ee301 answer3

What do the poles and zeros contribute to in the control system?
The transfer function provides a basis for determining important system response characteristics without solving the complete differential equation. As defined, the transfer function is a rational function in the complex variable s = σ + jω.

It is often convenient to factor the polynomials in the numerator and denominator, and to write
the transfer function in terms of those factors:

H(s) = N(s)/D(s)
= K(s − z1)(s − z2) . . . (s − zm−1)(s − zm)/((s − p1)(s − p2) . . . (s − pn−1)(s − pn) )
where the numerator and denominator polynomials, N(s) and D(s), have real coefficients defined
by the system’s differential equation . Here,
the zi’s are the roots of the equation
N(s) = 0
and are defined to be the system zeros, and the pi’s are the roots of the equation
D(s) = 0
and are defined to be the system poles. The factors in the numerator and denominator
are written so that when s = zi the numerator N(s) = 0 and the transfer function vanishes, that is

lim H(s) = 0.
s→zi

and similarly when s = pi the denominator polynomial D(s) = 0 and the value of the transfer
function becomes unbounded,

lim H(s) = ∞.
s→pi

All of the coefficients of polynomials N(s) and D(s) are real, therefore the poles and zeros must
be either purely real, or appear in complex conjugate pairs. In general for the poles,
either pi = σi, or else pi, pi+1 = σi±jωi. The existence of a single complex pole without a corresponding conjugate pole would generate complex coefficients in the polynomial D(s). Similarly, the system zeros are either real or appear in complex conjugate pairs.

Control systems, in the most simple sense, can be designed simply by assigning specific values to the poles and zeros of the system.Physically realizable control systems must have a number of poles greater than or equal to the number of zeros.The locations of the poles, and the values of the real and imaginary parts of the pole determine the response of the system.

Reference:
Control Systems Engineering, Nagrath & Gopal