I have a question! Applying linear controllers for nonlinear systems is not hard, but it can be difficult if the user want to control the nonlinear system in different position, for example a robotic arm.
For inverted pendulum, it's no problem to use a simple LQR control feedback. The model of the inverted pendulum is often linearized when the pendulum is standing straight up, where the reference position is.
If I want to change the reference position so the inverted pendulum slopping 20 degrees. Then I need to relinearize the inverted pendulum at 20 degrees.
Instead to include difficult mathematics and weird expressions, I got an idea that sounds like this:
What if I have a nonlinear state space model:
$$\dot{x} = f(x, u)$$ $$y = g(x, u)$$
And I want to linearize in the vectors $x_0$, which is the measured state vector from the nonlinear system and vector and $u_0$ is the current input signal. It can be estimated too. I juse the jacobian instead of Lie Derivative.
$$A = \left.\frac{\partial f(x, u) }{\partial x}\right|_{x_{0}} $$
$$B = \left.\frac{\partial f(x, u) }{\partial u}\right|_{u_{0}} $$
$$C = \left.\frac{\partial g(x, u) }{\partial x}\right|_{x_{0}} $$
$$D = \left.\frac{\partial g(x, u) }{\partial u}\right|_{u_{0}} $$
And then I got my linearized state space model in vector $x_0$ and $u_0$.
$$\dot{x} = Ax + Bu$$ $$y = Cx + Du$$
Of $A$ and $B$ I can create a state feedback system for optimal control such as LQR or compute the input signals for my system such as MPC.
Question:
Why use Lie Derivative when I can use Jacobian? Or is it me who using wrong method?