Effect of adding poles and zeros

Adding a pole or a zero changes the shape of the root locus and the time response. Poles slow the system down; zeros speed it up.

Addition of a pole

• Open-loop pole (to $G(s)H(s)$): the root locus bends to the right, towards the imaginary axis. The system becomes less stable and the range of $K$ for stability shrinks.
• Closed-loop response becomes slower: rise time and settling time increase.
• Peak overshoot usually increases (lower effective damping).
• A pole at the origin raises the system type, which reduces steady state error.
• Closed-loop pole: adds a slow mode; overshoot falls but rise time rises (more sluggish response).

Addition of a zero

• Open-loop zero: pulls the root locus to the left, away from the imaginary axis. Relative stability improves (this is the idea behind PD and lead compensation).
• Closed-loop response becomes faster: rise time and peak time decrease.
• Closed-loop zero: adds a derivative term $\frac{1}{a}\frac{dc}{dt}$ to the response, so overshoot increases, especially when the zero is close to the origin.
• Bandwidth increases.

 jw         pole added            zero added
  |  locus bends right      locus bends left
  |      \                         /
--+-------x-----             -----o-------

A pole or zero far from the dominant poles (more than about 5 times further left) has little effect.