# Why does Z≠0 correspond to instability?

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First recall some definitions:

• the "loop transfer function" is $L(s)=P(s)C(s)$
• the closed loop system is described by ${\frac {L(s)}{1+L(s)}}$
• Z = #RHP zeros of $1+L(s)$

So we see that if a point is a zero of $1+L(s)$, then it is a pole of the closed-loop system. Now, if that pole lies in the right half-plane, then the closed-loop system will be unstable. Thus if the function $1+L(s)$ has any RHP zeros, the closed loop system around the loop transfer function is unstable.

George Hines 17:12, 12 November 2007 (PST)

[[Category:CDS101/110