Finding Open Loop Transfer Functions and Critical Stability in Root Loci

In summary, the conversation discusses finding an open loop transfer function and critical stability in a system. The first problem involves finding a transfer function with poles at given values and understanding the difference between open and closed loop functions. The second problem involves finding the open loop gain for a system to have critical stability and using the root locus method to calculate the gain. A helpful resource is provided for further understanding.
  • #1
greg997
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2
Hi, I ve got two problems.
1) The problem i have is how to find an open loop transfer function that gives poles at -3, -6, -7+_ j5 and zero at -5.
So i think that the numerator is (s+5), and denominator has (s+3),(s+6). and now I don't know how to carry on. Is it ( s+7+j5) and (s+7-j5)?. Am I right so far? It would be now a closed loop transfer function, so how to go back to open loop one?

2.)I need to fine open loop transfer function, gain that gives critical stability, what is the frequency at that critical staility. I can only see that there are 3 poles and no zeros. How to proceede.?


Thanks
 

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  • #2
That's right so far. With your first question, they might want you to rearrange your complex poles. To do that, you'd just expand (s+7+j5)*(s+7-j5). Otherwise, you'd be correct in using those as part of your denominator to have it as (s+3)*(s+6)*(s+7+j5)*(s+7-j5)

How do you know that that's the closed loop transfer function? If you were reading the poles and zeros from a root locus, then those poles and zeros are the open loop ones.

When you've got a root locus, the visible X's and O's are your open loop poles and zeros. The lines are what happens to the closed loop poles as the open loop gain changes.

For your second question, do you know where the closed loop poles need to be for the system to be stable? Hint - one side of the s-plane gives a stable response, and if there are any on the other side it becomes unstable.
Once you've worked out that, you can look at your root locus and see where the pole is when it reaches that axis. Now to calculate the gain, you find the distance on the s-plane to this chosen pole each open loop pole or zero (where the distance is sqrt(IM{s}^2+RE{s}^2) ...). Then K = (distances of the poles all multiplied together)/(distances of the zeros all multiplied together).

This website has a really good example:
http://www.facstaff.bucknell.edu/mastascu/eControlHTML/RootLocus/RLocus1A.html#Problems
 
  • #3
:) thnks, the system is marginnaly stable when pole(s) is at 0. So the real part must be zero. I will try to solve that. And thanks for the link.
 

FAQ: Finding Open Loop Transfer Functions and Critical Stability in Root Loci

What is a root locus?

A root locus is a graphical representation of the possible locations of the roots of a transfer function as a parameter (typically the gain) is varied. It shows how the closed-loop poles of a system move in the complex plane as a result of changing a specific parameter.

How is a root locus plot created?

To create a root locus plot, the transfer function of the system is first written in terms of its open-loop transfer function. Then, the open-loop transfer function is factored to determine the poles and zeros of the system. Next, the root locus points are calculated based on the gain and the pole/zero locations. Finally, these points are plotted in the complex plane to create the root locus plot.

3. What information can be obtained from a root locus plot?

A root locus plot can provide information about the stability, damping ratio, and natural frequency of a system. It can also show the effects of changing the system parameters on the closed-loop poles and the system's overall performance.

4. How can a root locus plot be used to design a control system?

A root locus plot can be used to design a control system by selecting a desired location for the closed-loop poles, based on the system's performance requirements. The corresponding gain value at that point can then be determined from the root locus plot. This gain value can then be used to design a controller that will place the closed-loop poles at the desired locations, achieving the desired system performance.

5. What are the limitations of using a root locus plot?

A root locus plot assumes that the system is linear, time-invariant, and has a single input and output. It also does not take into account any nonlinearities or disturbances in the system. Therefore, it is only applicable for analyzing and designing linear control systems with single-input single-output (SISO) configurations.

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