Characteristic Equation Classical Control Pole Placement In order to prevent spam, users must register before they can edit or create articles.

1 Introduction to the Characteristic Equation

The characteristic equation is a means of determining the closed loop poles of the system. For a system to be stable all the solutions to the characteristic equation must satisfy

$LaTeX: \mbox{Re} \left \{ s \right \} \le 0$

Condition for Stability

The characteristic equation is essential for pole placement controller design.

2 Characteristic Equation

The characteristic equation is defined by

$LaTeX: C\left(s\right)=\frac{C_{num}\left(s\right)}{C_{den}\left(s\right)}$

Controller

$LaTeX: G\left(s\right)=\frac{G_{num}\left(s\right)}{G_{den}\left(s\right)}$

Plant

$LaTeX: \Phi_{CL}\left(s\right)=C_{den}\left(s\right)G_{den}\left(s\right) + C_{num}\left(s\right)G_{num}\left(s\right)=0$

Characteristic Equation

To find the closed loop poles of the system solve for $LaTeX: s$ where $LaTeX: \Phi_{CL}\left(s\right)=0$.