Equations in LaTex of the Udacity's Self-Drive Car Nanodegree lectures.
\alpha = - J_P * cte - J_D * \frac{\mathrm{d} }{\mathrm{d} t} cte - J_I * \sum cte
State
[ x_1, y_1, \psi_1, v_1, cte_1, e\psi_1]
Constraints
\begin{align}
\delta & \in [-25^{\circ}, 25^{\circ} ] \nonumber \\
a & \in [-1,1] \nonumber
\end{align}
Model
\begin{align}
x_{t+1} & = x_t + v_t * cos(\psi_t) * dt \nonumber \\
y_{t+1} & = y_t + v_t * sin(\psi_t) * dt \nonumber \\
\psi_{t+1} & = \psi_t + \frac{v_t}{L_f} * \delta_t * dt \nonumber \\
v_{t+1} & = v_t + a_t * dt \nonumber \\
cte_{t+1} & = f(x_t) - y_t + v_t * sin(e\psi_t) * dt \nonumber \\
e\psi_{t+1} &= \psi_t - {\psi}des_t + \frac{v_t}{L_f} * \delta_t * dt \nonumber
\end{align}
Cost
J = \sum_{t=1}^{N}(cte_t - cte.ref)^2 + (e\psi_t - e\psi.ref)^2 + ...
Latency
\begin{align}
x & = x + v * cos(\psi) * dt \nonumber \\
y & = y + v * sin(\psi) * dt \nonumber \\
\psi & = \psi + \frac{v}{L_f} * \delta * dt \nonumber \\
v & = v + a * dt \nonumber
\end{align}