# LQR Controller for a nonlinear system - how to split the SS model as A and B matrices?

Say we have a nonlinear system that is represented by the following state space model:

$$\dot{x}_1=x_2$$ $$\dot{x}_2=f_1(x)+b_1(x)u$$ $$\dot{x}_3=x_4$$ $$\dot{x}_4=f_2(x)+b_2(x)u$$

The inputs to the LQR controller are ($A$, $B$, $Q$, $R$) but the nonlinear system does not have an $A$ matrix nor a $B$ matrix. I remember reading something as lie matrix method to split the above state space model into two matrices as $A$ and $B$, but I cannot seem to find this source.

Can someone suggest a way to apply the LQR controller to the given nonlinear system?

• Just looking at these dynamics, I think you should use backstepping control. – Preston Roy Dec 9 '17 at 15:23