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Discrete time state space model: solving for input

  1. Dec 12, 2008 #1
    Preface: This homework assignment was due long ago. At this point I am only trying to understand the problem (or really if the posted solution follows the problem) before my exam. I have no real indication that this problem (or even one like it) will be on my final, but I feel that my inability to solve it shows a weakness in my understanding of the material that I should fix before I sit down to take the test. The PDF for the assignment is here (Q VII-VIII) and the posted solution here. So here goes...

    1. The problem statement, all variables and given/known data
    This the exact wording of the problem:
    2. Relevant equations
    [tex]\mathbf{x}[n+1] = A\mathbf{x}[n] + Bu[n][/tex]
    [tex]\mathbf{y}[n] = C\mathbf{x}[n] + Du[n][/tex]



    3. The attempt at a solution
    Well... first I tried to setup a difference equation of the form:
    [tex]\-1y[n+1] + \left(1+\frac{.09}{12}\right) y[n]=u[n][/tex]
    and set
    [tex]x_{1}[n]=y[n][/tex]
    [tex]x_{2}[n]=y[n+1][/tex]
    and
    [tex]x_{1}[n+1]=y[n+1]=x_{2}[n][/tex]


    However, I came to realize that this is either a) false or b) going to yield a state space representation with 1 state (A is a 1x1 matrix).

    I'd be thrilled with a lead to chase on this one; I can't tell if the question as-asked makes sense insomuch as I'm going to find a system matrix of dimensions m x m (m > 1) which are the only state spaces I can remember working in class.



    The posted solutions from the TA show a difference equation of the form:
    [tex]x[n+1] = ax[n]-ku[n][/tex]
    Which is then Z transformed to yield:
    [tex]X(z) = -k\left(\frac{z}{\left(z-1\right)\left(z-a\right)}\right)+\frac{z}{z-a}x[0][/tex]
    At which point I believe the TA made a superficial error (but am probably wrong about that).

    So, any guidance? Is there a state space solution that makes sense for this problem? And if so, does it essentially present itself like scalars?

    Many thanks for any pointers.
     
  2. jcsd
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