1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: State Space Controllability & Observability with Variable Terms in a Transfer Functio

  1. Nov 30, 2011 #1
    1. The problem statement, all variables and given/known data
    The following two gain equations are given (G is the forward-path gain; H is presumably the feedback gain):

    G(s) = [itex]\frac{s + 3}{s^{3} + 2s^{2} + as + 1}[/itex] ;

    H(s) = [itex]\frac{s + b}{s + 5}[/itex]

    The question asks that the range of values for a and b be determined such that the system is completely controllable and observable.

    2. Relevant equations/theory

    Closed-loop transfer function for no controller, M(s) = [itex]\frac{G(s)}{1 + G(s)H(s)}[/itex]

    State space vectors:
    [itex]\dot{x}[/itex] = Ax(t) + Bu(t)
    y(t) = Cx(t) + Du(t)

    If for these state vectors, the matrix S has full rank, the system is controllable; if V has full rank then the system is observable. S and V are defined as:

    S = [B A*B A*A*B ...]
    V = Transpose([C C*A C*A*A ...]) (just more convenient to write the transpose)

    3. The attempt at a solution

    I calculated the closed-loop transfer function as:

    M(s) = [itex]\frac{s^{2} + 8s + 5}{s^{4} + 7s^{3} + (11 + a)s^{2} + (5a + b + 4)s + (3b + 5)}[/itex]

    From this I multiplied out the highest order of s and transcribed the coefficients into the OCF and CCF forms. However, the S for CCF is always non-singular and a similar case occurs with the matrix V for OCF. This is because those matrices are always triangular (as far as I can tell). As such, it appears as though the variables a and b have no effect on the system, which obviously seems erroneous.

    Now perhaps I should calculate the S vector for the OCF form - I haven't done this as it seems absolutely messy and I know for a fact that this question is supposed to be done under time pressure (it is in preparation for an exam). On the other hand, I saw at least one question where the forward-path gain was considered alone, and the feedback treated separately - but in that case it was some unrecognised vector form.

    Note: my bad if this is in the wrong forum space - first post and seemed most relevant to engineering homework.
    Last edited: Nov 30, 2011
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted