# Using Jacobian insted of Lie Derivative

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?

• Where have you seen Lie Derivative for linearization of control systems? – Arash Apr 19 '18 at 2:21
• Keep in mind that this method for controlling a nonlinear system does not guarantee stability, because you kind of turn it into a switched linear system problem. – Kwin van der Veen Apr 19 '18 at 3:39
• So by using Lie Derivative, i can guarantee a stable system ? – Daniel Mårtensson Apr 19 '18 at 4:48
• The only place in control theory where I have seen Lie derivatives being used is to determine observability of nonlinear systems. Or are you referring to Lie algebra? – Kwin van der Veen Apr 19 '18 at 14:14
• Ok. But why not use Jacobian then the observability method for linear systems? – Daniel Mårtensson Apr 19 '18 at 14:49