Find Kp for 10% Error in Control Engineering Homework

In summary, to determine a suitable value for Kp in a proportional-plus-derivative-plus acceleration system with a steady-state error less than 10%, you can use the equation (96 / (96 + 23 Kp)) < 0.1 to solve for Kp, making sure to keep Kp positive.
  • #1
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Homework Statement


If the steady-state error of the proportional-plus-derivative-plus acceleration system is to be less than 10% determine a suitable value for Kp.

Homework Equations


The input of the system is: v
The output of the system is: y

The transfer function of the system including the controller is:
(23(Kp+4s+s^2)) / (s^3+19s^2+48s+96+23Kp)

The Attempt at a Solution


With steady state: s = 0
Therefore: y steady state = v x (23Kp / (96 + 23Kp))
The error of the system is: v-y
Therefore: error = v - (23Kp / (96+23Kp))v
error = 96v / (96 + 23Kp)

How can I correctly substitute the error of 10% now in this equation.
I think (v-y)/y = 0.1 [error]

But this will leave 3 variables in the equation...

Thanks in advance!
 
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  • #2
Hi,

The error is a ratio. So when you get to

error = ( 96 / ( 96 + 23 Kp ) ) v,

The error in percentage is:

%error = ( 96 / ( 96 + 23 Kp ) ) < 10%

There you can then solve for the minimum Kp.
 
  • #3
Cheers.

So that means that:
error = v - y
error [%] = ((v-y)/v)*100%

So:

error = ( 96 / ( 96 + 23 Kp ) ) v
error [%] = ( 96 / ( 96 + 23 Kp ) ) * 100%

( 96 / ( 96 + 23 Kp ) ) * 100% < 10%
( 96 / ( 96 + 23 Kp ) ) < 0.1

Then I get: Kp > 37.565

Is that correct? Thanks in advance!
 
Last edited:
  • #4
Hi, you must have done something wrong in the algebra, because Kp should be positive.

( 96 / ( 96 + 23 Kp ) ) < 0.1

Conceptually, when the denominator is 960, you will have exactly 0.1 error.
As Kp increases, the error decreases further. When you solve it using algebra
you can verify that Kp to give you a denominator that is greater than 960.
 
  • #5
I see, I just edited my reply before your reply.

Thanks a lot!
 

1. What is Kp in control engineering?

Kp, also known as the proportional gain, is a tuning parameter in control engineering that determines the strength of the response to an error signal. It is used in a proportional-integral-derivative (PID) controller to adjust the output of a system in response to changes in the input.

2. How is Kp calculated in control engineering?

Kp is typically calculated by trial and error, also known as tuning. The goal is to find the value of Kp that results in the desired response of the system. This can be done through simulations or by using mathematical models of the system.

3. What is the significance of a 10% error in control engineering homework?

A 10% error in control engineering homework is often used as a benchmark for acceptable performance of a control system. This means that the system's output can deviate by 10% from the desired value without negatively impacting the overall performance of the system.

4. How can Kp be adjusted to achieve a 10% error in control engineering homework?

To achieve a 10% error, Kp can be adjusted by increasing or decreasing its value until the desired error is achieved. This involves carefully tuning the other parameters of the PID controller as well, such as the integral and derivative gains.

5. What are the consequences of an incorrect Kp value in control engineering?

An incorrect Kp value can result in poor performance of the control system, leading to instability, oscillations, or overshoot. It can also cause the system to be unresponsive or have a slow response time. Therefore, it is crucial to accurately tune Kp to achieve the desired performance of the system.

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