Poles are values of Laplace transform variable, s, that make denominator of transfer function zero. Zeros are values of Laplace transform variable, s, that make numerator of transfer function zero. The gain, K, is constant that multiplies entire numerator and denominator. MATLAB function, tf2zp, obtains transfer function zeros, poles, and gain given its numerator and denominator. For example, [z, p,K] = tf2zp(num, den) given a numerator, 4s2 + 165 + 12 and denominator s4 + 1253 +44s2 + 485, determine the zeros, poles, and gain of the transfer function using the tf2zp function. Poles and zeros should be ordered from largest to smallest to match the order given below. Larger number, meaning -2 > -4, NOT largest magnitude. zero(1) zero(2) pole(1) pole(2) pole(3) pole(4) gain

Zero 1 = -1

Zero 2 = -3

Pole 1 = 0

Pole 2 = -2

Pole 3 = -4

Pole 4 = -6

Gain = 4

Explanation:

For any given transfer function, the general form is given as

T.F = k [N(s)] ÷ [D(s)]

where k = gain of the transfer function

N(s) is the numerator polynomial of the transfer function whose roots are the zeros of the transfer function.

D(s) is the denominator polynomial of the transfer function whose roots are the poles of the transfer function.

k [N(s)] = 4s² + 16s + 12 = 4[s² + 4s + 3]

it is evident that

Gain = k = 4

N(s) = (s² + 4s + 3) = (s² + s + 3s + 3)

= s(s + 1) + 3 (s + 1) = (s + 1)(s + 3)

The zeros are -1 and -3

D(s) = s⁴ + 12s³ + 44s² + 48s

= s(s³ + 12s² + 44s + 48)

= s(s + 2)(s + 4)(s + 6)

The roots are then, 0, -2, -4 and -6.

Hope this Helps!!!

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