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)}}
  • Failed to parse (lexing error): Z=#\text{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.