Help with a partially decoupled system.

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In summary, to solve the partially decoupled system for t > 0, we can use the equations x=ke^(1/t) and y'=y-3(ke^(1/t)^2) to find the values of x and y. However, we may encounter difficulties when t=1, as the value of x becomes undefined. Further assistance may be needed to solve the problem.
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manonfire315
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Homework Statement


Solve the following partially decoupled system defined for t > 0
dx/dt=x/t
dy/dt=y-3x^2
x(1)=1, y(1)=1


Homework Equations





The Attempt at a Solution


I have started the solution the way that a regular partial or fully decoupled system is started, but I have run into problems with t. I usually use x=ke^t, but in this case what would x equal when x prime is x/t.

I have...

x=ke^(1/t)
y prime=y-3(ke^(1/t)^2)
 
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y=3/2(ke^(1/t)^2)+Ce^tFrom here I am not sure how to proceed. I know that when t=1, x=1 and y=1, but I am not sure how to use this information to solve the problem. Any help would be greatly appreciated.
 

1. How can I determine if my system is partially decoupled?

To determine if a system is partially decoupled, you can perform a decoupling test. This involves measuring the response of one input while holding the other inputs constant. If the output is affected by the other inputs, then the system is partially decoupled.

2. What are the advantages of a partially decoupled system?

A partially decoupled system has the advantage of reduced complexity and improved stability. By reducing the coupling between inputs, it becomes easier to control each input independently and improves the overall stability of the system.

3. How can I decouple a partially decoupled system?

To fully decouple a partially decoupled system, you can use a control strategy such as feedback control or feedforward control. These methods use mathematical models to manipulate the inputs and compensate for the coupling between them.

4. What are the limitations of a partially decoupled system?

One limitation of a partially decoupled system is that it may not be possible to fully decouple all inputs. In some cases, there may still be some residual coupling between inputs that cannot be eliminated. Additionally, decoupling can also increase the complexity and cost of the system.

5. Can I use a partially decoupled system for all types of systems?

No, a partially decoupled system is not suitable for all types of systems. It is most commonly used for linear time-invariant systems, and may not be effective for nonlinear or time-varying systems. It is important to carefully consider the characteristics of the system before implementing a partially decoupled control strategy.

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