Effect of adding a zero

Adding a zero to a system, either in the open-loop $G(s)H(s)$ or in the closed-loop transfer function, generally makes the response faster and changes the stability.

Zero added to the open-loop transfer function

• The root locus is pulled to the left, away from the imaginary axis.
• Relative stability improves. A system that was unstable for high $K$ can become stable for all $K$. Example: $K/[s^2(s+a)]$ is always unstable, but $K(s+b)/[s^2(s+a)]$ with $b < a$ is stable for all $K > 0$.
• Damping increases, so overshoot reduces. This is the basis of PD control and lead compensation.

Zero added to the closed-loop transfer function

If $T(s)$ becomes $T(s)(1 + s/z)$, the new output is

$$c_{new}(t) = c(t) + \frac{1}{z}\frac{dc(t)}{dt}$$

• The derivative term adds to the response while it is rising, so rise time and peak time decrease.
• Peak overshoot increases. The closer the zero is to the origin (small $z$), the larger the overshoot.
• Bandwidth increases, so the system lets more high-frequency noise through.
• A zero far to the left (about 5 times further than the dominant poles) has little effect.

 c(t)       with zero (faster, more overshoot)
  |       _/\_
  |     /  _--\___________
  |    / /                original
  |   //
  +--------------------------- t