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: 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

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

  6. Nov 28, 2016 #5

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted