Help with partial fraction in control systems

[Huumah]

I'm having problem with task (a) in this problem

The question

My attempt

The solution

Why don't I get the same after I take the partial fraction using my calculator? And where does this R(s)=1/s^2 come from

 

[gneill]

1/s² is the Laplace transform of the driving signal R(s). The Laplace transform of a ramp r(t)=t is R(s) = 1/s².

Remember that a transfer function F(s) is the Laplace domain equivalent of the ratio of the output signal to the input. So if the input is R(s) then:

Y(s) = R(s)*F(s)

The transfer function that you've wrapped in "expand" doesn't appear to correspond to the one given in the problem statement.

 

[Huumah]

Thanks. I copied the problem from a pdf i found online to post here. But I guess they changed the numbers in the problem cause it's not the same numbers as in my book. So the solution manual doesn't show the correct answers to my book even though they are the same editon.

 

[SteamKing]

You can always check a PFD by adding the different fractions together to see if you get the original expression.

 

[rezeew]

As a scientist with expertise in control systems, I can offer some insights on the issue you are facing with partial fraction decomposition. Firstly, it is important to note that partial fraction decomposition is a mathematical technique used to simplify complex rational expressions by breaking them down into simpler fractions. It is commonly used in control systems to solve problems involving transfer functions.

Regarding your specific problem, it is essential to carefully follow the steps of partial fraction decomposition to ensure accurate results. I suggest reviewing the process and double-checking your calculations to identify any errors that may have led to a different result on your calculator.

As for the term R(s)=1/s^2, it is a residue term that appears in the partial fraction decomposition of a second-order transfer function. It represents the inverse Laplace transform of a pole at the origin, and it is used to determine the system's response to a unit step input. It is a crucial component in control systems analysis and plays a significant role in understanding system behavior.

In conclusion, I recommend revisiting the steps of partial fraction decomposition and seeking assistance from a tutor or colleague if needed. It is also helpful to understand the significance of the residue term in control systems analysis. I hope this information helps you in solving task (a) and gaining a better understanding of partial fraction decomposition in control systems.