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A proportional-integral-derivative controller (PID controller) is a generic control loop feedback mechanism widely used in industrial control systems. A PID controller attempts to correct the error between a measured process variable and a desired setpoint by calculating and then outputting a corrective action that can adjust the process accordingly.

The PID controller calculation (algorithm) involves three separate parameters; the Proportional, the Integral and Derivative values. The Proportional value determines the reaction to the current error, the Integral determines the reaction based on the sum of recent errors and the Derivative determines the reaction to the rate at which the error has been changing. The weighted sum of these three actions is used to adjust the process via a control element such as the position of a control valve or the power supply of a heating element.

By "tuning" the three constants in the PID controller algorithm the PID can provide control action designed for specific process requirements. The response of the controller can be described in terms of the responsiveness of the controller to an error, the degree to which the controller overshoots the setpoint and the degree of system oscillation. Note that the use of the PID algorithm for control does not guarantee optimal control of the system.

Some applications may require using only one or two modes to provide the appropriate system control. This is achieved by setting the gain of undesired control outputs to zero. A PID controller will be called a PI, PD, P or I controller in the absence of the respective control actions. PI controllers are particularly common, since derivative action is very sensitive to measurement noise, and the absence of an integral value prevents the system from reaching its target value due to the control action.



Note: Due to the diversity of the field of control theory and application, many naming conventions for the relevant variables are in common use.

Control loop basics A familiar example of a control loop is the action taken to keep one's shower water at the ideal temperature. The person feels the water to estimate its temperature. Based on this measurement they perform a control action: use the hot water tap to adjust the process. The person would repeat this input-output control loop, adjusting the hot water flow until the process temperature stabilized at the desired value.

Feeling the water temperature is taking a measurement of the process value or process variable (PV). The desired temperature is called the setpoint (SP). The output from the controller and input to the process (the tap position) is called the manipulated variable (MV). The difference between the measurement and the setpoint is the error (e), too hot or too cold and by how much.

As a controller, one decides roughly how much to change the tap position (MV) after one determines the temperature (PV), and therefore the error. This first estimate is the equivalent of the proportional action of a PID controller. The integral action of a PID controller can be thought of as gradually adjusting the temperature when it is almost right. Derivative action can be thought of as making smaller and smaller changes as one gets close to the right temperature and stopping when it is just right, rather than going too far.

Making a change that is too large when the error is small is equivalent to a high gain controller and will lead to overshoot. If the controller were to repeatedly make changes that were too large and repeatedly overshoot the target, this control loop would be termed unstable and the output would oscillate around the setpoint in either a constant, a growing or a decaying sinusoid. A human would not do this because we are adaptive controllers, learning from the process history, but PID controllers do not have the ability to learn and must be set up correctly. Selecting the correct gains for effective control is known as tuning the controller.

If a controller starts from a stable state at zero error (PV = SP), then further changes by the controller will be in response to changes in other measured or unmeasured inputs to the process that impact on the process, and hence on the PV. Variables that impact on the process other than the MV are known as disturbances and generally controllers are used to reject disturbances and/or implement setpoint changes. Changes in feed water temperature constitute a disturbance to the shower process.

In theory, a controller can be used to control any process which has a measurable output (PV), a known ideal value for that output (SP) and an input to the process (MV) that will affect the relevant PV. Controllers are used in industry to regulate temperature, pressure, flow rate, chemical composition, level in a tank containing fluid, speed and practically every other variable for which a measurement exists. Automobile cruise control is an example of a process outside of industry which utilizes automated control.

Due to their long history, simplicity, well grounded theory and simple setup and maintenance requirements, PID controllers are the controllers of choice for many of these applications.

PID controller theory Note: This section describes the ideal parallel or non-interacting form of the PID controller. For other forms please see the Section "Alternative notation and PID forms".

The PID control scheme is named after its three correcting terms, whose sum constitutes the manipulated variable (MV). Hence: \mathrm{MV(t)}=\,P_{\mathrm{out--> + I_{\mathrm{out--> + D_{\mathrm{out-->

where P_{\mathrm{out-->, I_{\mathrm{out-->, and D_{\mathrm{out--> are the contributions to the output from the PID controller from each of the three terms, as defined below.

Proportional term The proportional term makes a change to the output that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant Kp, called the proportional gain.

The proportional term is given by: P_{\mathrm{out-->=K_p\,{e(t)}

Where



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 sensitive) controller. If the proportional gain is too low, the control action may be too small when responding to system disturbances.

In the absence of disturbances pure proportional control will not settle at its target value, but will retain a steady state error that is a function of the proportional gain and the process gain. Despite the steady-state offset, both tuning theory and industrial practice indicate that it is the proportional term that should contribute the bulk of the output change.

Integral term The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error. Summing the instantaneous error over time (integrating the error) gives the accumulated offset that should have been corrected previously. The accumulated error is then multiplied by the integral gain and added to the controller output. The magnitude of the contribution of the integral term to the overall control action is determined by the integral gain, K_i.

The integral term is given by: I_{\mathrm{out-->=K_{i}\int_{0}^{t}{e(\tau)}\,{d\tau}



Where

The integral term (when added to the proportional term) accelerates the movement of the process towards setpoint and eliminates the residual steady-state error that occurs with a proportional only controller. However, since the integral term is responding to accumulated errors from the past, it can cause the present value to overshoot the setpoint value (cross over the setpoint and then create a deviation in the other direction). For further notes regarding integral gain tuning and controller stability, see the section on Loop Tuning.

Derivative term The rate of change of the process error is calculated by determining the slope of the error over time (i.e. its first derivative with respect to 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: D_{\mathrm{out-->=K_d\frac{de}{dt}



Where

The derivative term slows the rate of change of the controller output and this effect is most noticeable close to the controller setpoint. Hence, derivative control is used to reduce the magnitude of the overshoot produced by the integral component and improve the combined controller-process stability. However, differentiation of a signal amplifies noise in the signal and thus this term in the controller is highly sensitive to noise in the error term, and can cause a process to become unstable if the noise and the derivative gain are sufficiently large.

Summary The output from the three terms, the proportional, the integral and the 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: :\mathrm{MV(t)}=K_p{e(t)} + K_{i}\int_{0}^{t}{e(\tau)}\,{d\tau} + K_{d}\frac{de}{dt} and the tuning parameters are
  • K_p: Proportional Gain - Larger K_p typically means faster response since the larger the error, the larger the feedback to compensate. An excessively large proportional gain will lead to process instability.
  • K_i: Integral Gain - Larger K_i implies steady state errors are eliminated quicker. The trade-off is larger overshoot: any negative error integrated during transient response must be integrated away by positive error before we reach steady state.
  • K_d: Derivative Gain - Larger K_d decreases overshoot, but slows down transient response and may lead to instability.


    • Loop tuning 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, and is limited only by saturation or mechanical breakage. Tuning a control loop is the adjustment of its control parameters (gain/proportional band, integral gain/reset, derivative gain/rate) to the optimum values for the desired control response.

    The optimum behavior on a process change or setpoint change varies depending on the application. 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. Generally, stability of response (the reverse of instability) is required and the process must not oscillate for any combination of process conditions and setpoints. Some processes have a degree of non-linearity and so parameters that work well at full-load conditions don't work when the process is starting up from no-load. This section describes some traditional manual methods for loop tuning.

    There are several methods for tuning a PID loop. The most effective methods generally involve the development of some form of process model, then choosing P, I, and D based on the dynamic model parameters. Manual "tune by feel" methods can be inefficient.

    The choice of method will depend largely on whether or not the loop can be taken "offline" for tuning, and 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.

    {| class="wikitable"|-----| align="center" colspan="3" border="1" | Choosing a Tuning Method|-----| align="center" | Method| align="center" | Advantages| align="center" | Disadvantages|-----| align="center" | Ziegler-Nichols| align="center" | Proven Method. Online method.| align="center" | Process upset, some trial-and-error, very aggressive tuning|-----| align="center" | Tune By Feel| align="center" | No math required. Online method.| align="center" | Erratic, not repeatable|-----| align="center" | Software Tools| align="center" | Consistent tuning. Online or offline method. May include valve and sensor analysis. Allow simulation before downloading.| align="center" | Some cost and training involved.|-----| align="center" | Cohen-Coon| align="center" | Good process models.| align="center" | Some math. Offline method. Only good for first-order processes.|}

    If the system must remain online, one tuning method is to first set the I and D values to zero. Increase the P until the output of the loop oscillates, then the P should be left set to be approximately half of that value for a "quarter amplitude decay" type response. Then increase I until any offset is correct in sufficient time for the process. However, too much I will cause instability. Finally, increase D, if required, until the loop is acceptably quick to reach its reference after a load disturbance. However, too much D will cause 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 a "critically damped" tune is required, which will require a P setting significantly less than half that of the P setting causing oscillation.

    {| class="wikitable"|-----| align="center" colspan="5" border="1" | Effects of increasing parameters|-----| align="center" | Parameter| align="center" | Rise Time| align="center" | Overshoot| align="center" | Settling Time| align="center" | S.S. Error|-----| align="center" | K_p| align="center" | Decrease| align="center" | Increase| align="center" | Small Change| align="center" | Decrease|-----| align="center" | K_i| align="center" | Decrease| align="center" | Increase| align="center" | Increase| align="center" | Eliminate|-----| align="center" | K_d| align="center" | Small Change| align="center" | Decrease| align="center" | Decrease| align="center" | None|}

    Ziegler-Nichols method Another tuning method is formally known as the Ziegler-Nichols method, introduced by John G. Ziegler and Nathaniel B. Nichols. As in the method above, the I and D gains are first set to zero. The "P" gain is increased until it reaches the "critical gain" K_c at which the output of the loop starts to oscillate. K_c and the oscillation period P_c are used to set the gains as shown:

    {| class="wikitable"|-----| align="center" colspan="5" border="1" | Ziegler-Nichols method|-----| align="center" | Control Type| align="center" | K_p| align="center" | K_i| align="center" | K_d|-----| align="center" | P| align="center" | 0.5·K_c| align="center" | -| align="center" | -|-----| align="center" | PI| align="center" | 0.45·K_c| align="center" | 1.2 K_p/P_c| align="center" | -|-----| align="center" | PID| align="center" | 0.6·K_c| align="center" | 2 K_p/P_c| align="center" | K_p P_c/8|}

    PID 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 will gather the 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 literally 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.

    Other formulas are available to tune the loop according to different performance criteria.

    Modifications to the PID algorithm The basic PID algorithm presents some challenges in control applications that have been addressed by minor modifications to the PID form.

    One common problem resulting from the ideal PID implementations is integral windup. This can be addressed by:

    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 or a deadband 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 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.

    The proportional and differential 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 the following modifications:

    Limitations of PID control While PID controllers are applicable to many control problems, they can perform poorly in some applications.

    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. The control system performance can be improved by combining the PID controller functionality with that of a Feed-Forward control output as described in Control Theory. Any information or intelligence derived from the system state can be "fed forward" or combined with the PID output to improve the overall system performance. The Feed-Forward value alone can often provide a major portion of the controller output. The PID controller can then be used to respond to whatever difference or "error" that remains between the controller setpoint and the feedback value. Since the Feed-Forward output is not a function of the process feedback, it can never cause the control system to oscillate, thus improving the system response and stability.

    For example, in most motion control systems, in order to accelerate a mechanical load under control, more force or torque is required from the prime mover, motor or actuator. If a velocity loop PID controller is being used to control the speed of the load and command the force or torque being applied by the prime mover, then it is beneficial to take the instantaneous acceleration desired for the load, 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 prime mover regardless of the feedback value. The PID loop in this situation uses the feedback information to effect any increase or decrease of the combined output in order to reduce the remaining difference between the process setpoint and the feedback value. Working together, the combined Feed-Forward open loop controller and closed loop PID controller can provide more responsive, stable and reliable control systems.

    Another problem faced with PID controllers is that they are linear. Thus, performance of PID controllers in non-linear systems (such as HVAC control system) is variable. Often PID controllers are enhanced through methods such as gain scheduling or fuzzy logic.Further practical application issues can arise from instrumentation connected to the controller.A high enough sampling rate and measurement precision and measurement accuracy (more relevant to FF and MPC).

    A problem with the differential term is that small amounts of measurement or process noise can cause large amounts of change in the output. Sometimes it is helpful to filter the measurements, with a running average, also known as a low-pass filter. However, low-pass filtering and derivative control cancel each other out, so reducing noise by instrumentation means is a much better choice. Alternatively, the differential band can be turned off in most systems with little loss of control. This is equivalent to using the PID controller as a PI controller.

    Physical implementation of PID control In the early history of automatic process control the PID controller was implemented as a mechanical device. These mechanical controllers used a lever, spring (device) and a mass and were often energized by compressed air. These pneumatic controllers were once the industry standard.

    Electronic analog circuit controllers can be made from a transistor or vacuum tube Operational amplifier, a capacitor and a electrical resistance. Electronic analog PID control loops were often found within more complex electronic systems, for example, the head positioning of a disk drive, the power conditioning of a power supply, or even the movement-detection circuit of a modern seismometer. Nowadays, electronic controllers have largely been replaced by digital controllers implemented with microcontrollers or FPGAs.

    Most modern PID controllers in industry are implemented in software in programmable logic controllers (PLCs) or as a panel-mounted digital controller. Software implementations have the advantages that they are relatively cheap, highly reliable and reasonably flexible with respect to the implementation of the PID algorithm.

    Alternative Nomenclature and PID forms Pseudocode Here is a simple software loop that implements the PID algorithm:

    start: previous_error = error or 0 if undefined error = setpoint - actual_position P = Kp * error I = I + Ki * error * dt D = Kd * (error - previous_error) / dt output = P + I + D wait(dt) goto start

    Ideal vs Standard PID form Note that 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 K_p gain is applied to the I_{\mathrm{out-->, and D_{\mathrm{out--> terms, yielding:

    \mathrm{MV(t)}=K_p\left(\,{e(t)} + \frac{1}{T_i}\int_{0}^{t}{e(\tau)}\,{d\tau} + T_d\frac{de}{dt}\right) Where T_i is the Integral Time T_d is the Derivative Time

    In the ideal parallel form, shown in the Controller Theory section \mathrm{MV(t)}=K_p{e(t)} + K_i\int_{0}^{t}{e(\tau)}\,{d\tau} + K_d\frac{de}{dt}

    the gain parameters are related to the parameters of the standard form through K_i = \frac{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 least physical interpretation 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.

    Series / interacting form Another representation of the PID controller is the series, or "interacting" form. This form essentially consists of a PD and PI controller in series, and it made early (analog) controllers easier to build. When the controllers later became digital, many kept using the interacting form.

    Often, one deals with discrete time intervals instead of the continuity. Thus, the PID controller may also be dealt with recursively:

    \mathrm{MV}_n=\mathrm{MV}_{n-1} + (K_p + hK_i + \frac{K_d}{h})\,e_n - (K_p + 2 \frac{K_d}{h})\,e_{n-1} + \frac{K_d}{h}\,e_{n-2}.

    Where h is the length of the discrete intervals often called the sampling period.

    See also

    External links PID tutorials

    Simulations

    Special topics and PID control applications

    References | last =Liptak | first =Bela | authorlink = | coauthors = | title =Instrument Engineers' Handbook: Process Control | publisher =Chilton Book Company | date =1995 | location =Radnor, Pennsylvania | pages =20-29 | url = | doi = | id = ISBN 0-8019-8242-1 -->
     
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