A proportional–integral–derivative controller (PID controller or three-term controller) is a control loop mechanism employing feedback that is widely used in industrial control systems and a variety of other applications requiring continuously modulated control. A PID controller continuously calculates an error value as the difference between a desired setpoint (SP) and a measured process variable (PV) and applies a correction based on proportional, integral, and derivative terms (denoted P, I, and D respectively), hence the name.

mathematical form

The overall control function: $U(t)=k_{p}e(t)+k_{i}\int_{0}^{t}e(\tau)d\tau+k_{d}\frac{de(t)}{dt}$

举例调节水温：

通过误差e来调节输入u PD控制：提高稳定性，改善瞬态响应
PI控制：改善稳态误差
PD控制与PI控制组成了PID控制

例题1

纯比例控制
结果
无法完全消除稳态误差，误差大约在1左右
加上积分项
纯比例控制和比例积分控制在误差e和输入u的区别
再加上微分项
PID控制结果

链接