Question

A linear system at rest is subject to an input signal r(t)=1-e^-t. The response of the system for t>0 is given by c(t)=1-e^-2t. The transfer function of the system is:

(a) (s+2)/(s+1)

(b) (s+1)/(s+2)

(c) 2(s+1)/(s+2)

(d) (s+1)/2(s+2)

The question was asked during an interview for a job.

The doubt is from Transfer Functions in chapter Mathematical Models of Physical Systems of Control Systems

Answer 1

To find the transfer function $H(s)$ of the system, we analyze the given input and output signals in the Laplace domain.

1. Input r(t)=1−e−tr(t) = 1 - e^{-t}r(t)=1−e−t has the Laplace transform:

   R(s)=1s−1s+1=1s(s+1)R(s) = \frac{1}{s} - \frac{1}{s+1} = \frac{1}{s(s+1)}R(s)=s1​−s+11​=s(s+1)1​

2. Output c(t)=1−e−2tc(t) = 1 - e^{-2t}c(t)=1−e−2t has the Laplace transform:

   C(s)=1s−1s+2=2s(s+2)C(s) = \frac{1}{s} - \frac{1}{s+2} = \frac{2}{s(s+2)}C(s)=s1​−s+21​=s(s+2)2​

The transfer function $H(s)$ is given by H(s)=C(s)R(s)H(s) = \frac{C(s)}{R(s)}H(s)=R(s)C(s)​:

H(s)=2s(s+2)1s(s+1)=2(s+1)s+2H(s) = \frac{\frac{2}{s(s+2)}}{\frac{1}{s(s+1)}} = \frac{2(s+1)}{s+2}H(s)=s(s+1)1​s(s+2)2​​=s+22(s+1)​

So, the correct answer is:

(c) 2(s+1)s+2\frac{2(s+1)}{s+2}s+22(s+1)​