P-Controller Design for Steady State Error

In summary, the conversation involves finding the closed loop transfer function and designing a proportional controller for a twin water tank system with a reference level of 40mm. The method for finding the proportional controller involves setting the numerator of the characteristic equation to zero and solving for S. The poles can also be found by setting the denominator to zero. The steady-state error can be found by taking the limit of the transfer function at s=0.
  • #1
Spimon
25
0

Homework Statement



So this is a bit of a two-part question and I'm unsure which part I'm not doing right (or both!).
i) Find the closed loop transfer function of the system shown
ii) Design a proportional controller for the system to give a 10% steady state error

Any help, hints, suggestions would be greatly appreciated.
Thanks!

2lp4q8.jpg


Homework Equations


Go = 4.0
α = 0.168
β = 0.0047
Z1 = 8.9
G(sen2) = 5.33*10^-5
The reference level is 40mm (it's a twin water tank system)

The Attempt at a Solution



Firstly, the CLTF
fkbmvr.jpg


And the P-controller
21amr9x.jpg


Thanks for any help :D
 
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  • #2
1. In your CLTF panel. what's with the last two equations? You didn't try to isolate the pole locations... not that there was any reason to ... but your 1st equation is correct. I'm too lazy to work thru the rest of that panel.

2. As for the kp computation: Assuming your final T(s) is correct, just take
lim s → 0 of T(s). Using your derivation of T(s) I make that to be
T(0) = kpG0Gsens/(β + GsensCG0z1)
which you set to 0.1 & solve for kp.
 
  • #3
Oh, ok. Thanks guys. Maybe it is correct, or at least fundamentally on track - I'm not so worried about the actual numbers as the method.

Rude Boy, the final 2 lines of the CLTF are the characteristic equation - just the denominator of the transfer function.
To isolate the pole locations, do I make set the numerator to equal zero and solve for S?
Similarly, to isolate zero locations I set the denominator to zero and solve for S?

Thanks! :)
 
Last edited:
  • #4
Spimon said:
Oh, ok. Thanks guys. Maybe it is correct, or at least fundamentally on track - I'm not so worried about the actual numbers as the method.

Rude Boy, the final 2 lines of the CLTF are the characteristic equation - just the denominator of the transfer function.
To isolate the pole locations, do I make set the numerator to equal zero and solve for S?
Similarly, to isolate zero locations I set the denominator to zero and solve for S?

Thanks! :)

Well, there was really no need to find the poles (yes, they are found by setting the denominator to zero) since you weren't asked to find the time response, just the steady-state error. And that, for a unity step input, is just T(0).
(Reason: step input transform is 1/s but the final-value theorem says lim s→ 0 of sT(s) so the s's cancel).

If you had been asked to find the time response to a step input you would, after finding the poles, have to do a fractional expansion of the entire T(s), includng the numerator, and then done the inverse transform on each of the terms of that expansion.

Or, if you got lucky, you might have found the entire T(s) in a table of transforms, then there would have been no need to find the poles.
 
  • #5


I would like to provide some suggestions for your P-controller design for steady state error.

1. Understand the concept of steady state error: Steady state error is the difference between the desired output and the actual output of a system when it reaches a stable state. It is important to understand this concept in order to design an effective P-controller.

2. Use the closed loop transfer function: The closed loop transfer function is a mathematical representation of the system's response to a given input. In order to design a P-controller, you need to find the closed loop transfer function of the system.

3. Analyze the system's response: Once you have the closed loop transfer function, analyze the system's response to different inputs. This will help you understand the system's behavior and identify any potential problems.

4. Choose an appropriate gain value: The gain value of a P-controller determines the amount of correction applied to the system's output. It is important to choose an appropriate gain value to ensure stable and accurate control of the system.

5. Consider the desired steady state error: In this case, you are asked to design a P-controller that will result in a 10% steady state error. This means that the difference between the desired output (40mm) and the actual output should be 4mm. Keep this in mind when choosing the gain value.

6. Use trial and error: P-controller design is often an iterative process. It is recommended to start with a small gain value and gradually increase it until the desired steady state error is achieved. You may also need to adjust other parameters, such as the reference level and the system's time constant, to achieve the desired response.

7. Validate your design: Once you have designed your P-controller, it is important to validate its performance through simulations or experiments. This will help you identify any issues and fine-tune your design if needed.

I hope these suggestions will help you in designing an effective P-controller for your system. Good luck!
 

1. What is a P-controller and how does it work?

A P-controller is a type of proportional controller used in control systems. It works by taking the error between the desired output and the actual output and multiplying it by a constant gain (Kp) to produce a control signal. This control signal is then applied to the system to reduce the error and bring the output closer to the desired value.

2. What is steady state error and how does a P-controller help reduce it?

Steady state error is the difference between the desired output and the actual output when the system has reached a stable state. A P-controller helps reduce steady state error by continuously adjusting the control signal based on the error. As the error decreases, the control signal decreases, and vice versa, until the error is minimized and the system reaches the desired output.

3. How do you choose the appropriate value for Kp in a P-controller design?

The appropriate value for Kp depends on the characteristics of the system and the desired response. It is usually determined through experimentation and tuning. A higher value of Kp will result in a faster response, but may also lead to oscillations and instability. A lower value of Kp will result in a slower response, but may also lead to a larger steady state error. It is important to find a balance between speed and stability when choosing the value of Kp.

4. Can a P-controller eliminate steady state error completely?

No, a P-controller can only reduce steady state error but cannot eliminate it completely. This is because the proportional control signal is directly proportional to the error, meaning there will always be some residual error present. To further reduce steady state error, other controllers such as integral and derivative controllers can be used in conjunction with a P-controller.

5. What are some common challenges in P-controller design for steady state error?

Some common challenges in P-controller design for steady state error include instability and oscillations due to a high value of Kp, and slow response and large steady state error due to a low value of Kp. It can also be challenging to choose the appropriate value of Kp as it depends on the characteristics of the system and the desired response. Tuning and experimentation are often necessary to find the optimal value of Kp for a specific system.

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