Why does Z≠0 correspond to instability?

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)