Controller Design Homework: Stuck on Problem 1a - UCSD Robotics

[AeroGuy86]

Homework Statement

http://robotics.ucsd.edu/final_143b_13_version1.pdf

I am stuck on problem 1a. I understand all of the concepts I am just unsure of how to determine the best possible location to place the unknown zero.

The Attempt at a Solution

I know that the zero must be located to the left of z1 and to the right of the origin.
From approximate design guides I have determined that wn=1.8/tr=10rad/s
the damping ratio should be zeta>=.5 meaning theta>=30 degrees (from imaginary axis)
I thought maybe there is a way of proving that this root locus plot forms a perfect circle in its rounded region(i am not sure of this but it looks like it from plotting it in MATLAB with axis equal). Since we know our damping and our frequency, we could determine the radius of the circle, and then find the ideal location to place za but that seems to be a bit stretched and unnecessary to obtain a solution,

This was my final from last quarter by the way and it is our first homework assignment for the quarter.

Any help would be great! Thanks

 

[KataKoniK]

in advance



Thank you for your question. It seems like you have a good understanding of the concepts involved in this problem, but are unsure of how to determine the best possible location for the unknown zero. I would suggest approaching this problem by looking at the desired closed-loop system response and working backwards from there.

First, consider what you want the closed-loop system to look like. Based on the given design guides, we know that the system should have a natural frequency of 10 rad/s and a damping ratio of at least 0.5. This means that we want our closed-loop poles to be at a distance of 10 from the origin and at an angle of at least 30 degrees from the negative real axis.

Next, think about how the unknown zero will affect the location of the poles. Since the zero will affect the overall transfer function, it will also impact the location of the poles on the root locus plot. In general, we want the unknown zero to be placed in a location that will bring the poles closer to the desired location.

One approach you could take is to use the angle criterion for root locus plots. This states that the angle of departure from a pole on the root locus plot should be the same as the angle between the pole and the zero. You could use this criterion to determine the ideal location for the unknown zero, keeping in mind the desired location of the poles.

I hope this helps. Let me know if you have any further questions or need clarification. Good luck with your assignment!