TECH SAVVVYY!!!: EFFECT OF ADDING ZERO

Answer 4

EFFECTS OF ADDING A ZERO ON THE ROOT LOCUS FOR A SECOND-ORDER SYSTEM

We can put the zero at three different positions with respect to the poles:

1. To the right of s = –p1

2. Between s = –p2 and s = –p1

3. To the left of s = –p2

We now discuss the effect of changing the gain K on the position of closed-loop poles

and type of responses.

(a) The zero s = –z1 is not present.

For different values of K, the system can have two real poles or a pair of complex

conjugate poles. This means that we can choose K for the system to be overdamped,

critically damped or underdamped.

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

In this case, the system can have only real poles and hence we can only find a value

for K to make the system overdamped. Thus the pole–zero configuration is even more

restricted than in case (a). Therefore this may not be a good location for our zero,

since the time response will become slower.

This case provides a root locus on the real axis. The responses are therefore limited to

overdamped responses. It is a slightly better location than (b), since faster responses

are possible due to the dominant pole (pole nearest to jaxis) lying further from the j

axis than the dominant pole in (b).

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

This is the most interesting case. Note that by placing the zero to the left of both

poles, the vertical branches of case (a) are bent backward and one end approaches the

zero and the other moves to infinity on the real axis. With this configuration, we can

now change the damping ratio and the natural frequency (to some extent). The

closed-loop pole locations can lie further to the left than s = –p2, which will provide

faster time responses. This structure therefore gives a more flexible configuration for

control design.

We can see that the resulting closed-loop pole positions are considerably influenced by

the position of this zero. Since there is a relationship between the position of closed-loop

poles and the system time domain performance, we can therefore modify the behaviour of

closed-loop system by introducing appropriate zeros in the controller.

Reference:

Web.mit.edu

TECH SAVVVYY!!!