Parameters have been initially set to those derived in Project's Lesson by Sebastian Thrun using twiddle:

Kp = 0.2
Ki = 0.004
kd = 3.0

I implemented backpropagation in the following function:

/**
 * @param Kx: Kp, Ki or Kd
 * @param dx: partial derivative for Kp, Ki or Kd respectively
 * @param dE: total delta error over the whole epoch (previous epoch - current epoch)
 */
void PID::adjust(double &Kx, double dx, double dE) {
    double partialDKx = Kx * dx * dE * learnRate_;
    Kx -= partialDKx;
}

It computes individual (one of three among Kp, Ki and Kd) parameters' contribution to the total error and updates it accordingly.

The function that updates the steering value is defined as:

f(p_err, i_err, d_err) = -Kp * p_err - Ki * i_err - Kd * d_err

Since I am tuning Kp, Ki and Kd, to obtain partial derivatives with respect to Kp, Ki and Kd, that function must be viewed as f(Kp, Ki, Kd) with the partial derivatives equal to -p_error, -i_error and -d_error accordingly.

The tricky part is with i_error. Since I evaluate cumulative error through some number of iterations (say, 200), and each iteration's CTE might be both positive or negative, their raw summation (which is i_error) doesn't represent the magnitude of that parameter as positives and negatives cancel each other out. To address the issue, I've introduced the new variable (creatively called it i_e_fabs_), that accumulates absolute values of CTE throughout the epoch, and use it as a measure of the partial derivative for Ki.

The video illustrates the training process.

It can be seen that the backpropagation updates are mostly concerned about Ki, with Kp and Kd staying pretty much unchanged and oscillating around 0.2 and 3.0, while Ki rapidly drops by one order of magnitude to some value around 0.0003 or 3e-4, having even gone to the negative territory for a brief period of time.

1. Clone this repo.
2. Make a build directory: mkdir build && cd build
3. Compile: cmake .. && make
4. Run it: ./pid.

Alternatively, there are convenient scripts provided by Tiffany Huang for Particle Filter project: clean.sh, build.sh.