Characteristic Equation

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Characteristic Equation
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Classical Control Pole Placement
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Contents

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.

3 References

3.1 Notes