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

Question

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?

Answer 1

LQR stands for linear quadratic regulator, where: linear refers to the linear dynamics of the system (which can both be invariant or variant in time); quadratic refers to the cost function which is an integral of a quadratic form, which the LQR minimizes; regulator refers to the goal of the control input to bring the system to zero.

Since your system is not linear you can't directly use LQR for that system. You would either have to resort to linearizing your model around an equilibrium point, or use another technique like some model predictive control (MPC). Linearization will in general only stabilize the system locally around the equilibrium point, while nonlinear MPC might be none convex and therefore potentially really hard to solve.