Question

Discuss how the locations of the zeros of a system impact its properties and characteristics.

Answer 1

1)Addition of zeros to the transfer function has the effect of pulling the root locus to the left, making the system more stable.

2)Zeros with positive real part give a negative phase contribution, reducing the phase margin (which is bad) thus limits the performance of the system.

3)Time delay in the system can also be approximated as a zero with positive real part.

4)Blocking property of zeros, If you have a transfer function with a zero in the right hand plane, and an input tuned to that zero, then the output is at 0 for any time t.

5)Zero in the right-hand side of the s plane can cause undershoot in the time response of the system, and this can be very very dangerous in some cases.

Zeros are very import for the system behavior. They influence the stability and the transient behavior of the system. The referenced document is a good start.

When dealing with transfer functions it is important to understand that we are usually interested in the stability of a closed loop feedback system. In order for the closed loop system to be stable, the poles have to be located in the left half plane. The zeros have no importance, since the stability of a linear system is solely determined by the position of the poles.

When designing a closed loop system (i.e. a circuit), this is usually done by analyzing the open loop system. Because for the open loop system it is easier to understand how the circuit parameters are going to influence the system behavior.

It can be shown that the position of zeros of the open loop system are important for the stability of the closed loop system. When closing the loop slowly by increasing the feedback while monitoring the poles, it can be seen that the poles are attracted by the zeros. The poles move towards the zeros and if there are zeros in the right half plane, the tendency for the system to become unstable is higher because finally the pole will assume the position of the zero. Such a system would be called a non-minimum phase system, and they are quite common.