Linearize...

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?

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?

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)

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.

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.

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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.