Closed Loop Stability Criterion: Find the values for closed loop stability

In summary, the conversation discusses a closed loop stability problem in Control Theory and the closed loop transfer function that is being used. The problem requires finding the range of values for a real constant K to ensure closed loop stability. The conversation also includes an attempt at solving the problem and a discussion on how to apply the Nyquist theorem correctly. The main questions raised are how the solution was arrived at so quickly and where the attempt at solving the problem went wrong.
  • #1
Master1022
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Homework Statement
We are working with the closed loop transfer function: $$ P(s) = \frac{K(s) G(s)}{1+K(s)G(s)} $$, where $$ G(s) = \frac{3s^2 - 2s + 3}{(s - 1)^2} $$ and ## K(s) = K ## where K is a real constant. Find the range of values of K to ensure closed loop stability
Relevant Equations
Closed loop transfer function
Hi,

I was just working through a closed loop stability problem in Control Theory and I don't really understand how the answer has arrived at the solution so quickly.

Problem: We are working with the closed loop transfer function:

$$ P(s) = \frac{K(s) G(s)}{1+K(s)G(s)} $$, where $$ G(s) = \frac{3s^2 - 2s + 3}{(s - 1)^2} $$ and ## K(s) = K ## where K is a real constant. Find the range of values of K to ensure closed loop stability

My attempt:
For stability, we want the poles to be in the left hand side (< 0)

I have tried to calculate the poles and just enforce the condition that they are < 0 (in the left half plane). Also note that this question is 3 marks so I don't really see how one can be expected to do lots and lots of algebra.

Nonetheless, if we consider ## 1 + KG(s) = 0 ## to find the open loop poles, we get the equation:
$$ 1 + K \frac{3s^2 - 2s + 3}{(s - 1)^2} = 0 $$
which leads to:
$$ s^2 (3K + 1) + s( -2 - 2K) + (1 + 3K) = 0 $$

At this point the solution just says: "## 1 + 3K > 0 \rightarrow K > \frac{-1}{3} ## or ## 1 + K < 0 \rightarrow K < -1 ##" and then combines those ranges for the answer

if we use the quadratic equation, we get:
$$ s = \frac{(2 + 2K) \pm \sqrt {(2+2K)^2 - 4(3K+1)^2}}{2(3K+1)} $$
For now, I just choose to look at the first part of the term ## \frac{2 + 2K}{2(1 + 3K)} ##

if we want this to be < 0, then either the numerator or denominator have to be negative, but not both:
Case 1: numerator is negative
$$ 2 + 2K < 0 \rightarrow 1 + K < 0 \rightarrow K < -1 $$
and
$$ 2(1 + 3K) > 0 \rightarrow K > \frac{-1}{3} $$
thus, there are no values of K for this solution which meet both constraints

Case 2: denominator is negative
$$ 2(1 + 3K) < 0 \rightarrow K < \frac{-1}{3} $$
and $$ 2 + 2K > 0 \rightarrow K > -1 $$
which means that the following range ensures closed loop stability: ## -1 < K < \frac{-1}{3} ## (after I need do check that those values work for the discriminant of the pole in the quadratic equation)

However, this is different than the quoted solution. They have the closed loop transfer function the same as me.

Questions:
1) How did they get the answer so quickly? Is there some additional theory that allows us to do this?
2) Where did I go wrong such that my range is the opposite of theirs?

Any help is greatly appreciated. Thanks.
 
Last edited:
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  • #2
I am afraid, you did not apply the Nyquist theorem correctly.
The theorem applies to the product L(s)=K(s)*G(s) only .
And - for stability of the closed loop - the theorem requires that the function L(s) encircles the point (-1, j0) counter-clockwise k times., where k is the number of poles of L(s) in the right half of the s-plane.
 
  • #3
LvW said:
I am afraid, you did not apply the Nyquist theorem correctly.
The theorem applies to the product L(s)=K(s)*G(s) only .
And - for stability of the closed loop - the theorem requires that the function L(s) encircles the point (-1, j0) counter-clockwise k times., where k is the number of poles of L(s) in the right half of the s-plane.
Apologies, you are right. I seemed to just assign the nyquist stability criterion title when I was typing up this post. However, the question was about using the closed loop transfer function so the working won't change. I have edited the post now
 

Related to Closed Loop Stability Criterion: Find the values for closed loop stability

1. What is the Closed Loop Stability Criterion?

The Closed Loop Stability Criterion is a mathematical method used to determine the stability of a control system. It involves finding the values of the system's parameters that will result in a stable closed loop system.

2. Why is it important to find the values for closed loop stability?

Finding the values for closed loop stability is important because it ensures that the control system will operate in a stable manner. This means that the output of the system will not oscillate or diverge, and the system will be able to achieve its desired goal.

3. What factors affect the closed loop stability of a control system?

The closed loop stability of a control system is affected by the system's parameters, such as the gain, time constants, and feedback coefficients. The type of controller used, whether it is proportional, integral, or derivative, also plays a role in determining the stability of the system.

4. How do you determine the values for closed loop stability?

The values for closed loop stability can be determined by analyzing the system's transfer function and using mathematical techniques such as the Routh-Hurwitz stability criterion or the Nyquist stability criterion. These methods involve finding the roots of the characteristic equation of the system and determining their location in the complex plane.

5. What happens if the values for closed loop stability are not found?

If the values for closed loop stability are not found, the control system may operate in an unstable manner. This can result in oscillations, overshoot, or even complete system failure. It is important to find these values to ensure the stability and proper functioning of the control system.

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