I want to design a LQR controller for my two wheeled mobile robot. While designing the controller, I need to find gain, K for the LQR which this required the state space model of wheeled mobile robot. My question is how to find the matrix A, B, C and D for this nonlinear system by using the equations below.
The outputs are x, y and theta while inputs are v and w.
Unfortunately, you're trying to apply a linear controller (the L of LQR) to a nonlinear system. In general, this doesn't work.
As an approximation, you could linearize the system. If the nonlinearities are small it's common to linearize about a nominal point such as $\theta=0$, but in your case this might not work well since it would eliminate any lateral dynamics. $$ \begin{align} \dot{x} = v \\ \dot{y} = 0 \\ \dot{\theta} = \omega \end{align} $$
Another option would be to linearize every timestep. This could get expensive though as your feedback gain matrix, $K$, also needs to be recalculated each time.
Finally, a middle ground could be to linearize about several predefined points, e.g. $\theta=[-20,-15,-10,0,10,15,20]$. At each timestep you would separately compute the control action using the $K_j$ computed for each of these linearization points and then interpolate between those values based on the current $\theta$.
