PID Control added

doc/research/paf24/planning/ACC.md

@@ -1,5 +1,7 @@
 # ACC
 
+This file gives a general overview on possible ACC implementations and reflects the current state of the ACC.
+
 ## General
 
 The main goal of an ACC (Adaptive Cruise Control) is to follow a car driving in front while keeping a safe distance to it. This can be achieved by adjusting the speed to the speed of the car in front.
@@ -8,6 +10,15 @@ There are basically three different techniques that can be used to implement an
 
 ### PID Control
 
+The PID Controller consists of three terms: the proportional term, the integral term and the derivative term. One possible simple controller model looks as follows:
+
+$$ v_f(t) = v_f(t - t_s) + k_p e (t-t_s) + k_d \dot{e}(t - t_s) $$
+$$ e(t-t_s) = \Delta x(t - t_s) - t_{hw,d} v_f (t - t_s) $$
+
+- $t_s$: sampling time
+- $k_p$ and $k_d$: coefficients for proportional and derivative terms
+- $e$: distance error (difference between actual distance $\Delta x$ and desired distance $\Delta x_d$)
+
 ### Model Predictive Control (MPC)
 
 It calculates the current control action by solving an online, iterative and finite-horizon optimization of the model.
@@ -32,7 +43,7 @@ Provides a unified control framework to offer both functions: ACC and Stop & Go.
 - Which output should be transfered to the Acting component? Only the desired speed?
 - Which input do we get from the Perception component? The distance and velocity of the car in front?
 
-## ACC Algorithm
+## ACC in our project
 
 ### Current implementation
 
@@ -86,4 +97,4 @@ Start implementing safety distance and general target speed logic. Subscriber lo
 
 ## Sources
 
-[ACC](https://www.researchgate.net/publication/335934496_Adaptive_Cruise_Control_Strategies_Implemented_on_Experimental_Vehicles_A_Review)
+[He, Yinglong et al. (2019). Adaptive Cruise Control Strategies Implemented on Experimental Vehicles: A Review. IFAC-PapersOnLine. 52. 21-27. 10.1016/j.ifacol.2019.09.004. ](https://www.researchgate.net/publication/335934496_Adaptive_Cruise_Control_Strategies_Implemented_on_Experimental_Vehicles_A_Review)