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Riccati Optimal Control

  1. Nov 26, 2016 #1
    1. The problem statement, all variables and given/known data

    I was wondering if I can get some help on a Linear Regulator Problem for an Optimal Control Problem. Given a state equation and performance measure I am trying to solve using the Riccati equation on MATLAB. This is a sample example I got from a book Optimal Control Donald Kirk. I don't understand how they derived three separate differential equations from the Riccati equation:

    The problem goes as followed:

    Consider the system https://www.physicsforums.com/attachments/upload_2016-11-26_20-25-28-png.109458/ [Broken]
    https://www.physicsforums.com/attachments/upload_2016-11-26_20-26-3-png.109459/ [Broken]

    How did they go from a single equation K to three separate equations. I keep looking for resources but many other examples seem to skip this part. Thank for any help or input.
    2. Relevant equations
    https://www.physicsforums.com/attachments/upload_2016-11-26_20-26-46-png.109460/ [Broken]

    3. The attempt at a solution

    upload_2016-11-26_20-43-29.png
    upload_2016-11-26_20-43-49.png
     
    Last edited by a moderator: May 8, 2017
  2. jcsd
  3. Nov 26, 2016 #2
    https://www.physicsforums.com/attachments/upload_2016-11-26_20-39-19-png.109463/ [Broken]

    I found a paper on this online that gives somewhat of an example of this problem.
    https://www.physicsforums.com/attachments/upload_2016-11-26_20-40-26-png.109465/ [Broken]
    https://www.physicsforums.com/attachments/upload_2016-11-26_20-41-4-png.109467/ [Broken]
    It seems slightly different though. Sorry for any inconvenience.
     
    Last edited by a moderator: May 8, 2017
  4. Nov 26, 2016 #3

    Ray Vickson

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    The differential equations for ##\mathbf{K}## and its transpose ##\mathbf{K}^T## are the same; and (as your attachment in post #2 states), ##\mathbf{K}(t_f) = \mathbf{S}##, where ##\mathbf{S}## is a symmetric matrix. Therefore, the solution ##\mathbf{K}(t)## is a symmetric matrix as well. Now just write the Ricatti equation for the symmetric matrix ##\mathbf{K}## in terms of its components:
    $$\mathbf{K} = \pmatrix{K_{11}&K_{12}\\K_{12} & K_{22}} $$
     
    Last edited by a moderator: May 8, 2017
  5. Nov 27, 2016 #4
    I'm still a little confused. I understand that the matrix is symmetric. I just don't understand how they have it equal on the LHS a row of three 3x1 differential equations when K seems to be a 2x2. That's where I'm confused. I'm thinking this is some linear algebra property that's going over my head. I don't know if the notes I provided actually answer my question. Thanks for the response

    upload_2016-11-27_23-2-37.png
     
  6. Nov 28, 2016 #5

    Ray Vickson

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    I have not checked the picture: it is too messy and unstructured. However, both sides of your differential equation are 2x2 matrices, so you get 4 coupled differential equations. Since the matrix is symmetric, only three of the equations are different
     
  7. Nov 28, 2016 #6
    I think I understand now. I was just confused on multiplying the X Matrix by another 2x2 Matrix. I was thinking the equations would combine X11 + X12 as in the case of a 2x2 and 2x1 but it makes sense now. Thanks.
     
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