Determining the range of a controller for a stable system

In summary, the question is asking for the range of Kp values that will make the closed loop system stable. The transfer function of the system is Y(s)/V(s)=Kp/(1.5s^2 + 2.5s - 1 + Kp). To solve for the range of Kp, the roots of the denominator (which is a second order polynomial) must be found using the quadratic formula. Treating Kp as a constant, the formula will include it. By analyzing how Kp affects the roots, the appropriate range of values can be determined for the system to be stable.
  • #1
MattH150197
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Homework Statement


The controller K(s) = Kp, determine the range of Kp over which the closed loop system is stable.

Homework Equations


I found the transfer function of system = Y(s)/V(s)=Kp/(1.5s^2 + 2.5s - 1 + Kp)

The Attempt at a Solution


So i know (1.5s^2 + 2.5s - 1 + Kp) is the characteristic equation is been a while since I've done these i think i remember that all roots of the polynomial for a stable system must be negative real parts and i know its probably obvious but i just can't remember how to find the range of Kp. Thanks!
 
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  • #2
Solve for the roots of the denominator. It's only second order, so you can use a formula.
 
  • #3
yeah it's just the Kp being part of the denominator that's throwing me a bit, how do I include that in the quadratic formula
 
  • #4
MattH150197 said:
yeah it's just the Kp being part of the denominator that's throwing me a bit, how do I include that in the quadratic formula
Um, by including it in the quadratic formula? Treat Kp as a constant and write out the formula. Then see how the value of Kp affects the roots. What range of values for Kp will satisfy your requirements? Be discriminating :smile:
 
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  • #5
MattH150197 said:
... how do I include that in the quadratic formula

As far as I can see IT IS included in the formula (see your own post).
 
  • #6
Yeah I got it now, cheers guys.
 

1. How do you determine the range of a controller for a stable system?

The range of a controller for a stable system can be determined by analyzing the system's transfer function. This involves understanding the system's poles and zeros, and using methods such as the Routh-Hurwitz stability criterion or the Nyquist stability criterion to determine the range of controller gains that will keep the system stable.

2. What is the significance of determining the range of a controller for a stable system?

Determining the range of a controller for a stable system is important because it ensures that the system will not become unstable due to excessive controller gains. It also helps to optimize the performance of the system by finding the appropriate range of controller gains that will achieve the desired response.

3. Can the range of a controller for a stable system change over time?

Yes, the range of a controller for a stable system can change over time. This can happen due to changes in the system's parameters, external disturbances, or changes in the desired response. Therefore, it is important to regularly evaluate and adjust the controller gains to keep the system stable.

4. Are there any limitations to determining the range of a controller for a stable system?

Yes, there are some limitations to determining the range of a controller for a stable system. These may include non-linearities in the system, uncertainty in the system's parameters, and limitations in the chosen stability analysis method. It is important to consider these limitations and their potential impact on the system's stability when determining the controller range.

5. How do you ensure that the determined range of a controller will keep the system stable?

To ensure that the determined range of a controller will keep the system stable, it is important to validate the results through simulations or experiments. This involves testing the system's response with different controller gains within the determined range to verify that it remains stable. Additionally, it is important to regularly monitor and adjust the controller gains to maintain stability as the system may change over time.

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