Consider the second-order system given by

G(s) =1/[(s+p1)(s+p2)] p1 > 0, p2 > 0

The poles are given by s = –p1 and s = –p2 and the simple root locus plot for this system is shown in the figure(a).

When we add a zero at s = –z1 to the controller, the open-loop transfer function will change to:

G1(s) =K( s+ z1)/[(s+ p1)( s+ p2)] , z1 > 0

We can put the zero at different positions and see the effects.

(a) The zero s = –z1 is not present.

Here, we can choose K for the system to be over damped, critically damped or under damped.

(b) The zero s = –z1 is located to the right of both poles, s = – p2 and s = –p1.

Here we can only find a value for K to make the system over damped.

(c) The zero s = –z1 is located between s = –p2 and s = –p1.

The responses here are limited to over damped responses but, with faster responses.

(d) The zero s = –z1 is located to the left of s = –p2.

Here, we can change the damping ratio and the natural frequency. This structure gives a more flexible configuration for control design.

Hence we conclude that there is a relationship between the positions of closed-loop poles and the system time domain performance. **We can therefore modify the behavior of closed-loop system by introducing appropriate zeros in the controller.**

**The effect of the zero is to contribute a pronounced early peak to the system’s response whereby the peak overshoot may increase appreciably.** The smaller the value of z,the closer the zero to origin, the more pronounced is the peaking phenomenon. Thus, the zeros on the real axis near the origin are generally avoided in design. However in a sluggish system the introduction of a zero at proper position can improve the transient response.

Reference:

Control Systems Engineering, Nagrath & Gopal