# Effect of RHP zeros on stability

Why do zeros in the right half of the $$\s\$$ plane reduce stability (i.e., gain / phase margin)?

• What research have you made that needs clarifying? – Peter Smith Feb 23 at 17:26
• A screenshot of some notes link – TVV Feb 23 at 17:47
• I don't understand why boosting the gain and lagging the phase is bad for stability. – TVV Feb 23 at 17:52

Instability issues concern us when a system is used in a negative-feedback connection. Remember a negative-feedback system may oscillate at frequency $$\\omega\$$ if the phase shift around the loop at this frequency is more negative than -180 that the feedback becomes positive (because negative feedback itself introduces 180 degrees of phase shift) while the loop gain is still greater than one.
By the same token, a left plane zero can counteract the destabilising effect of a right plane pole. For example, an open loop TF, $$\\frac{1}{s-1}\$$, can be stabilised by adding a zero at, say, $$\\small s=-2\$$, i.e OLTF = $$\\frac{s+2}{s-1}\$$ gives a CLTF = $$\\frac{s+2}{2s+1}\$$. So the unstable pole at $$\\small s=1\$$ is dragged into the left plane; becoming a stable pole at $$\\small s=-0.5\$$