Effect of adding a zero to a system

Consider the expression for time response  for a second order closed loop transfer function.

Let a zero at s=-z be added to this transfer function.The multiplication term in the numerator of this expression has been adjusted so that steady state gain C/R(0) of the system is unity.

C(s)/R(s)=(s+z)(w_n²/z)/(s²+2zw_ns+w_n²)

This gives the steady state value of output Css=1 when input is unit step.Thus the system will track the step input with zero steady state error.

We have:

C(s)/R(s)=w_n²/(s²+2zw_ns+w_n²)+ (s/z)(w_n²)/(s²+2zw_ns+w_n²)

Let C_z(t) be the response of the system with a zero at s=-z.

Therefore

C_z(t)=c(t)+1/z*d(c(t))/dt

Where c(t) is the response .

The effect of added derivative term can be understood from the figure where a case for a typical value of ζ (less than 1) is considered.

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.

As z increases(the zero moves farther into the left half of s-plane),its effect becomes less pronounced ie,the effect of zero on transient response may become negligible.

Reference:

Control System engineering -Nagrath and Gopal