# Remarks on Pole Assignment by Constant Output Feedback

• Edward Jezierski
Conference paper

## Abstract

Consider the linear dynamical system described by the equations
$${\rm{\dot x}}\left( {\rm{t}} \right){\rm{ = Ax}}\left( {\rm{t}} \right){\rm{ + Bu}}\left( {\rm{t}} \right)$$
(1)
$${\rm{y}}\left( {\rm{t}} \right) = {\rm{Cx}}\left( {\rm{t}} \right)$$
(2)
where $${\rm{x}}\left( {\rm{t}} \right) \in {{\rm{R}}^{\rm{n}}},{\rm{u}}\left( {\rm{t}} \right) \in {{\rm{R}}^{\rm{m}}},\,{\rm{y}}\left( {\rm{t}} \right) \in {{\rm{R}}^{\rm{r}}}$$ and A,B,C are constant matrices of appropriate dimensions. The Wonham’s result states that if the system is controllable then it is also pole-assignable using a constant state feedback. Unfortunately, in practical situations usually the state vector is not accessible but only the output vector. In that case the problem of pole assignment consists in finding the constant output feedback
$${\rm{u}}\left( {\rm{t}} \right){\rm{ = Ky}}\left( {{\rm{\dot t}}} \right)$$
(3)
to assign the prespecified location of the closed-loop system poles
$$\sigma \left( {{\rm{A + BKC}}} \right) = \left( {{{\hat \lambda }_1},{{\hat \lambda }_2}, \ldots ,{{\hat \lambda }_{\rm{n}}}} \right)$$
(4)
which are real valued or complex valued in conjugate pairs.

## Keywords

Output Feedback Linear Dynamical System Pole Placement Root Locus Pole Assignment
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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