PID controller theory v2.1

Transfer function of the PID controller this library implements is:

Continuos PID is transformed to the discrete domain and can be described as a sum of three components:

proportional:

integral:

derivative:

Where the u(k) is the control signal (voltage U_q in our case) in moment k, e(k),e(k-1) is the tracking error in current moment k and previous step k-1. Tracking error presents the difference in between the target velocity value v_d and measured velocity v.

Implementation details

The PID algorithm is implemented in the SimpleFOClibrary in the PIDController class. The class is instantiated by specifying the parameters:

PIDController(float P, float I, float D, float ramp, float limit);

And the class has only one function:

// PID controller function
float PIDController::operator() (float error){
    // calculate the time from the last call
    unsigned long timestamp_now = _micros();
    float Ts = (timestamp_now - timestamp_prev) * 1e-6;
    // quick fix for strange cases (micros overflow)
    if(Ts <= 0 || Ts > 0.5) Ts = 1e-3; 

    // u(s) = (P + I/s + Ds)e(s)
    // Discrete implementations
    // proportional part 
    // u_p  = P *e(k)
    float proportional = P * error;
    // Tustin transform of the integral part
    // u_ik = u_ik_1  + I*Ts/2*(ek + ek_1)
    float integral = integral_prev + I*Ts*0.5*(error + error_prev);
    // antiwindup - limit the output voltage_q
    integral = _constrain(integral, -limit, limit);
    // Discrete derivation
    // u_dk = D(ek - ek_1)/Ts
    float derivative = D*(error - error_prev)/Ts;

    // sum all the components
    float output = proportional + integral + derivative;
    // antiwindup - limit the output variable
    output = _constrain(output, -limit, limit);

    // limit the acceleration by ramping the output
    float output_rate = (output - output_prev)/Ts;
    if (output_rate > output_ramp)
        output = output_prev + output_ramp*Ts;
    else if (output_rate < -output_ramp)
        output = output_prev - output_ramp*Ts;

    // saving for the next pass
    integral_prev = integral;
    output_prev = output;
    error_prev = error;
    timestamp_prev = timestamp_now;
    return output;
}

Therefore you can integrate the PID into your code very easily by just calling:

void setup(){
  ...
  PIDController some_pid = PIDController{.....};
  ...
}
void loop(){
  float control = some_pid(target-measurement);
} 

This PID class is implemented in the BLDCMotor and StepperMotor class for handling the motion control velocity (motor.PID_velocity) and position (motor.P_angle). You can change the values parameters of these PID controllers by changing their public variables

// PID controller configuration structure
class PIDController
{
  .....
  float P; //!< Proportional gain 
  float I; //!< Integral gain 
  float D; //!< Derivative gain 
  ....
};

For example:

motor.PID_velocity.P = 1;
motor.P_angle.P = 10;