DziNe
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Given the dynamic system shown in the image, how can we show how it moves towards its steady state. I believe the new steady state is y=((o-e)/(1-p)) which it moves towards asymptotically, but am not sure how to show this.

Is there a general expression in terms of y_0 (which I believe is equal to y_-1 - e) which can be derived?

Is it also possible to find the half-life of the process by which the system moves towards its steady state?

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mqb2766
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(Original post by DziNe)
Given the dynamic system shown in the image, how can we show how it moves towards its steady state. I believe the new steady state is y=((o-e)/(1-p)) which it moves towards asymptotically, but am not sure how to show this.

Is there a general expression in terms of y_0 (which I believe is equal to y_-1 - e) which can be derived?

Is it also possible to find the half-life of the process by which the system moves towards its steady state?

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What have you covered? There are two parts to the dynamic response:
* The transient, exponential phase which will go to zero if the base is less than 1. This will determine the half life.
* The steady state phase, ie what the response converged to in the limit.
In a level speak, the first is the complementary function, the second is the particular integral.

Have you been taught to solve these systems? If vso, what have you covered?
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DziNe
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(Original post by mqb2766)
What have you covered? There are two parts to the dynamic response:
* The transient, exponential phase which will go to zero if the base is less than 1. This will determine the half life.
* The steady state phase, ie what the response converged to in the limit.
In a level speak, the first is the complementary function, the second is the particular integral.

Have you been taught to solve these systems? If vso, what have you covered?
I'm in sort of a weird situation with this - I'm an economics student and we have never covered anything really like this. The module for which this assignment is set has not gone over anything similar - the lectures are mainly qualitative, but we have been given this to solve in the assignment under the assumption we can do it based on prior mathematical knowledge I suppose.

The relevant parts of the question (shortened) are "Derive an expression for values y takes. What can be said about adjustment path y takes from t = 0? Draw diagram to illustrate." and then "What is the half-life of the process if p = 0.95?"
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mqb2766
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(Original post by DziNe)
I'm in sort of a weird situation with this - I'm an economics student and we have never covered anything really like this. The module for which this assignment is set has not gone over anything similar - the lectures are mainly qualitative, but we have been given this to solve in the assignment under the assumption we can do it based on prior mathematical knowledge I suppose.

The relevant parts of the question (shortened) are "Derive an expression for values q takes. What can be said about adjustment path q takes from t = 0? Draw diagram to illustrate." and then "What is the half-life of the process if p = 0.95?"
Ok but are they assuming you know about convolution, z-transforms, ...? What does the module spec say for solving discrete time systems.
If it's an assignment, you'll need to do a bit reading up about the techniques. Tbh, the maths is only a few lines as it's a linear first order difference equation. Have you covered Complementary Functions and Particular Integral at a level?
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DziNe
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(Original post by mqb2766)
Ok but are they assuming you know about convolution, z-transforms, ...? What does the module spec say for solving discrete time systems.
If it's an assignment, you'll need to do a bit reading up about the techniques. Tbh, the maths is only a few lines as it's a linear first order difference equation.
I've never heard of convolution or z-transforms, hah. We haven't got a module spec beyond a brief description of what concepts we're meant to understand by the end of the module (all economic in nature - no reference to mathematical techniques).


And none of the module lecture notes go through any mathematics. So I was expecting to have to to some reading to figure out the assignment - there are three questions, I've done two and a half and what I'm asking about is the second half of the final question which I've been unable to figure out through reading as yet.
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mqb2766
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Ok. Assuming you've covered CFs and PIs for ordinary differential equations at a level. The solution of a linear difference equation is similar.

For the complementary function of transient response, consider the homogeneous system
y(t) - py(t-1) = 0
And assume a response
y1(t) = a^t
Sub it in and solve for a. When |a| < 1, the dynamics will converge to the steady state and the value of a determines the half life.

For the Particular integral or steady state solution, consider the original system and assume a solution
y2(t) = C
Sub it in and solve for C.

The overall solution is
y(t) = Ay1(t) + y2(t)
Where A is determined by the initial conditions y(-1)

Have a go and post progress / questions?
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DziNe
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(Original post by mqb2766)
Ok. Assuming you've covered CFs and PIs for ordinary differential equations at a level. The solution of a linear difference equation is similar.

For the complementary function of transient response, consider the homogeneous system
y(t) - py(t-1) = 0
And assume a response
y1(t) = a^t
Sub it in and solve for a. When |a| < 1, the dynamics will converge to the steady state and the value of a determines the half life.

For the Particular integral or steady state solution, consider the original system and assume a solution
y2(t) = C
Sub it in and solve for C.

The overall solution is
y(t) = Ay1(t) + y2(t)
Where A is determined by the initial conditions y(-1)

Have a go and post progress / questions?
Thanks! Will have a go and report back. What's the notation there with y1(t) and y2(t) - are those just arbitrary numbers added to distinguish the functions from y(t)?
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mqb2766
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Thanks! Will have a go and report back. What's the notation there with y1(t) and y2(t) - are those just arbitrary numbers added to distinguish the functions from y(t)?
Yes, you could call them Bill and Ben if you wanted to.
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