Root Locus Help (Break in/Away pts)

Discussion in 'Homework Help' started by Whaler, Mar 28, 2011.

  1. Whaler

    Thread Starter New Member

    Mar 28, 2011
    i have a feed-forward tf
    G(s) = K(s+2) / s(s+8)(s+1+3j)(s+1-3j)

    Poles: 0,-8,-1+j,-1-j (four fintite)
    zeros: -2 (Three @ infinity)

    (-8-1+3j-1-3j)-(-2) / (4-1) = -8/3 = -2.6667
    (2k+1)Pi / 4-1 = (Pi/3)(2k+1)
    k=0, (Pi/3);
    K=1, (Pi);
    k=2, (5/3)Pi;

    Break In/Away points:
    CLTF: K(s+2)/s(s+8)(s+1+3j)(s+1-3j)+k(s+2)=-1
    then do i solve for k and take the derivative wrt to s?

    jw-axis crossing do i use the denominator from CLTF
    and do Routh Hurwitz and manipulate till get row a zero's and figure out what values of k the system is stable and use above row to figure out where it crosses the jw axis?

    Any help would be appreciated, been a while and don’t remember dealing with imaginaries in the denominator of a feed-forward-loop.

  2. Georacer


    Nov 25, 2009
    Unfortunately Matlab doesn't have a feedforward root locus plotter, so we 're going to have to go with math only.

    What is CTLF that you mention? The way I see it, the new system will have a TF of:
    not the one you write right after you first mention CTLF.

    Can we clear this before going on?
  3. Whaler

    Thread Starter New Member

    Mar 28, 2011
    CLTF is my closed-loop transfer function which not sure about.

    i figure tf=G/1+GH which is equal to:

    denH*numG / denH*denG+numG*numH and h =1 so denH and nomH both equal 1 which is how i got my CLTF.

    I have to sketch root locus for feed-forward transfer function (given) in a negative, unity feedback loop.
    i thought that you set KGH=-1 solve for k then differentiate, but the imaginaries throw me off when trying to differentiate. i am also not sure if KGH is what i have it set to as CLTF.

    Again sorry for not being very helpful but first time with root locus.
    Last edited: Mar 28, 2011
  4. Georacer


    Nov 25, 2009
    Wow, wait a minute. Do you want to do a feedback loop with positive feedback? In this diagram you take the output of G and add it to its input.

    In the feedforward mode, you take the input of G and add it directly to its output.

    Which of the two you want to do?
  5. Whaler

    Thread Starter New Member

    Mar 28, 2011
    no negative feedback loop. unfortunately my hw didn't came with a figure, but all my examples/notes are for negative feedback so i am pretty sure he wouldn't switch it on us now.

    I am just not sure which equation to use when finding the break away pt's. do i use the given feed-forward tf in negative unity feedback

    G(s) = K(s+2) / s(s+8)(s+1+3j)(s+1-3j)

    or do i use "my" closed loop tf


    thats where i'm not to sure, then i have the added problem of differentiating a equation with imaginary numbers, which was years ago, but will deal with that when i get to it.
  6. Georacer


    Nov 25, 2009
    Can you post the homework question as-is, because it seems we have a problem of communication here?
  7. Whaler

    Thread Starter New Member

    Mar 28, 2011
    sure can, hope this helps.

    Sketch the root locus for the following feed-forward transfer functions in a negative, unity feedback loop. Include:
    (a) asymptotes
    (b) break-away/break-in points
    (c) jw-axis crossings
    (d) angles of arrival/departure
    (e) label the gain, K, at the poles, zeros, break-away/break-in points, and jw-axis crossings
    (f) For what range of K are the systems stable?

  8. Georacer


    Nov 25, 2009
    It looks weird in the place it is, but I don't think that feed-forward here means anything else than that the TF carries the signal forward. As I see it, it is a common unity feedback loop with a feedback gain of K.

    If you haven't talked in class about feed-forward control at all, chances are that I am right, so just do your analysis as usual. I will try to do it myself to tomorrow (maybe) and come back to you with the results for comparison.
  9. Whaler

    Thread Starter New Member

    Mar 28, 2011
    sweet, thanks. I will continue with my efforts and post my solutions to both problems. thanks.
  10. Whaler

    Thread Starter New Member

    Mar 28, 2011
    for the first one i simplified the equation too:

    so then get intial equation of
    from which i get

    poles: (0,-8,-1+3i,-1-3i) four total
    zeros: (-2) one finite, so three @ infinity.


    k=0 add 60
    k=1 add 180
    k=2 add 300, increments of 120

    Break-away/in points

    set initial equaion to -1 solve for k, then differeneciate wrt to s and solve for s with dirivative set to zero.

    k= -[s^4/(s+2)^2] - [10s^3/(s+2)^2] - [26s^2/(s+2)^2] - [80s/(s+2)^2]
    which reduces to s^4-24s^3-34s^2-134s-160/(s+2)^2 set to zero and s equals

    25.5459, 0
    -0.1589, +- 2.2528
    -1.228, 0

    for jω-axis crossing i use CLTF
    with den of
    then thru routh-hurwitz i get
    k=160 with crossing at +-4.89898i
    so sys stable for k<160 and > than 0.

    angles of departure/arrival i am not sure.

    hope this looks right, any help would be appreciated.
    Last edited: Mar 29, 2011
  11. ryanw202

    New Member

    Mar 29, 2011
    yea man your doing everything i would do and btw i believe we are in the same class