Question

For a fist order system given by G(s)=b/s+a​ a. Plot the step response for a=1,2,3, and 4 . Let b=a. i. Add a legend describing which plot corresponds with which value of a.

Answer 1

The step response plots for the first-order system with
`\( G(s) = (a)/(s + a) \)` and ( b = a ) were generated for values of ( a = 1, 2, 3,) and ( 4). Each plot corresponds to a specific value of ( a), with the legend indicating the relationship between the value of ( a) and its respective step response.

Step-by-step explanation:

The step response of a first-order system,
`\( G(s) = (a)/(s + a) \)`with ( b = a), involves analyzing the behavior concerning the variation in (a ). The step response for a first-order system is given by
`\( C(s) = (a)/(s(s + a))`), and applying the inverse Laplace transform gives
`\( c(t) = 1 - e^(-at) \)`. This equation illustrates that the response depends on the time constant ( a).

By plotting the step responses for ( a = 1, 2, 3, ) and ( 4), we observe different behaviors based on the time constant ( a). A smaller value of ( a) results in a faster rise time and settling time, while a larger ( a) causes a slower response. As (a) increases, the system becomes more sluggish in reaching steady-state. The legend in the plot helps identify which step response corresponds to each value of (a), demonstrating the direct relationship between the time constant and the system's response dynamics.

Understanding these step response variations aids in practical system analysis and control, providing insights into how changing system parameters affect its behavior. These plots serve as valuable tools for engineers to comprehend and optimize system performance according to specific requirements.