Proportional–integral–derivative controller
A proportional–integral–derivative controller is a feedback-based control loop mechanism commonly used to manage machines and processes that require continuous control and automatic adjustment. It is typically used in industrial control systems and various other applications where constant control through modulation is necessary without human intervention. The PID controller automatically compares the desired target value with the actual value of the system. The difference between these two values is called the error value, denoted as.
It then applies corrective actions automatically to bring the PV to the same value as the SP using three methods: The proportional component responds to the current error value by producing an output that is directly proportional to the magnitude of the error. This provides immediate correction based on how far the system is from the desired setpoint. The integral component, in turn, considers the cumulative sum of past errors to address any residual steady-state errors that persist over time, eliminating lingering discrepancies. Lastly, the derivative component predicts future error by assessing the rate of change of the error, which helps to mitigate overshoot and enhance system stability, particularly when the system undergoes rapid changes. The PID output signal can directly control actuators through voltage, current, or other modulation methods, depending on the application. The PID controller reduces the likelihood of human error and improves automation.
A common example is a vehicle’s cruise control system. For instance, when a vehicle encounters a hill, its speed will decrease if the engine power output is kept constant. The PID controller adjusts the engine's power output to restore the vehicle to its desired speed, doing so efficiently with minimal delay and overshoot.
The theoretical foundation of PID controllers dates back to the early 1920s with the development of automatic steering systems for ships. This concept was later adopted for automatic process control in manufacturing, first appearing in pneumatic actuators and evolving into electronic controllers. PID controllers are widely used in numerous applications requiring accurate, stable, and optimized automatic control, such as temperature regulation, motor speed control, and industrial process management.
Fundamental operation
The distinguishing feature of the PID controller is the ability to use the three control terms of proportional, integral and derivative influence on the controller output to apply accurate and optimal control. The block diagram on the right shows the principles of how these terms are generated and applied. It shows a PID controller, which continuously calculates an error value as the difference between a desired setpoint and a measured process variable :, and applies a correction based on proportional, integral, and derivative terms. The controller attempts to minimize the error over time by adjustment of a control variable, such as the opening of a control valve, to a new value determined by a weighted sum of the control terms.The PID controller directly generates a continuous control signal based on error, without discrete modulation.
In this model:
- Term P is proportional to the current value of the SP − PV error. For example, if the error is large, the control output will be proportionately large by using the gain factor "Kp". Using proportional control alone will result in an error between the set point and the process value because the controller requires an error to generate the proportional output response. In steady state process conditions an equilibrium is reached, with a steady SP-PV "offset".
- Term I accounts for past values of the SP − PV error and integrates them over time to produce the I term. For example, if there is a residual SP − PV error after the application of proportional control, the integral term seeks to eliminate the residual error by adding a control effect due to the historic cumulative value of the error. When the error is eliminated, the integral term will cease to grow. This will result in the proportional effect diminishing as the error decreases, but this is compensated for by the growing integral effect.
- Term D is a best estimate of the future trend of the SP − PV error, based on its current rate of change. It is sometimes called "anticipatory control", as it is effectively seeking to reduce the effect of the SP − PV error by exerting a control influence generated by the rate of error change. The more rapid the change, the greater the controlling or damping effect.
Control action – The mathematical model and practical loop above both use a direct control action for all the terms, which means an increasing positive error results in an increasing positive control output correction. This is because the "error" term is not the deviation from the setpoint but is in fact the correction needed. The system is called reverse acting if it is necessary to apply negative corrective action. For instance, if the valve in the flow loop was 100–0% valve opening for 0–100% control output, meaning that the controller action has to be reversed. Some process control schemes and final control elements require this reverse action. An example would be a valve for cooling water, where the fail-safe mode, in the case of signal loss, would be 100% opening of the valve; therefore 0% controller output needs to cause 100% valve opening.
Control function
The overall control function iswhere,, and, all non-negative, denote the coefficients for the proportional, integral, and derivative terms respectively.
Standard form
In the standard form of the equation, and are respectively replaced by and ; the advantage of this being that and have some understandable physical meaning, as they represent an integration time and a derivative time respectively. is the time constant with which the controller will attempt to approach the set point. determines how long the controller will tolerate the output being consistently above or below the set point.where
Selective use of control terms
Although a PID controller has three control terms, some applications need only one or two terms to provide appropriate control. This is achieved by setting the unused parameters to zero and is called a PI, PD, P, or I controller in the absence of the other control actions. PI controllers are fairly common in applications where derivative action would be sensitive to measurement noise, but the integral term is often needed for the system to reach its target value.Applicability
The use of the PID algorithm does not guarantee optimal control of the system or its control stability. Situations may occur where there are excessive delays: the measurement of the process value is delayed, or the control action does not apply quickly enough. In these cases, lead–lag compensation is required to be effective. The response of the controller can be described in terms of its responsiveness to an error, the degree to which the system overshoots a setpoint, and the degree of any system oscillation. But the PID controller is broadly applicable since it relies only on the response of the measured process variable, not on knowledge or a model of the underlying process.History
Origins
The centrifugal governor was invented by Christiaan Huygens in the 17th century to regulate the gap between millstones in windmills depending on the speed of rotation, and thereby compensate for the variable speed of grain feed.With the invention of the low-pressure stationary steam engine, there was a need for automatic speed control, and James Watt's self-designed "conical pendulum" governor, a set of revolving steel balls attached to a vertical spindle by link arms, came to be an industry standard. This was based on the millstone-gap control concept.
Rotating-governor speed control, however, was still variable under conditions of varying load, where the shortcoming of what is now known as proportional control alone was evident. The error between the desired speed and the actual speed would increase with increasing load. In the 19th century, the theoretical basis for the operation of governors was first described by James Clerk Maxwell in 1868 in his now-famous paper On Governors. He explored the mathematical basis for control stability and progressed a good way towards a solution, but made an appeal for mathematicians to examine the problem. The problem was examined further in 1874 by Edward Routh, Charles Sturm, and in 1895, Adolf Hurwitz, all of whom contributed to the establishment of control stability criteria.
In subsequent applications, speed governors were further refined, notably by American scientist Willard Gibbs, who in 1872 theoretically analyzed Watt's conical pendulum governor.
About this time, the invention of the Whitehead torpedo posed a control problem that required accurate control of the running depth. Use of a depth pressure sensor alone proved inadequate, and a pendulum that measured the fore and aft pitch of the torpedo was combined with depth measurement to become the pendulum-and-hydrostat control. Pressure control provided only a proportional control that, if the control gain was too high, would become unstable and go into overshoot with considerable instability of depth-holding. The pendulum added what is now known as derivative control, which damped the oscillations by detecting the torpedo dive/climb angle and thereby the rate-of-change of depth. This development was around 1868.
Another early example of a PID-type controller was developed by Elmer Sperry in 1911 for ship steering, though his work was intuitive rather than mathematically-based.
It was not until 1922, however, that a formal control law for what we now call PID or three-term control was first developed using theoretical analysis, by Russian American engineer Nicolas Minorsky. Minorsky was researching and designing automatic ship steering for the US Navy and based his analysis on observations of a helmsman. He noted the helmsman steered the ship based not only on the current course error but also on past error, as well as the current rate of change; this was then given a mathematical treatment by Minorsky.
His goal was stability, not general control, which simplified the problem significantly. While proportional control provided stability against small disturbances, it was insufficient for dealing with a steady disturbance, notably a stiff gale, which required adding the integral term. Finally, the derivative term was added to improve stability and control.
Trials were carried out on the USS New Mexico, with the controllers controlling the angular velocity of the rudder. PI control yielded sustained yaw of ±2°. Adding the D element yielded a yaw error of ±1/6°, better than most helmsmen could achieve.
The Navy ultimately did not adopt the system due to resistance by personnel. Similar work was carried out and published by several others in the 1930s.