ee301 assignment 3

What do the poles and zeros contribute in the control system?
The open loop transfer function of a general linear feedback system can be written as
G(s)H(s)=k(s+x 1)(s+x2)…………/(s+p1)(s+p2)……..→→→→→→→→→→→→(1)
Where - x 1 ,-x 2, -x 3…………..are the zeros and -p1 ,-p2 ,-p3………..are the poles of open-loop transfer function.
The characterstic equation in terms of open loop transfer function is formed as
1+G(s)H(s)= 1 + k(s+x 1)(s+x2)…………/(s+p1)(s+p2)……..
= (s+p1)(s+p2 )…k(s+x1)(s+x2)…/(s+p1)(s+p2)….→→→→→→→(2)
Equation (2) can also be written as
1+G(s)H(s)= k(s+z 1)(s+z2)………/(s+p1)(s+p2)……..→→→→→→→→→(3)
The roots of charactersatic equation are obtained from the expression
1+G(s)H(s)=0
Hence ,for this condition equation (3)yields that
(s+z 1)(s+z2)……=0
These roots are (-z1 ,-z2 ,-z3 .........................) called as zeros of the characterstic equation .It canbe observed that the poles of open loop transfer function are also the poles of characterstic equation (2)
Criterion for stability:The linear feedback control system represented by characterstic equation (1+G(s)H(s))will be stable only if zeros - z1 ,-z2 ,-z3……………….lie in left half of s plane ie roots of the characterstic equation must contain negative real part.This indicates that no zero of charactrstic equation should lie in right half of the s plane for the system to be stable


Reference:
(1)Control Systems Engineering-I.J Nagarath,M.Gopal
(2)Control Systems Engineering-S.N Sivanandam