I. PID Controller
The concept of PID control was first proposed in 1932 by H. Nyquist, a Swedish physicist who later immigrated to the United States. In one of his papers, he proposed using a graphical method to determine the stability of a system. Building on his work, H.W. Bode (the founder of the "Bode plot/Bode plot") and others developed a complete set of methods for designing feedback amplifiers in the frequency domain, which were later used in the analysis and design of automatic control systems. This marked the earliest transition of the PID algorithm from theory to practice.
Meanwhile, feedback control principles began to be applied to industrial processes. In 1936, British scientists A. Callender and A. Stevenson presented the method for the PID controller, thus formally establishing the PID algorithm, which has since occupied a very important position in automatic control technology.
Everyone has surely seen practical applications of PID.
For example, quadcopters, balance cars, cruise control in cars, and temperature controllers in 3D printers are all similar applications where a certain physical quantity needs to be kept stable (such as maintaining balance, stabilizing temperature, rotation speed, etc.). PID controllers are very useful in these situations.
II. Integral Control with PID Controller
The controller's output is proportional to the integral of the input error signal. It is primarily used to eliminate steady-state error and improve the system's accuracy. The strength of the integral action depends on the integral time constant T; the larger T is, the weaker the integral action, and vice versa.
Why introduce the role of integrals?
The output of proportional control is directly proportional to the magnitude of the error; the larger the error, the larger the output, and the smaller the error, the smaller the output. When the error is zero, the output is zero. Since the output is zero when there is no error, proportional control cannot completely eliminate the error and cannot make the controlled PV value reach the setpoint. A stable error must exist to maintain a stable output so that the system's PV value remains stable. This is what is commonly referred to as proportional control having a steady-state error; strengthening the proportional action can only reduce the steady-state error, not eliminate it.
To eliminate steady-state error, integral action must be introduced. Integral action can eliminate steady-state error, so that the controlled y(t) value eventually matches the setpoint. The purpose of introducing integral action is to eliminate steady-state error, so that the y(t) value reaches the setpoint and remains consistent.
The principle of integral action to eliminate steady-state error is that as long as an error exists, the error is integrated to make the output continue to increase or decrease until the error is zero. When integration stops, the output no longer changes, the PV value of the system remains stable, and the y(t) value is equal to the u(t) value, thus achieving the effect of zero-error regulation.
However, due to the inertia of real-world systems, the value of y(t) will not change immediately after the output changes; it will change slowly over a period of time. Therefore, the rate of integration must be matched to the inertia of the real-world system. A larger inertia requires a weaker integral action and a larger integration time I, and vice versa. If the integral action is too strong, the integral output will change too quickly, leading to overshoot and oscillations. Typically, the I parameter is adjusted from large to small, meaning the integral action is adjusted from small to large. The system response is observed to ensure that the error is quickly eliminated, the setpoint is reached, and oscillations are not caused.
For an automatic control system, if a steady-state error exists after reaching steady state, the control system is said to have a steady-state error, or simply a system with error. To eliminate steady-state error, an "integral term" must be introduced into the controller. The integral term depends on the integral of the error over time; as time increases, the integral term increases. Thus, even if the error is small, the integral term will increase with time, driving the controller output to increase and further reduce the steady-state error until it equals zero. Therefore, a proportional-integral (PI) controller can make the system error-free after reaching steady state. The PI controller not only retains the "memory function" of the integral controller in eliminating steady-state error but also overcomes the insensitive response when using integral control alone to eliminate error.
