Calculating the transfer function from Laplace Transforms

In summary, the control system for a spacecraft platform is described by the equations: d²p/dt² + 2 dp/dt + 4p = θ, v1 = r - p, dθ/dt = .6*v2, and v2 = 7*v1. The variables v1, v2, r, p, θ are all functions in the time domain. The system transfer function P(s)/R(s) is equal to 4.2/(s³ + 2s² + 8.2s).
  • #1
Lolsauce
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0

Homework Statement


A Control system for a spacecraft platform is governed by the following equations:

d²p/dt² + 2 dp/dt + 4p = θ
v1 = r - p
dθ/dt = .6*v2
v2 = 7*v1

All the variavles v1, v2, r, p, θ are functions in the time domain

Determine the system transfer function P(s)/R(s)

Homework Equations



d²x/dt² -> Laplace -> s² * X(s)
dx/dt -> Laplace -> s * X(s)
x(t) -> Laplace -> X(s)

The Attempt at a Solution



So first I took the Laplace of all the given equations:

d²p/dt² + 2 dp/dt + 4p = θ -> s²*P(s) +2s*P(s) + 4*P(s) = θ(s) )
dθ/dt = .6*v2 -> θ(s) = .6*V2(s)

v1 = r - p -> V1(s) = R(s) - P(s)
v2 = 7*v1 -> V2(s) = 7*V1(s)

I now have four equations I can substitute:
1. s²*P(s) +2s*P(s) + 4*P(s) = θ(s)
2. θ(s) = .6*V2(s)
3. V1(s) = R(s) - P(s)
4. V2(s) = 7*V1(s)

I substitute Eq 3 into Eq 4 and I obtain V2(s) = 7*R(s) - 7*P(s)
then I substitute my new V2(s) equation into Eq 2, which gives me

θ(s) = 4.2*R(s) - 4.2*P(s)

Which I then substitute into Eq 1.

I get a Transfer Function

P(s)/R(s) = 4.2/(s³ + 2s² + 8.2s)

I think I did all my calculations right, but the solution in the book is this:
ItMBm.jpg


Can somebody tell me what I did wrong?
 
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  • #2
Lolsauce said:
dθ/dt = .6*v2 -> θ(s) = .6*V2(s)

You'll want to double check this part :wink:
 

1. What are Laplace Transforms and why are they used in calculating transfer functions?

Laplace Transforms are mathematical operations that convert a function of time into a function of complex frequency. They are used in calculating transfer functions because they simplify the analysis of linear systems by transforming differential equations into algebraic equations, making it easier to solve for the transfer function.

2. How do I calculate the transfer function from a Laplace Transform?

To calculate the transfer function from a Laplace Transform, you need to first take the Laplace Transform of the input and output signals. Then, divide the output transform by the input transform to get the transfer function. This can be done using algebraic manipulation and the properties of Laplace Transforms.

3. What is the significance of the poles and zeros in the transfer function?

Poles and zeros in the transfer function represent the frequencies at which the system responds strongly or weakly, respectively. Poles indicate the natural frequencies of the system, while zeros represent frequencies where the system does not respond at all. The location of these poles and zeros can provide insights into the stability and behavior of the system.

4. Can I use Laplace Transforms to calculate transfer functions for non-linear systems?

No, Laplace Transforms can only be used for linear systems. Non-linear systems require different mathematical techniques, such as state-space representation, to calculate their transfer functions.

5. Are there any limitations or drawbacks to using Laplace Transforms in calculating transfer functions?

While Laplace Transforms are a powerful tool for analyzing linear systems, they do have some limitations. For example, they can only be used for systems with constant coefficients and cannot handle discontinuous inputs. Additionally, the calculations can become complex and time-consuming for higher-order systems.

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