values can easily be negated for [[negative feedback]].
The presence of a negative sign (the common notation) is merely a notational one and its absence has no impact on the end results.

<math display="block">\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)</math>
<math display="block">\mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t)</math>

becomes

<math display="block">\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B K \mathbf{y}(t)</math>
<math display="block">\mathbf{y}(t) = C \mathbf{x}(t) + D K \mathbf{y}(t)</math>

solving the output equation for <math>\mathbf{y}(t)</math> and substituting in the state equation results in

<math display="block">\dot{\mathbf{x}}(t) = \left(A + B K \left(I - D K\right)^{-1} C \right) \mathbf{x}(t)</math>
<math display="block">\mathbf{y}(t) = \left(I - D K\right)^{-1} C \mathbf{x}(t)</math>

The advantage of this is that the [[eigenvalues]] of ''A'' can be controlled by setting ''K'' appropriately through [[Eigendecomposition of a matrix|eigendecomposition]] of <math>\left(A + B K \left(I - D K\right)^{-1} C \right)</math>.
This assumes that the closed-loop system is [[controllability|controllable]] or that the unstable eigenvalues of ''A'' can be made stable through appropriate choice of ''K''.

==== Example ====
For a strictly proper system ''D'' equals zero. Another fairly common situation is when all states are outputs, i.e. ''y'' = ''x'', which yields ''C'' = ''I'', the [[identity matrix]]. This would then result in the simpler equations

<math display="block">\dot{\mathbf{x}}(t) = \left(A + B K \right) \mathbf{x}(t)</math>
<math display="block">\mathbf{y}(t) = \mathbf{x}(t)</math>

This reduces the necessary eigendecomposition to just <math>A + B K</math>.