So i have the following system:
$$ \dot x_1 = x_2 $$
$$ \dot x_2 = -x_1 + \frac{x^5_1}{16} - x_2 + u $$
And I need to design a controller of the form using pole placement method:
$$ u = -Kx $$
From what I have read I need to use the Ackerman formula or write the system in controllable canonical form. However, I am not sure how I can use the Ackerman formula as I cannot figure out how to write the states in controllable canonical form.
The question is also asking for the dynamics of the closed loop system. Are the states above not already in the closed-loop as a result of the control feedback, u?
Any help would really be appreciated; i've been banging my head against a wall for the last few days trying to figure this out!
As mentioned in the comments, this system is nonlinear. So we can't talk about poles or pole-placement. Also note that the open-loop nonlinear system $$\begin{align}\dot{x}_1&=x_2\\ \dot{x}_2&=-x_1+\frac{x_1^5}{16}-x_2 \end{align}$$ has three equilibrium points: $$x=(0,0)\quad\text{and}\quad x=(\pm 2,0)$$ where the origin is stable and the two other points are unstable. Here is what the phase portrait looks like:
So if you want to linearize the system and do the pole-placement, you have to do the linearization around an equlibrium point. By the way, that doesn't seem to be useful and doesn't make any sense as well.
I suspect that your problem wasn't about the pole-placement, but rather moving the unstable equilibrium points far enough from the origin, thus widening the region of attraction of the stable equilibrium point. With this assumption, let the state feedback vector be $K=\pmatrix{k_1&k_2}$. Then the closed-loop equations are: $$\begin{align}\dot{x}_1&=x_2\\ \dot{x}_2&=-(k_1+1)x_1+\frac{x_1^5}{16}-(k_2+1)x_2 \end{align}$$ so the unstable equilibrium points will move to: $$\bar{x}^1=\pmatrix{-2\sqrt[4]{k_1+1}\\0},\;\bar{x}^2=\pmatrix{2\sqrt[4]{k_1+1}\\0}$$ As you see, we don't need $k_2$.