Proportional–integral–derivative controller

5325:

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.

1405: 1456: 1464: 9721:

alpha := dt / (2*tau) d0 := 0 d1 := 0 fd0 := 0 fd1 := 0 loop: error := error error := error error := setpoint − measured_value // PI output := output + A0 * error + A1 * error // Filtered D d1 := d0 d0 := A0d * error + A1d * error + A2d * error fd1 := fd0 fd0 := ((alpha) / (alpha + 1)) * (d0 + d1) - ((alpha - 1) / (alpha + 1)) * fd1 output := output + fd0 wait(dt) goto loop

1267: 5363:

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..

5371:

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.

1373:. 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 5393: 1255: 4902: 6130:. 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. 145: 2256: 2359:). 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. 4821:

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.

7935: 5350:, 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 5316:(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 9276: 2656:, 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. 8228: 9544: 7587: 1199: 484:

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 introducing a setpoint change and observing the system response.

8827:

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

5324:

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

4820:

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

1642:

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

1616:

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

1587:

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

471:

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

9720:

A0 := Kp + Ki*dt A1 := -Kp error := 0 // e(t-2) error := 0 // e(t-1) error := 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

9583:

and Kd is expressed in s. When doing a regulation where the actuator and the measured value are not in the same unit (ex. 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

5298:

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

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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

3109:

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

1574:

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

6225:

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

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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

6138:

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

5374:

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

5320:

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

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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

1309:

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

1209:

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

499:

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.

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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

9549:

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

5333:

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

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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

4716:

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,

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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

483:

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 physical system, external to the controller. These are dependent on the behavior of the measuring sensor, the

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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

5792:

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

4746:

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

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control action for all the terms, which means an increasing positive error results in an increasing positive control output correction. This because the "error" term is not the deviation from the setpoint (actual-desired) but is in fact the correction needed (desired-actual). The system is called

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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

3105:

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

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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

1432:

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

6904:

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.

1629:. 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 1424:

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

5370:

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

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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

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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

131:

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 pneumatic and then

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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

1350:

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.

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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

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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.

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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.

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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

6406: 452:". 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". 4873:. 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: 184:

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

1904: 127:

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.

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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.

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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

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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.

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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.

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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.

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With these controllers, a pneumatic industry signaling standard of 3–15 psi (0.2–1.0 bar) was established, which had an elevated zero to ensure devices were working within their linear characteristic and represented the control range of 0-100%.

4738:) 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. 1591:

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

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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

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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

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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,

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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.

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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.

1038: 6569: 1707:

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

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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

4893:

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.

4372:, 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. 1588:

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).

6204:, 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. 4837:

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

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Kebriaei, Reza; Frischkorn, Jan; Reese, Stefanie; Husmann, Tobias; Meier, Horst; Moll, Heiko; Theisen, Werner (2013). "Numerical modelling of powder metallurgical coatings on ring-shaped parts integrated with ring rolling".

7930:{\displaystyle {\frac {u(t_{k})-u(t_{k-1})}{\Delta t}}=K_{p}{\frac {e(t_{k})-e(t_{k-1})}{\Delta t}}+K_{i}e(t_{k})+K_{d}{\frac {{\frac {e(t_{k})-e(t_{k-1})}{\Delta t}}-{\frac {e(t_{k-1})-e(t_{k-2})}{\Delta t}}}{\Delta t}}} 7148: 8641: 4769:

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

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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

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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

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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.

6692: 1318:. 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 9708: 6232: 2417:

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

3097:. 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. 2647: 2222: 5238:

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.

9271:{\displaystyle C(z)={\frac {\left(K_{p}+K_{i}\Delta _{t}+{\dfrac {K_{d}}{\Delta _{t}}}\right)+\left(-K_{p}-{\dfrac {2K_{d}}{\Delta _{t}}}\right)z^{-1}+{\dfrac {K_{d}}{\Delta _{t}}}z^{-2}}{1-z^{-1}}}} 6898: 1743: 8223:{\displaystyle u(t_{k})=u(t_{k-1})+\left(K_{p}+K_{i}\Delta t+{\dfrac {K_{d}}{\Delta t}}\right)e(t_{k})+\left(-K_{p}-{\dfrac {2K_{d}}{\Delta t}}\right)e(t_{k-1})+{\dfrac {K_{d}}{\Delta t}}e(t_{k-2})} 2825: 1452:

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.

8634: 2057: 9539:{\displaystyle u=u+\left(K_{p}+K_{i}\Delta _{t}+{\dfrac {K_{d}}{\Delta _{t}}}\right)\epsilon +\left(-K_{p}-{\dfrac {2K_{d}}{\Delta _{t}}}\right)\epsilon +{\dfrac {K_{d}}{\Delta _{t}}}\epsilon } 6839: 4973: 2675:. 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. 3418:

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

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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.

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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).

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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.

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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.

6213:

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.

3480: 2350: 5128: 1600:. 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 8535: 6790: 5414: 5565: 1448:

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

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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.

1425:

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

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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

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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.

1393:(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. 10855:

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

7582:{\displaystyle {\frac {u(t_{k})-u(t_{k-1})}{\Delta t}}=K_{p}{\frac {e(t_{k})-e(t_{k-1})}{\Delta t}}+K_{i}e(t_{k})+K_{d}{\frac {{\dot {e}}(t_{k})-{\dot {e}}(t_{k-1})}{\Delta t}}} 5557: 5522: 4220: 2284:

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

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Analytically derived, works on time delayed processes, has an additional tuning parameter that allows additional flexibility. Tuning can be performed with step-response model.

1234:). 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 6128: 4167: 295: 255: 8901: 5281:

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

4454: 4350: 1330:, all of whom contributed to the establishment of control stability criteria. In subsequent applications, speed governors were further refined, notably by American scientist 1306:" 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. 9703: 5052: 2671:

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

1246:. 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. 354: 3064: 2355:

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

3017: 1995: 1966: 1937: 951: 924: 818: 791: 745: 718: 691: 9793: 9671: 8893: 5251:

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

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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

1194:{\displaystyle u(t)=K_{\text{p}}\left(e(t)+{\frac {1}{T_{\text{i}}}}\int _{0}^{t}e(\tau )\,\mathrm {d} \tau +T_{\text{d}}{\frac {\mathrm {d} e(t)}{\mathrm {d} t}}\right)} 7158: 6202: 1210:

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.

7008: 4613: 4352:. 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. 5167: 4996: 3138:

This is an iterative, experience-based, trial-and-error procedure that can be relatively time consuming. Operators may find "bad" parameters without proper training.

9581: 6429: 9628: 8825: 6164: 5873: 5846: 5818: 5783: 5751: 5487: 4404: 4309: 4282: 4255: 3942: 3914: 3886: 3848: 3821: 3794: 3767: 3740: 3684: 3643: 3598: 3553: 3474: 3447: 3416: 3389: 3362: 3335: 3308: 3281: 3254: 2105: 6985: 6956: 2886: 2857: 1735: 446: 398: 215: 86: 5884: 5233: 4788:

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

4376:

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.

3089:(implementing setpoint changes). These terms refer to how well the controlled variable tracks the desired value. Specific criteria for command tracking include 2910: 2245: 2125: 2083: 4781:

parameters, and no direct knowledge of the process, and thus overall performance is reactive and a compromise. While PID control is the best controller for an

7940:

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:

2713:

Generally, stabilization of response is required and the process must not oscillate for any combination of process conditions and setpoints, though sometimes

1346:. Pressure control provided only a proportional control that, if the control gain was too high, would become unstable and go into overshoot with considerable 10044: 2557:

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

4809:

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.

1361:

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

1467:

Current loops used for sensing and control signals. A modern electronic "smart" valve positioner is shown, which will incorporate its own PID controller.

4754:

Another approach calculates initial values via the Ziegler–Nichols method, and uses a numerical optimization technique to find better PID coefficients.

4481: 1396:

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.

1238:

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

8793:{\displaystyle u(t)=K_{\text{p}}e(t)+K_{\text{i}}\int _{0}^{t}e(\tau )\,\mathrm {d} \tau +K_{\text{d}}{\frac {\mathrm {d} e(t)}{\mathrm {d} t}}+u_{0}} 10902: 7022: 11265: 9914: 4230:

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

4785:

without a model of the process, better performance can be obtained by overtly modeling the actor of the process without resorting to an observer.

10171: 657:{\displaystyle u(t)=K_{\text{p}}e(t)+K_{\text{i}}\int _{0}^{t}e(\tau )\,\mathrm {d} \tau +K_{\text{d}}{\frac {\mathrm {d} e(t)}{\mathrm {d} t}},} 464:

grow. This will result in the proportional effect diminishing as the error decreases, but this is compensated for by the growing integral effect.

3207:

May give excessive derivative action and sluggish response. Later extensions resolve these issues, but require a more complex tuning procedure.

2434: 1625:

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

6592: 6401:{\displaystyle \mathrm {MV(t)} =K_{p}\left(\,{-PV(t)}+{\frac {1}{T_{i}}}\int _{0}^{t}{e(\tau )}\,{d\tau }-T_{d}{\frac {d}{dt}}PV(t)\right)} 1533:

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

10925: 4855:

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

2686:, 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 1412:

The wide use of feedback controllers did not become feasible until the development of wideband high-gain amplifiers to use the concept of

11067: 10151: 3364:

until any offset is corrected in sufficient time for the process, but not until too great a value causes instability. Finally, increase

504:

mode, in the case of signal loss, would be 100% opening of the valve; therefore 0% controller output needs to cause 100% valve opening.

11234: 10456: 9876: 3178:) techniques; may include valve and sensor analysis; allows simulation before downloading; can support non-steady-state (NSS) tuning. 11068:"An Overview of Proportional plus Integral plus Derivative Control and Suggestions for Its Successful Application and Implementation" 2580: 2145: 1558:

in the motor) is the output from the PID controller. It is called either the manipulated variable (MV) or the control variable (CV).

1491:

Electronic analog PID control loops were often found within more complex electronic systems, for example, the head positioning of a

10195: 2652:

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

4913:(proportional-integral controller) is a special case of the PID controller in which the derivative (D) of the error is not used. 1899:{\displaystyle u(t)=\mathrm {MV} (t)=K_{\text{p}}e(t)+K_{\text{i}}\int _{0}^{t}e(\tau )\,d\tau +K_{\text{d}}{\frac {de(t)}{dt}},} 1434: 747:, all non-negative, denote the coefficients for the proportional, integral, and derivative terms respectively (sometimes denoted 10688:

Li, Y. and Ang, K.H. and Chong, G.C.Y. (2006) PID control system analysis and design - Problems, remedies, and future directions

5334:

postmultiplying by the integral gain, which prevents discontinuous output when the I gain is changed, but not the P or D gains.

3204:

Unlike the Ziegler–Nichols method this will not introduce a risk of loop instability. Little prior process knowledge required.

1551:

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.

1537:

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.

6850: 2413:

The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error. The

11181: 11126: 11043: 11022: 11003: 10885: 10573: 10134: 10107: 9999: 4720:

One way to determine the parameters for the first-order process is using the 63.2% method. In this method, the process gain (

3151:

Process upsets may occur in the tuning, can yield very aggressive parameters. Does not work well with time-delay processes.

1369:. Minorsky was researching and designing automatic ship steering for the US Navy and based his analysis on observations of a 11113: 10379: 5274:

the case of an instantaneous step change. As a result, some PID algorithms incorporate some of the following modifications:

9735: 1270:

Pneumatic PID (three-term) controller. The magnitudes of the three terms (P, I and D) are adjusted by the dials at the top.

11093: 2733: 56:

and a variety of other applications requiring continuously modulated control. A PID controller continuously calculates an

10944: 10272: 8543: 5458:

The form of the PID controller most often encountered in industry, and the one most relevant to tuning algorithms is the

2002: 11219: 10753:

Yang, T. (June 2005). "Architectures of Computational Verb Controllers: Towards a New Paradigm of Intelligent Control".

10542: 6797: 4922: 4227:

The oscillation frequency is often measured instead, and the reciprocals of each multiplication yields the same result.

2564:. The magnitude of the contribution of the derivative term to the overall control action is termed the derivative gain, 10025: 9584:(integration time). The above implementation allows to perform an I-only controller which may be useful in some cases. 5299:

the response to load disturbances and measurement noise and can be tuned to improve the controller's setpoint response.

10399: 2300: 11287: 11272: 10975: 10795: 10788:

Proceedings of the 3rd International Conference on Anti-Counterfeiting, Security, and Identification in Communication

10256: 9886: 5440: 5075: 4801:

sampling rate, precision, and accuracy, and low-pass filtering if necessary), or cascading multiple PID controllers.

1377:), which required adding the integral term. Finally, the derivative term was added to improve stability and control. 953:

have some understandable physical meaning, as they represent an integration time and a derivative time respectively.

123:

PID systems automatically apply accurate and responsive correction to a control function. An everyday example is the

11207: 11153: 8239: 5713:{\displaystyle u(t)=K_{p}\left(e(t)+{\frac {1}{T_{i}}}\int _{0}^{t}e(\tau )\,d\tau +T_{d}{\frac {d}{dt}}e(t)\right)} 5422: 3162:

Process upsets may occur in the tuning; operator needs to select a parameter for the method which requires insight.

10048: 9754:

The only exception is where the target value is the same as the value obtained when the controller output is zero.

6749: 5379:

controls – heat transfer and thermal mass of the whole tank or of just the heater – giving better total response.

4731:

is the amount of time between when the step change occurred and when the output first changed. The time constant (

3218:

Offline method; cannot be applied to oscillatory processes. Operator must choose the additional tuning parameter.

448:. For example, if the error is large, the control output will be proportionately large by using the gain factor "K 1298:

With the invention of the low-pressure stationary steam engine there was a need for automatic speed control, and

1279:

Continuous control, before PID controllers were fully understood and implemented, has one of its origins in the

11302: 5418: 1433:

maintenance devices that operated well in harsh industrial environments and did not present explosion risks in

1343: 993: 823: 10590: 6703: 5174: 2668:

behavior, different applications have different requirements, and requirements may conflict with one another.

1698:

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

956: 865: 11054: 9550:

measured_value output := output + A0 * error + A1 * error + A2 * error wait(dt) goto loop

5366:

For example, a temperature-controlled circulating bath has two PID controllers in cascade, each with its own

2699: 1630: 1476: 1032:

determines how long the controller will tolerate the output being consistently above or below the set point.

17: 4833:

that can cause large amounts of change in the output. It is often helpful to filter the measurements with a

3391:, if required, until the loop is acceptably quick to reach its reference after a load disturbance. Too much 11297: 1437:. They were the industry standard for many decades until the advent of discrete electronic controllers and 136:. The PID concept has been used widely in applications requiring accurate and optimized automatic control. 11148: 9593: 9588: 5351: 5282: 3067: 1223: 133: 9937: 9632: 9019:{\displaystyle C(z)=K_{p}+K_{i}\Delta _{t}{\frac {z}{z-1}}+{\frac {K_{d}}{\Delta _{t}}}{\frac {z-1}{z}}} 5527: 5492: 4173: 10853: 6028: 4320: 4034: 3704: 3698: 3066:

for frequencies that suffer high phase shifts. A more general formalism of this effect is known as the

1438: 6083: 5321:

the setpoint and on extra measured disturbances. Setpoint weighting is a simple form of feed forward.

4125: 263: 223: 11158: 9714: 8858: 5848:

seconds (or samples). The resulting compensated single error value is then scaled by the single gain

4409: 4326: 218: 165: 89: 53: 11170: 11033: 10767: 10494: 10350: 10311: 10287: 9818: 9676: 8836:

Here is a very simple and explicit group of pseudocode that can be easily understood by the layman:

5016: 300: 11292: 9707: 7300:{\displaystyle {\dot {f}}(t_{k})={\dfrac {df(t_{k})}{dt}}={\dfrac {f(t_{k})-f(t_{k-1})}{\Delta t}}} 5403: 4319:

This method was developed in 1953 and is based on a first-order + time delay model. Similar to the

3769:

gains are first set to zero. The proportional gain is increased until it reaches the ultimate gain

3022: 1235: 11175: 4323:, a set of tuning parameters were developed to yield a closed-loop response with a decay ratio of 2965: 1973: 1944: 1915: 929: 902: 796: 769: 723: 696: 669: 9641: 9638:

A typical workaround is to filter the derivative action using a low pass filter of time constant

8871: 5407: 4703:{\displaystyle y(t)=k_{\text{p}}\Delta u\left(1-e^{\frac {-t-\theta }{\tau _{\text{p}}}}\right),} 4092: 4001: 3955: 3181:"Black box tuning" that requires specification of an objective describing the optimal behaviour. 3078:

The optimal behavior on a process change or setpoint change varies depending on the application.

2915: 1503:. Discrete electronic analog controllers have been largely replaced by digital controllers using 1295:

depending on the speed of rotation, and thereby compensate for the variable speed of grain feed.

11187: 10864: 10783: 10695: 9766: 4829:

A problem with the derivative term is that it amplifies higher frequency measurement or process

1617:

applied, and so reduces overshoot (error on the other side because of too great applied force).

1459:

Showing the evolution of analog control loop signaling from the pneumatic to the electronic eras

10762: 10345: 6564:{\displaystyle G(s)=K_{p}+{\frac {K_{i}}{s}}+K_{d}{s}={\frac {K_{d}{s^{2}}+K_{p}{s}+K_{i}}{s}}} 6172: 5313: 5256: 4361: 1381: 10619:"Patents, software, and hardware for PID control: An overview and analysis of the current art" 10126: 9630:

is the response to the derivative of a rising or falling edge of the setpoint as shown below:

6990: 5354:. The order of the integrator and differentiator add increased flexibility to the controller. 3337:

to approximately half that value for a "quarter amplitude decay"-type response. Then increase

3159:

Online tuning, an extension of the Ziegler–Nichols method, that is generally less aggressive.

11074: 10481: 10298: 10102: 5139: 4981: 1404: 11143: 9556: 5342:

In addition to feed-forward, PID controllers are often enhanced through methods such as PID

4883:

Back-calculating the integral term to constrain the regulator output within feasible bounds.

4777:. The fundamental difficulty with PID control is that it is a feedback control system, with 3172:

Consistent tuning; online or offline – can employ computer-automated control system design (

11241: 10687: 10463: 10337: 10201:. Hacettepe University Department of Electrical and Electronics Engineering. Archived from 9606: 8803: 7014: 6142: 5851: 5824: 5796: 5761: 5729: 5465: 4797: 4382: 4287: 4260: 4233: 3920: 3892: 3864: 3826: 3799: 3772: 3745: 3718: 3712: 3662: 3621: 3576: 3531: 3452: 3425: 3394: 3367: 3340: 3313: 3286: 3259: 3232: 2090: 1669: 1417: 1358:

in 1911 for ship steering, though his work was intuitive rather than mathematically-based.

1280: 1262:

in keeping a vessel on course in the face of varying influences such as wind and sea state.

459:

accounts for past values of the SP − PV error and integrates them over time to produce the

357: 97: 6961: 6932: 6015:{\displaystyle u(t)=K_{p}e(t)+K_{i}\int _{0}^{t}e(\tau )\,d\tau +K_{d}{\frac {d}{dt}}e(t)} 4607:) is a step change input. Converting this transfer function to the time domain results in 2862: 2833: 1711: 422: 374: 191: 62: 8: 10666: 9961: 9029:

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

4817: 4789: 1311: 11232: 10341: 10202: 9965: 990:

is the time constant with which the controller will attempt to approach the set point.

11213:

PID Tuning Guide: A Best-Practices Approach to Understanding and Tuning PID Controllers

10641: 10437: 10085: 5218: 2895: 2714: 2513: 2230: 2110: 2068: 1601: 1480: 1421: 1408:

Proportional control using nozzle and flapper high gain amplifier and negative feedback

1284: 1239: 11197: 11192: 10526: 11259: 11122: 11039: 11018: 10999: 10971: 10881: 10791: 10691: 10660: 10569: 10252: 10233: 10218: 10130: 9995: 9882: 6926: 6420: 6139:

dimensioned units such as temperature. It is common in this case to express the gain

5066: 4365: 2706:, and is limited only by saturation or mechanical breakage. Instability is caused by 2132: 1681: 1626: 1472: 1413: 1338: 10645: 4406:

is assumed to be equal to the observed period, and the ultimate gain is computed as

1429:. Later the derivative term was added by a further bellows and adjustable orifice. 10834: 10824: 10633: 10592:

A Method for automatic tuning of PID controller following Luus-Jaakola optimization

10521: 10429: 10355: 10229: 10081: 10069: 9929: 4880:

Preventing the integral term from accumulating above or below pre-determined bounds

1555: 1542: 1504: 1389: 1366: 1362: 1303: 258: 161: 93: 11225: 10441: 9596:

new values for the PV and the setpoint and calculating a new value for the error.

4571:{\displaystyle y(s)={\frac {k_{\text{p}}e^{-\theta s}}{\tau _{\text{p}}s+1}}u(s),} 1455: 368:

terms. The controller attempts to minimize the error over time by adjustment of a

10670: 10328:

Bechhoefer, John (2005). "Feedback for Physicists: A Tutorial Essay On Control".

6914: 5343: 4870: 4864: 4834: 4771: 4727:) is equal to the change in output divided by the change in input. The dead time 3708: 2687: 1219: 180:

The distinguishing feature of the PID controller is the ability to use the three

8857:

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

6697:

where the parameters are related to the parameters of the standard form through

5235:

is often a trade off between decreasing overshoot and increasing settling time.

2376:

to the setpoint AND output or corrected dynamically by adding an integral term.

1463: 11097: 10903:"Discrete PI and PID Controller Design and Analysis for Digital Implementation" 10618: 9730: 6025:

the gain parameters are related to the parameters of the standard form through

4782: 2721: 2136: 1530: 1520: 124: 11202: 10708: 10598:(Master's Thesis ed.). Tampere, Finland: Tampere University of Technology 10543:"A Review of Relay Auto-tuning Methods for the Tuning of PID-type Controllers" 10400:"A Review of Relay Auto-tuning Methods for the Tuning of PID-type Controllers" 10359: 9281:

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

7143:{\displaystyle {\dot {u}}(t)=K_{p}{\dot {e}}(t)+K_{i}e(t)+K_{d}{\ddot {e}}(t)} 11281: 10637: 10433: 9992:

Iron Men and Tin Fish: The Race to Build a Better Torpedo during World War II

6913:

The analysis for designing a digital implementation of a PID controller in a

4839: 4368:

and measures the resultant oscillations. The output is switched (as if by a

3094: 2653: 1673: 1331: 1327: 1323: 401: 149: 10414: 3479: 5367: 4747:

packages can even develop tuning by gathering data from reference changes.

4364:

and Tore Hägglund, the relay method temporarily operates the process using

2683: 2107:

is the variable of integration (takes on values from time 0 to the present

1496: 1449: 1355: 1319: 405: 45: 10829: 10812: 4877:

Disabling the integration until the PV has entered the controllable region

10839: 10729: 8865: 5347: 2703: 2501:{\displaystyle I_{\text{out}}=K_{\text{i}}\int _{0}^{t}e(\tau )\,d\tau .} 1653: 1534: 1526: 1500: 1347: 1283:, which uses rotating weights to control a process. This was invented by 1243: 11233:

Jinghua Zhong, Mechanical Engineering, Purdue University (Spring 2006).

1310:

theoretical basis for the operation of governors was first described by

1266: 10509: 6687:{\displaystyle G(s)=K_{c}({\frac {1}{\tau _{i}{s}}}+1)(\tau _{d}{s}+1)} 3419: 1688:, and practically every other variable for which a measurement exists. 1492: 1334:, who in 1872 theoretically analyzed Watt's conical pendulum governor. 1299: 365: 105: 10674: 9933: 11159:

Introduction to the key terms associated with PID Temperature Control

10380:"Simple analytic rules for model reduction and PID controller tuning" 9819:"Control Systems, Robotics and Automation – Volume VII - PID Control" 3090: 1665: 1593: 1288: 501: 11182:

Shows how to build a PID controller with basic electronic components

10784:"Controlling fuel annealer using computational verb PID controllers" 5392: 3229:

If the system must remain online, one tuning method is to first set

10709:"Integral (Reset) Windup, Jacketing Logic and the Velocity PI Form" 10032:

which constitutes what is termed as the secret of the fish torpedo.

10010:

The Devil's Device: Robert Whitehead and the History of the Torpedo

6166:

not as "output per degree", but rather in the reciprocal form of a

5309: 5261: 5252: 5058: 4869:

One common problem resulting from the ideal PID implementations is

2414: 1657: 1597: 1370: 1292: 1259: 361: 101: 49: 10927:

Hardware Implimentation [sic] of FPGA based PID Controller

6925:. Approximations for first-order derivatives are made by backward 4816:

An asymmetric application, for example, is temperature control in

4475:

The transfer function for a first-order process with dead time is

3148:

Online tuning, with no tuning parameter therefore easy to deploy.

1416:. This had been developed in telephone engineering electronics by 1254: 10072:(1922). "Directional stability of automatically steered bodies". 2532:

Response of PV to step change of SP vs time, for three values of

2528: 2388:

Response of PV to step change of SP vs time, for three values of

2384: 2373: 2259:

Response of PV to step change of SP vs time, for three values of

1737:

as the controller output, the final form of the PID algorithm is

11012: 10249:

Instrument Engineers' Handbook: Process control and optimization

5308:

The control system performance can be improved by combining the

4901: 3796:

at which the output of the loop starts to oscillate constantly.

1354:

Another early example of a PID-type controller was developed by

10866:| The Benefits of Cascade Control | September 22, 2020 | Watlow 10617:

Li, Yun; Ang, Kiam Heong; Chong, Gregory C.Y. (February 2006).

6582:

Another representation of the PID controller is the series, or

4464:

is the amplitude of the control output change which caused it.

3174: 3085:(disturbance rejection – staying at a given setpoint) and 2642:{\displaystyle D_{\text{out}}=K_{\text{d}}{\frac {de(t)}{dt}}.} 2255: 2217:{\displaystyle L(s)=K_{\text{p}}+K_{\text{i}}/s+K_{\text{d}}s,} 1677: 1596:

around the setpoint in either a constant, growing, or decaying

10998:. Radnor, Pennsylvania: Chilton Book Company. pp. 20–29. 10568:. Upper Saddle River, New Jersey: Prentice Hall. p. 129. 6921:

device requires the standard form of the PID controller to be

2720:

Mathematically, the origins of instability can be seen in the

5878:

In the parallel form, shown in the controller theory section

4830: 4717:

try to minimize disturbances when performing the step test).

4369: 2690:(using different parameters in different operating regions). 1685: 1661: 490:– The mathematical model and practical loop above both use a 6133: 6918: 5062: 5057:

A PI controller can be modelled easily in software such as

1508: 1258:

Early PID theory was developed by observing the actions of

4460:

is the amplitude of the process variable oscillation, and

2892:

where the closed-loop transfer function diverges for some

472:

the change, the greater the controlling or damping effect.

419:

is proportional to the current value of the SP − PV error

10813:"Some Applications of Fractional Calculus in Engineering" 10412: 5346:(changing parameters in different operating conditions), 3422:

closed-loop system is required, which in turn requires a

3110:

using this response to determine the control parameters.

1575:

where the integral and derivative terms play their part.

500:

An example would be a valve for cooling water, where the

144: 11273:

Introduction to P,PI,PD & PID Controller with MATLAB

11013:

Tan, Kok Kiong; Wang Qing-Guo; Hang Chang Chieh (1999).

6893:{\displaystyle \alpha =1+{\frac {\tau _{d}}{\tau _{i}}}} 5453: 1471:

Most modern PID controls in industry are implemented as

9553:

Here, Kp is a dimensionless number, Ki is expressed in

4887: 2710:

gain, particularly in the presence of significant lag.

1699: 1529:

that can be moved and positioned by a control loop. An

9769: 8854:

previous_error := error wait(dt) goto loop

4331: 3019:

with a 180° phase shift. Stability is guaranteed when

10415:"PID control system analysis, design, and technology" 9679: 9644: 9609: 9559: 9493: 9436: 9369: 9290: 9206: 9154: 9099: 9038: 8904: 8874: 8806: 8644: 8546: 8474: 8412: 8347: 8320: 8242: 8172: 8110: 8038: 7949: 7601: 7319: 7235: 7194: 7161: 7025: 6993: 6964: 6935: 6853: 6800: 6752: 6706: 6595: 6432: 6419:

Sometimes it is useful to write the PID regulator in

6235: 6175: 6145: 6086: 6031: 5887: 5854: 5827: 5799: 5764: 5732: 5568: 5530: 5495: 5468: 5382: 5221: 5177: 5142: 5078: 5019: 4984: 4925: 4616: 4484: 4412: 4385: 4329: 4290: 4263: 4236: 4176: 4128: 4095: 4037: 4004: 3958: 3923: 3895: 3867: 3829: 3802: 3775: 3748: 3721: 3665: 3624: 3579: 3534: 3455: 3428: 3397: 3370: 3343: 3316: 3289: 3262: 3235: 3025: 2968: 2918: 2898: 2865: 2836: 2736: 2583: 2437: 2303: 2233: 2148: 2113: 2093: 2071: 2005: 1976: 1947: 1918: 1746: 1714: 1499:, or even the movement-detection circuit of a modern 1041: 996: 959: 932: 905: 868: 826: 799: 772: 726: 699: 672: 518: 425: 377: 303: 266: 226: 194: 65: 11171:

PID Control in MATLAB/Simulink and Python with TCLab

10690:. IEEE Control Systems Magazine, 26 (1). pp. 32-41. 10212: 4998:

is the error or deviation of actual measured value (

2820:{\displaystyle H(s)={\frac {K(s)G(s)}{1+K(s)G(s)}},} 2372:

gain. SSE may be mitigated by adding a compensating

10074:

Journal of the American Society for Naval Engineers

10030:, Portsmouth: Griffin & Co., pp. 137–138, 8629:{\displaystyle T_{i}=K_{p}/K_{i},T_{d}=K_{d}/K_{p}} 6216: 4796:The most significant improvement is to incorporate 4470: 3199: 2052:{\displaystyle e(t)=\mathrm {SP} -\mathrm {PV} (t)} 10901: 9874: 9787: 9697: 9665: 9622: 9575: 9538: 9270: 9018: 8887: 8849:dt - loop interval time (assumes reasonable scale) 8819: 8792: 8628: 8529: 8222: 7929: 7581: 7299: 7142: 7002: 6979: 6950: 6892: 6834:{\displaystyle T_{d}={\frac {\tau _{d}}{\alpha }}} 6833: 6784: 6738: 6686: 6563: 6400: 6196: 6158: 6122: 6072: 6014: 5867: 5840: 5812: 5777: 5745: 5712: 5551: 5516: 5481: 5312:(or closed-loop) control of a PID controller with 5227: 5203: 5161: 5122: 5046: 4990: 4968:{\displaystyle K_{P}\Delta +K_{I}\int \Delta \,dt} 4967: 4702: 4570: 4448: 4398: 4344: 4303: 4276: 4249: 4214: 4161: 4113: 4070: 4022: 3976: 3936: 3908: 3880: 3842: 3815: 3788: 3761: 3734: 3678: 3637: 3592: 3547: 3468: 3441: 3410: 3383: 3356: 3329: 3310:until the output of the loop oscillates; then set 3302: 3275: 3248: 3058: 3011: 2954: 2904: 2880: 2851: 2819: 2641: 2500: 2344: 2239: 2216: 2119: 2099: 2077: 2051: 1989: 1960: 1931: 1898: 1729: 1193: 1024: 982: 945: 918: 891: 854: 812: 785: 739: 712: 685: 656: 440: 392: 348: 289: 249: 209: 80: 11203:Proven Methods and Best Practices for PID Control 11111: 11091: 10968:Process Control: Modeling, Design, and Simulation 10945:"PID process control, a "Cruise Control" example" 10566:Process Control: Modeling, Design, and Simulation 10012:, Annapolis, MD: U.S. Naval Institute, p. 33 7013:Differentiating both sides of PID equation using 6207: 4850: 4467:There are numerous variants on the relay method. 4355: 3449:setting significantly less than half that of the 1548:The desired position is called the setpoint (SP). 1204: 11279: 10810: 10755:International Journal of Computational Cognition 10394: 10392: 8828:null, it prevents from sending the output to 0. 6411:King describes an effective chart-based method. 3703:Another heuristic tuning method is known as the 2717:(bounded oscillation) is acceptable or desired. 2345:{\displaystyle P_{\text{out}}=K_{\text{p}}e(t).} 2085:is the time or instantaneous time (the present), 1486: 1287:in the 17th century to regulate the gap between 1218:The use of the PID algorithm does not guarantee 10996:Instrument Engineers' Handbook: Process Control 10422:IEEE Transactions on Control Systems Technology 5123:{\displaystyle C={\frac {G(1+\tau s)}{\tau s}}} 11224:, Embedded Systems Programming, archived from 11115:Handbook of PI and PID Controller Tuning Rules 10507: 9763:Note that for very small intervals (e.g. 60Hz/ 8530:{\displaystyle u(t_{k})=u(t_{k-1})+K_{p}\left} 3194:Offline; only good for first-order processes. 479:– The balance of these effects is achieved by 11038:. Chichester, UK: John Wiley & Sons Ltd. 10454: 10389: 10323: 10321: 10172:"Diode Laser Locking and Linewidth Narrowing" 10106:harvnb error: no target: CITEREFBennett1993 ( 9796:holds true for all pseudocode presented here. 3100: 1939:is the proportional gain, a tuning parameter, 11264:: CS1 maint: multiple names: authors list ( 11217: 6785:{\displaystyle T_{i}=\tau _{i}\cdot \alpha } 2888:is the plant transfer function. A system is 10937: 10285: 10123:A history of control engineering, 1800-1930 5421:. Unsourced material may be challenged and 4824: 3143: 1997:is the derivative gain, a tuning parameter, 1637: 34:proportional–integral–derivative controller 10616: 10413:Kiam Heong Ang; Chong, G.; Yun Li (2005). 10327: 10318: 9713:A variant of the above algorithm using an 7592:Applying backward difference again gives, 6908: 6577: 6221:Most commercial control systems offer the 3715:in the 1940s. As in the method above, the 3186: 11235:"PID Controller Tuning: A Short Tutorial" 11052: 10838: 10828: 10811:Tenreiro Machado JA, et al. (2009). 10781: 10766: 10734:Practical Process Control by Control Guru 10662:On Automation of the PID Tuning Procedure 10525: 10457:"PID Controller Tuning: A Short Tutorial" 10377: 10349: 9878:Adaptive Control Processes: A Guided Tour 8722: 6341: 6271: 6134:Reciprocal gain, a.k.a. proportional band 5965: 5658: 5441:Learn how and when to remove this message 4958: 4804: 3692: 2488: 2291:, called the proportional gain constant. 2059:is the error (SP is the setpoint, and PV( 1968:is the integral gain, a tuning parameter, 1844: 1700:§ Alternative nomenclature and forms 1242:a setpoint, and the degree of any system 1131: 1025:{\displaystyle K_{\text{p}}/T_{\text{i}}} 855:{\displaystyle K_{\text{p}}/T_{\text{i}}} 596: 11121:(3rd ed.). Imperial College Press. 10965: 10563: 10508:Åström, K.J.; Hägglund, T. (July 1984). 10251:(4th ed.). CRC Press. p. 108. 10149: 10068: 9989: 6739:{\displaystyle K_{p}=K_{c}\cdot \alpha } 5204:{\displaystyle {\frac {G}{\tau }}=K_{I}} 4900: 4314: 4284:are dependent on the oscillation period 3478: 2527: 2383: 2254: 1462: 1454: 1403: 1265: 1253: 983:{\displaystyle K_{\text{p}}T_{\text{d}}} 892:{\displaystyle K_{\text{p}}T_{\text{d}}} 766:of the equation (see later in article), 152:of a PID controller in a feedback loop. 143: 139: 11092:Graham, Ron; Mike McHugh (2005-10-03). 10658: 10540: 10510:"Automatic Tuning of Simple Regulators" 10373: 10371: 10369: 10163: 10120: 10098: 9960: 9912: 9908: 9906: 9904: 9902: 9900: 9898: 9875:Richard E. Bellman (December 8, 2015). 9836:"9.3: PID Tuning via Classical Methods" 6987:are discretized with a sampling period 5268: 4713:using the same parameters found above. 1514: 1387:, with the controllers controlling the 1374: 96:(PV) and applies a correction based on 14: 11280: 10993: 10246: 9915:"A brief history of automatic control" 7152:Derivative terms are approximated as, 3850:are used to set the gains as follows: 3476:setting that was causing oscillation. 2425:) and added to the controller output. 1337:About this time, the invention of the 11198:PID with single Operational Amplifier 11035:Process Control: A Practical Approach 10878:Process Control: A Practical Approach 10273:"Introduction: PID Controller Design" 10045:"A Brief Building Automation History" 9859: 9816: 5454:Standard versus parallel (ideal) form 5337: 5328: 2727:The closed-loop transfer function is 2362: 1399: 11208:Principles of PID Control and Tuning 11193:PID Control with MATLAB and Simulink 11176:What's All This P-I-D Stuff, Anyhow? 11154:Principles of PID Control and Tuning 11053:Van Doren, Vance J. (July 1, 2003). 11031: 10875: 10817:Mathematical Problems in Engineering 10752: 10675:847ca38e-93e8-4188-b3d5-8ec6c23f2132 10582: 10541:Hornsey, Stephen (29 October 2012). 10366: 10153:The power of external-reset feedback 9895: 9736:Active disturbance rejection control 5419:adding citations to reliable sources 5386: 4888:Overshooting from known disturbances 3483:Effects of varying PID parameters (K 3167: 2962:. In other words, this happens when 2250: 1691: 507: 356:, and applies a correction based on 217:as the difference between a desired 88:as the difference between a desired 11221:Introduction to Closed-Loop Control 11065: 10169: 5065:using a "flow chart" box involving 3073: 2912:. This happens in situations where 899:; the advantage of this being that 24: 10986: 10923: 10730:"PI Control of the Heat Exchanger" 10727: 10706: 10659:Soltesz, Kristian (January 2012). 10086:10.1111/j.1559-3584.1922.tb04958.x 9994:. Praeger Security International. 9506: 9454: 9382: 9356: 9219: 9172: 9112: 9086: 8987: 8944: 8876: 8767: 8748: 8724: 8487: 8430: 8360: 8323: 8185: 8128: 8051: 8028: 7918: 7907: 7837: 7728: 7654: 7570: 7446: 7372: 7287: 6994: 6246: 6240: 6237: 5552:{\displaystyle D_{\mathrm {out} }} 5543: 5540: 5537: 5517:{\displaystyle I_{\mathrm {out} }} 5508: 5505: 5502: 5383:Alternative nomenclature and forms 5357: 5288:Derivative of the process variable 5020: 4985: 4955: 4936: 4916:The controller output is given by 4858: 4844: 4741: 4642: 4379:Specifically, the ultimate period 4215:{\displaystyle 3{K_{u}}{T_{u}}/40} 3495:) on the step response of a system 3130: 2859:is the PID transfer function, and 2523: 2294:The proportional term is given by 2036: 2033: 2025: 2022: 1766: 1763: 1672:, chemical composition (component 1620: 1176: 1157: 1133: 641: 622: 598: 25: 11314: 11137: 10782:Liang, Yilong; Yang, Tao (2009). 9717:(IIR) filter for the derivative: 8638:Note: This method solves in fact 6073:{\displaystyle K_{i}=K_{p}/T_{i}} 5875:to compute the control variable. 4071:{\displaystyle 0.54{K_{u}}/T_{u}} 3135:No mathematics required; online. 404:, to a new value determined by a 11164: 10234:10.1016/j.jmatprotec.2013.05.023 9973:Proceedings of the Royal Society 9706: 9631: 6217:Basing proportional action on PV 6123:{\displaystyle K_{d}=K_{p}T_{d}} 5391: 5285:present in a simple step change. 4896: 4471:First-order model with dead time 4162:{\displaystyle 1.2{K_{u}}/T_{u}} 3224: 2682:Some processes have a degree of 2672: 2574:The derivative term is given by 2517: 2379: 2356: 1314:in 1868 in his now-famous paper 1229: 1213: 480: 290:{\displaystyle {\text{PV}}=y(t)} 250:{\displaystyle {\text{SP}}=r(t)} 11017:. London, UK: Springer-Verlag. 10917: 10894: 10869: 10858: 10847: 10804: 10775: 10746: 10721: 10700: 10681: 10652: 10610: 10589:Heinänen, Eero (October 2018). 10557: 10534: 10501: 10448: 10406: 10279: 10265: 10240: 10188: 10143: 10114: 10092: 10062: 9757: 9748: 6414: 5303: 4449:{\displaystyle K_{u}=4b/\pi a,} 4345:{\displaystyle {\tfrac {1}{4}}} 1646: 1565: 1380:Trials were carried out on the 120:respectively), hence the name. 27:Control loop feedback mechanism 11094:"FAQ on PID controller tuning" 10222:Material Processing Technology 10037: 10017: 9983: 9954: 9881:. Princeton University Press. 9868: 9853: 9828: 9810: 9779: 9698:{\displaystyle 3<=N<=10} 9533: 9521: 9486: 9474: 9408: 9402: 9321: 9309: 9300: 9294: 9048: 9042: 8914: 8908: 8761: 8755: 8719: 8713: 8679: 8673: 8654: 8648: 8519: 8500: 8467: 8448: 8391: 8378: 8287: 8268: 8259: 8246: 8217: 8198: 8165: 8146: 8082: 8069: 7994: 7975: 7966: 7953: 7902: 7883: 7874: 7855: 7832: 7813: 7804: 7791: 7766: 7753: 7723: 7704: 7695: 7682: 7649: 7630: 7621: 7608: 7565: 7546: 7528: 7515: 7484: 7471: 7441: 7422: 7413: 7400: 7367: 7348: 7339: 7326: 7282: 7263: 7254: 7241: 7216: 7203: 7187: 7174: 7137: 7131: 7103: 7097: 7078: 7072: 7044: 7038: 6974: 6968: 6945: 6939: 6681: 6657: 6654: 6621: 6605: 6599: 6442: 6436: 6390: 6384: 6337: 6331: 6288: 6282: 6249: 6243: 6208:Basing derivative action on PV 6009: 6003: 5962: 5956: 5922: 5916: 5897: 5891: 5702: 5696: 5655: 5649: 5608: 5602: 5578: 5572: 5106: 5091: 5047:{\displaystyle \Delta =SP-PV.} 4905:Basic block of a PI controller 4851:Modifications to the algorithm 4764: 4626: 4620: 4562: 4556: 4494: 4488: 4356:Relay (Åström–Hägglund) method 3047: 3041: 3035: 3029: 2999: 2995: 2989: 2983: 2977: 2970: 2940: 2934: 2928: 2922: 2875: 2869: 2846: 2840: 2808: 2802: 2796: 2790: 2776: 2770: 2764: 2758: 2746: 2740: 2659: 2622: 2616: 2485: 2479: 2428:The integral term is given by 2336: 2330: 2158: 2152: 2046: 2040: 2015: 2009: 1879: 1873: 1841: 1835: 1801: 1795: 1776: 1770: 1756: 1750: 1724: 1718: 1554:The input to the process (the 1495:, the power conditioning of a 1477:programmable logic controllers 1344:pendulum-and-hydrostat control 1205:Selective use of control terms 1170: 1164: 1128: 1122: 1081: 1075: 1051: 1045: 635: 629: 593: 587: 553: 547: 528: 522: 435: 429: 387: 381: 349:{\displaystyle e(t)=r(t)-y(t)} 343: 337: 328: 322: 313: 307: 284: 278: 244: 238: 204: 198: 75: 69: 13: 1: 11096:. Mike McHugh. Archived from 10626:IEEE Control Systems Magazine 10527:10.1016/S1474-6670(17)61248-5 10455:Jinghua Zhong (Spring 2006). 10121:Bennett, Stuart (June 1986). 10027:Torpedoes and Torpedo Warfare 9922:IEEE Control Systems Magazine 9803: 8831: 3283:values to zero. Increase the 3059:{\displaystyle K(s)G(s)<1} 1607: 1487:Electronic analog controllers 820:are respectively replaced by 512:The overall control function 11178:Article in Electronic Design 11144:PID tuning using Mathematica 10286:Tim Wescott (October 2000). 9864:, Cambridge University Press 8868:of a PID can be written as ( 3012:{\displaystyle |K(s)G(s)|=1} 2693: 1990:{\displaystyle K_{\text{d}}} 1961:{\displaystyle K_{\text{i}}} 1932:{\displaystyle K_{\text{p}}} 946:{\displaystyle T_{\text{d}}} 919:{\displaystyle T_{\text{i}}} 813:{\displaystyle K_{\text{d}}} 786:{\displaystyle K_{\text{i}}} 740:{\displaystyle K_{\text{d}}} 713:{\displaystyle K_{\text{i}}} 686:{\displaystyle K_{\text{p}}} 7: 11218:Michael Barr (2002-07-30), 10966:Bequette, B. Wayne (2006). 10564:Bequette, B. Wayne (2003). 9788:{\textstyle .016{\bar {6}}} 9724: 9666:{\displaystyle \tau _{d}/N} 9587:In the real world, this is 8888:{\displaystyle \Delta _{t}} 5246: 4114:{\displaystyle 0.60{K_{u}}} 4023:{\displaystyle 0.45{K_{u}}} 3977:{\displaystyle 0.50{K_{u}}} 3823:and the oscillation period 3212:Simple control rule (SIMC) 3081:Two basic requirements are 3068:Nyquist stability criterion 2955:{\displaystyle K(s)G(s)=-1} 2063:) is the process variable), 1578: 1541:The sensed position is the 1439:distributed control systems 1227: 400:, such as the opening of a 10: 11319: 11055:"Loop Tuning Fundamentals" 10378:Skogestad, Sigurd (2003). 9990:Newpower, Anthony (2006). 9824:. Japan: Kyoto University. 9603:A common issue when using 4862: 3696: 3504:a parameter independently 3101:Overview of tuning methods 2673:limitations of PID control 2518:the section on loop tuning 2357:the section on loop tuning 2247:is the complex frequency. 1518: 1274: 1249: 54:industrial control systems 10880:. Wiley. pp. 52–78. 10588: 10360:10.1103/revmodphys.77.783 10330:Reviews of Modern Physics 10275:. University of Michigan. 10196:"Position control system" 10150:Shinskey, F Greg (2004), 9860:Hills, Richard L (1996), 9715:infinite impulse response 7010:, k is the sample index. 6197:{\displaystyle 100/K_{p}} 3114:Choosing a tuning method 2139:of the PID controller is 11288:Classical control theory 10638:10.1109/MCS.2006.1580153 10514:IFAC Proceedings Volumes 10434:10.1109/TCST.2005.847331 10247:Lipták, Béla G. (2003). 9913:Bennett, Stuart (1996). 9741: 7003:{\displaystyle \Delta t} 4825:Noise in derivative term 2516:the setpoint value (see 1638:Response to disturbances 11149:PID tuning using Python 11015:Advances in PID Control 10024:Sleeman, C. W. (1880), 8895:is the sampling time): 6909:Discrete implementation 6578:Series/interacting form 5489:gain is applied to the 5162:{\displaystyle G=K_{P}} 4991:{\displaystyle \Delta } 3854:Ziegler–Nichols method 52:that is widely used in 11112:Aidan O'Dwyer (2009). 10489:Cite journal requires 10306:Cite journal requires 9840:Engineering LibreTexts 9789: 9699: 9667: 9624: 9577: 9576:{\displaystyle s^{-1}} 9540: 9272: 9020: 8889: 8840:Kp - proportional gain 8821: 8794: 8630: 8531: 8224: 7931: 7583: 7301: 7144: 7004: 6981: 6952: 6894: 6835: 6786: 6740: 6688: 6565: 6402: 6198: 6160: 6124: 6074: 6016: 5869: 5842: 5814: 5779: 5747: 5714: 5553: 5518: 5483: 5229: 5205: 5163: 5124: 5048: 4992: 4969: 4906: 4805:Linearity and symmetry 4704: 4599:is the dead time, and 4595:is the time constant, 4572: 4450: 4400: 4346: 4321:Ziegler–Nichols method 4305: 4278: 4251: 4216: 4163: 4115: 4072: 4024: 3978: 3938: 3910: 3882: 3844: 3817: 3790: 3763: 3736: 3705:Ziegler–Nichols method 3699:Ziegler–Nichols method 3693:Ziegler–Nichols method 3680: 3639: 3594: 3549: 3496: 3470: 3443: 3412: 3385: 3358: 3331: 3304: 3277: 3250: 3060: 3013: 2956: 2906: 2882: 2853: 2821: 2643: 2554: 2502: 2410: 2346: 2281: 2241: 2218: 2121: 2101: 2079: 2053: 1991: 1962: 1933: 1900: 1731: 1570:The obvious method is 1468: 1460: 1409: 1271: 1263: 1195: 1026: 984: 947: 920: 893: 856: 814: 787: 741: 714: 687: 658: 442: 408:of the control terms. 394: 350: 291: 251: 211: 177: 82: 11303:Industrial automation 10994:Liptak, Bela (1995). 10970:. Prentice Hall PTR. 9790: 9700: 9668: 9625: 9623:{\displaystyle K_{d}} 9578: 9541: 9273: 9021: 8890: 8822: 8820:{\displaystyle u_{0}} 8795: 8631: 8532: 8225: 7932: 7584: 7302: 7145: 7005: 6982: 6953: 6895: 6836: 6787: 6741: 6689: 6566: 6403: 6199: 6161: 6159:{\displaystyle K_{p}} 6125: 6075: 6017: 5870: 5868:{\displaystyle K_{p}} 5843: 5841:{\displaystyle T_{i}} 5815: 5813:{\displaystyle T_{d}} 5780: 5778:{\displaystyle T_{d}} 5748: 5746:{\displaystyle T_{i}} 5715: 5554: 5519: 5484: 5482:{\displaystyle K_{p}} 5230: 5206: 5164: 5125: 5049: 5004:) from the setpoint ( 4993: 4970: 4904: 4863:Further information: 4705: 4588:is the process gain, 4573: 4451: 4401: 4399:{\displaystyle T_{u}} 4360:Published in 1984 by 4347: 4315:Cohen–Coon parameters 4306: 4304:{\displaystyle T_{u}} 4279: 4277:{\displaystyle K_{d}} 4252: 4250:{\displaystyle K_{i}} 4217: 4164: 4116: 4073: 4025: 3979: 3939: 3937:{\displaystyle K_{d}} 3911: 3909:{\displaystyle K_{i}} 3883: 3881:{\displaystyle K_{p}} 3845: 3843:{\displaystyle T_{u}} 3818: 3816:{\displaystyle K_{u}} 3791: 3789:{\displaystyle K_{u}} 3764: 3762:{\displaystyle K_{d}} 3737: 3735:{\displaystyle K_{i}} 3697:Further information: 3681: 3679:{\displaystyle K_{d}} 3640: 3638:{\displaystyle K_{d}} 3595: 3593:{\displaystyle K_{i}} 3550: 3548:{\displaystyle K_{p}} 3482: 3471: 3469:{\displaystyle K_{p}} 3444: 3442:{\displaystyle K_{p}} 3413: 3411:{\displaystyle K_{p}} 3386: 3384:{\displaystyle K_{d}} 3359: 3357:{\displaystyle K_{i}} 3332: 3330:{\displaystyle K_{p}} 3305: 3303:{\displaystyle K_{p}} 3278: 3276:{\displaystyle K_{d}} 3251: 3249:{\displaystyle K_{i}} 3191:Good process models. 3061: 3014: 2957: 2907: 2883: 2854: 2822: 2644: 2531: 2503: 2387: 2347: 2258: 2242: 2219: 2122: 2102: 2100:{\displaystyle \tau } 2080: 2054: 1992: 1963: 1934: 1901: 1732: 1519:Further information: 1466: 1458: 1407: 1269: 1257: 1236:lead–lag compensation 1222:of the system or its 1196: 1027: 985: 948: 921: 894: 857: 815: 788: 742: 715: 688: 659: 443: 395: 351: 292: 252: 212: 176:) is the measured PV. 147: 140:Fundamental operation 83: 42:three-term controller 10790:. Asid'09: 417–420. 10008:Gray, Edwyn (1991), 9767: 9677: 9642: 9607: 9557: 9288: 9036: 8902: 8872: 8846:Kd - derivative gain 8804: 8642: 8544: 8240: 7947: 7599: 7317: 7159: 7023: 6991: 6980:{\displaystyle e(t)} 6962: 6951:{\displaystyle u(t)} 6933: 6851: 6798: 6750: 6704: 6593: 6430: 6233: 6173: 6143: 6084: 6029: 5885: 5852: 5825: 5797: 5762: 5730: 5566: 5528: 5493: 5466: 5415:improve this section 5269:Setpoint step change 5219: 5215:Setting a value for 5175: 5140: 5076: 5017: 4982: 4923: 4798:feed-forward control 4614: 4482: 4410: 4383: 4327: 4288: 4261: 4234: 4174: 4126: 4093: 4035: 4002: 3956: 3921: 3893: 3865: 3827: 3800: 3773: 3746: 3719: 3713:Nathaniel B. Nichols 3663: 3656:No effect in theory 3622: 3577: 3532: 3453: 3426: 3395: 3368: 3341: 3314: 3287: 3260: 3233: 3023: 2966: 2916: 2896: 2881:{\displaystyle G(s)} 2863: 2852:{\displaystyle K(s)} 2834: 2734: 2581: 2435: 2301: 2231: 2146: 2111: 2091: 2069: 2003: 1974: 1945: 1916: 1744: 1730:{\displaystyle u(t)} 1712: 1515:Control loop example 1479:(PLCs), or discrete 1281:centrifugal governor 1039: 994: 957: 930: 903: 866: 824: 797: 770: 724: 697: 670: 516: 441:{\displaystyle e(t)} 423: 393:{\displaystyle u(t)} 375: 301: 264: 224: 210:{\displaystyle e(t)} 192: 92:(SP) and a measured 81:{\displaystyle e(t)} 63: 48:mechanism employing 11298:Control engineering 11100:on February 6, 2005 11059:Control Engineering 11032:King, Myke (2010). 10947:. CodeProject. 2009 10876:King, Myke (2011). 10830:10.1155/2010/639801 10342:2005RvMP...77..783B 10288:"PID without a PhD" 9862:Power From the Wind 8709: 6326: 5952: 5645: 5462:. In this form the 5169:= proportional gain 3855: 3521:Steady-state error 3505: 3115: 2475: 1831: 1481:compact controllers 1435:hazardous locations 1312:James Clerk Maxwell 1118: 583: 10924:Thakur, Bhushana. 9817:Araki, M. (2009). 9785: 9695: 9663: 9620: 9573: 9536: 9516: 9464: 9392: 9268: 9229: 9182: 9122: 9016: 8885: 8843:Ki - integral gain 8817: 8790: 8695: 8626: 8527: 8495: 8438: 8368: 8341: 8220: 8193: 8136: 8059: 7927: 7579: 7297: 7295: 7229: 7140: 7000: 6977: 6948: 6927:finite differences 6890: 6831: 6782: 6736: 6684: 6561: 6398: 6312: 6194: 6156: 6120: 6070: 6012: 5938: 5865: 5838: 5810: 5775: 5743: 5710: 5631: 5549: 5514: 5479: 5338:Other improvements 5329:Bumpless operation 5295:Setpoint weighting 5225: 5201: 5159: 5120: 5044: 4988: 4965: 4907: 4700: 4568: 4446: 4396: 4342: 4340: 4301: 4274: 4247: 4212: 4159: 4111: 4068: 4020: 3974: 3934: 3906: 3878: 3853: 3840: 3813: 3786: 3759: 3732: 3676: 3635: 3590: 3545: 3499: 3497: 3466: 3439: 3408: 3381: 3354: 3327: 3300: 3273: 3246: 3113: 3056: 3009: 2952: 2902: 2878: 2849: 2817: 2715:marginal stability 2702:, with or without 2639: 2555: 2498: 2461: 2411: 2369:steady-state error 2363:Steady-state error 2342: 2282: 2237: 2214: 2131:Equivalently, the 2117: 2097: 2075: 2049: 1987: 1958: 1929: 1896: 1817: 1727: 1469: 1461: 1422:nozzle and flapper 1410: 1400:Industrial control 1375:steady-state error 1302:'s self-designed " 1285:Christiaan Huygens 1272: 1264: 1230:§ Limitations 1191: 1104: 1022: 980: 943: 916: 889: 852: 810: 783: 737: 710: 683: 654: 569: 438: 390: 346: 287: 247: 207: 178: 78: 11188:PID Without a PhD 11128:978-1-84816-242-6 11045:978-0-470-97587-9 11024:978-1-85233-138-2 11005:978-0-8019-8242-2 10887:978-0-470-97587-9 10728:Cooper, Douglas. 10707:Cooper, Douglas. 10575:978-0-13-353640-9 10293:. EE Times-India. 10170:Neuhaus, Rudolf. 10136:978-0-86341-047-5 10070:Minorsky, Nicolas 10001:978-0-275-99032-9 9934:10.1109/37.506394 9782: 9515: 9463: 9391: 9266: 9228: 9181: 9121: 9014: 8996: 8969: 8775: 8741: 8692: 8667: 8494: 8437: 8367: 8340: 8192: 8135: 8058: 7925: 7914: 7844: 7735: 7661: 7577: 7543: 7512: 7453: 7379: 7294: 7228: 7171: 7128: 7069: 7035: 7015:Newton's notation 6888: 6829: 6646: 6559: 6476: 6421:Laplace transform 6376: 6310: 6168:proportional band 5998: 5691: 5629: 5559:terms, yielding: 5451: 5450: 5443: 5228:{\displaystyle G} 5186: 5118: 4689: 4686: 4639: 4551: 4538: 4510: 4366:bang-bang control 4362:Karl Johan Åström 4339: 4225: 4224: 3690: 3689: 3222: 3221: 2905:{\displaystyle s} 2812: 2634: 2604: 2591: 2458: 2445: 2324: 2311: 2251:Proportional term 2240:{\displaystyle s} 2205: 2184: 2171: 2133:transfer function 2120:{\displaystyle t} 2078:{\displaystyle t} 1984: 1955: 1926: 1891: 1861: 1814: 1789: 1692:Controller theory 1627:critically damped 1602:marginally stable 1473:computer software 1414:negative feedback 1339:Whitehead torpedo 1224:control stability 1184: 1150: 1102: 1099: 1064: 1019: 1004: 977: 967: 940: 913: 886: 876: 849: 834: 807: 780: 734: 707: 680: 649: 615: 566: 541: 508:Mathematical form 270: 230: 160:) is the desired 16:(Redirected from 11310: 11269: 11263: 11255: 11253: 11252: 11246: 11240:. Archived from 11239: 11229: 11132: 11120: 11108: 11106: 11105: 11088: 11086: 11085: 11080:on March 7, 2007 11079: 11073:. Archived from 11072: 11066:Sellers, David. 11062: 11049: 11028: 11009: 10981: 10957: 10956: 10954: 10952: 10941: 10935: 10934: 10932: 10921: 10915: 10914: 10912: 10911: 10905: 10898: 10892: 10891: 10873: 10867: 10862: 10856: 10851: 10845: 10844: 10842: 10832: 10808: 10802: 10801: 10779: 10773: 10772: 10770: 10750: 10744: 10743: 10741: 10740: 10725: 10719: 10718: 10716: 10715: 10704: 10698: 10685: 10679: 10678: 10656: 10650: 10649: 10623: 10614: 10608: 10607: 10605: 10603: 10597: 10586: 10580: 10579: 10561: 10555: 10554: 10538: 10532: 10531: 10529: 10520:(2): 1867–1872. 10505: 10499: 10498: 10492: 10487: 10485: 10477: 10475: 10474: 10468: 10462:. Archived from 10461: 10452: 10446: 10445: 10419: 10410: 10404: 10403: 10396: 10387: 10386: 10384: 10375: 10364: 10363: 10353: 10325: 10316: 10315: 10309: 10304: 10302: 10294: 10292: 10283: 10277: 10276: 10269: 10263: 10262: 10244: 10238: 10237: 10228:(1): 2015–2032. 10216: 10210: 10209: 10207: 10200: 10192: 10186: 10185: 10183: 10181: 10176: 10167: 10161: 10160: 10159:, Control Global 10158: 10147: 10141: 10140: 10125:. IET. pp. 10118: 10112: 10111: 10096: 10090: 10089: 10066: 10060: 10059: 10057: 10056: 10047:. Archived from 10041: 10035: 10034: 10021: 10015: 10013: 10006:p. citing 10005: 9987: 9981: 9980: 9970: 9958: 9952: 9951: 9949: 9948: 9942: 9936:. Archived from 9919: 9910: 9893: 9892: 9872: 9866: 9865: 9857: 9851: 9850: 9848: 9847: 9832: 9826: 9825: 9823: 9814: 9797: 9794: 9792: 9791: 9786: 9784: 9783: 9775: 9761: 9755: 9752: 9710: 9704: 9702: 9701: 9696: 9672: 9670: 9669: 9664: 9659: 9654: 9653: 9635: 9629: 9627: 9626: 9621: 9619: 9618: 9589:D-to-A converted 9582: 9580: 9579: 9574: 9572: 9571: 9545: 9543: 9542: 9537: 9517: 9514: 9513: 9504: 9503: 9494: 9470: 9466: 9465: 9462: 9461: 9452: 9451: 9450: 9437: 9431: 9430: 9398: 9394: 9393: 9390: 9389: 9380: 9379: 9370: 9364: 9363: 9354: 9353: 9341: 9340: 9277: 9275: 9274: 9269: 9267: 9265: 9264: 9263: 9244: 9243: 9242: 9230: 9227: 9226: 9217: 9216: 9207: 9201: 9200: 9188: 9184: 9183: 9180: 9179: 9170: 9169: 9168: 9155: 9149: 9148: 9128: 9124: 9123: 9120: 9119: 9110: 9109: 9100: 9094: 9093: 9084: 9083: 9071: 9070: 9055: 9025: 9023: 9022: 9017: 9015: 9010: 8999: 8997: 8995: 8994: 8985: 8984: 8975: 8970: 8968: 8954: 8952: 8951: 8942: 8941: 8929: 8928: 8894: 8892: 8891: 8886: 8884: 8883: 8826: 8824: 8823: 8818: 8816: 8815: 8799: 8797: 8796: 8791: 8789: 8788: 8776: 8774: 8770: 8764: 8751: 8745: 8743: 8742: 8739: 8727: 8708: 8703: 8694: 8693: 8690: 8669: 8668: 8665: 8635: 8633: 8632: 8627: 8625: 8624: 8615: 8610: 8609: 8597: 8596: 8584: 8583: 8574: 8569: 8568: 8556: 8555: 8536: 8534: 8533: 8528: 8526: 8522: 8518: 8517: 8496: 8493: 8485: 8484: 8475: 8466: 8465: 8444: 8440: 8439: 8436: 8428: 8427: 8426: 8413: 8390: 8389: 8374: 8370: 8369: 8366: 8358: 8357: 8348: 8342: 8339: 8338: 8329: 8321: 8302: 8301: 8286: 8285: 8258: 8257: 8229: 8227: 8226: 8221: 8216: 8215: 8194: 8191: 8183: 8182: 8173: 8164: 8163: 8142: 8138: 8137: 8134: 8126: 8125: 8124: 8111: 8105: 8104: 8081: 8080: 8065: 8061: 8060: 8057: 8049: 8048: 8039: 8027: 8026: 8014: 8013: 7993: 7992: 7965: 7964: 7936: 7934: 7933: 7928: 7926: 7924: 7916: 7915: 7913: 7905: 7901: 7900: 7873: 7872: 7850: 7845: 7843: 7835: 7831: 7830: 7803: 7802: 7786: 7783: 7781: 7780: 7765: 7764: 7749: 7748: 7736: 7734: 7726: 7722: 7721: 7694: 7693: 7677: 7675: 7674: 7662: 7660: 7652: 7648: 7647: 7620: 7619: 7603: 7588: 7586: 7585: 7580: 7578: 7576: 7568: 7564: 7563: 7545: 7544: 7536: 7527: 7526: 7514: 7513: 7505: 7501: 7499: 7498: 7483: 7482: 7467: 7466: 7454: 7452: 7444: 7440: 7439: 7412: 7411: 7395: 7393: 7392: 7380: 7378: 7370: 7366: 7365: 7338: 7337: 7321: 7306: 7304: 7303: 7298: 7296: 7293: 7285: 7281: 7280: 7253: 7252: 7236: 7230: 7227: 7219: 7215: 7214: 7195: 7186: 7185: 7173: 7172: 7164: 7149: 7147: 7146: 7141: 7130: 7129: 7121: 7118: 7117: 7093: 7092: 7071: 7070: 7062: 7059: 7058: 7037: 7036: 7028: 7009: 7007: 7006: 7001: 6986: 6984: 6983: 6978: 6957: 6955: 6954: 6949: 6899: 6897: 6896: 6891: 6889: 6887: 6886: 6877: 6876: 6867: 6840: 6838: 6837: 6832: 6830: 6825: 6824: 6815: 6810: 6809: 6791: 6789: 6788: 6783: 6775: 6774: 6762: 6761: 6745: 6743: 6742: 6737: 6729: 6728: 6716: 6715: 6693: 6691: 6690: 6685: 6674: 6669: 6668: 6647: 6645: 6644: 6639: 6638: 6625: 6620: 6619: 6570: 6568: 6567: 6562: 6560: 6555: 6554: 6553: 6541: 6536: 6535: 6523: 6522: 6521: 6511: 6510: 6500: 6495: 6490: 6489: 6477: 6472: 6471: 6462: 6457: 6456: 6407: 6405: 6404: 6399: 6397: 6393: 6377: 6375: 6364: 6362: 6361: 6349: 6340: 6325: 6320: 6311: 6309: 6308: 6296: 6291: 6265: 6264: 6252: 6203: 6201: 6200: 6195: 6193: 6192: 6183: 6165: 6163: 6162: 6157: 6155: 6154: 6129: 6127: 6126: 6121: 6119: 6118: 6109: 6108: 6096: 6095: 6079: 6077: 6076: 6071: 6069: 6068: 6059: 6054: 6053: 6041: 6040: 6021: 6019: 6018: 6013: 5999: 5997: 5986: 5984: 5983: 5951: 5946: 5937: 5936: 5912: 5911: 5874: 5872: 5871: 5866: 5864: 5863: 5847: 5845: 5844: 5839: 5837: 5836: 5819: 5817: 5816: 5811: 5809: 5808: 5784: 5782: 5781: 5776: 5774: 5773: 5752: 5750: 5749: 5744: 5742: 5741: 5719: 5717: 5716: 5711: 5709: 5705: 5692: 5690: 5679: 5677: 5676: 5644: 5639: 5630: 5628: 5627: 5615: 5593: 5592: 5558: 5556: 5555: 5550: 5548: 5547: 5546: 5523: 5521: 5520: 5515: 5513: 5512: 5511: 5488: 5486: 5485: 5480: 5478: 5477: 5446: 5439: 5435: 5432: 5426: 5395: 5387: 5352:fractional order 5278:Setpoint ramping 5234: 5232: 5231: 5226: 5210: 5208: 5207: 5202: 5200: 5199: 5187: 5179: 5168: 5166: 5165: 5160: 5158: 5157: 5129: 5127: 5126: 5121: 5119: 5117: 5109: 5086: 5053: 5051: 5050: 5045: 4997: 4995: 4994: 4989: 4974: 4972: 4971: 4966: 4951: 4950: 4935: 4934: 4709: 4707: 4706: 4701: 4696: 4692: 4691: 4690: 4688: 4687: 4684: 4678: 4664: 4641: 4640: 4637: 4577: 4575: 4574: 4569: 4552: 4550: 4540: 4539: 4536: 4529: 4528: 4527: 4512: 4511: 4508: 4501: 4463: 4459: 4455: 4453: 4452: 4447: 4436: 4422: 4421: 4405: 4403: 4402: 4397: 4395: 4394: 4351: 4349: 4348: 4343: 4341: 4332: 4310: 4308: 4307: 4302: 4300: 4299: 4283: 4281: 4280: 4275: 4273: 4272: 4256: 4254: 4253: 4248: 4246: 4245: 4221: 4219: 4218: 4213: 4208: 4203: 4202: 4201: 4191: 4190: 4189: 4168: 4166: 4165: 4160: 4158: 4157: 4148: 4143: 4142: 4141: 4120: 4118: 4117: 4112: 4110: 4109: 4108: 4077: 4075: 4074: 4069: 4067: 4066: 4057: 4052: 4051: 4050: 4029: 4027: 4026: 4021: 4019: 4018: 4017: 3983: 3981: 3980: 3975: 3973: 3972: 3971: 3943: 3941: 3940: 3935: 3933: 3932: 3915: 3913: 3912: 3907: 3905: 3904: 3887: 3885: 3884: 3879: 3877: 3876: 3856: 3852: 3849: 3847: 3846: 3841: 3839: 3838: 3822: 3820: 3819: 3814: 3812: 3811: 3795: 3793: 3792: 3787: 3785: 3784: 3768: 3766: 3765: 3760: 3758: 3757: 3741: 3739: 3738: 3733: 3731: 3730: 3707:, introduced by 3685: 3683: 3682: 3677: 3675: 3674: 3644: 3642: 3641: 3636: 3634: 3633: 3599: 3597: 3596: 3591: 3589: 3588: 3554: 3552: 3551: 3546: 3544: 3543: 3506: 3498: 3475: 3473: 3472: 3467: 3465: 3464: 3448: 3446: 3445: 3440: 3438: 3437: 3417: 3415: 3414: 3409: 3407: 3406: 3390: 3388: 3387: 3382: 3380: 3379: 3363: 3361: 3360: 3355: 3353: 3352: 3336: 3334: 3333: 3328: 3326: 3325: 3309: 3307: 3306: 3301: 3299: 3298: 3282: 3280: 3279: 3274: 3272: 3271: 3255: 3253: 3252: 3247: 3245: 3244: 3116: 3112: 3087:command tracking 3074:Optimal behavior 3065: 3063: 3062: 3057: 3018: 3016: 3015: 3010: 3002: 2973: 2961: 2959: 2958: 2953: 2911: 2909: 2908: 2903: 2887: 2885: 2884: 2879: 2858: 2856: 2855: 2850: 2826: 2824: 2823: 2818: 2813: 2811: 2779: 2753: 2648: 2646: 2645: 2640: 2635: 2633: 2625: 2608: 2606: 2605: 2602: 2593: 2592: 2589: 2507: 2505: 2504: 2499: 2474: 2469: 2460: 2459: 2456: 2447: 2446: 2443: 2351: 2349: 2348: 2343: 2326: 2325: 2322: 2313: 2312: 2309: 2246: 2244: 2243: 2238: 2223: 2221: 2220: 2215: 2207: 2206: 2203: 2191: 2186: 2185: 2182: 2173: 2172: 2169: 2126: 2124: 2123: 2118: 2106: 2104: 2103: 2098: 2084: 2082: 2081: 2076: 2058: 2056: 2055: 2050: 2039: 2028: 1996: 1994: 1993: 1988: 1986: 1985: 1982: 1967: 1965: 1964: 1959: 1957: 1956: 1953: 1938: 1936: 1935: 1930: 1928: 1927: 1924: 1905: 1903: 1902: 1897: 1892: 1890: 1882: 1865: 1863: 1862: 1859: 1830: 1825: 1816: 1815: 1812: 1791: 1790: 1787: 1769: 1736: 1734: 1733: 1728: 1556:electric current 1543:process variable 1505:microcontrollers 1390:angular velocity 1367:Nicolas Minorsky 1363:Russian American 1304:conical pendulum 1233: 1200: 1198: 1197: 1192: 1190: 1186: 1185: 1183: 1179: 1173: 1160: 1154: 1152: 1151: 1148: 1136: 1117: 1112: 1103: 1101: 1100: 1097: 1088: 1066: 1065: 1062: 1031: 1029: 1028: 1023: 1021: 1020: 1017: 1011: 1006: 1005: 1002: 989: 987: 986: 981: 979: 978: 975: 969: 968: 965: 952: 950: 949: 944: 942: 941: 938: 925: 923: 922: 917: 915: 914: 911: 898: 896: 895: 890: 888: 887: 884: 878: 877: 874: 861: 859: 858: 853: 851: 850: 847: 841: 836: 835: 832: 819: 817: 816: 811: 809: 808: 805: 792: 790: 789: 784: 782: 781: 778: 746: 744: 743: 738: 736: 735: 732: 719: 717: 716: 711: 709: 708: 705: 692: 690: 689: 684: 682: 681: 678: 663: 661: 660: 655: 650: 648: 644: 638: 625: 619: 617: 616: 613: 601: 582: 577: 568: 567: 564: 543: 542: 539: 447: 445: 444: 439: 399: 397: 396: 391: 370:control variable 355: 353: 352: 347: 296: 294: 293: 288: 271: 268: 259:process variable 256: 254: 253: 248: 231: 228: 216: 214: 213: 208: 162:process variable 94:process variable 87: 85: 84: 79: 21: 11318: 11317: 11313: 11312: 11311: 11309: 11308: 11307: 11293:Control devices 11278: 11277: 11257: 11256: 11250: 11248: 11244: 11237: 11167: 11140: 11135: 11129: 11118: 11103: 11101: 11083: 11081: 11077: 11070: 11046: 11025: 11006: 10989: 10987:Further reading 10984: 10978: 10961: 10960: 10950: 10948: 10943: 10942: 10938: 10930: 10922: 10918: 10909: 10907: 10900: 10899: 10895: 10888: 10874: 10870: 10863: 10859: 10852: 10848: 10809: 10805: 10798: 10780: 10776: 10768:10.1.1.152.9564 10751: 10747: 10738: 10736: 10726: 10722: 10713: 10711: 10705: 10701: 10686: 10682: 10671:Lund university 10657: 10653: 10621: 10615: 10611: 10601: 10599: 10595: 10587: 10583: 10576: 10562: 10558: 10539: 10535: 10506: 10502: 10490: 10488: 10479: 10478: 10472: 10470: 10466: 10459: 10453: 10449: 10417: 10411: 10407: 10398: 10397: 10390: 10382: 10376: 10367: 10351:10.1.1.124.7043 10326: 10319: 10307: 10305: 10296: 10295: 10290: 10284: 10280: 10271: 10270: 10266: 10259: 10245: 10241: 10217: 10213: 10205: 10198: 10194: 10193: 10189: 10179: 10177: 10174: 10168: 10164: 10156: 10148: 10144: 10137: 10119: 10115: 10105: 10097: 10093: 10067: 10063: 10054: 10052: 10043: 10042: 10038: 10023: 10022: 10018: 10007: 10002: 9988: 9984: 9968: 9959: 9955: 9946: 9944: 9940: 9917: 9911: 9896: 9889: 9873: 9869: 9858: 9854: 9845: 9843: 9834: 9833: 9829: 9821: 9815: 9811: 9806: 9801: 9800: 9774: 9773: 9768: 9765: 9764: 9762: 9758: 9753: 9749: 9744: 9727: 9722: 9678: 9675: 9674: 9655: 9649: 9645: 9643: 9640: 9639: 9614: 9610: 9608: 9605: 9604: 9564: 9560: 9558: 9555: 9554: 9551: 9509: 9505: 9499: 9495: 9492: 9457: 9453: 9446: 9442: 9438: 9435: 9426: 9422: 9418: 9414: 9385: 9381: 9375: 9371: 9368: 9359: 9355: 9349: 9345: 9336: 9332: 9331: 9327: 9289: 9286: 9285: 9256: 9252: 9245: 9235: 9231: 9222: 9218: 9212: 9208: 9205: 9193: 9189: 9175: 9171: 9164: 9160: 9156: 9153: 9144: 9140: 9136: 9132: 9115: 9111: 9105: 9101: 9098: 9089: 9085: 9079: 9075: 9066: 9062: 9061: 9057: 9056: 9054: 9037: 9034: 9033: 9000: 8998: 8990: 8986: 8980: 8976: 8974: 8958: 8953: 8947: 8943: 8937: 8933: 8924: 8920: 8903: 8900: 8899: 8879: 8875: 8873: 8870: 8869: 8855: 8834: 8811: 8807: 8805: 8802: 8801: 8784: 8780: 8766: 8765: 8747: 8746: 8744: 8738: 8734: 8723: 8704: 8699: 8689: 8685: 8664: 8660: 8643: 8640: 8639: 8620: 8616: 8611: 8605: 8601: 8592: 8588: 8579: 8575: 8570: 8564: 8560: 8551: 8547: 8545: 8542: 8541: 8507: 8503: 8486: 8480: 8476: 8473: 8455: 8451: 8429: 8422: 8418: 8414: 8411: 8401: 8397: 8385: 8381: 8359: 8353: 8349: 8346: 8334: 8330: 8322: 8319: 8312: 8308: 8307: 8303: 8297: 8293: 8275: 8271: 8253: 8249: 8241: 8238: 8237: 8205: 8201: 8184: 8178: 8174: 8171: 8153: 8149: 8127: 8120: 8116: 8112: 8109: 8100: 8096: 8092: 8088: 8076: 8072: 8050: 8044: 8040: 8037: 8022: 8018: 8009: 8005: 8004: 8000: 7982: 7978: 7960: 7956: 7948: 7945: 7944: 7917: 7906: 7890: 7886: 7862: 7858: 7851: 7849: 7836: 7820: 7816: 7798: 7794: 7787: 7785: 7784: 7782: 7776: 7772: 7760: 7756: 7744: 7740: 7727: 7711: 7707: 7689: 7685: 7678: 7676: 7670: 7666: 7653: 7637: 7633: 7615: 7611: 7604: 7602: 7600: 7597: 7596: 7569: 7553: 7549: 7535: 7534: 7522: 7518: 7504: 7503: 7502: 7500: 7494: 7490: 7478: 7474: 7462: 7458: 7445: 7429: 7425: 7407: 7403: 7396: 7394: 7388: 7384: 7371: 7355: 7351: 7333: 7329: 7322: 7320: 7318: 7315: 7314: 7286: 7270: 7266: 7248: 7244: 7237: 7234: 7220: 7210: 7206: 7196: 7193: 7181: 7177: 7163: 7162: 7160: 7157: 7156: 7120: 7119: 7113: 7109: 7088: 7084: 7061: 7060: 7054: 7050: 7027: 7026: 7024: 7021: 7020: 6992: 6989: 6988: 6963: 6960: 6959: 6934: 6931: 6930: 6915:microcontroller 6911: 6882: 6878: 6872: 6868: 6866: 6852: 6849: 6848: 6820: 6816: 6814: 6805: 6801: 6799: 6796: 6795: 6770: 6766: 6757: 6753: 6751: 6748: 6747: 6724: 6720: 6711: 6707: 6705: 6702: 6701: 6670: 6664: 6660: 6640: 6634: 6630: 6629: 6624: 6615: 6611: 6594: 6591: 6590: 6580: 6549: 6545: 6537: 6531: 6527: 6517: 6513: 6512: 6506: 6502: 6501: 6499: 6491: 6485: 6481: 6467: 6463: 6461: 6452: 6448: 6431: 6428: 6427: 6417: 6368: 6363: 6357: 6353: 6342: 6327: 6321: 6316: 6304: 6300: 6295: 6272: 6270: 6266: 6260: 6256: 6236: 6234: 6231: 6230: 6219: 6210: 6188: 6184: 6179: 6174: 6171: 6170: 6150: 6146: 6144: 6141: 6140: 6136: 6114: 6110: 6104: 6100: 6091: 6087: 6085: 6082: 6081: 6064: 6060: 6055: 6049: 6045: 6036: 6032: 6030: 6027: 6026: 5990: 5985: 5979: 5975: 5947: 5942: 5932: 5928: 5907: 5903: 5886: 5883: 5882: 5859: 5855: 5853: 5850: 5849: 5832: 5828: 5826: 5823: 5822: 5804: 5800: 5798: 5795: 5794: 5787:derivative time 5769: 5765: 5763: 5760: 5759: 5737: 5733: 5731: 5728: 5727: 5683: 5678: 5672: 5668: 5640: 5635: 5623: 5619: 5614: 5598: 5594: 5588: 5584: 5567: 5564: 5563: 5536: 5535: 5531: 5529: 5526: 5525: 5501: 5500: 5496: 5494: 5491: 5490: 5473: 5469: 5467: 5464: 5463: 5456: 5447: 5436: 5430: 5427: 5412: 5396: 5385: 5360: 5358:Cascade control 5344:gain scheduling 5340: 5331: 5306: 5271: 5249: 5220: 5217: 5216: 5211:= integral gain 5195: 5191: 5178: 5176: 5173: 5172: 5153: 5149: 5141: 5138: 5137: 5110: 5087: 5085: 5077: 5074: 5073: 5018: 5015: 5014: 4983: 4980: 4979: 4946: 4942: 4930: 4926: 4924: 4921: 4920: 4899: 4890: 4871:integral windup 4867: 4865:Integral windup 4861: 4859:Integral windup 4853: 4835:low-pass filter 4827: 4807: 4767: 4744: 4742:Tuning software 4737: 4726: 4683: 4679: 4665: 4663: 4659: 4652: 4648: 4636: 4632: 4615: 4612: 4611: 4594: 4587: 4535: 4531: 4530: 4517: 4513: 4507: 4503: 4502: 4500: 4483: 4480: 4479: 4473: 4461: 4457: 4432: 4417: 4413: 4411: 4408: 4407: 4390: 4386: 4384: 4381: 4380: 4358: 4330: 4328: 4325: 4324: 4317: 4295: 4291: 4289: 4286: 4285: 4268: 4264: 4262: 4259: 4258: 4241: 4237: 4235: 4232: 4231: 4204: 4197: 4193: 4192: 4185: 4181: 4180: 4175: 4172: 4171: 4153: 4149: 4144: 4137: 4133: 4132: 4127: 4124: 4123: 4104: 4100: 4099: 4094: 4091: 4090: 4062: 4058: 4053: 4046: 4042: 4041: 4036: 4033: 4032: 4013: 4009: 4008: 4003: 4000: 3999: 3967: 3963: 3962: 3957: 3954: 3953: 3928: 3924: 3922: 3919: 3918: 3900: 3896: 3894: 3891: 3890: 3872: 3868: 3866: 3863: 3862: 3834: 3830: 3828: 3825: 3824: 3807: 3803: 3801: 3798: 3797: 3780: 3776: 3774: 3771: 3770: 3753: 3749: 3747: 3744: 3743: 3726: 3722: 3720: 3717: 3716: 3709:John G. Ziegler 3701: 3695: 3670: 3666: 3664: 3661: 3660: 3629: 3625: 3623: 3620: 3619: 3584: 3580: 3578: 3575: 3574: 3539: 3535: 3533: 3530: 3529: 3494: 3490: 3486: 3460: 3456: 3454: 3451: 3450: 3433: 3429: 3427: 3424: 3423: 3402: 3398: 3396: 3393: 3392: 3375: 3371: 3369: 3366: 3365: 3348: 3344: 3342: 3339: 3338: 3321: 3317: 3315: 3312: 3311: 3294: 3290: 3288: 3285: 3284: 3267: 3263: 3261: 3258: 3257: 3240: 3236: 3234: 3231: 3230: 3227: 3200:Åström-Hägglund 3144:Ziegler–Nichols 3103: 3076: 3024: 3021: 3020: 2998: 2969: 2967: 2964: 2963: 2917: 2914: 2913: 2897: 2894: 2893: 2864: 2861: 2860: 2835: 2832: 2831: 2780: 2754: 2752: 2735: 2732: 2731: 2696: 2688:gain scheduling 2662: 2626: 2609: 2607: 2601: 2597: 2588: 2584: 2582: 2579: 2578: 2570: 2563: 2552: 2545: 2538: 2526: 2524:Derivative term 2470: 2465: 2455: 2451: 2442: 2438: 2436: 2433: 2432: 2424: 2408: 2401: 2394: 2382: 2365: 2321: 2317: 2308: 2304: 2302: 2299: 2298: 2290: 2279: 2272: 2265: 2253: 2232: 2229: 2228: 2202: 2198: 2187: 2181: 2177: 2168: 2164: 2147: 2144: 2143: 2112: 2109: 2108: 2092: 2089: 2088: 2070: 2067: 2066: 2032: 2021: 2004: 2001: 2000: 1981: 1977: 1975: 1972: 1971: 1952: 1948: 1946: 1943: 1942: 1923: 1919: 1917: 1914: 1913: 1883: 1866: 1864: 1858: 1854: 1826: 1821: 1811: 1807: 1786: 1782: 1762: 1745: 1742: 1741: 1713: 1710: 1709: 1694: 1649: 1640: 1623: 1621:Control damping 1610: 1581: 1568: 1523: 1517: 1489: 1402: 1326:, and in 1895, 1277: 1252: 1220:optimal control 1216: 1207: 1175: 1174: 1156: 1155: 1153: 1147: 1143: 1132: 1113: 1108: 1096: 1092: 1087: 1071: 1067: 1061: 1057: 1040: 1037: 1036: 1016: 1012: 1007: 1001: 997: 995: 992: 991: 974: 970: 964: 960: 958: 955: 954: 937: 933: 931: 928: 927: 910: 906: 904: 901: 900: 883: 879: 873: 869: 867: 864: 863: 846: 842: 837: 831: 827: 825: 822: 821: 804: 800: 798: 795: 794: 777: 773: 771: 768: 767: 731: 727: 725: 722: 721: 704: 700: 698: 695: 694: 677: 673: 671: 668: 667: 640: 639: 621: 620: 618: 612: 608: 597: 578: 573: 563: 559: 538: 534: 517: 514: 513: 510: 451: 424: 421: 420: 411:In this model: 376: 373: 372: 302: 299: 298: 267: 265: 262: 261: 257:and a measured 227: 225: 222: 221: 193: 190: 189: 142: 108:terms (denoted 64: 61: 60: 28: 23: 22: 15: 12: 11: 5: 11316: 11306: 11305: 11300: 11295: 11290: 11276: 11275: 11270: 11230: 11215: 11210: 11205: 11200: 11195: 11190: 11185: 11179: 11173: 11166: 11163: 11162: 11161: 11156: 11151: 11146: 11139: 11138:External links 11136: 11134: 11133: 11127: 11109: 11089: 11063: 11050: 11044: 11029: 11023: 11010: 11004: 10990: 10988: 10985: 10983: 10982: 10976: 10962: 10959: 10958: 10936: 10916: 10893: 10886: 10868: 10857: 10846: 10803: 10796: 10774: 10745: 10720: 10699: 10680: 10651: 10609: 10581: 10574: 10556: 10533: 10500: 10491:|journal= 10447: 10428:(4): 559–576. 10405: 10388: 10365: 10336:(3): 783–835. 10317: 10308:|journal= 10278: 10264: 10257: 10239: 10211: 10208:on 2014-05-13. 10187: 10162: 10142: 10135: 10113: 10091: 10080:(2): 280–309. 10061: 10036: 10016: 10000: 9982: 9966:"On Governors" 9962:Maxwell, J. C. 9953: 9894: 9887: 9867: 9852: 9827: 9808: 9807: 9805: 9802: 9799: 9798: 9781: 9778: 9772: 9756: 9746: 9745: 9743: 9740: 9739: 9738: 9733: 9731:Control theory 9726: 9723: 9719: 9694: 9691: 9688: 9685: 9682: 9662: 9658: 9652: 9648: 9617: 9613: 9570: 9567: 9563: 9548: 9547: 9546: 9535: 9532: 9529: 9526: 9523: 9520: 9512: 9508: 9502: 9498: 9491: 9488: 9485: 9482: 9479: 9476: 9473: 9469: 9460: 9456: 9449: 9445: 9441: 9434: 9429: 9425: 9421: 9417: 9413: 9410: 9407: 9404: 9401: 9397: 9388: 9384: 9378: 9374: 9367: 9362: 9358: 9352: 9348: 9344: 9339: 9335: 9330: 9326: 9323: 9320: 9317: 9314: 9311: 9308: 9305: 9302: 9299: 9296: 9293: 9279: 9278: 9262: 9259: 9255: 9251: 9248: 9241: 9238: 9234: 9225: 9221: 9215: 9211: 9204: 9199: 9196: 9192: 9187: 9178: 9174: 9167: 9163: 9159: 9152: 9147: 9143: 9139: 9135: 9131: 9127: 9118: 9114: 9108: 9104: 9097: 9092: 9088: 9082: 9078: 9074: 9069: 9065: 9060: 9053: 9050: 9047: 9044: 9041: 9027: 9026: 9013: 9009: 9006: 9003: 8993: 8989: 8983: 8979: 8973: 8967: 8964: 8961: 8957: 8950: 8946: 8940: 8936: 8932: 8927: 8923: 8919: 8916: 8913: 8910: 8907: 8882: 8878: 8852: 8851: 8850: 8847: 8844: 8841: 8833: 8830: 8814: 8810: 8787: 8783: 8779: 8773: 8769: 8763: 8760: 8757: 8754: 8750: 8737: 8733: 8730: 8726: 8721: 8718: 8715: 8712: 8707: 8702: 8698: 8688: 8684: 8681: 8678: 8675: 8672: 8663: 8659: 8656: 8653: 8650: 8647: 8623: 8619: 8614: 8608: 8604: 8600: 8595: 8591: 8587: 8582: 8578: 8573: 8567: 8563: 8559: 8554: 8550: 8538: 8537: 8525: 8521: 8516: 8513: 8510: 8506: 8502: 8499: 8492: 8489: 8483: 8479: 8472: 8469: 8464: 8461: 8458: 8454: 8450: 8447: 8443: 8435: 8432: 8425: 8421: 8417: 8410: 8407: 8404: 8400: 8396: 8393: 8388: 8384: 8380: 8377: 8373: 8365: 8362: 8356: 8352: 8345: 8337: 8333: 8328: 8325: 8318: 8315: 8311: 8306: 8300: 8296: 8292: 8289: 8284: 8281: 8278: 8274: 8270: 8267: 8264: 8261: 8256: 8252: 8248: 8245: 8231: 8230: 8219: 8214: 8211: 8208: 8204: 8200: 8197: 8190: 8187: 8181: 8177: 8170: 8167: 8162: 8159: 8156: 8152: 8148: 8145: 8141: 8133: 8130: 8123: 8119: 8115: 8108: 8103: 8099: 8095: 8091: 8087: 8084: 8079: 8075: 8071: 8068: 8064: 8056: 8053: 8047: 8043: 8036: 8033: 8030: 8025: 8021: 8017: 8012: 8008: 8003: 7999: 7996: 7991: 7988: 7985: 7981: 7977: 7974: 7971: 7968: 7963: 7959: 7955: 7952: 7938: 7937: 7923: 7920: 7912: 7909: 7904: 7899: 7896: 7893: 7889: 7885: 7882: 7879: 7876: 7871: 7868: 7865: 7861: 7857: 7854: 7848: 7842: 7839: 7834: 7829: 7826: 7823: 7819: 7815: 7812: 7809: 7806: 7801: 7797: 7793: 7790: 7779: 7775: 7771: 7768: 7763: 7759: 7755: 7752: 7747: 7743: 7739: 7733: 7730: 7725: 7720: 7717: 7714: 7710: 7706: 7703: 7700: 7697: 7692: 7688: 7684: 7681: 7673: 7669: 7665: 7659: 7656: 7651: 7646: 7643: 7640: 7636: 7632: 7629: 7626: 7623: 7618: 7614: 7610: 7607: 7590: 7589: 7575: 7572: 7567: 7562: 7559: 7556: 7552: 7548: 7542: 7539: 7533: 7530: 7525: 7521: 7517: 7511: 7508: 7497: 7493: 7489: 7486: 7481: 7477: 7473: 7470: 7465: 7461: 7457: 7451: 7448: 7443: 7438: 7435: 7432: 7428: 7424: 7421: 7418: 7415: 7410: 7406: 7402: 7399: 7391: 7387: 7383: 7377: 7374: 7369: 7364: 7361: 7358: 7354: 7350: 7347: 7344: 7341: 7336: 7332: 7328: 7325: 7308: 7307: 7292: 7289: 7284: 7279: 7276: 7273: 7269: 7265: 7262: 7259: 7256: 7251: 7247: 7243: 7240: 7233: 7226: 7223: 7218: 7213: 7209: 7205: 7202: 7199: 7192: 7189: 7184: 7180: 7176: 7170: 7167: 7139: 7136: 7133: 7127: 7124: 7116: 7112: 7108: 7105: 7102: 7099: 7096: 7091: 7087: 7083: 7080: 7077: 7074: 7068: 7065: 7057: 7053: 7049: 7046: 7043: 7040: 7034: 7031: 6999: 6996: 6976: 6973: 6970: 6967: 6947: 6944: 6941: 6938: 6910: 6907: 6902: 6901: 6885: 6881: 6875: 6871: 6865: 6862: 6859: 6856: 6842: 6841: 6828: 6823: 6819: 6813: 6808: 6804: 6793: 6781: 6778: 6773: 6769: 6765: 6760: 6756: 6735: 6732: 6727: 6723: 6719: 6714: 6710: 6695: 6694: 6683: 6680: 6677: 6673: 6667: 6663: 6659: 6656: 6653: 6650: 6643: 6637: 6633: 6628: 6623: 6618: 6614: 6610: 6607: 6604: 6601: 6598: 6579: 6576: 6572: 6571: 6558: 6552: 6548: 6544: 6540: 6534: 6530: 6526: 6520: 6516: 6509: 6505: 6498: 6494: 6488: 6484: 6480: 6475: 6470: 6466: 6460: 6455: 6451: 6447: 6444: 6441: 6438: 6435: 6416: 6413: 6409: 6408: 6396: 6392: 6389: 6386: 6383: 6380: 6374: 6371: 6367: 6360: 6356: 6352: 6348: 6345: 6339: 6336: 6333: 6330: 6324: 6319: 6315: 6307: 6303: 6299: 6294: 6290: 6287: 6284: 6281: 6278: 6275: 6269: 6263: 6259: 6255: 6251: 6248: 6245: 6242: 6239: 6218: 6215: 6209: 6206: 6191: 6187: 6182: 6178: 6153: 6149: 6135: 6132: 6117: 6113: 6107: 6103: 6099: 6094: 6090: 6067: 6063: 6058: 6052: 6048: 6044: 6039: 6035: 6023: 6022: 6011: 6008: 6005: 6002: 5996: 5993: 5989: 5982: 5978: 5974: 5971: 5968: 5964: 5961: 5958: 5955: 5950: 5945: 5941: 5935: 5931: 5927: 5924: 5921: 5918: 5915: 5910: 5906: 5902: 5899: 5896: 5893: 5890: 5862: 5858: 5835: 5831: 5807: 5803: 5790: 5789: 5772: 5768: 5757: 5740: 5736: 5721: 5720: 5708: 5704: 5701: 5698: 5695: 5689: 5686: 5682: 5675: 5671: 5667: 5664: 5661: 5657: 5654: 5651: 5648: 5643: 5638: 5634: 5626: 5622: 5618: 5613: 5610: 5607: 5604: 5601: 5597: 5591: 5587: 5583: 5580: 5577: 5574: 5571: 5545: 5542: 5539: 5534: 5510: 5507: 5504: 5499: 5476: 5472: 5455: 5452: 5449: 5448: 5399: 5397: 5390: 5384: 5381: 5375:of the system 5359: 5356: 5339: 5336: 5330: 5327: 5305: 5302: 5301: 5300: 5296: 5293: 5289: 5286: 5279: 5270: 5267: 5248: 5245: 5224: 5213: 5212: 5198: 5194: 5190: 5185: 5182: 5170: 5156: 5152: 5148: 5145: 5131: 5130: 5116: 5113: 5108: 5105: 5102: 5099: 5096: 5093: 5090: 5084: 5081: 5055: 5054: 5043: 5040: 5037: 5034: 5031: 5028: 5025: 5022: 4987: 4976: 4975: 4964: 4961: 4957: 4954: 4949: 4945: 4941: 4938: 4933: 4929: 4898: 4895: 4889: 4886: 4885: 4884: 4881: 4878: 4860: 4857: 4852: 4849: 4826: 4823: 4806: 4803: 4766: 4763: 4743: 4740: 4735: 4724: 4711: 4710: 4699: 4695: 4682: 4677: 4674: 4671: 4668: 4662: 4658: 4655: 4651: 4647: 4644: 4635: 4631: 4628: 4625: 4622: 4619: 4592: 4585: 4579: 4578: 4567: 4564: 4561: 4558: 4555: 4549: 4546: 4543: 4534: 4526: 4523: 4520: 4516: 4506: 4499: 4496: 4493: 4490: 4487: 4472: 4469: 4445: 4442: 4439: 4435: 4431: 4428: 4425: 4420: 4416: 4393: 4389: 4357: 4354: 4338: 4335: 4316: 4313: 4298: 4294: 4271: 4267: 4244: 4240: 4223: 4222: 4211: 4207: 4200: 4196: 4188: 4184: 4179: 4169: 4156: 4152: 4147: 4140: 4136: 4131: 4121: 4107: 4103: 4098: 4088: 4082: 4081: 4078: 4065: 4061: 4056: 4049: 4045: 4040: 4030: 4016: 4012: 4007: 3997: 3991: 3990: 3987: 3984: 3970: 3966: 3961: 3951: 3945: 3944: 3931: 3927: 3916: 3903: 3899: 3888: 3875: 3871: 3860: 3837: 3833: 3810: 3806: 3783: 3779: 3756: 3752: 3729: 3725: 3694: 3691: 3688: 3687: 3673: 3669: 3657: 3654: 3651: 3648: 3645: 3632: 3628: 3616: 3615: 3612: 3609: 3606: 3603: 3600: 3587: 3583: 3571: 3570: 3567: 3564: 3561: 3558: 3555: 3542: 3538: 3526: 3525: 3522: 3519: 3518:Settling time 3516: 3513: 3510: 3492: 3488: 3484: 3463: 3459: 3436: 3432: 3405: 3401: 3378: 3374: 3351: 3347: 3324: 3320: 3297: 3293: 3270: 3266: 3243: 3239: 3226: 3223: 3220: 3219: 3216: 3213: 3209: 3208: 3205: 3202: 3196: 3195: 3192: 3189: 3183: 3182: 3179: 3170: 3168:Software tools 3164: 3163: 3160: 3157: 3156:Tyreus Luyben 3153: 3152: 3149: 3146: 3140: 3139: 3136: 3133: 3127: 3126: 3125:Disadvantages 3123: 3120: 3102: 3099: 3075: 3072: 3055: 3052: 3049: 3046: 3043: 3040: 3037: 3034: 3031: 3028: 3008: 3005: 3001: 2997: 2994: 2991: 2988: 2985: 2982: 2979: 2976: 2972: 2951: 2948: 2945: 2942: 2939: 2936: 2933: 2930: 2927: 2924: 2921: 2901: 2877: 2874: 2871: 2868: 2848: 2845: 2842: 2839: 2828: 2827: 2816: 2810: 2807: 2804: 2801: 2798: 2795: 2792: 2789: 2786: 2783: 2778: 2775: 2772: 2769: 2766: 2763: 2760: 2757: 2751: 2748: 2745: 2742: 2739: 2722:Laplace domain 2695: 2692: 2661: 2658: 2650: 2649: 2638: 2632: 2629: 2624: 2621: 2618: 2615: 2612: 2600: 2596: 2587: 2568: 2561: 2553:held constant) 2550: 2543: 2536: 2525: 2522: 2509: 2508: 2497: 2494: 2491: 2487: 2484: 2481: 2478: 2473: 2468: 2464: 2454: 2450: 2441: 2422: 2409:held constant) 2406: 2399: 2392: 2381: 2378: 2364: 2361: 2353: 2352: 2341: 2338: 2335: 2332: 2329: 2320: 2316: 2307: 2288: 2280:held constant) 2277: 2270: 2263: 2252: 2249: 2236: 2225: 2224: 2213: 2210: 2201: 2197: 2194: 2190: 2180: 2176: 2167: 2163: 2160: 2157: 2154: 2151: 2137:Laplace domain 2129: 2128: 2116: 2096: 2086: 2074: 2064: 2048: 2045: 2042: 2038: 2035: 2031: 2027: 2024: 2020: 2017: 2014: 2011: 2008: 1998: 1980: 1969: 1951: 1940: 1922: 1907: 1906: 1895: 1889: 1886: 1881: 1878: 1875: 1872: 1869: 1857: 1853: 1850: 1847: 1843: 1840: 1837: 1834: 1829: 1824: 1820: 1810: 1806: 1803: 1800: 1797: 1794: 1785: 1781: 1778: 1775: 1772: 1768: 1765: 1761: 1758: 1755: 1752: 1749: 1726: 1723: 1720: 1717: 1705: 1704: 1693: 1690: 1674:concentrations 1648: 1645: 1639: 1636: 1622: 1619: 1609: 1606: 1580: 1577: 1567: 1564: 1560: 1559: 1552: 1549: 1546: 1531:electric motor 1521:Control system 1516: 1513: 1488: 1485: 1401: 1398: 1276: 1273: 1251: 1248: 1215: 1212: 1206: 1203: 1202: 1201: 1189: 1182: 1178: 1172: 1169: 1166: 1163: 1159: 1146: 1142: 1139: 1135: 1130: 1127: 1124: 1121: 1116: 1111: 1107: 1095: 1091: 1086: 1083: 1080: 1077: 1074: 1070: 1060: 1056: 1053: 1050: 1047: 1044: 1015: 1010: 1000: 973: 963: 936: 909: 882: 872: 845: 840: 830: 803: 776: 730: 703: 676: 653: 647: 643: 637: 634: 631: 628: 624: 611: 607: 604: 600: 595: 592: 589: 586: 581: 576: 572: 562: 558: 555: 552: 549: 546: 537: 533: 530: 527: 524: 521: 509: 506: 488:Control action 474: 473: 465: 453: 449: 437: 434: 431: 428: 389: 386: 383: 380: 345: 342: 339: 336: 333: 330: 327: 324: 321: 318: 315: 312: 309: 306: 286: 283: 280: 277: 274: 246: 243: 240: 237: 234: 206: 203: 200: 197: 141: 138: 125:cruise control 77: 74: 71: 68: 38:PID controller 26: 9: 6: 4: 3: 2: 11315: 11304: 11301: 11299: 11296: 11294: 11291: 11289: 11286: 11285: 11283: 11274: 11271: 11267: 11261: 11247:on 2015-04-21 11243: 11236: 11231: 11228:on 2010-02-09 11227: 11223: 11222: 11216: 11214: 11211: 11209: 11206: 11204: 11201: 11199: 11196: 11194: 11191: 11189: 11186: 11183: 11180: 11177: 11174: 11172: 11169: 11168: 11165:PID tutorials 11160: 11157: 11155: 11152: 11150: 11147: 11145: 11142: 11141: 11130: 11124: 11117: 11116: 11110: 11099: 11095: 11090: 11076: 11069: 11064: 11060: 11056: 11051: 11047: 11041: 11037: 11036: 11030: 11026: 11020: 11016: 11011: 11007: 11001: 10997: 10992: 10991: 10979: 10977:9789861544779 10973: 10969: 10964: 10963: 10946: 10940: 10929: 10928: 10920: 10904: 10897: 10889: 10883: 10879: 10872: 10865: 10861: 10854: 10850: 10841: 10840:10400.22/4306 10836: 10831: 10826: 10822: 10818: 10814: 10807: 10799: 10797:9781424438839 10793: 10789: 10785: 10778: 10769: 10764: 10761:(2): 74–101. 10760: 10756: 10749: 10735: 10731: 10724: 10710: 10703: 10697: 10693: 10689: 10684: 10676: 10672: 10668: 10664: 10663: 10655: 10647: 10643: 10639: 10635: 10631: 10627: 10620: 10613: 10594: 10593: 10585: 10577: 10571: 10567: 10560: 10552: 10548: 10544: 10537: 10528: 10523: 10519: 10515: 10511: 10504: 10496: 10483: 10469:on 2015-04-21 10465: 10458: 10451: 10443: 10439: 10435: 10431: 10427: 10423: 10416: 10409: 10401: 10395: 10393: 10381: 10374: 10372: 10370: 10361: 10357: 10352: 10347: 10343: 10339: 10335: 10331: 10324: 10322: 10313: 10300: 10289: 10282: 10274: 10268: 10260: 10258:0-8493-1081-4 10254: 10250: 10243: 10235: 10231: 10227: 10223: 10215: 10204: 10197: 10191: 10173: 10166: 10155: 10154: 10146: 10138: 10132: 10128: 10124: 10117: 10109: 10104: 10100: 10095: 10087: 10083: 10079: 10075: 10071: 10065: 10051:on 2011-07-08 10050: 10046: 10040: 10033: 10029: 10028: 10020: 10011: 10003: 9997: 9993: 9986: 9978: 9974: 9967: 9963: 9957: 9943:on 2016-08-09 9939: 9935: 9931: 9927: 9923: 9916: 9909: 9907: 9905: 9903: 9901: 9899: 9890: 9888:9781400874668 9884: 9880: 9879: 9871: 9863: 9856: 9841: 9837: 9831: 9820: 9813: 9809: 9776: 9770: 9760: 9751: 9747: 9737: 9734: 9732: 9729: 9728: 9718: 9716: 9711: 9709: 9692: 9689: 9686: 9683: 9680: 9660: 9656: 9650: 9646: 9636: 9634: 9615: 9611: 9601: 9597: 9595: 9590: 9585: 9568: 9565: 9561: 9530: 9527: 9524: 9518: 9510: 9500: 9496: 9489: 9483: 9480: 9477: 9471: 9467: 9458: 9447: 9443: 9439: 9432: 9427: 9423: 9419: 9415: 9411: 9405: 9399: 9395: 9386: 9376: 9372: 9365: 9360: 9350: 9346: 9342: 9337: 9333: 9328: 9324: 9318: 9315: 9312: 9306: 9303: 9297: 9291: 9284: 9283: 9282: 9260: 9257: 9253: 9249: 9246: 9239: 9236: 9232: 9223: 9213: 9209: 9202: 9197: 9194: 9190: 9185: 9176: 9165: 9161: 9157: 9150: 9145: 9141: 9137: 9133: 9129: 9125: 9116: 9106: 9102: 9095: 9090: 9080: 9076: 9072: 9067: 9063: 9058: 9051: 9045: 9039: 9032: 9031: 9030: 9011: 9007: 9004: 9001: 8991: 8981: 8977: 8971: 8965: 8962: 8959: 8955: 8948: 8938: 8934: 8930: 8925: 8921: 8917: 8911: 8905: 8898: 8897: 8896: 8880: 8867: 8862: 8860: 8848: 8845: 8842: 8839: 8838: 8837: 8829: 8812: 8808: 8785: 8781: 8777: 8771: 8758: 8752: 8735: 8731: 8728: 8716: 8710: 8705: 8700: 8696: 8686: 8682: 8676: 8670: 8661: 8657: 8651: 8645: 8636: 8621: 8617: 8612: 8606: 8602: 8598: 8593: 8589: 8585: 8580: 8576: 8571: 8565: 8561: 8557: 8552: 8548: 8523: 8514: 8511: 8508: 8504: 8497: 8490: 8481: 8477: 8470: 8462: 8459: 8456: 8452: 8445: 8441: 8433: 8423: 8419: 8415: 8408: 8405: 8402: 8398: 8394: 8386: 8382: 8375: 8371: 8363: 8354: 8350: 8343: 8335: 8331: 8326: 8316: 8313: 8309: 8304: 8298: 8294: 8290: 8282: 8279: 8276: 8272: 8265: 8262: 8254: 8250: 8243: 8236: 8235: 8234: 8212: 8209: 8206: 8202: 8195: 8188: 8179: 8175: 8168: 8160: 8157: 8154: 8150: 8143: 8139: 8131: 8121: 8117: 8113: 8106: 8101: 8097: 8093: 8089: 8085: 8077: 8073: 8066: 8062: 8054: 8045: 8041: 8034: 8031: 8023: 8019: 8015: 8010: 8006: 8001: 7997: 7989: 7986: 7983: 7979: 7972: 7969: 7961: 7957: 7950: 7943: 7942: 7941: 7921: 7910: 7897: 7894: 7891: 7887: 7880: 7877: 7869: 7866: 7863: 7859: 7852: 7846: 7840: 7827: 7824: 7821: 7817: 7810: 7807: 7799: 7795: 7788: 7777: 7773: 7769: 7761: 7757: 7750: 7745: 7741: 7737: 7731: 7718: 7715: 7712: 7708: 7701: 7698: 7690: 7686: 7679: 7671: 7667: 7663: 7657: 7644: 7641: 7638: 7634: 7627: 7624: 7616: 7612: 7605: 7595: 7594: 7593: 7573: 7560: 7557: 7554: 7550: 7540: 7537: 7531: 7523: 7519: 7509: 7506: 7495: 7491: 7487: 7479: 7475: 7468: 7463: 7459: 7455: 7449: 7436: 7433: 7430: 7426: 7419: 7416: 7408: 7404: 7397: 7389: 7385: 7381: 7375: 7362: 7359: 7356: 7352: 7345: 7342: 7334: 7330: 7323: 7313: 7312: 7311: 7290: 7277: 7274: 7271: 7267: 7260: 7257: 7249: 7245: 7238: 7231: 7224: 7221: 7211: 7207: 7200: 7197: 7190: 7182: 7178: 7168: 7165: 7155: 7154: 7153: 7150: 7134: 7125: 7122: 7114: 7110: 7106: 7100: 7094: 7089: 7085: 7081: 7075: 7066: 7063: 7055: 7051: 7047: 7041: 7032: 7029: 7018: 7016: 7011: 6997: 6971: 6965: 6942: 6936: 6928: 6924: 6920: 6916: 6906: 6883: 6879: 6873: 6869: 6863: 6860: 6857: 6854: 6847: 6846: 6845: 6826: 6821: 6817: 6811: 6806: 6802: 6794: 6779: 6776: 6771: 6767: 6763: 6758: 6754: 6733: 6730: 6725: 6721: 6717: 6712: 6708: 6700: 6699: 6698: 6678: 6675: 6671: 6665: 6661: 6651: 6648: 6641: 6635: 6631: 6626: 6616: 6612: 6608: 6602: 6596: 6589: 6588: 6587: 6585: 6575: 6556: 6550: 6546: 6542: 6538: 6532: 6528: 6524: 6518: 6514: 6507: 6503: 6496: 6492: 6486: 6482: 6478: 6473: 6468: 6464: 6458: 6453: 6449: 6445: 6439: 6433: 6426: 6425: 6424: 6422: 6412: 6394: 6387: 6381: 6378: 6372: 6369: 6365: 6358: 6354: 6350: 6346: 6343: 6334: 6328: 6322: 6317: 6313: 6305: 6301: 6297: 6292: 6285: 6279: 6276: 6273: 6267: 6261: 6257: 6253: 6229: 6228: 6227: 6224: 6214: 6205: 6189: 6185: 6180: 6176: 6169: 6151: 6147: 6131: 6115: 6111: 6105: 6101: 6097: 6092: 6088: 6065: 6061: 6056: 6050: 6046: 6042: 6037: 6033: 6006: 6000: 5994: 5991: 5987: 5980: 5976: 5972: 5969: 5966: 5959: 5953: 5948: 5943: 5939: 5933: 5929: 5925: 5919: 5913: 5908: 5904: 5900: 5894: 5888: 5881: 5880: 5879: 5876: 5860: 5856: 5833: 5829: 5805: 5801: 5788: 5770: 5766: 5758: 5756: 5755:integral time 5738: 5734: 5726: 5725: 5724: 5706: 5699: 5693: 5687: 5684: 5680: 5673: 5669: 5665: 5662: 5659: 5652: 5646: 5641: 5636: 5632: 5624: 5620: 5616: 5611: 5605: 5599: 5595: 5589: 5585: 5581: 5575: 5569: 5562: 5561: 5560: 5532: 5497: 5474: 5470: 5461: 5460:standard form 5445: 5442: 5434: 5431:December 2012 5424: 5420: 5416: 5410: 5409: 5405: 5400:This section 5398: 5394: 5389: 5388: 5380: 5378: 5372: 5369: 5364: 5355: 5353: 5349: 5345: 5335: 5326: 5322: 5319: 5315: 5311: 5297: 5294: 5290: 5287: 5284: 5283:discontinuity 5280: 5277: 5276: 5275: 5266: 5263: 5258: 5254: 5244: 5240: 5236: 5222: 5196: 5192: 5188: 5183: 5180: 5171: 5154: 5150: 5146: 5143: 5136: 5135: 5134: 5114: 5111: 5103: 5100: 5097: 5094: 5088: 5082: 5079: 5072: 5071: 5070: 5068: 5064: 5060: 5041: 5038: 5035: 5032: 5029: 5026: 5023: 5013: 5012: 5011: 5009: 5008: 5003: 5002: 4962: 4959: 4952: 4947: 4943: 4939: 4931: 4927: 4919: 4918: 4917: 4914: 4912: 4911:PI controller 4903: 4897:PI controller 4894: 4882: 4879: 4876: 4875: 4874: 4872: 4866: 4856: 4848: 4846: 4845:PI controller 4841: 4840:median filter 4836: 4832: 4822: 4819: 4814: 4810: 4802: 4799: 4794: 4791: 4786: 4784: 4780: 4776: 4774: 4762: 4758: 4755: 4752: 4748: 4739: 4734: 4730: 4723: 4718: 4714: 4697: 4693: 4680: 4675: 4672: 4669: 4666: 4660: 4656: 4653: 4649: 4645: 4633: 4629: 4623: 4617: 4610: 4609: 4608: 4606: 4602: 4598: 4591: 4584: 4565: 4559: 4553: 4547: 4544: 4541: 4532: 4524: 4521: 4518: 4514: 4504: 4497: 4491: 4485: 4478: 4477: 4476: 4468: 4465: 4443: 4440: 4437: 4433: 4429: 4426: 4423: 4418: 4414: 4391: 4387: 4377: 4373: 4371: 4367: 4363: 4353: 4336: 4333: 4322: 4312: 4296: 4292: 4269: 4265: 4242: 4238: 4228: 4209: 4205: 4198: 4194: 4186: 4182: 4177: 4170: 4154: 4150: 4145: 4138: 4134: 4129: 4122: 4105: 4101: 4096: 4089: 4087: 4084: 4083: 4079: 4063: 4059: 4054: 4047: 4043: 4038: 4031: 4014: 4010: 4005: 3998: 3996: 3993: 3992: 3988: 3985: 3968: 3964: 3959: 3952: 3950: 3947: 3946: 3929: 3925: 3917: 3901: 3897: 3889: 3873: 3869: 3861: 3859:Control type 3858: 3857: 3851: 3835: 3831: 3808: 3804: 3781: 3777: 3754: 3750: 3727: 3723: 3714: 3710: 3706: 3700: 3671: 3667: 3658: 3655: 3652: 3649: 3647:Minor change 3646: 3630: 3626: 3618: 3617: 3613: 3610: 3607: 3604: 3601: 3585: 3581: 3573: 3572: 3568: 3565: 3563:Small change 3562: 3559: 3556: 3540: 3536: 3528: 3527: 3523: 3520: 3517: 3514: 3511: 3508: 3507: 3503: 3481: 3477: 3461: 3457: 3434: 3430: 3421: 3403: 3399: 3376: 3372: 3349: 3345: 3322: 3318: 3295: 3291: 3268: 3264: 3241: 3237: 3225:Manual tuning 3217: 3214: 3211: 3210: 3206: 3203: 3201: 3198: 3197: 3193: 3190: 3188: 3185: 3184: 3180: 3177: 3176: 3171: 3169: 3166: 3165: 3161: 3158: 3155: 3154: 3150: 3147: 3145: 3142: 3141: 3137: 3134: 3132: 3131:Manual tuning 3129: 3128: 3124: 3121: 3118: 3117: 3111: 3107: 3098: 3096: 3095:settling time 3092: 3088: 3084: 3079: 3071: 3069: 3053: 3050: 3044: 3038: 3032: 3026: 3006: 3003: 2992: 2986: 2980: 2974: 2949: 2946: 2943: 2937: 2931: 2925: 2919: 2899: 2891: 2872: 2866: 2843: 2837: 2814: 2805: 2799: 2793: 2787: 2784: 2781: 2773: 2767: 2761: 2755: 2749: 2743: 2737: 2730: 2729: 2728: 2725: 2723: 2718: 2716: 2711: 2709: 2705: 2701: 2691: 2689: 2685: 2680: 2676: 2674: 2669: 2666: 2657: 2655: 2636: 2630: 2627: 2619: 2613: 2610: 2598: 2594: 2585: 2577: 2576: 2575: 2572: 2567: 2560: 2549: 2542: 2535: 2530: 2521: 2519: 2515: 2495: 2492: 2489: 2482: 2476: 2471: 2466: 2462: 2452: 2448: 2439: 2431: 2430: 2429: 2426: 2421: 2416: 2405: 2398: 2391: 2386: 2380:Integral term 2377: 2375: 2370: 2360: 2358: 2339: 2333: 2327: 2318: 2314: 2305: 2297: 2296: 2295: 2292: 2287: 2276: 2269: 2262: 2257: 2248: 2234: 2211: 2208: 2199: 2195: 2192: 2188: 2178: 2174: 2165: 2161: 2155: 2149: 2142: 2141: 2140: 2138: 2134: 2114: 2094: 2087: 2072: 2065: 2062: 2043: 2029: 2018: 2012: 2006: 1999: 1978: 1970: 1949: 1941: 1920: 1912: 1911: 1910: 1893: 1887: 1884: 1876: 1870: 1867: 1855: 1851: 1848: 1845: 1838: 1832: 1827: 1822: 1818: 1808: 1804: 1798: 1792: 1783: 1779: 1773: 1759: 1753: 1747: 1740: 1739: 1738: 1721: 1715: 1703: 1701: 1696: 1695: 1689: 1687: 1683: 1679: 1675: 1671: 1667: 1663: 1659: 1655: 1644: 1635: 1632: 1628: 1618: 1615: 1605: 1603: 1599: 1595: 1589: 1586: 1576: 1573: 1563: 1557: 1553: 1550: 1547: 1544: 1540: 1539: 1538: 1536: 1535:inertial mass 1532: 1528: 1522: 1512: 1510: 1506: 1502: 1498: 1494: 1484: 1482: 1478: 1474: 1465: 1457: 1453: 1451: 1446: 1442: 1440: 1436: 1430: 1428: 1423: 1419: 1415: 1406: 1397: 1394: 1392: 1391: 1386: 1385: 1378: 1376: 1372: 1368: 1364: 1359: 1357: 1352: 1349: 1345: 1340: 1335: 1333: 1332:Willard Gibbs 1329: 1328:Adolf Hurwitz 1325: 1324:Charles Sturm 1321: 1317: 1313: 1307: 1305: 1301: 1296: 1294: 1290: 1286: 1282: 1268: 1261: 1256: 1247: 1245: 1241: 1237: 1231: 1225: 1221: 1214:Applicability 1211: 1187: 1180: 1167: 1161: 1144: 1140: 1137: 1125: 1119: 1114: 1109: 1105: 1093: 1089: 1084: 1078: 1072: 1068: 1058: 1054: 1048: 1042: 1035: 1034: 1033: 1013: 1008: 998: 971: 961: 934: 907: 880: 870: 843: 838: 828: 801: 774: 765: 764:standard form 760: 758: 754: 750: 728: 701: 674: 664: 651: 645: 632: 626: 609: 605: 602: 590: 584: 579: 574: 570: 560: 556: 550: 544: 535: 531: 525: 519: 505: 503: 498: 493: 489: 485: 482: 478: 470: 466: 462: 458: 454: 432: 426: 418: 414: 413: 412: 409: 407: 403: 402:control valve 384: 378: 371: 367: 363: 359: 340: 334: 331: 325: 319: 316: 310: 304: 281: 275: 272: 260: 241: 235: 232: 220: 201: 195: 188: 183: 182:control terms 175: 171: 167: 163: 159: 155: 151: 150:block diagram 146: 137: 135: 129: 126: 121: 119: 115: 111: 107: 103: 99: 95: 91: 72: 66: 59: 55: 51: 47: 43: 39: 35: 30: 19: 18:PI controller 11249:. Retrieved 11242:the original 11226:the original 11220: 11114: 11102:. Retrieved 11098:the original 11082:. Retrieved 11075:the original 11058: 11034: 11014: 10995: 10967: 10949:. Retrieved 10939: 10926: 10919: 10908:. Retrieved 10906:. Scribd.com 10896: 10877: 10871: 10860: 10849: 10820: 10816: 10806: 10787: 10777: 10758: 10754: 10748: 10737:. Retrieved 10733: 10723: 10712:. Retrieved 10702: 10683: 10661: 10654: 10632:(1): 42–54. 10629: 10625: 10612: 10600:. Retrieved 10591: 10584: 10565: 10559: 10550: 10546: 10536: 10517: 10513: 10503: 10482:cite journal 10471:. Retrieved 10464:the original 10450: 10425: 10421: 10408: 10333: 10329: 10299:cite journal 10281: 10267: 10248: 10242: 10225: 10221: 10214: 10203:the original 10190: 10178:. Retrieved 10165: 10152: 10145: 10122: 10116: 10099:Bennett 1993 10094: 10077: 10073: 10064: 10053:. Retrieved 10049:the original 10039: 10031: 10026: 10019: 10009: 9991: 9985: 9976: 9972: 9956: 9945:. Retrieved 9938:the original 9928:(3): 17–25. 9925: 9921: 9877: 9870: 9861: 9855: 9844:. Retrieved 9842:. 2020-05-19 9839: 9830: 9812: 9759: 9750: 9712: 9637: 9602: 9598: 9586: 9552: 9280: 9028: 8863: 8856: 8835: 8637: 8539: 8232: 7939: 7591: 7309: 7151: 7019: 7012: 6922: 6912: 6903: 6843: 6696: 6583: 6581: 6573: 6418: 6415:Laplace form 6410: 6222: 6220: 6211: 6167: 6137: 6024: 5877: 5791: 5786: 5754: 5722: 5459: 5457: 5437: 5428: 5413:Please help 5401: 5376: 5373: 5368:thermocouple 5365: 5361: 5341: 5332: 5323: 5317: 5314:feed-forward 5307: 5304:Feed-forward 5272: 5250: 5241: 5237: 5214: 5132: 5056: 5006: 5005: 5000: 4999: 4977: 4915: 4910: 4908: 4891: 4868: 4854: 4828: 4818:HVAC systems 4815: 4811: 4808: 4795: 4787: 4778: 4772: 4768: 4759: 4756: 4753: 4749: 4745: 4732: 4728: 4721: 4719: 4715: 4712: 4604: 4600: 4596: 4589: 4582: 4580: 4474: 4466: 4378: 4374: 4359: 4318: 4229: 4226: 4085: 3994: 3948: 3702: 3501: 3228: 3173: 3108: 3104: 3086: 3082: 3080: 3077: 2889: 2829: 2726: 2719: 2712: 2707: 2697: 2684:nonlinearity 2681: 2677: 2670: 2664: 2663: 2651: 2573: 2565: 2558: 2556: 2547: 2540: 2533: 2510: 2427: 2419: 2412: 2403: 2396: 2389: 2368: 2366: 2354: 2293: 2285: 2283: 2274: 2267: 2260: 2226: 2130: 2060: 1908: 1706: 1697: 1650: 1647:Applications 1641: 1624: 1613: 1611: 1590: 1584: 1582: 1572:proportional 1571: 1569: 1566:Proportional 1561: 1524: 1497:power supply 1490: 1470: 1450:current loop 1447: 1443: 1431: 1426: 1418:Harold Black 1411: 1395: 1388: 1383: 1379: 1360: 1356:Elmer Sperry 1353: 1336: 1320:Edward Routh 1316:On Governors 1315: 1308: 1297: 1278: 1217: 1208: 763: 761: 756: 752: 748: 665: 511: 496: 491: 487: 486: 476: 475: 468: 460: 456: 416: 410: 406:weighted sum 369: 358:proportional 186: 181: 179: 173: 169: 157: 153: 130: 122: 117: 113: 109: 98:proportional 57: 46:control loop 41: 37: 33: 31: 29: 10547:Reinvention 8866:Z-transform 6923:discretized 6584:interacting 5348:fuzzy logic 5069:operators: 4765:Limitations 3659:Improve if 3500:Effects of 3122:Advantages 2704:oscillation 2660:Loop tuning 1654:temperature 1527:robotic arm 1525:Consider a 1501:seismometer 1348:instability 1244:oscillation 481:loop tuning 187:error value 134:controllers 132:electronic 58:error value 11282:Categories 11251:2013-12-04 11104:2009-01-05 11084:2007-05-05 10951:4 November 10910:2011-04-04 10739:2014-02-27 10714:2014-02-18 10667:Licentiate 10473:2011-04-04 10055:2011-04-04 9947:2014-08-21 9846:2024-05-31 9804:References 9594:reading in 8832:Pseudocode 5292:weighting. 3611:Eliminate 3524:Stability 3515:Overshoot 3512:Rise time 3509:Parameter 3502:increasing 3420:overdamped 3187:Cohen–Coon 3083:regulation 1614:derivative 1608:Derivative 1493:disk drive 1384:New Mexico 1300:James Watt 1289:millstones 1240:overshoots 366:derivative 168:(SP), and 106:derivative 10763:CiteSeerX 10696:0272-1708 10346:CiteSeerX 9780:¯ 9647:τ 9566:− 9528:− 9519:ϵ 9507:Δ 9481:− 9472:ϵ 9455:Δ 9433:− 9420:− 9400:ϵ 9383:Δ 9357:Δ 9316:− 9258:− 9250:− 9237:− 9220:Δ 9195:− 9173:Δ 9151:− 9138:− 9113:Δ 9087:Δ 9005:− 8988:Δ 8963:− 8945:Δ 8877:Δ 8729:τ 8717:τ 8697:∫ 8512:− 8488:Δ 8460:− 8431:Δ 8409:− 8403:− 8361:Δ 8324:Δ 8280:− 8210:− 8186:Δ 8158:− 8129:Δ 8107:− 8094:− 8052:Δ 8029:Δ 7987:− 7919:Δ 7908:Δ 7895:− 7878:− 7867:− 7847:− 7838:Δ 7825:− 7808:− 7729:Δ 7716:− 7699:− 7655:Δ 7642:− 7625:− 7571:Δ 7558:− 7541:˙ 7532:− 7510:˙ 7447:Δ 7434:− 7417:− 7373:Δ 7360:− 7343:− 7288:Δ 7275:− 7258:− 7169:˙ 7126:¨ 7067:˙ 7033:˙ 6995:Δ 6917:(MCU) or 6880:τ 6870:τ 6855:α 6827:α 6818:τ 6780:α 6777:⋅ 6768:τ 6734:α 6731:⋅ 6662:τ 6632:τ 6351:− 6347:τ 6335:τ 6314:∫ 6274:− 5970:τ 5960:τ 5940:∫ 5793:value at 5663:τ 5653:τ 5633:∫ 5402:does not 5184:τ 5112:τ 5101:τ 5033:− 5021:Δ 4986:Δ 4956:Δ 4953:∫ 4937:Δ 4681:τ 4676:θ 4673:− 4667:− 4657:− 4643:Δ 4533:τ 4522:θ 4519:− 4438:π 3653:Decrease 3650:Decrease 3608:Increase 3605:Increase 3602:Decrease 3566:Decrease 3560:Increase 3557:Decrease 3091:rise time 2947:− 2694:Stability 2514:overshoot 2493:τ 2483:τ 2463:∫ 2374:bias term 2095:τ 2030:− 1849:τ 1839:τ 1819:∫ 1670:flow rate 1666:feed rate 1594:oscillate 1475:in DCSs, 1365:engineer 1293:windmills 1138:τ 1126:τ 1106:∫ 603:τ 591:τ 571:∫ 502:fail-safe 332:− 11260:cite web 11184:(pg. 22) 10823:: 1–34. 10669:theis). 10646:18461921 9964:(1868). 9725:See also 8861:filter: 5310:feedback 5262:deadband 5257:backlash 5253:stiction 5247:Deadband 5059:Simulink 4783:observer 4779:constant 3614:Degrade 3569:Degrade 2890:unstable 2700:diverges 2415:integral 1682:position 1658:pressure 1598:sinusoid 1585:integral 1579:Integral 1441:(DCSs). 1371:helmsman 1260:helmsmen 362:integral 219:setpoint 166:setpoint 164:(PV) or 102:integral 90:setpoint 50:feedback 10338:Bibcode 10180:June 8, 10127:142–148 7017:gives: 5785:is the 5753:is the 5423:removed 5408:sources 5067:Laplace 4775:control 4773:optimal 3119:Method 3106:times. 2135:in the 1275:Origins 1250:History 1232:, below 762:In the 497:reverse 44:) is a 11125: 11042: 11021: 11002: 10974: 10884: 10794: 10765: 10694: 10644: 10602:Feb 1, 10572: 10442:921620 10440: 10348: 10255: 10133: 9998: 9885: 9673:where 8800:where 6423:form: 6223:option 5723:where 5524:, and 5133:where 4978:where 4581:where 4456:where 3686:small 3175:CAutoD 2830:where 2708:excess 2665:Tuning 2654:causal 2227:where 1909:where 1678:weight 755:, and 720:, and 666:where 492:direct 477:Tuning 364:, and 116:, and 104:, and 11245:(PDF) 11238:(PDF) 11119:(PDF) 11078:(PDF) 11071:(PDF) 10931:(PDF) 10642:S2CID 10622:(PDF) 10596:(PDF) 10467:(PDF) 10460:(PDF) 10438:S2CID 10418:(PDF) 10383:(PDF) 10291:(PDF) 10206:(PDF) 10199:(PDF) 10175:(PDF) 10157:(PDF) 10103:p. 67 9969:(PDF) 9941:(PDF) 9918:(PDF) 9822:(PDF) 9742:Notes 9690:<= 9684:<= 8540:s.t. 6844:with 6792:, and 6586:form 5318:error 4831:noise 4370:relay 1686:speed 1662:force 1545:(PV). 1509:FPGAs 1427:Reset 467:Term 455:Term 415:Term 11266:link 11123:ISBN 11040:ISBN 11019:ISBN 11000:ISBN 10972:ISBN 10953:2012 10882:ISBN 10821:2010 10792:ISBN 10692:ISSN 10604:2019 10570:ISBN 10553:(2). 10495:help 10312:help 10253:ISBN 10182:2015 10131:ISBN 10108:help 9996:ISBN 9883:ISBN 9771:.016 8864:The 8233:or: 7310:So, 6958:and 6919:FPGA 6080:and 5406:any 5404:cite 5063:Xcos 4790:hunt 4257:and 4097:0.60 4039:0.54 4006:0.45 3960:0.50 3742:and 3711:and 3256:and 3093:and 3051:< 2546:and 2402:and 2367:The 2273:and 1382:USS 1228:see 926:and 862:and 793:and 10835:hdl 10825:doi 10634:doi 10522:doi 10430:doi 10356:doi 10230:doi 10226:213 10082:doi 9977:100 9930:doi 8859:IIR 6177:100 5417:by 5255:or 5061:or 5010:). 4130:1.2 4086:PID 2590:out 2520:). 2444:out 2310:out 1676:), 1631:PLC 1583:An 1507:or 1291:in 759:). 40:or 11284:: 11262:}} 11258:{{ 11057:. 10833:. 10819:. 10815:. 10786:. 10757:. 10732:. 10673:. 10640:. 10630:26 10628:. 10624:. 10549:. 10545:. 10518:17 10516:. 10512:. 10486:: 10484:}} 10480:{{ 10436:. 10426:13 10424:. 10420:. 10391:^ 10368:^ 10354:. 10344:. 10334:77 10332:. 10320:^ 10303:: 10301:}} 10297:{{ 10224:. 10129:. 10101:, 10078:34 10076:. 9975:. 9971:. 9926:16 9924:. 9920:. 9897:^ 9838:. 9705:: 9693:10 6929:. 6746:, 5377:it 5007:SP 5001:PV 4909:A 4847:. 4311:. 4210:40 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9322:] 9319:1 9313:n 9310:[ 9307:u 9304:= 9301:] 9298:n 9295:[ 9292:u 9261:1 9254:z 9247:1 9240:2 9233:z 9224:t 9214:d 9210:K 9203:+ 9198:1 9191:z 9186:) 9177:t 9166:d 9162:K 9158:2 9146:p 9142:K 9134:( 9130:+ 9126:) 9117:t 9107:d 9103:K 9096:+ 9091:t 9081:i 9077:K 9073:+ 9068:p 9064:K 9059:( 9052:= 9049:) 9046:z 9043:( 9040:C 9012:z 9008:1 9002:z 8992:t 8982:d 8978:K 8972:+ 8966:1 8960:z 8956:z 8949:t 8939:i 8935:K 8931:+ 8926:p 8922:K 8918:= 8915:) 8912:z 8909:( 8906:C 8881:t 8813:0 8809:u 8786:0 8782:u 8778:+ 8772:t 8768:d 8762:) 8759:t 8756:( 8753:e 8749:d 8740:d 8736:K 8732:+ 8725:d 8720:) 8714:( 8711:e 8706:t 8701:0 8691:i 8687:K 8683:+ 8680:) 8677:t 8674:( 8671:e 8666:p 8662:K 8658:= 8655:) 8652:t 8649:( 8646:u 8622:p 8618:K 8613:/ 8607:d 8603:K 8599:= 8594:d 8590:T 8586:, 8581:i 8577:K 8572:/ 8566:p 8562:K 8558:= 8553:i 8549:T 8524:] 8520:) 8515:2 8509:k 8505:t 8501:( 8498:e 8491:t 8482:d 8478:T 8471:+ 8468:) 8463:1 8457:k 8453:t 8449:( 8446:e 8442:) 8434:t 8424:d 8420:T 8416:2 8406:1 8399:( 8395:+ 8392:) 8387:k 8383:t 8379:( 8376:e 8372:) 8364:t 8355:d 8351:T 8344:+ 8336:i 8332:T 8327:t 8317:+ 8314:1 8310:( 8305:[ 8299:p 8295:K 8291:+ 8288:) 8283:1 8277:k 8273:t 8269:( 8266:u 8263:= 8260:) 8255:k 8251:t 8247:( 8244:u 8218:) 8213:2 8207:k 8203:t 8199:( 8196:e 8189:t 8180:d 8176:K 8169:+ 8166:) 8161:1 8155:k 8151:t 8147:( 8144:e 8140:) 8132:t 8122:d 8118:K 8114:2 8102:p 8098:K 8090:( 8086:+ 8083:) 8078:k 8074:t 8070:( 8067:e 8063:) 8055:t 8046:d 8042:K 8035:+ 8032:t 8024:i 8020:K 8016:+ 8011:p 8007:K 8002:( 7998:+ 7995:) 7990:1 7984:k 7980:t 7976:( 7973:u 7970:= 7967:) 7962:k 7958:t 7954:( 7951:u 7922:t 7911:t 7903:) 7898:2 7892:k 7888:t 7884:( 7881:e 7875:) 7870:1 7864:k 7860:t 7856:( 7853:e 7841:t 7833:) 7828:1 7822:k 7818:t 7814:( 7811:e 7805:) 7800:k 7796:t 7792:( 7789:e 7778:d 7774:K 7770:+ 7767:) 7762:k 7758:t 7754:( 7751:e 7746:i 7742:K 7738:+ 7732:t 7724:) 7719:1 7713:k 7709:t 7705:( 7702:e 7696:) 7691:k 7687:t 7683:( 7680:e 7672:p 7668:K 7664:= 7658:t 7650:) 7645:1 7639:k 7635:t 7631:( 7628:u 7622:) 7617:k 7613:t 7609:( 7606:u 7574:t 7566:) 7561:1 7555:k 7551:t 7547:( 7538:e 7529:) 7524:k 7520:t 7516:( 7507:e 7496:d 7492:K 7488:+ 7485:) 7480:k 7476:t 7472:( 7469:e 7464:i 7460:K 7456:+ 7450:t 7442:) 7437:1 7431:k 7427:t 7423:( 7420:e 7414:) 7409:k 7405:t 7401:( 7398:e 7390:p 7386:K 7382:= 7376:t 7368:) 7363:1 7357:k 7353:t 7349:( 7346:u 7340:) 7335:k 7331:t 7327:( 7324:u 7291:t 7283:) 7278:1 7272:k 7268:t 7264:( 7261:f 7255:) 7250:k 7246:t 7242:( 7239:f 7232:= 7225:t 7222:d 7217:) 7212:k 7208:t 7204:( 7201:f 7198:d 7191:= 7188:) 7183:k 7179:t 7175:( 7166:f 7138:) 7135:t 7132:( 7123:e 7115:d 7111:K 7107:+ 7104:) 7101:t 7098:( 7095:e 7090:i 7086:K 7082:+ 7079:) 7076:t 7073:( 7064:e 7056:p 7052:K 7048:= 7045:) 7042:t 7039:( 7030:u 6998:t 6975:) 6972:t 6969:( 6966:e 6946:) 6943:t 6940:( 6937:u 6900:. 6884:i 6874:d 6864:+ 6861:1 6858:= 6822:d 6812:= 6807:d 6803:T 6772:i 6764:= 6759:i 6755:T 6726:c 6722:K 6718:= 6713:p 6709:K 6682:) 6679:1 6676:+ 6672:s 6666:d 6658:( 6655:) 6652:1 6649:+ 6642:s 6636:i 6627:1 6622:( 6617:c 6613:K 6609:= 6606:) 6603:s 6600:( 6597:G 6557:s 6551:i 6547:K 6543:+ 6539:s 6533:p 6529:K 6525:+ 6519:2 6515:s 6508:d 6504:K 6497:= 6493:s 6487:d 6483:K 6479:+ 6474:s 6469:i 6465:K 6459:+ 6454:p 6450:K 6446:= 6443:) 6440:s 6437:( 6434:G 6395:) 6391:) 6388:t 6385:( 6382:V 6379:P 6373:t 6370:d 6366:d 6359:d 6355:T 6344:d 6338:) 6332:( 6329:e 6323:t 6318:0 6306:i 6302:T 6298:1 6293:+ 6289:) 6286:t 6283:( 6280:V 6277:P 6268:( 6262:p 6258:K 6254:= 6250:) 6247:t 6244:( 6241:V 6238:M 6190:p 6186:K 6181:/ 6152:p 6148:K 6116:d 6112:T 6106:p 6102:K 6098:= 6093:d 6089:K 6066:i 6062:T 6057:/ 6051:p 6047:K 6043:= 6038:i 6034:K 6010:) 6007:t 6004:( 6001:e 5995:t 5992:d 5988:d 5981:d 5977:K 5973:+ 5967:d 5963:) 5957:( 5954:e 5949:t 5944:0 5934:i 5930:K 5926:+ 5923:) 5920:t 5917:( 5914:e 5909:p 5905:K 5901:= 5898:) 5895:t 5892:( 5889:u 5861:p 5857:K 5834:i 5830:T 5806:d 5802:T 5771:d 5767:T 5739:i 5735:T 5707:) 5703:) 5700:t 5697:( 5694:e 5688:t 5685:d 5681:d 5674:d 5670:T 5666:+ 5660:d 5656:) 5650:( 5647:e 5642:t 5637:0 5625:i 5621:T 5617:1 5612:+ 5609:) 5606:t 5603:( 5600:e 5596:( 5590:p 5586:K 5582:= 5579:) 5576:t 5573:( 5570:u 5544:t 5541:u 5538:o 5533:D 5509:t 5506:u 5503:o 5498:I 5475:p 5471:K 5444:) 5438:( 5433:) 5429:( 5425:. 5411:. 5223:G 5197:I 5193:K 5189:= 5181:G 5155:P 5151:K 5147:= 5144:G 5115:s 5107:) 5104:s 5098:+ 5095:1 5092:( 5089:G 5083:= 5080:C 5042:. 5039:V 5036:P 5030:P 5027:S 5024:= 4963:t 4960:d 4948:I 4944:K 4940:+ 4932:P 4928:K 4736:p 4733:τ 4729:θ 4725:p 4722:k 4698:, 4694:) 4685:p 4670:t 4661:e 4654:1 4650:( 4646:u 4638:p 4634:k 4630:= 4627:) 4624:t 4621:( 4618:y 4605:s 4603:( 4601:u 4597:θ 4593:p 4590:τ 4586:p 4583:k 4566:, 4563:) 4560:s 4557:( 4554:u 4548:1 4545:+ 4542:s 4537:p 4525:s 4515:e 4509:p 4505:k 4498:= 4495:) 4492:s 4489:( 4486:y 4462:b 4458:a 4444:, 4441:a 4434:/ 4430:b 4427:4 4424:= 4419:u 4415:K 4392:u 4388:T 4337:4 4334:1 4297:u 4293:T 4270:d 4266:K 4243:i 4239:K 4206:/ 4199:u 4195:T 4187:u 4183:K 4178:3 4155:u 4151:T 4146:/ 4139:u 4135:K 4106:u 4102:K 4064:u 4060:T 4055:/ 4048:u 4044:K 4015:u 4011:K 3969:u 3965:K 3949:P 3930:d 3926:K 3902:i 3898:K 3874:p 3870:K 3836:u 3832:T 3809:u 3805:K 3782:u 3778:K 3755:d 3751:K 3728:i 3724:K 3672:d 3668:K 3631:d 3627:K 3586:i 3582:K 3541:p 3537:K 3493:d 3489:i 3485:p 3462:p 3458:K 3435:p 3431:K 3404:p 3400:K 3377:d 3373:K 3350:i 3346:K 3323:p 3319:K 3296:p 3292:K 3269:d 3265:K 3242:i 3238:K 3054:1 3048:) 3045:s 3042:( 3039:G 3036:) 3033:s 3030:( 3027:K 3007:1 3004:= 3000:| 2996:) 2993:s 2990:( 2987:G 2984:) 2981:s 2978:( 2975:K 2971:| 2950:1 2944:= 2941:) 2938:s 2935:( 2932:G 2929:) 2926:s 2923:( 2920:K 2900:s 2876:) 2873:s 2870:( 2867:G 2847:) 2844:s 2841:( 2838:K 2815:, 2809:) 2806:s 2803:( 2800:G 2797:) 2794:s 2791:( 2788:K 2785:+ 2782:1 2777:) 2774:s 2771:( 2768:G 2765:) 2762:s 2759:( 2756:K 2750:= 2747:) 2744:s 2741:( 2738:H 2637:. 2631:t 2628:d 2623:) 2620:t 2617:( 2614:e 2611:d 2603:d 2599:K 2595:= 2586:D 2569:d 2566:K 2562:d 2559:K 2551:i 2548:K 2544:p 2541:K 2539:( 2537:d 2534:K 2496:. 2490:d 2486:) 2480:( 2477:e 2472:t 2467:0 2457:i 2453:K 2449:= 2440:I 2423:i 2420:K 2418:( 2407:d 2404:K 2400:p 2397:K 2395:( 2393:i 2390:K 2340:. 2337:) 2334:t 2331:( 2328:e 2323:p 2319:K 2315:= 2306:P 2289:p 2286:K 2278:d 2275:K 2271:i 2268:K 2266:( 2264:p 2261:K 2235:s 2212:, 2209:s 2204:d 2200:K 2196:+ 2193:s 2189:/ 2183:i 2179:K 2175:+ 2170:p 2166:K 2162:= 2159:) 2156:s 2153:( 2150:L 2115:t 2073:t 2061:t 2047:) 2044:t 2041:( 2037:V 2034:P 2026:P 2023:S 2019:= 2016:) 2013:t 2010:( 2007:e 1983:d 1979:K 1954:i 1950:K 1925:p 1921:K 1894:, 1888:t 1885:d 1880:) 1877:t 1874:( 1871:e 1868:d 1860:d 1856:K 1852:+ 1846:d 1842:) 1836:( 1833:e 1828:t 1823:0 1813:i 1809:K 1805:+ 1802:) 1799:t 1796:( 1793:e 1788:p 1784:K 1780:= 1777:) 1774:t 1771:( 1767:V 1764:M 1760:= 1757:) 1754:t 1751:( 1748:u 1725:) 1722:t 1719:( 1716:u 1702:. 1226:( 1188:) 1181:t 1177:d 1171:) 1168:t 1165:( 1162:e 1158:d 1149:d 1145:T 1141:+ 1134:d 1129:) 1123:( 1120:e 1115:t 1110:0 1098:i 1094:T 1090:1 1085:+ 1082:) 1079:t 1076:( 1073:e 1069:( 1063:p 1059:K 1055:= 1052:) 1049:t 1046:( 1043:u 1018:i 1014:T 1009:/ 1003:p 999:K 976:d 972:T 966:p 962:K 939:d 935:T 912:i 908:T 885:d 881:T 875:p 871:K 848:i 844:T 839:/ 833:p 829:K 806:d 802:K 779:i 775:K 757:D 753:I 749:P 733:d 729:K 706:i 702:K 679:p 675:K 652:, 646:t 642:d 636:) 633:t 630:( 627:e 623:d 614:d 610:K 606:+ 599:d 594:) 588:( 585:e 580:t 575:0 565:i 561:K 557:+ 554:) 551:t 548:( 545:e 540:p 536:K 532:= 529:) 526:t 523:( 520:u 469:D 461:I 457:I 450:p 436:) 433:t 430:( 427:e 417:P 388:) 385:t 382:( 379:u 344:) 341:t 338:( 335:y 329:) 326:t 323:( 320:r 317:= 314:) 311:t 308:( 305:e 285:) 282:t 279:( 276:y 273:= 245:) 242:t 239:( 236:r 233:= 205:) 202:t 199:( 196:e 174:t 172:( 170:y 158:t 156:( 154:r 118:D 114:I 110:P 76:) 73:t 70:( 67:e 36:( 20:)

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