A proportional–integral–derivative controller (PID controller or three-term controller) is a
control loop
A control loop is the fundamental building block of industrial control systems. It consists of all the physical components and control functions necessary to automatically adjust the value of a measured process variable (PV) to equal the value of ...
mechanism employing
feedback that is widely used in
industrial control system
An industrial control system (ICS) is an electronic control system and associated instrumentation used for industrial process control. Control systems can range in size from a few modular panel-mounted controllers to large interconnected and in ...
s 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
In control theory, a process variable (PV; also process value or process parameter) is the current measured value of a particular part of a process which is being monitored or controlled. An example of this would be the temperature of a furnace ...
(PV) and applies a correction based on
proportional,
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
, and
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
terms (denoted ''P'', ''I'', and ''D'' respectively), hence the name.
In practical terms, PID automatically applies an accurate and responsive correction to a control function. An everyday example is the
cruise control
Cruise control (also known as speed control, cruise command, autocruise, or tempomat) is a system that automatically controls the speed of a motor vehicle. The system is a servomechanism that takes over the throttle of the car to maintain a s ...
on a car, where ascending a hill would lower speed if constant engine power were applied. The controller's PID algorithm restores the measured speed to the desired speed with minimal delay and overshoot by increasing the power output of the engine in a controlled manner.
The first theoretical analysis and practical application of PID was in the field of automatic steering systems for ships, developed from the early 1920s onwards. It was then used for automatic process control in the manufacturing industry, where it was widely implemented in at first pneumatic and then electronic
controllers. Today the PID concept is used universally in applications requiring accurate and optimized automatic control.
Fundamental operation
![PID en](https://upload.wikimedia.org/wikipedia/commons/4/43/PID_en.svg)
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
In control theory, a process variable (PV; also process value or process parameter) is the current measured value of a particular part of a process which is being monitored or controlled. An example of this would be the temperature of a furnace ...
:
, and applies a correction based on
proportional,
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
, and
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
terms. The controller attempts to minimize the error over time by adjustment of a ''control variable''
, such as the opening of a
control valve
A control valve is a valve used to control fluid flow by varying the size of the flow passage as directed by a signal from a controller. This enables the direct control of flow rate and the consequential control of process quantities such as pressu ...
, to a new value determined by a
weighted sum
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is ...
of the control terms.
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 "K
p". 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.
Tuning – The balance of these effects is achieved by
loop tuning to produce the optimal control function. The tuning constants are shown below as "K" and must be derived for each control application, as they depend on the response characteristics of the complete loop external to the controller. These are dependent on the behavior of the measuring sensor, the final control element (such as a control valve), any control signal delays and the process itself. Approximate values of constants can usually be initially entered knowing the type of application, but they are normally refined, or tuned, by "bumping" the process in practice by introducing a setpoint change and observing the system response.
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. 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
In engineering, a fail-safe is a design feature or practice that in the event of a specific type of failure, inherently responds in a way that will cause minimal or no harm to other equipment, to the environment or to people. Unlike inherent safe ...
mode, in the case of loss of signal, would be 100% opening of the valve; therefore 0% controller output needs to cause 100% valve opening.
Mathematical form
The overall control function
where
,
, and
, all non-negative, denote the coefficients for the proportional, integral, and derivative terms respectively (sometimes denoted ''P'', ''I'', and ''D'').
In the ''standard form'' of the equation (see later in article),
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.
:
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
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
. 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
Continuous control, before PID controllers were fully understood and implemented, has one of its origins in the
centrifugal governor
A centrifugal governor is a specific type of governor with a feedback system that controls the speed of an engine by regulating the flow of fuel or working fluid, so as to maintain a near-constant speed. It uses the principle of proportional con ...
, which uses rotating weights to control a process. This was invented by
Christiaan Huygens in the 17th century to regulate the gap between
millstones in
windmill
A windmill is a structure that converts wind power into rotational energy using vanes called sails or blades, specifically to mill grain (gristmills), but the term is also extended to windpumps, wind turbines, and other applications, in some ...
s 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
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and li ...
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
Edward John Routh (; 20 January 18317 June 1907), was an English mathematician, noted as the outstanding coach of students preparing for the Mathematical Tripos examination of the University of Cambridge in its heyday in the middle of the ninet ...
,
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
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in ...
, who in 1872 theoretically analyzed Watt's conical pendulum governor.
About this time, the invention of the
Whitehead torpedo
The Whitehead torpedo was the first self-propelled or "locomotive" torpedo ever developed. It was perfected in 1866 by Robert Whitehead from a rough design conceived by Giovanni Luppis of the Austro-Hungarian Navy in Fiume. It was driven by a t ...
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
Pendulum-and-hydrostat control is a control mechanism developed originally for depth control of the Whitehead torpedo. It is an early example of what is now known as proportional and derivative control.
The hydrostat is a mechanism that senses p ...
. 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
In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be mar ...
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 (named by Whitehead as "The Secret" to give no clue to its action) was around 1868.
Another early example of a PID-type controller was developed by
Elmer Sperry
Elmer Ambrose Sperry Sr. (October 12, 1860 – June 16, 1930) was an American inventor and entrepreneur, most famous for construction, two years after Hermann Anschütz-Kaempfe, of the gyrocompass and as founder of the Sperry Gyroscope Company. ...
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
Russian Americans ( rus, русские американцы, r=russkiye amerikantsy, p= ˈruskʲɪje ɐmʲɪrʲɪˈkant͡sɨ) are Americans of full or partial Russian ancestry. The term can apply to recent Russian immigrants to the United Stat ...
engineer
Nicolas Minorsky. Minorsky was researching and designing automatic ship steering for the US Navy and based his analysis on observations of a
helmsman
A helmsman or helm (sometimes driver) is a person who steering, steers a ship, sailboat, submarine, other type of maritime vessel, or spacecraft. The rank and seniority of the helmsman may vary: on small vessels such as fishing vessels and yacht ...
. 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 (due to
steady-state error), 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
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
'' (not the angle) of the rudder. PI control yielded sustained yaw (angular error) 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.
Industrial control
![Nozzle and flapper proportional controller](https://upload.wikimedia.org/wikipedia/commons/6/69/Nozzle_and_flapper_proportional_controller.png)
The wide use of feedback controllers did not become feasible until the development of wideband high-gain amplifiers to use the concept of negative feedback. This had been developed in telephone engineering electronics by
Harold Black
Harold Stephen Black (April 14, 1898 – December 11, 1983) was an American electrical engineer, who revolutionized the field of applied electronics by discovering the negative feedback amplifier in 1927. To some, his discovery is considered the ...
in the late 1920s, but not published until 1934.
Independently, Clesson E Mason of the Foxboro Company in 1930 invented a wide-band pneumatic controller by combining the
nozzle and flapper The nozzle and flapper mechanism is a displacement type detector which converts mechanical movement into a pressure signal by covering the opening of a nozzle with a flat plate called the flapper. This restricts fluid flow through the nozzle and gen ...
high-gain pneumatic amplifier, which had been invented in 1914, with negative feedback from the controller output. This dramatically increased the linear range of operation of the nozzle and flapper amplifier, and integral control could also be added by the use of a precision bleed valve and a bellows generating the integral term. The result was the "Stabilog" controller which gave both proportional and integral functions using feedback bellows.
The integral term was called ''Reset''. Later the derivative term was added by a further bellows and adjustable orifice.
From about 1932 onwards, the use of wideband pneumatic controllers increased rapidly in a variety of control applications. Air pressure was used for generating the controller output, and also for powering process modulating devices such as diaphragm-operated control valves. They were simple low maintenance devices that operated well in harsh industrial environments and did not present explosion risks in
hazardous locations. They were the industry standard for many decades until the advent of discrete electronic controllers and
distributed control system
A distributed control system (DCS) is a computerised control system for a process or plant usually with many control loops, in which autonomous controllers are distributed throughout the system, but there is no central operator supervisory contro ...
s (DCSs).
With these controllers, a pneumatic industry signaling standard of was established, which had an elevated zero to ensure devices were working within their linear characteristic and represented the control range of 0-100%.
In the 1950s, when high gain electronic amplifiers became cheap and reliable, electronic PID controllers became popular, and the pneumatic standard was emulated by 10-50 mA and 4–20 mA
current loop
In electrical signalling an analog current loop is used where a device must be monitored or controlled remotely over a pair of conductors. Only one current level can be present at any time.
A major application of current loops is the industry de ...
signals (the latter became the industry standard). Pneumatic field actuators are still widely used because of the advantages of pneumatic energy for control valves in process plant environments.
![Smart current loop positioner](https://upload.wikimedia.org/wikipedia/commons/9/93/Smart_current_loop_positioner.png)
Most modern PID controls in industry are implemented as
computer software
Software is a set of computer programs and associated documentation and data. This is in contrast to hardware, from which the system is built and which actually performs the work.
At the lowest programming level, executable code consists ...
in DCSs,
programmable logic controller
A programmable logic controller (PLC) or programmable controller is an industrial computer that has been ruggedized and adapted for the control of manufacturing processes, such as assembly lines, machines, robotic devices, or any activity tha ...
s (PLCs), or discrete
compact controller
A compact controller is a generic name given to a small autonomous controller which can control one or several control loops. They are also known as panel mounted, discrete, dedicated, or universal process controllers. The controllers can be easily ...
s.
Electronic analog controllers
Electronic analog PID control loops were often found within more complex electronic systems, for example, the head positioning of a
disk drive
Disk storage (also sometimes called drive storage) is a general category of storage mechanisms where data is recorded by various electronic, magnetic, optical, or mechanical changes to a surface layer of one or more rotating disks. A disk drive is ...
, the power conditioning of a
power supply
A power supply is an electrical device that supplies electric power to an electrical load. The main purpose of a power supply is to convert electric current from a source to the correct voltage, current, and frequency to power the load. As a r ...
, or even the movement-detection circuit of a modern
seismometer
A seismometer is an instrument that responds to ground noises and shaking such as caused by earthquakes, volcanic eruptions, and explosions. They are usually combined with a timing device and a recording device to form a seismograph. The outpu ...
. Discrete electronic analog controllers have been largely replaced by digital controllers using
microcontrollers
A microcontroller (MCU for ''microcontroller unit'', often also MC, UC, or μC) is a small computer on a single VLSI integrated circuit (IC) chip. A microcontroller contains one or more CPUs ( processor cores) along with memory and programmabl ...
or
FPGA
A field-programmable gate array (FPGA) is an integrated circuit designed to be configured by a customer or a designer after manufacturinghence the term '' field-programmable''. The FPGA configuration is generally specified using a hardware de ...
s to implement PID algorithms. However, discrete analog PID controllers are still used in niche applications requiring high-bandwidth and low-noise performance, such as laser-diode controllers.
Control loop example
Consider a
robotic arm
A robotic arm is a type of mechanical arm, usually programmable, with similar functions to a human arm; the arm may be the sum total of the mechanism or may be part of a more complex robot. The links of such a manipulator are connected by joints ...
that can be moved and positioned by a control loop. An
electric motor
An electric motor is an Electric machine, electrical machine that converts electrical energy into mechanical energy. Most electric motors operate through the interaction between the motor's magnetic field and electric current in a Electromagneti ...
may lift or lower the arm, depending on forward or reverse power applied, but power cannot be a simple function of position because of the
inertial mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
of the arm, forces due to gravity, external forces on the arm such as a load to lift or work to be done on an external object.
* The sensed position is the
process variable
In control theory, a process variable (PV; also process value or process parameter) is the current measured value of a particular part of a process which is being monitored or controlled. An example of this would be the temperature of a furnace ...
(PV).
* The desired position is called the setpoint (SP).
* The difference between the PV and SP is the error (e), which quantifies whether the arm is too low or too high and by how much.
* The input to the process (the
electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
in the motor) is the output from the PID controller. It is called either the manipulated variable (MV) or the control variable (CV).
By measuring the position (PV), and subtracting it from the setpoint (SP), the error (e) is found, and from it the controller calculates how much electric current to supply to the motor (MV).
Proportional
The obvious method is proportional control: the motor current is set in proportion to the existing error. However, this method fails if, for instance, the arm has to lift different weights: a greater weight needs a greater force applied for the same error on the down side, but a smaller force if the error is low on the upside. That's where the integral and derivative terms play their part.
Integral
An integral term increases action in relation not only to the error but also the time for which it has persisted. So, if the applied force is not enough to bring the error to zero, this force will be increased as time passes. A pure "I" controller could bring the error to zero, but it would be both slow reacting at the start (because the action would be small at the beginning, depending on time to get significant) and brutal at the end (the action increases as long as the error is positive, even if the error has started to approach zero).
Applying too much integral when the error is small and decreasing will lead to overshoot. After overshooting, if the controller were to apply a large correction in the opposite direction and repeatedly overshoot the desired position, the output would
oscillate
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
around the setpoint in either a constant, growing, or decaying
sinusoid
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ma ...
. If the amplitude of the oscillations increases with time, the system is unstable. If they decrease, the system is stable. If the oscillations remain at a constant magnitude, the system is
marginally stable.
Derivative
A derivative term does not consider the magnitude of the error (meaning it cannot bring it to zero: a pure D controller cannot bring the system to its setpoint), but the rate of change of error, trying to bring this rate to zero. It aims at flattening the error trajectory into a horizontal line, damping the force applied, and so reduces overshoot (error on the other side because of too great applied force).
Control damping
In the interest of achieving a controlled arrival at the desired position (SP) in a timely and accurate way, the controlled system needs to be
critically damped
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...
. A well-tuned position control system will also apply the necessary currents to the controlled motor so that the arm pushes and pulls as necessary to resist external forces trying to move it away from the required position. The setpoint itself may be generated by an external system, such as a
PLC or other computer system, so that it continuously varies depending on the work that the robotic arm is expected to do. A well-tuned PID control system will enable the arm to meet these changing requirements to the best of its capabilities.
Response to disturbances
If a controller starts from a stable state with 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 affect the process, and hence the PV. Variables that affect the process other than the MV are known as disturbances. Generally, controllers are used to reject disturbances and to implement setpoint changes. A change in load on the arm constitutes a disturbance to the robot arm control process.
Applications
In theory, a controller can be used to control any process that 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
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
,
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
,
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
,
feed rate
The phrase speeds and feeds or feeds and speeds refers to two separate velocities in machine tool practice, cutting speed and feed rate. They are often considered as a pair because of their combined effect on the cutting process. Each, however, ...
,
flow rate, chemical composition (component
concentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', ''molar concentration'', ''number concentration'', an ...
s),
weight
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
,
position
Position often refers to:
* Position (geometry), the spatial location (rather than orientation) of an entity
* Position, a job or occupation
Position may also refer to:
Games and recreation
* Position (poker), location relative to the dealer
* ...
,
speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quanti ...
, and practically every other variable for which a measurement exists.
Controller theory
:''This section describes the parallel or non-interacting form of the PID controller. For other forms please see the section
Alternative nomenclature and forms.''
The PID control scheme is named after its three correcting terms, whose sum constitutes the manipulated variable (MV). The proportional, integral, and derivative terms are summed to calculate the output of the PID controller. Defining
as the controller output, the final form of the PID algorithm is
:
where
:
is the proportional gain, a tuning parameter,
:
is the integral gain, a tuning parameter,
:
is the derivative gain, a tuning parameter,
:
is the error (SP is the setpoint, and PV(''t'') is the process variable),
:
is the time or instantaneous time (the present),
:
is the variable of integration (takes on values from time 0 to the present
).
Equivalently, the
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
in the
Laplace domain
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the ...
of the PID controller is
:
where
is the complex frequency.
Proportional term
![PID varyingP](https://upload.wikimedia.org/wikipedia/commons/a/a3/PID_varyingP.jpg)
The proportional term produces an output value that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant ''K''
p, called the proportional gain constant.
The proportional term is given by
:
A 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 less sensitive controller. If the proportional gain is too low, the control action may be too small when responding to system disturbances. Tuning theory and industrial practice indicate that the proportional term should contribute the bulk of the output change.
Steady-state error
The steady-state error is the difference between the desired final output and the actual one. Because a non-zero error is required to drive it, a proportional controller generally operates with a steady-state error. Steady-state error (SSE) is proportional to the process gain and inversely proportional to proportional gain. SSE may be mitigated by adding a compensating
bias term to the setpoint AND output or corrected dynamically by adding an integral term.
Integral term
![Change with Ki](https://upload.wikimedia.org/wikipedia/commons/c/c0/Change_with_Ki.png)
The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error. The
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
in a PID controller is the sum of the instantaneous error over time and gives the accumulated offset that should have been corrected previously. The accumulated error is then multiplied by the integral gain (''K''
i) and added to the controller output.
The integral term is given by
:
The integral term accelerates the movement of the process towards setpoint and eliminates the residual steady-state error that occurs with a pure proportional controller. However, since the integral term responds to accumulated errors from the past, it can cause the present value to
overshoot the setpoint value (see
the section on loop tuning).
Derivative term
![Change with Kd](https://upload.wikimedia.org/wikipedia/commons/c/c7/Change_with_Kd.png)
The derivative of the process error is calculated by determining the slope of the error over time and multiplying this rate of change by the derivative gain ''K''
d. The magnitude of the contribution of the derivative term to the overall control action is termed the derivative gain, ''K''
d.
The derivative term is given by
:
Derivative action predicts system behavior and thus improves settling time and stability of the system. An ideal derivative is not
causal
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
, so that implementations of PID controllers include an additional low-pass filtering for the derivative term to limit the high-frequency gain and noise. Derivative action is seldom used in practice though – by one estimate in only 25% of deployed controllers – because of its variable impact on system stability in real-world applications.
Loop tuning
''Tuning'' a control loop is the adjustment of its control parameters (proportional band/gain, integral gain/reset, derivative gain/rate) to the optimum values for the desired control response. Stability (no unbounded oscillation) is a basic requirement, but beyond that, different systems have different behavior, different applications have different requirements, and requirements may conflict with one another.
Even though there are only three parameters and it is simple to describe in principle, PID tuning is a difficult problem because it must satisfy complex criteria within the
limitations of PID control. Accordingly, there are various methods for loop tuning, and more sophisticated techniques are the subject of patents; this section describes some traditional, manual methods for loop tuning.
Designing and tuning a PID controller appears to be conceptually intuitive, but can be hard in practice, if multiple (and often conflicting) objectives, such as short transient and high stability, are to be achieved. PID controllers often provide acceptable control using default tunings, but performance can generally be improved by careful tuning, and performance may be unacceptable with poor tuning. Usually, initial designs need to be adjusted repeatedly through computer simulations until the closed-loop system performs or compromises as desired.
Some processes have a degree of
nonlinearity
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
, so parameters that work well at full-load conditions do not work when the process is starting up from no load. This can be corrected by
gain scheduling
In control theory, gain scheduling is an approach to control of non-linear systems that uses a family of linear controllers, each of which provides satisfactory control for a different operating point of the system.
One or more observable variable ...
(using different parameters in different operating regions).
Stability
If the PID controller parameters (the gains of the proportional, integral and derivative terms) are chosen incorrectly, the controlled process input can be unstable; i.e., its output
diverges, with or without
oscillation
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
, and is limited only by saturation or mechanical breakage. Instability is caused by ''excess'' gain, particularly in the presence of significant lag.
Generally, stabilization of response is required and the process must not oscillate for any combination of process conditions and setpoints, though sometimes
marginal stability
In the theory of dynamical systems and control theory, a linear time-invariant system is marginally stable if it is neither asymptotically stable nor unstable. Roughly speaking, a system is stable if it always returns to and stays near a particul ...
(bounded oscillation) is acceptable or desired.
Mathematically, the origins of instability can be seen in the
Laplace domain
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the ...
.
The total loop transfer function is:
:
where
is the PID transfer function and
is the plant transfer function. A system is ''unstable'' where the closed loop transfer function diverges for some
.
This happens in situations where
. Typically, this happens when
with a 180-degree phase shift. Stability is guaranteed when
for frequencies that suffer high phase shifts. A more general formalism of this effect is known as the
Nyquist stability criterion
In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 and the Swedish-American electrical engineer Harry ...
.
Optimal behavior
The optimal behavior on a process change or setpoint change varies depending on the application.
Two basic requirements are ''regulation'' (disturbance rejection – staying at a given setpoint) and ''command tracking'' (implementing setpoint changes). These terms refer to how well the controlled variable tracks the desired value. Specific criteria for command tracking include
rise time In electronics, when describing a voltage or current step function, rise time is the time taken by a signal to change from a specified low value to a specified high value. These values may be expressed as ratiosSee for example , and . or, equivalen ...
and
settling time
In control theory the settling time of a dynamical system such as an amplifier or other output device is the time elapsed from the application of an ideal instantaneous step input to the time at which the amplifier output has entered and remained ...
. Some processes must not allow an overshoot of the process variable beyond the setpoint if, for example, this would be unsafe. Other processes must minimize the energy expended in reaching a new setpoint.
Overview of tuning methods
There are several methods for tuning a PID loop. The most effective methods generally involve developing some form of process model and then choosing P, I, and D based on the dynamic model parameters. Manual tuning methods can be relatively time-consuming, particularly for systems with long loop times.
The choice of method depends largely on whether the loop can be taken offline for tuning, and on the response time of the system. If the system can be taken offline, the best tuning method often involves subjecting the system to a step change in input, measuring the output as a function of time, and using this response to determine the control parameters.
Manual tuning
If the system must remain online, one tuning method is to first set
and
values to zero. Increase the
until the output of the loop oscillates; then set
to approximately half that value for a "quarter amplitude decay"-type response. Then increase
until any offset is corrected in sufficient time for the process, but not until too great a value causes instability. Finally, increase
, if required, until the loop is acceptably quick to reach its reference after a load disturbance. Too much
causes excessive response and overshoot. A fast PID loop tuning usually overshoots slightly to reach the setpoint more quickly; however, some systems cannot accept overshoot, in which case an
overdamped
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
closed-loop system is required, which in turn requires a
setting significantly less than half that of the
setting that was causing oscillation.
Ziegler–Nichols method
Another heuristic tuning method is known as the
Ziegler–Nichols method
The Ziegler–Nichols tuning method is a heuristic method of tuning a PID controller. It was developed by John G. Ziegler and Nathaniel B. Nichols. It is performed by setting the ''I'' (integral) and ''D'' (derivative) gains to zero. The "P" (prop ...
, introduced by
John G. Ziegler
John G. Ziegler (August 21, 1909 – December 9, 1997) was an American control engineer who made significant contributions to the field of control theory. He is well known for his research paper on Ziegler-Nichols method, co-authored with Natha ...
and
Nathaniel B. Nichols
Nathaniel B. Nichols (1914–1997) was an American control engineer who made significant contributions to the field of control theory. He is well known for his book ''Theory of Servomechanisms'', one of the most widely read books in control engin ...
in the 1940s. As in the method above, the
and
gains are first set to zero. The proportional gain is increased until it reaches the ultimate gain,
, at which the output of the loop starts to oscillate constantly.
and the oscillation period
are used to set the gains as follows:
These gains apply to the ideal, parallel form of the PID controller. When applied to the standard PID form, only the integral and derivative gains
and
are dependent on the oscillation period
.
Cohen–Coon parameters
This method was developed in 1953 and is based on a first-order + time delay model. Similar to the
Ziegler–Nichols method
The Ziegler–Nichols tuning method is a heuristic method of tuning a PID controller. It was developed by John G. Ziegler and Nathaniel B. Nichols. It is performed by setting the ''I'' (integral) and ''D'' (derivative) gains to zero. The "P" (prop ...
, a set of tuning parameters were developed to yield a closed-loop response with a decay ratio of
. Arguably the biggest problem with these parameters is that a small change in the process parameters could potentially cause a closed-loop system to become unstable.
Relay (Åström–Hägglund) method
Published in 1984 by
Karl Johan Åström
Karl Johan Åström (born August 5, 1934) is a Swedish control theorist, who has made contributions to the fields of control theory and control engineering, computer control and adaptive control. In 1965, he described a general framework of Marko ...
and Tore Hägglund, the relay method temporarily operates the process using
bang-bang control and measures the resultant oscillations. The output is switched (as if by a
relay
A relay
Electromechanical relay schematic showing a control coil, four pairs of normally open and one pair of normally closed contacts
An automotive-style miniature relay with the dust cover taken off
A relay is an electrically operated switch ...
, hence the name) between two values of the control variable. The values must be chosen so the process will cross the setpoint, but they need not be 0% and 100%; by choosing suitable values, dangerous oscillations can be avoided.
As long as the process variable is below the setpoint, the control output is set to the higher value. As soon as it rises above the setpoint, the control output is set to the lower value. Ideally, the output waveform is nearly square, spending equal time above and below the setpoint. The period and amplitude of the resultant oscillations are measured, and used to compute the ultimate gain and period, which are then fed into the Ziegler–Nichols method.
Specifically, the ultimate period
is assumed to be equal to the observed period, and the ultimate gain is computed as
where is the amplitude of the process variable oscillation, and is the amplitude of the control output change which caused it.
There are numerous variants on the relay method.
First order with dead time model
The transfer function for a first-order process, with dead time, is:
where k
p is the process gain, τ
p is the time constant, θ is the dead time, and u(s) is a step change input. Converting this transfer function to the time domain results in:
using the same parameters found above.
It is important when using this method to apply a large enough step change input that the output can be measured; however, too large of a step change can affect the process stability. Additionally, a larger step change ensures that the output does not change due to a disturbance (for best results, try to minimize disturbances when performing the step test).
One way to determine the parameters for the first-order process is using the 63.2% method. In this method, the process gain (k
p) is equal to the change in output divided by the change in input. The dead time (θ) is the amount of time between when the step change occurred and when the output first changed. The time constant (τ
p) is the amount of time it takes for the output to reach 63.2% of the new steady-state value after the step change. One downside to using this method is that it can take a while to reach a new steady-state value if the process has large time constants.
Tuning software
Most modern industrial facilities no longer tune loops using the manual calculation methods shown above. Instead, PID tuning and loop optimization software are used to ensure consistent results. These software packages gather data, develop process models, and suggest optimal tuning. Some software packages can even develop tuning by gathering data from reference changes.
Mathematical PID loop tuning induces an impulse in the system and then uses the controlled system's frequency response to design the PID loop values. In loops with response times of several minutes, mathematical loop tuning is recommended, because trial and error can take days just to find a stable set of loop values. Optimal values are harder to find. Some digital loop controllers offer a self-tuning feature in which very small setpoint changes are sent to the process, allowing the controller itself to calculate optimal tuning values.
Another approach calculates initial values via the Ziegler–Nichols method, and uses a numerical optimization technique to find better PID coefficients.
Other formulas are available to tune the loop according to different performance criteria. Many patented formulas are now embedded within PID tuning software and hardware modules.
Advances in automated PID loop tuning software also deliver algorithms for tuning PID Loops in a dynamic or non-steady state (NSS) scenario. The software models the dynamics of a process, through a disturbance, and calculate PID control parameters in response.
Limitations
While PID controllers are applicable to many control problems, and often perform satisfactorily without any improvements or only coarse tuning, they can perform poorly in some applications and do not in general provide
''optimal'' control. The fundamental difficulty with PID control is that it is a feedback control system, with ''constant'' parameters, and no direct knowledge of the process, and thus overall performance is reactive and a compromise. While PID control is the best controller in an
observer
An observer is one who engages in observation or in watching an experiment.
Observer may also refer to:
Computer science and information theory
* In information theory, any system which receives information from an object
* State observer in co ...
without a model of the process, better performance can be obtained by overtly modeling the actor of the process without resorting to an observer.
PID controllers, when used alone, can give poor performance when the PID loop gains must be reduced so that the control system does not overshoot, oscillate or
hunt
Hunting is the human practice of seeking, pursuing, capturing, or killing wildlife or feral animals. The most common reasons for humans to hunt are to harvest food (i.e. meat) and useful animal products (fur/ hide, bone/tusks, horn/antler, e ...
about the control setpoint value. They also have difficulties in the presence of non-linearities, may trade-off regulation versus response time, do not react to changing process behavior (say, the process changes after it has warmed up), and have lag in responding to large disturbances.
The most significant improvement is to incorporate
feed-forward control
A feed forward (sometimes written feedforward) is an element or pathway within a control system that passes a controlling signal from a source in its external environment to a load elsewhere in its external environment. This is often a comman ...
with knowledge about the system, and using the PID only to control error. Alternatively, PIDs can be modified in more minor ways, such as by changing the parameters (either gain scheduling in different use cases or adaptively modifying them based on performance), improving measurement (higher sampling rate, precision, and accuracy, and low-pass filtering if necessary), or cascading multiple PID controllers.
Linearity and symmetry
PID controllers work best when the loop to be controlled is linear and symmetric. Thus, their performance in non-linear and asymmetric systems is degraded.
A non-linear valve, for instance, in a flow control application, will result in variable loop sensitivity, requiring dampened action to prevent instability. One solution is the use of the valve's non-linear characteristic in the control algorithm to compensate for this.
An asymmetric application, for example, is temperature control in
HVAC systems using only active heating (via a heating element), where there is only passive cooling available. When it is desired to lower the controlled temperature the heating output is off, but there is no active cooling due to control output. Any overshoot of rising temperature can therefore only be corrected slowly; it cannot be forced downward by the control output. In this case the PID controller could be tuned to be over-damped, to prevent or reduce overshoot, but this reduces performance by increasing the settling time of a rising temperature to the set point. The inherent degradation of control quality in this application could be solved by application of active cooling.
Noise in derivative term
A problem with the derivative term is that it amplifies higher frequency measurement or process
noise
Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arise ...
that can cause large amounts of change in the output. It is often helpful to filter the measurements with a
low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
in order to remove higher-frequency noise components. As low-pass filtering and derivative control can cancel each other out, the amount of filtering is limited. Therefore, low noise instrumentation can be important. A nonlinear
median filter
The median filter is a non-linear digital filtering technique, often used to remove noise from an image or signal. Such noise reduction is a typical pre-processing step to improve the results of later processing (for example, edge detection on an ...
may be used, which improves the filtering efficiency and practical performance. In some cases, the differential band can be turned off with little loss of control. This is equivalent to using the PID controller as a
PI controller
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 continuousl ...
.
Modifications to the algorithm
The basic PID algorithm presents some challenges in control applications that have been addressed by minor modifications to the PID form.
Integral windup
One common problem resulting from the ideal PID implementations is
integral windup
Integral windup, also known as integrator windup or reset windup, refers to the situation in a PID feedback controller where a large change in setpoint occurs (say a positive change) and the integral term accumulates a significant error during the ...
. Following a large change in setpoint the integral term can accumulate an error larger than the maximal value for the regulation variable (windup), thus the system overshoots and continues to increase until this accumulated error is unwound. This problem can be addressed by:
*Disabling the integration until the PV has entered the controllable region
*Preventing the integral term from accumulating above or below pre-determined bounds
*Back-calculating the integral term to constrain the regulator output within feasible bounds.
Overshooting from known disturbances
For example, a PID loop is used to control the temperature of an electric resistance furnace where the system has stabilized. Now when the door is opened and something cold is put into the furnace the temperature drops below the setpoint. The integral function of the controller tends to compensate for error by introducing another error in the positive direction. This overshoot can be avoided by freezing of the integral function after the opening of the door for the time the control loop typically needs to reheat the furnace.
PI controller
![PI controller](https://upload.wikimedia.org/wikipedia/commons/a/af/PI_controller.svg)
A PI controller (proportional-integral controller) is a special case of the PID controller in which the derivative (D) of the error is not used.
The controller output is given by
:
where
is the error or deviation of actual measured value (''PV'') from the setpoint (''SP'').
:
A PI controller can be modelled easily in software such as
Simulink
Simulink is a MATLAB-based graphical programming environment for modeling, simulating and analyzing multidomain dynamical systems. Its primary interface is a graphical block diagramming tool and a customizable set of block libraries. It offers t ...
or
Xcos
Scilab is a free and open-source, cross-platform numerical computational package and a high-level, numerically oriented programming language. It can be used for signal processing, statistical analysis, image enhancement, fluid dynamics simu ...
using a "flow chart" box involving
Laplace operators:
:
where
:
= proportional gain
:
= integral gain
Setting a value for
is often a trade off between decreasing overshoot and increasing settling time.
The lack of derivative action may make the system more steady in the steady state in the case of noisy data. This is because derivative action is more sensitive to higher-frequency terms in the inputs.
Without derivative action, a PI-controlled system is less responsive to real (non-noise) and relatively fast alterations in state and so the system will be slower to reach setpoint and slower to respond to perturbations than a well-tuned PID system may be.
Deadband
Many PID loops control a mechanical device (for example, a valve). Mechanical maintenance can be a major cost and wear leads to control degradation in the form of either
stiction
Stiction is the static friction that needs to be overcome to enable relative motion of stationary objects in contact. The term is a portmanteau of the words ''static'' and ''friction'', and is perhaps also influenced by the verb '' to stick''.
Any ...
or
backlash
Backlash may refer to:
Literature
* '' Backlash: The Undeclared War Against American Women'', a 1991 book by Susan Faludi
* ''Backlash'' (Star Wars novel), a 2010 novel by Aaron Allston
* Backlash (Marc Slayton), comic book character
* ''Backla ...
in the mechanical response to an input signal. The rate of mechanical wear is mainly a function of how often a device is activated to make a change. Where wear is a significant concern, the PID loop may have an output
deadband
A deadband or dead-band (also known as a dead zone or a neutral zone) is a band of input values in the domain of a transfer function in a control system or signal processing system where the output is zero (the output is 'dead' - no action occurs) ...
to reduce the frequency of activation of the output (valve). This is accomplished by modifying the controller to hold its output steady if the change would be small (within the defined deadband range). The calculated output must leave the deadband before the actual output will change.
Setpoint step change
The proportional and derivative terms can produce excessive movement in the output when a system is subjected to an instantaneous step increase in the error, such as a large setpoint change. In the case of the derivative term, this is due to taking the derivative of the error, which is very large in the case of an instantaneous step change. As a result, some PID algorithms incorporate some of the following modifications:
;Setpoint ramping
:In this modification, the setpoint is gradually moved from its old value to a newly specified value using a linear or first-order differential ramp function. This avoids the
discontinuity present in a simple step change.
;Derivative of the process variable
:In this case the PID controller measures the derivative of the measured
process variable
In control theory, a process variable (PV; also process value or process parameter) is the current measured value of a particular part of a process which is being monitored or controlled. An example of this would be the temperature of a furnace ...
(PV), rather than the derivative of the error. This quantity is always continuous (i.e., never has a step change as a result of changed setpoint). This modification is a simple case of setpoint weighting.
;Setpoint weighting
:Setpoint weighting adds adjustable factors (usually between 0 and 1) to the setpoint in the error in the proportional and derivative element of the controller. The error in the integral term must be the true control error to avoid steady-state control errors. These two extra parameters do not affect the response to load disturbances and measurement noise and can be tuned to improve the controller's setpoint response.
Feed-forward
The control system performance can be improved by combining the
feedback (or closed-loop) control of a PID controller with
feed-forward
Feedforward is the provision of context of what one wants to communicate prior to that communication. In purposeful activity, feedforward creates an expectation which the actor anticipates. When expected experience occurs, this provides confirmato ...
(or open-loop) control. Knowledge about the system (such as the desired acceleration and inertia) can be fed forward and combined with the PID output to improve the overall system performance. The feed-forward value alone can often provide the major portion of the controller output. The PID controller primarily has to compensate for whatever difference or ''error'' remains between the setpoint (SP) and the system response to the open-loop control. Since the feed-forward output is not affected by the process feedback, it can never cause the control system to oscillate, thus improving the system response without affecting stability. Feed forward can be based on the setpoint and on extra measured disturbances. Setpoint weighting is a simple form of feed forward.
For example, in most motion control systems, in order to accelerate a mechanical load under control, more force is required from the actuator. If a velocity loop PID controller is being used to control the speed of the load and command the force being applied by the actuator, then it is beneficial to take the desired instantaneous acceleration, scale that value appropriately and add it to the output of the PID velocity loop controller. This means that whenever the load is being accelerated or decelerated, a proportional amount of force is commanded from the actuator regardless of the feedback value. The PID loop in this situation uses the feedback information to change the combined output to reduce the remaining difference between the process setpoint and the feedback value. Working together, the combined open-loop feed-forward controller and closed-loop PID controller can provide a more responsive control system.
Bumpless operation
PID controllers are often implemented with a "bumpless" initialization feature that recalculates the integral accumulator term to maintain a consistent process output through parameter changes. A partial implementation is to store the integral gain times the error rather than storing the error and postmultiplying by the integral gain, which prevents discontinuous output when the I gain is changed, but not the P or D gains.
Other improvements
In addition to feed-forward, PID controllers are often enhanced through methods such as PID
gain scheduling
In control theory, gain scheduling is an approach to control of non-linear systems that uses a family of linear controllers, each of which provides satisfactory control for a different operating point of the system.
One or more observable variable ...
(changing parameters in different operating conditions),
fuzzy logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
, or computational verb logic. Further practical application issues can arise from instrumentation connected to the controller. A high enough sampling rate, measurement precision, and measurement accuracy are required to achieve adequate control performance. Another new method for improvement of PID controller is to increase the degree of freedom by using
fractional order. The order of the integrator and differentiator add increased flexibility to the controller.
Cascade control
One distinctive advantage of PID controllers is that two PID controllers can be used together to yield better dynamic performance. This is called cascaded PID control. Two controllers are in cascade when they are arranged so that one regulates the set point of the other. A PID controller acts as outer loop controller, which controls the primary physical parameter, such as fluid level or velocity. The other controller acts as inner loop controller, which reads the output of outer loop controller as setpoint, usually controlling a more rapid changing parameter, flowrate or acceleration. It can be mathematically proven that the working frequency of the controller is increased and the time constant of the object is reduced by using cascaded PID controllers..
For example, a temperature-controlled circulating bath has two PID controllers in cascade, each with its own thermocouple temperature sensor. The outer controller controls the temperature of the water using a thermocouple located far from the heater, where it accurately reads the temperature of the bulk of the water. The error term of this PID controller is the difference between the desired bath temperature and measured temperature. Instead of controlling the heater directly, the outer PID controller sets a heater temperature goal for the inner PID controller. The inner PID controller controls the temperature of the heater using a thermocouple attached to the heater. The inner controller's error term is the difference between this heater temperature setpoint and the measured temperature of the heater. Its output controls the actual heater to stay near this setpoint.
The proportional, integral, and differential terms of the two controllers will be very different. The outer PID controller has a long time constant – all the water in the tank needs to heat up or cool down. The inner loop responds much more quickly. Each controller can be tuned to match the physics of the system ''it'' controls – heat transfer and thermal mass of the whole tank or of just the heater – giving better total response.
Alternative nomenclature and forms
Standard versus parallel (ideal) form
The form of the PID controller most often encountered in industry, and the one most relevant to tuning algorithms is the ''standard form''. In this form the
gain is applied to the
, and
terms, yielding:
:
where
:
is the ''integral time''
:
is the ''derivative time''
In this standard form, the parameters have a clear physical meaning. In particular, the inner summation produces a new single error value which is compensated for future and past errors. The proportional error term is the current error. The derivative components term attempts to predict the error value at
seconds (or samples) in the future, assuming that the loop control remains unchanged. The integral component adjusts the error value to compensate for the sum of all past errors, with the intention of completely eliminating them in
seconds (or samples). The resulting compensated single error value is then scaled by the single gain
to compute the control variable.
In the parallel form, shown in the controller theory section
:
the gain parameters are related to the parameters of the standard form through
and
. This parallel form, where the parameters are treated as simple gains, is the most general and flexible form. However, it is also the form where the parameters have the weakest relationship to physical behaviors and is generally reserved for theoretical treatment of the PID controller. The standard form, despite being slightly more complex mathematically, is more common in industry.
Reciprocal gain, a.k.a. proportional band
In many cases, the manipulated variable output by the PID controller is a dimensionless fraction between 0 and 100% of some maximum possible value, and the translation into real units (such as pumping rate or watts of heater power) is outside the PID controller. The process variable, however, is in dimensioned units such as temperature. It is common in this case to express the gain
not as "output per degree", but rather in the reciprocal form of a ''proportional band''
, which is "degrees per full output": the range over which the output changes from 0 to 1 (0% to 100%). Beyond this range, the output is saturated, full-off or full-on. The narrower this band, the higher the proportional gain.
Basing derivative action on PV
In most commercial control systems, derivative action is based on process variable rather than error. That is, a change in the setpoint does not affect the derivative action. This is because the digitized version of the algorithm produces a large unwanted spike when the setpoint is changed. If the setpoint is constant then changes in the PV will be the same as changes in error. Therefore, this modification makes no difference to the way the controller responds to process disturbances.
Basing proportional action on PV
Most commercial control systems offer the ''option'' of also basing the proportional action solely on the process variable. This means that only the integral action responds to changes in the setpoint. The modification to the algorithm does not affect the way the controller responds to process disturbances.
Basing proportional action on PV eliminates the instant and possibly very large change in output caused by a sudden change to the setpoint. Depending on the process and tuning this may be beneficial to the response to a setpoint step.
:
King describes an effective chart-based method.
Laplace form
Sometimes it is useful to write the PID regulator in
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
form:
:
Having the PID controller written in Laplace form and having the transfer function of the controlled system makes it easy to determine the closed-loop transfer function of the system.
Series/interacting form
Another representation of the PID controller is the series, or ''interacting'' form
:
where the parameters are related to the parameters of the standard form through
:
,
, and
:
with
:
.
This form essentially consists of a PD and PI controller in series. As the integral is required to calculate the controller's bias this form provides the ability to track an external bias value which is required to be used for proper implementation of multi-controller advanced control schemes.
Discrete implementation
The analysis for designing a digital implementation of a PID controller in a
microcontroller
A microcontroller (MCU for ''microcontroller unit'', often also MC, UC, or μC) is a small computer on a single VLSI integrated circuit (IC) chip. A microcontroller contains one or more CPUs (processor cores) along with memory and programmable i ...
(MCU) or
FPGA
A field-programmable gate array (FPGA) is an integrated circuit designed to be configured by a customer or a designer after manufacturinghence the term '' field-programmable''. The FPGA configuration is generally specified using a hardware de ...
device requires the standard form of the PID controller to be ''discretized''.
Approximations for first-order derivatives are made by backward
finite difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...
s.
and
are discretized with a sampling period
, k is the sample index.
Differentiating both sides of PID equation using
Newton's notation
In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with t ...
gives:
Derivative terms are approximated as,
:
So,
:
Applying backward difference again gives,
:
By simplifying and regrouping terms of the above equation, an algorithm for an implementation of the discretized PID controller in a MCU is finally obtained:
:
or:
: