Proportional–integral–derivative controller

 ( Classical Control Theory ) 

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 systems (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 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.

Most modern PID controls in industry are implemented as computer software in DCSs, programmable logic controllers (PLCs), or discrete compact controllers.

• The sensed position is the process variable (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 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).

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 around the setpoint in either a constant, growing, or decaying sinusoid. If the amplitude of the oscillations increases with time, the system is unstable. If it decreases, the system is stable. If the oscillations remain at a constant magnitude, the system is marginally stable.

This section describes the parallel or non-interacting form of the PID controller. For other forms please see .

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 $u(t)$ as the controller output, the final form of the PID algorithm is

$u(t)\; =\; \backslash mathrm\{MV\}(t)\; =\; K\_\backslash text\{p\}\; e(t)\; +\; K\_\backslash text\{i\}\; \backslash int\_0^t\; e(\backslash tau)\; \backslash ,d\backslash tau\; +\; K\_\backslash text\{d\}\; \backslash frac\{de(t)\}\{dt\},$

where

$K\_\backslash text\{p\}$ is the proportional gain, a tuning parameter,

$K\_\backslash text\{i\}$ is the integral gain, a tuning parameter,

$K\_\backslash text\{d\}$ is the derivative gain, a tuning parameter,

$e(t)\; =\; \backslash mathrm\{SP\}\; -\; \backslash mathrm\{PV\}(t)$ is the error (SP is the setpoint, and PV( t) is the process variable),

$t$ is the time or instantaneous time (the present),

$\backslash tau$ is the variable of integration (takes on values from time 0 to the present $t$).

Equivalently, the transfer function in the Laplace domain of the PID controller is

$L(s)\; =\; K\_\backslash text\{p\}\; +\; K\_\backslash text\{i\}/s\; +\; K\_\backslash text\{d\}\; s$

::$=\; \{\; K\_\backslash text\{d\}\; s^2\; +\; K\_\backslash text\{p\}\; s\; +\; K\_\backslash text\{i\}\; \backslash over\; s\; \}$

where $s$ is the complex angular frequency.

The proportional term is given by

$P\_\backslash text\{out\}\; =\; K\_\backslash text\{p\}\; e(t).$

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.

The integral term is given by

$I\_\backslash text\{out\}\; =\; K\_\backslash text\{i\}\; \backslash int\_0^t\; e(\backslash tau)\; \backslash ,d\backslash tau.$

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).

The derivative term is given by

$D\_\backslash text\{out\}\; =\; K\_\backslash text\{d\}\; \backslash frac\{de(t)\}\{dt\}.$

Derivative action predicts system behavior and thus improves settling time and stability of the system. An ideal derivative is not causal system, 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.

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, 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 (using different parameters in different operating regions).

Generally, stabilization of response is required and the process must not oscillate for any combination of process conditions and setpoints, though sometimes marginal stability (bounded oscillation) is acceptable or desired.

Mathematically, the origins of instability can be seen in the Laplace domain.

The closed-loop transfer function is

$H(s)\; =\; \backslash frac\{K(s)G(s)\}\{1\; +\; K(s)G(s)\},$

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.

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 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 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.

A nonlinear valve in a flow control application, for instance, will result in variable loop sensitivity that requires damping to prevent instability. One solution is to include a model of the valve's nonlinearity in the control algorithm to compensate for this.

An asymmetric application, for example, is temperature control in HVAC systems that use only active heating (via a heating element), whereas only passive cooling is available. Overshoot of rising temperature can only be corrected slowly; active cooling is not available to force temperature downward as a function of 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.

• 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.

The controller output is given by

$K\_P\; \backslash Delta\; +\; K\_I\; \backslash int\; \backslash Delta\backslash ,dt$

$\backslash Delta\; =\; SP\; -\; PV.$

A PI controller can be modelled easily in software such as Simulink or Xcos using a "flow chart" box involving Laplace operators:

$C=\backslash frac\{G(1+\backslash tau\; s)\}\{\backslash tau\; s\}$

$G\; =\; K\_P$ = proportional gain

$\backslash frac\; G\; \backslash tau\; =\; K\_I$ = integral gain

Setting a value for $G$ 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.

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 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.

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.

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.[2] Fundamentals of cascade control | Sometimes two controllers can do a better job of keeping one process variable where you want it. | By Vance VanDoren, PHD, PE | AUGUST 17, 2014[3] | The Benefits of Cascade Control | September 22, 2020 | Watlow

$u(t)\; =\; K\_p\; \backslash left(\; e(t)\; +\; \backslash frac\{1\}\{T\_i\}\backslash int\_0^t\; e(\backslash tau)\backslash ,d\backslash tau\; +\; T\_d\backslash frac\{d\}\{dt\}e(t)\; \backslash right)$

$T\_i$ is the integral time

$T\_d$ 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 $T\_d$ 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 $T\_i$ seconds (or samples). The resulting compensated single error value is then scaled by the single gain $K\_p$ to compute the control variable.

In the parallel form, shown in the controller theory section

$u(t)\; =\; K\_p\; e(t)\; +\; K\_i\; \backslash int\_0^t\; e(\backslash tau)\backslash ,d\backslash tau\; +\; K\_d\backslash frac\{d\}\{dt\}e(t)$

the gain parameters are related to the parameters of the standard form through $K\_i\; =\; K\_p/T\_i$ and $K\_d\; =\; K\_p\; T\_d$. 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.

$\backslash mathrm\{MV(t)\}=K\_p\backslash left(\backslash ,\{-PV(t)\}\; +\; \backslash frac\{1\}\{T\_i\}\backslash int\_\{0\}^\{t\}\{e(\backslash tau)\}\backslash ,\{d\backslash tau\}\; -\; T\_d\backslash frac\{d\}\{dt\}PV(t)\backslash right)$

$G(s)=K\_p\; +\; \backslash frac\{K\_i\}\{s\}\; +\; K\_d\{s\}=\backslash frac\{K\_d\{s^2\}\; +\; K\_p\{s\}\; +\; K\_i\}\{s\}$

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.

$G(s)\; =\; K\_c\; (\backslash frac\{1\}\{\backslash tau\_i\{s\}\}+1)\; (\backslash tau\_d\{s\}+1)$

$K\_p\; =\; K\_c\; \backslash cdot\; \backslash alpha$, $T\_i\; =\; \backslash tau\_i\; \backslash cdot\; \backslash alpha$, and

$T\_d\; =\; \backslash frac\{\backslash tau\_d\}\{\backslash alpha\}$

$\backslash alpha\; =\; 1\; +\; \backslash frac\{\backslash tau\_d\}\{\backslash tau\_i\}$.

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.

Differentiating both sides of PID equation using Newton's notation gives:

$\backslash dot\{u\}(t)\; =\; K\_p\backslash dot\{e\}(t)\; +\; K\_ie(t)\; +\; K\_d\backslash ddot\{e\}(t)$

Derivative terms are approximated as,

$\backslash dot\{f\}(t\_k)\; =\; \backslash dfrac\{df(t\_k)\}\{dt\}=\backslash dfrac\{f(t\_\{k\})-f(t\_\{k-1\})\}\{\backslash Delta\; t\}$

So,

$\backslash frac\{u(t\_\{k\})-u(t\_\{k-1\})\}\{\backslash Delta\; t\}\; =\; K\_p\backslash frac\{e(t\_\{k\})-e(t\_\{k-1\})\}\{\backslash Delta\; t\}\; +\; K\_i\; e(t\_\{k\})\; +\; K\_d\; \backslash frac\{\backslash dot\{e\}(t\_\{k\})\; -\; \backslash dot\{e\}(t\_\{k-1\})\}\{\backslash Delta\; t\}$

Applying backward difference again gives,

$\backslash frac\{u(t\_\{k\})-u(t\_\{k-1\})\}\{\backslash Delta\; t\}\; =\; K\_p\backslash frac\{e(t\_\{k\})-e(t\_\{k-1\})\}\{\backslash Delta\; t\}\; +\; K\_i\; e(t\_\{k\})\; +\; K\_d\; \backslash frac\{\; \backslash frac\{e(t\_\{k\})-e(t\_\{k-1\})\}\{\backslash Delta\; t\}\; -\; \backslash frac\{e(t\_\{k-1\})-e(t\_\{k-2\})\}\{\backslash Delta\; t\}\; \}\{\backslash Delta\; t\}$

By simplifying and regrouping terms of the above equation, an algorithm for an implementation of the discretized PID controller in an MCU is finally obtained:

$u(t\_\{k\})=u(t\_\{k-1\})+\backslash left(K\_p+K\_i\backslash Delta\; t+\backslash dfrac\{K\_d\}\{\backslash Delta\; t\}\backslash right)\; e(t\_\{k\})+\backslash left(-K\_p-\backslash dfrac\{2K\_d\}\{\backslash Delta\; t\}\backslash right)\; e(t\_\{k-1\})\; +\; \backslash dfrac\{K\_d\}\{\backslash Delta\; t\}e(t\_\{k-2\})$

or:

$u(t\_k)=u(t\_\{k-1\})+K\_p\backslash left\backslash left(1+\backslash dfrac\{\backslash Delta$

s.t. $T\_i\; =\; K\_p/K\_i,\; T\_d\; =\; K\_d/K\_p$

Note: This method solves in fact $u(t)\; =\; K\_\backslash text\{p\}\; e(t)\; +\; K\_\backslash text\{i\}\; \backslash int\_0^t\; e(\backslash tau)\; \backslash ,\backslash mathrm\{d\}\backslash tau\; +\; K\_\backslash text\{d\}\; \backslash frac\{\backslash mathrm\{d\}e(t)\}\{\backslash mathrm\{d\}t\}\; +\; u\_0$ where $u\_0$ is a constant independent of t. This constant is useful when you want to have a start and stop control on the regulation loop. For instance, setting Kp,Ki and Kd to 0 will keep u(t) constant. Likewise, when you want to start a regulation on a system where the error is already close to 0 with u(t) non null, it prevents from sending the output to 0.

• Kp - proportional gain
• Ki - integral gain
• Kd - derivative gain
• dt - loop interval time (assumes reasonable scale)Note that for very small intervals (e.g. 60Hz/$.016\backslash bar\{6\}$ seconds), the resulting derivative value will be extremely large, and orders of magnitude larger than the proportional or integral components. Adjusting this value for the derivative (e.g. multiplying by 1000) or changing the division to multiplication is likely to yield the intended results. This holds true for all pseudocode presented here.

previous_error := 0
integral := 0
loop:
   error := setpoint − measured_value
   proportional := error;
   integral := integral + error × dt
   derivative := (error - previous_error) / dt
   output := Kp × proportional + Ki × integral + Kd × derivative
   previous_error := error
   wait(dt)
   goto loop
     

Below a pseudocode illustrates how to implement a PID considering the PID as an IIR filter:

The Z-transform of a PID can be written as ($\backslash Delta\_t$ is the sampling time):

$C(z)=\; K\_p\; +\; K\_i\backslash Delta\_t\; \backslash frac\{z\}\{z-1\}\; +\; \backslash frac\{K\_d\}\{\backslash Delta\_t\}\; \backslash frac\{z-1\}\{z\}$

and expressed in a IIR form (in agreement with the discrete implementation shown above):

$C(z)=\backslash frac\{\backslash left(K\_p+K\_i\backslash Delta\_t+\backslash dfrac\{K\_d\}\{\backslash Delta\_t\}\backslash right)+\backslash left(-K\_p-\backslash dfrac\{2K\_d\}\{\backslash Delta\_t\}\backslash right)\; z^\{-1\}\; +\; \backslash dfrac\{K\_d\}\{\backslash Delta\_t\}z^\{-2\}\}\{1-z^\{-1\}\}$

We can then deduce the recursive iteration often found in FPGA implementation

$un\; =\; un-1\; +\; \backslash left(K\_p+\; K\_i\backslash Delta\_t+\backslash dfrac\{K\_d\}\{\backslash Delta\_t\}\backslash right)\backslash epsilonn\; +\; \backslash left(-K\_p-\backslash dfrac\{2K\_d\}\{\backslash Delta\_t\}\backslash right)\backslash epsilonn-1\; +\; \backslash dfrac\{K\_d\}\{\backslash Delta\_t\}\backslash epsilonn-2$

A0 := Kp + Ki*dt + Kd/dt
A1 := -Kp - 2*Kd/dt
A2 := Kd/dt
error[2] := 0  // e(t-2)
error[1] := 0  // e(t-1)
error[0] := 0  // e(t)
output   := u0 // Usually the current value of the actuator
     

loop:
    error[2] := error[1]
    error[1] := error[0]
    error[0] := setpoint − measured_value
    output   := output + A0 * error[0] + A1 * error[1] + A2 * error[2]
    wait(dt)
    goto loop
     

Here, Kp is a dimensionless number, Ki is expressed in $s^\{-1\}$ and Kd is expressed in s. When doing a regulation where the actuator and the measured value are not in the same unit (e.g., temperature regulation using a motor controlling a valve), Kp, Ki and Kd may be corrected by a unit conversion factor. It may also be interesting to use Ki in its reciprocal form (integration time). The above implementation allows to perform an I-only controller which may be useful in some cases.

In the real world, this is D-to-A converted and passed into the process under control as the manipulated variable (MV). The current error is stored elsewhere for re-use in the next differentiation, the program then waits until dt seconds have passed since start, and the loop begins again, reading in new values for the PV and the setpoint and calculating a new value for the error.

Note that for real code, the use of "wait(dt)" might be inappropriate because it doesn't account for time taken by the algorithm itself during the loop, or more importantly, any pre-emption delaying the algorithm.

A common issue when using $K\_d$ is the response to the derivative of a rising or falling edge of the setpoint, as shown below:

A typical workaround is to filter the derivative action using a low pass filter of time constant $\backslash tau\_d/N$ where :

A variant of the above algorithm using an infinite impulse response (IIR) filter for the derivative:

A0 := Kp + Ki*dt
A1 := -Kp
error[2] := 0 // e(t-2)
error[1] := 0 // e(t-1)
error[0] := 0 // e(t)
output := u0  // Usually the current value of the actuator
A0d := Kd/dt
A1d := - 2.0*Kd/dt
A2d := Kd/dt
N := 5
tau := Kd / (Kp*N) // IIR filter time constant
alpha := dt / (2*tau)
d0 := 0
d1 := 0
fd0 := 0
fd1 := 0
loop:
    error[2] := error[1]
    error[1] := error[0]
    error[0] := setpoint − measured_value
    // PI
    output := output + A0 * error[0] + A1 * error[1]
    // Filtered D
    d1 := d0
    d0 := A0d * error[0] + A1d * error[1] + A2d * error[2]
    fd1 := fd0
    fd0 := ((alpha) / (alpha + 1)) * (d0 + d1) - ((alpha - 1) / (alpha + 1)) * fd1
    output := output + fd0
    wait(dt)
    goto loop
     

• Control theory
• Active disturbance rejection control

• B. Wayne Bequette (2025). 9789861544779, Prentice Hall PTR. ISBN 9789861544779

• Bela Liptak (1995). 9780801982422, Chilton Book Company. ISBN 9780801982422
• Kok Kiong Tan (1999). 9781852331382, Springer-Verlag. ISBN 9781852331382
• Myke King (2025). 9780470975879, John Wiley & Sons Ltd.. . ISBN 9780470975879
• Aidan O'Dwyer (2025). 9781848162426, Imperial College Press. . ISBN 9781848162426

• PID tuning using Mathematica
• PID tuning using Python
• Principles of PID Control and Tuning
• Introduction to the key terms associated with PID Temperature Control

• PID Control in MATLAB/Simulink and Python with TCLab
• What's All This P-I-D Stuff, Anyhow? Article in Electronic Design
• Shows how to build a PID controller with basic electronic components (pg. 22)
• PID Without a PhD
• PID Control with MATLAB and Simulink
• PID with single Operational Amplifier
• Proven Methods and Best Practices for PID Control
• Principles of PID Control and Tuning
• PID Tuning Guide: A Best-Practices Approach to Understanding and Tuning PID Controllers
• Introduction to P,PI,PD & PID Controller with MATLAB
• Improving The Beginners PID