- 18 Jan '11 21:27Suppose 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? - 19 Jan '11 09:33

If u is a control, I'd say then just use v = u - p*pi*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?

Unless you have restrictions on u such that this is not possible. - 19 Jan '11 22:07

Isn't that a Lyapunov function? Or is that linear state feedback? I'm still somewhat new to these names.*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.

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. - 19 Jan '11 23:23

I don't think so, I mean just a simple change of variable.*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.

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?