If a controller starts from a stable state at zero error (PV = SP), then further changes by the controller will be in response to changes in other measured or unmeasured inputs to the process that impact on the process, and hence on the PV. Variables that impact on the process other than the MV are known as disturbances and generally controllers are used to reject disturbances and/or implement setpoint changes. Changes in feed water temperature constitute a disturbance to the shower process.
In theory, a controller can be used to control any process which has a measurable output (PV), a known ideal value for that output (SP) and an input to the process (MV) that will affect the relevant PV. Controllers are used in industry to regulate temperature, pressure, flow rate, chemical composition, speed and practically every other variable for which a measurement exists. 9
Automobile cruise control is an example of a process which utilizes automated control.
Due to their long history, simplicity, well grounded theory and simple setup and maintenance requirements, PID controllers are the controllers of choice for many of these applications.
2.PID controller theory
Note: This section describes the ideal parallel or non-interacting form of the PID controller. For other forms please see the Section "Alternative notation and PID forms".
The PID control scheme is named after its three correcting terms, whose sum constitutes the manipulated variable (MV). Hence:
 
where Pout, Iout, and Dout are the contributions to the output from the PID controller from each of the three terms, as defined below.
2.1. Proportional term
The proportional term makes a change to the output that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant Kp, called the proportional gain.
The proportional term is given by:
 
Where
Pout: Proportional output
Kp: Proportional Gain, a tuning parameter
e: Error = SP − PV
t: Time or instantaneous time (the present)
Change of response for varying KpA high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable (See the section on Loop Tuning). In contrast, a small gain results in a small output response to a large input error, and a less responsive (or sensitive) controller. If the proportional gain is too low, the control action may be too small when responding to system disturbances.
In the absence of disturbances, pure proportional control will not settle at its target value, but will retain a steady state error that is a function of the proportional gain and the process gain. Despite the steady-state offset, both tuning theory and industrial practice indicate that it is the proportional term that should contribute the bulk of the output change.
2.2.Integral term
The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error. Summing the instantaneous error over time (integrating the error) gives the accumulated offset that should have been corrected previously. The accumulated error is then multiplied by the integral gain and added to the controller output. The magnitude of the contribution of the integral term to the overall control action is determined by the integral gain, Ki.
The integral term is given by:
 
Iout: Integral output
Ki: Integral Gain, a tuning parameter
e: Error = SP − PV
τ: Time in the past contributing to the integral response
The integral term (when added to the proportional term) accelerates the movement of the process towards setpoint and eliminates the residual steady-state error that occurs with a proportional only controller. However, since the integral term is responding to accumulated errors from the past, it can cause the present value to overshoot the setpoint value (cross over the setpoint and then create a deviation in the other direction). For further notes regarding integral gain tuning and controller stability, see the section on loop tuning. PID控制器英文文献和中文翻译(2):http://www.751com.cn/fanyi/lunwen_13019.html