If the closed-loop dynamics can be represented by the state space equation (see State space (controls))

${\displaystyle {\dot {\underline {x}}}=\mathbf {A} {\underline {x}}+\mathbf {B} {\underline {u}},}$ 

with output equation

${\displaystyle {\underline {y}}=\mathbf {C} {\underline {x}}+\mathbf {D} {\underline {u}},}$ 

then the poles of the system transfer function are the roots of the characteristic equation given by

${\displaystyle \left|s{\textbf {I}}-{\textbf {A}}\right|=0.}$ 

Full state feedback is utilized by commanding the input vector ${\displaystyle {\underline {u}}}$ . Consider an input proportional (in the matrix sense) to the state vector,

${\displaystyle {\underline {u}}=-\mathbf {K} {\underline {x}}}$ .

Substituting into the state space equations above, we have

${\displaystyle {\dot {\underline {x}}}=(\mathbf {A} -\mathbf {B} \mathbf {K} ){\underline {x}}}$ 

${\displaystyle {\underline {y}}=(\mathbf {C} -\mathbf {D} \mathbf {K} ){\underline {x}}.}$ 

The poles of the FSF system are given by the characteristic equation of the matrix ${\displaystyle \mathbf {A} -\mathbf {B} \mathbf {K} }$ , ${\displaystyle \det \left[s{\textbf {I}}-\left({\textbf {A}}-{\textbf {B}}{\textbf {K}}\right)\right]=0}$ . Comparing the terms of this equation with those of the desired characteristic equation yields the values of the feedback matrix ${\displaystyle {\textbf {K}}}$  which force the closed-loop eigenvalues to the pole locations specified by the desired characteristic equation.^[2]

Consider a system given by the following state space equations:

${\displaystyle {\dot {\underline {x}}}={\begin{bmatrix}0&1\\-2&-3\end{bmatrix}}{\underline {x}}+{\begin{bmatrix}0\\1\end{bmatrix}}{\underline {u}}.}$ 

The uncontrolled system has open-loop poles at ${\displaystyle s=-1}$  and ${\displaystyle s=-2}$ . These poles are the eigenvalues of the ${\displaystyle \mathbf {A} }$  matrix and they are the roots of ${\displaystyle \left|s\mathbf {I} -\mathbf {A} \right|}$ . Suppose, for considerations of the response, we wish the controlled system eigenvalues to be located at ${\displaystyle s=-1}$  and ${\displaystyle s=-5}$ , which are not the poles we currently have. The desired characteristic equation is then ${\displaystyle s^{2}+6s+5=0}$ , from ${\displaystyle (s+1)(s+5)}$ .

Following the procedure given above, the FSF controlled system characteristic equation is

${\displaystyle \left|s\mathbf {I} -\left(\mathbf {A} -\mathbf {B} \mathbf {K} \right)\right|=\det {\begin{bmatrix}s&-1\\2+k_{1}&s+3+k_{2}\end{bmatrix}}=s^{2}+(3+k_{2})s+(2+k_{1}),}$ 

where

${\displaystyle \mathbf {K} ={\begin{bmatrix}k_{1}&k_{2}\end{bmatrix}}.}$ 

Upon setting this characteristic equation equal to the desired characteristic equation, we find

${\displaystyle \mathbf {K} ={\begin{bmatrix}3&3\end{bmatrix}}}$ .

Therefore, setting ${\displaystyle {\underline {u}}=-\mathbf {K} {\underline {x}}}$  forces the closed-loop poles to the desired locations, affecting the response as desired.

This only works for Single-Input systems. Multiple input systems will have a ${\displaystyle {\textbf {K}}}$  matrix that is not unique. Choosing, therefore, the best ${\displaystyle {\textbf {K}}}$  values is not trivial. A linear-quadratic regulator might be used for such applications^{[citation needed]}.

• Pole splitting
• Step response
• Ackermann's Formula
• Linear-quadratic regulator

• Mathematica function to compute the state feedback gains