Full state feedback (FSF), or pole placement, is a method employed in feedback control system theory to place the closed-loop poles of a plant in predetermined locations in the s-plane. Placing poles is desirable because the location of the poles corresponds directly to the eigenvalues of the system, which control the characteristics of the response of the system. The system must be considered controllable in order to implement this method.
Principle
If the closed-loop dynamics can be represented by the state space equation (see State space (controls))
x _ ˙ = A x _ + B u _ , {\displaystyle {\dot {\underline {x}}}=\mathbf {A} {\underline {x}}+\mathbf {B} {\underline {u}},}
with output equation
y _ = C x _ + D u _ , {\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
| s I − A | = 0. {\displaystyle \left|s{\textbf {I}}-{\textbf {A}}\right|=0.}
Full state feedback is utilized by commanding the input vector u _ {\displaystyle {\underline {u}}} . Consider an input proportional (in the matrix sense) to the state vector,
u _ = − K x _ {\displaystyle {\underline {u}}=-\mathbf {K} {\underline {x}}} . Substituting into the state space equations above, we have
x _ ˙ = ( A − B K ) x _ {\displaystyle {\dot {\underline {x}}}=(\mathbf {A} -\mathbf {B} \mathbf {K} ){\underline {x}}}
y _ = ( C − D K ) 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 A − B K {\displaystyle \mathbf {A} -\mathbf {B} \mathbf {K} } , det [ s I − ( A − B K ) ] = 0 {\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 K {\displaystyle {\textbf {K}}} which force the closed-loop eigenvalues to the pole locations specified by the desired characteristic equation.
Example of FSF Consider a system given by the following state space equations:
x _ ˙ = [ 0 1 − 2 − 3 ] x _ + [ 0 1 ] u _ . {\displaystyle {\dot {\underline {x}}}={\begin{bmatrix}0&1\\-2&-3\end{bmatrix}}{\underline {x}}+{\begin{bmatrix}0\\1\end{bmatrix}}{\underline {u}}.}
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