Question

Consider the step response, Y(s), for a first order system given by: G(s)=b/s+a​. Show/prove/derive the following: i. The step response is given by y(t)=b/a​(1−e⁻ᵃᵗ)u(t).

Answer 1

To prove the step response of a first order system G(s)=b/s+a, a unit step input is considered, and the output is found by multiplying the transfer function by 1/s and taking the inverse Laplace transform, resulting in y(t)=b/a(1-e^-at) u(t).

Step-by-step explanation:

The student has asked how to prove that the step response for a first order system with transfer function G(s)=b/s+a is given by y(t)=b/a(1-e-at) u(t). We will solve this by using Laplace transform methods.

Steps to Prove the Step Response

1. Consider a unit step input U(s) = 1/s. This is the Laplace transform of the step function u(t).
2. The output Y(s) of the system when subjected to this input is the product of the transfer function G(s) and the input U(s), which gives Y(s)=G(s)U(s)=b/s(s+a).
3. Perform partial fraction decomposition if necessary. In this case, it's not needed as the terms are already separated.
4. Take the inverse Laplace transform of Y(s) to find y(t). The inverse Laplace of b/s(s+a) is b/a(1-e-at) u(t).
5. Finally, recognize that u(t) is the unit-step function which completes the solution.

The key steps in this process are identifying the step input in Laplace form as 1/s, computing the output Y(s) from the product of G(s) and U(s), and then performing the inverse Laplace transform to determine the time-domain response.