question 3

A system is characterized by its poles and zeros in the sense that they allow reconstruction of the (Input /output) differential equation. The terms zeros and poles are chosen, because the transfer function is zero at and infinite at . In general, the poles and zeros of a transfer function may be complex and the system dynamics may be represented graphically by plotting their locations on the complex s-plane, whose axes represent the real and imaginary parts of the complex variable s. Such plots are known as pole-zero plots. It is usual to mark a zero location by a circle and a pole location, a cross. The location of the poles and zeros provide qualitative insights into the response characteristics of a system. A linear time-invariant systemwithout dead time is described completely by the distribution of its poles and zeros and the gain factor . Many computer programs are available to determine the poles and zeros of a system from either the transfer function or the system state equations.

Example of the pole and zero distribution of a rational transfer function in the complex plane

Real axis zeros tend to spread the loci faster and stabilize the system. Real-axis poles, on the other hand, tend to make the loci spread more slowly and curve toward instability. The root locus design method involves three steps: the closed loop function is determined, and then the open loop transfer function, and finally a compensation network are synthesized.

For stable systems, pole should be in right hand of S plane (negative real value) and if it is imaginary axis (non real), it indicates system is oscillatory.
If it is real and imaginary (a complex), then system response is damped oscillation. If poles are left in S plane then system is highly unstable.