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Posers and Puzzles

Posers and Puzzles

  1. 18 Jan '11 21:27
    Suppose x1_dot = x2, and x2_dot = -p sin(x1) + u, where u is a controlled input.
    I'm told to linearize around (pi, 0) using Jacobian linearization and linear state feedback.
    With Jacobian linearization, I get:

    x1_dot = x2
    x2_dot ~ p*x1 + u - p*pi

    I can't write this as x_dot = Ax + Bu because of the -p*pi part, but I don't know how to use linear state feedback to fix this. Any suggestions?
  2. Standard member Palynka
    Upward Spiral
    19 Jan '11 09:33
    Originally posted by Jirakon
    Suppose x1_dot = x2, and x2_dot = -p sin(x1) + u, where u is a controlled input.
    I'm told to linearize around (pi, 0) using Jacobian linearization and linear state feedback.
    With Jacobian linearization, I get:

    x1_dot = x2
    x2_dot ~ p*x1 + u - p*pi

    I can't write this as x_dot = Ax + Bu because of the -p*pi part, but I don't know how to use linear state feedback to fix this. Any suggestions?
    If u is a control, I'd say then just use v = u - p*pi

    Unless you have restrictions on u such that this is not possible.
  3. 19 Jan '11 09:40 / 1 edit
    Doesn't it go like this?

    (x1, x2) = (pi + y1, y2)

    =>
    y1' = y2

    y2' = -p sin(pi + y1) + u
    = p sin(y1) + u
    = p y1 + u + O(y1^2)
  4. 19 Jan '11 22:07
    Originally posted by Palynka
    If u is a control, I'd say then just use v = u - p*pi

    Unless you have restrictions on u such that this is not possible.
    Isn't that a Lyapunov function? Or is that linear state feedback? I'm still somewhat new to these names.

    The first problem is to linearize with Jacobian linearization and linear state feedback.
    The second is to linearize with feedback linearization and linear state feedback.
    The third is with a Lyapunov function.
  5. Standard member Palynka
    Upward Spiral
    19 Jan '11 23:23
    Originally posted by Jirakon
    Isn't that a Lyapunov function? Or is that linear state feedback? I'm still somewhat new to these names.

    The first problem is to linearize with Jacobian linearization and linear state feedback.
    The second is to linearize with feedback linearization and linear state feedback.
    The third is with a Lyapunov function.
    I don't think so, I mean just a simple change of variable.

    If you have the system:
    x1_dot = x2
    x2_dot ~ p*x1 + u - p*pi

    Then this is the same as
    x1_dot = x2
    x2_dot ~ p*x1 + v

    Where v = u-p*pi . If can then set a linear rule (the feedback) on v so that v = K*x2 (or whatever is your output vector). You now have a state-space representation for your system. I think that since p*pi is time-invariant there's nothing else you need to add.

    I think this should work, but I'm a bit confused by your system. Are x1 and x2 both inputs and there's no output vector?
  6. 19 Jan '11 23:45
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  7. 20 Jan '11 07:42
    The output is just y = x1. It's a simple pendulum with x1 the angular displacement, and x2 the angular velocity. I tried the feedback linearization today, and I got the same result you mentioned (u = v + p*pi).
    It turns out all I needed was that control law. Thanks.