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.
Effects of adding a zero on the root locus for a second-order system
Consider the second-order system given by
G(s)=
1
12 ()( ) spsp ++
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 Figure 13.13(a). When we add a zero at s =–z1 to the controller, the open-loop
transfer function will change to:
G1(s)=
Ks z
spsp
()
()( )
+
++
1
12
, z1 >0
13.6 Effects of adding a pole or a zero to the root locus of a second- order system 365
Imag. axis
Imag. axis Imag. axis
–6 –4 –2 0 2
–2
–1
0
1
2
Real axis
–6 –4 –2 02
–2
–1
0
1
2
Real axis
–8 –6 –4 –2 0 2
–2
–1
0
1
2
Real axis
(b)
(c) (d)
–p1
–p1 –p1
–p2
–p2 –p2
–6 –4 –2 02
–2
–1
0
1
2
Real axis
Imag. axis
(a)
–p1 –p2
Figure 13.13 Effect of adding a zero to a second-order system root locus.We can put the zero at three different positions with respect to the poles:
1. To the right of s =–p1 Figure 13.13(b)
2. Between s =–p2 and s =–p1 Figure 13.13(c)
3. To the left of s =–p2 Figure 13.13(d)
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 tomake the systemoverdamped. Thus the pole–zero configuration is evenmore
restricted than in case (a). Therefore this may not be a good location for our zero,
since the time response will become slower.
(c) The zero s =–z1 is located between s =–p2 and s =–p1.
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 jωaxis) lying further fromthe jω
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 amore 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 systemtime domain performance,we can thereforemodify the behaviour of
closed-loop system by introducing appropriate zeros in the controller.
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 given in the figure below. When we add a zero at s= -z1 to the controller, the open-loop transfer function will change to :
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