Discrete-Time PID https://jp.mathworks.com/help/control/ug/discrete-time-proportional-integral-derivative-pid-controller.html

pid2.h

#pragma once
#include <integrator.h>
template<typename T>
class PID2 {
	T Kp;
	T Ki;
	T Kd;
	T Tf;
	T Ts;
	T b;
	T c;
	DescreteTimeIntegrator<T> IF;
	DescreteTimeIntegrator<T> DF;
public:
	PID2(T kp, T ki, T kd, T tf, T b, T c, T ts, T reference=T(), T initial=T()) : Kp(kp), Ki(ki), Kd(kd), Tf(tf), b(b), c(c), IF(1, ts), DF(1, ts) {
		IF.reset(reference - initial);
		DF.reset(c * reference - initial);
	}
	T update(T reference, T input) {
		auto u = Kp * (b * reference - input) + Ki * IF.update(reference - input) + Kd / (Tf + DF.update(c * reference - input));
		return u;
	}
	void reset(T reference=T(), T initial=T())  {
		IF.reset(reference - initial);
		DF.reset(c * reference - initial);
	}
};