# Why does Z≠0 correspond to instability?

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Jump to navigationJump to searchFirst 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)