A control system transfer function is H(s) = 1/s³. Express its impulse response in terms of unit step signal

Solution

Convolution in the time domain implies multiplication in S(or frequency) domain

The Laplace transform of any signal h(t) is given by

\(\mathscr{L}\){h(t)} = H(s)

Here ' \(\mathscr{L}\) ' represents the Laplacian operator

H(s) is the frequency domain equivalent of h(t)

\(\mathscr{L}\){u(t)} = \(\frac{1}{s}\)

\(\mathscr{L}\){u(t) ⊗ u(t)} = \(\frac{1}{s}\)×\(\frac{1}{s}\) = \(\frac{1}{s^2}\)  ; here ⊗ denotes convolution

Similarly

\(\mathscr{L}\){u(t) ⊗ u(t) ⊗ u(t)} = \(\frac{1}{s}\) ×\(\frac{1}{s}\)× \(\frac{1}{s}\) = \(\frac{1}{s^3}\)​​​

\(\mathscr{L}\){u(t) ⊗ u(t) ⊗ u(t)⊗ u(t)} = \(\frac{1}{s^4}\)

So we observe that the H(s) = 1/s3, corresponds to a unit step signal convoluted with itself thrice

Therefore the correct answer is option 4