Proportional–integral–derivative controller

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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.

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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

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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..

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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.

1362:. 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 5382: 1244: 4891: 6119:. 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. 134: 2245: 2348:). 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. 4810:

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.

7924: 5339:, 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 5305:(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 9265: 2645:, 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. 8217: 9533: 7576: 1188: 473:

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.

8816:

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

5313:

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

4809:

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

1631:

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

1605:

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

1576:

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

460:

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

9709:

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

9572:

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

5287:

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

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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

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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

6214:

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

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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

5363:

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

5309:

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

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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

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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

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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

9538:

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 −

5322:

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

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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

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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

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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

4735:

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

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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

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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

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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.

1618:. 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 1413:

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

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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

120:

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

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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.

2500:

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

6395: 441:". 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". 4862:. 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: 173:

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

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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%.

4727:) 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. 1580:

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.

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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

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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.

4361:, 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. 1577:

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).

6193:, 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. 4826:

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".

7919:{\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}}} 7137: 8630: 4758:

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.

6681: 1307:. 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 9697: 6221: 2406:

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

3086:. 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. 2636: 2211: 5227:

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.

9260:{\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}}}} 6887: 1732: 8212:{\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})} 2814: 1441:

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.

8623: 2046: 9528:{\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 } 6828: 4962: 2664:. 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. 3407:

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.

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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.

3469: 2339: 5117: 1589:. 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 8524: 6779: 5403: 5554: 1437:

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.

1414:

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.

1382:(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. 10844:

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

7571:{\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}}} 5546: 5511: 4209: 2273:

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.

1223:). 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 6117: 4156: 284: 244: 8890: 5270:

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

4443: 4339: 1319:, all of whom contributed to the establishment of control stability criteria. In subsequent applications, speed governors were further refined, notably by American scientist 1295:" 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. 9692: 5041: 2660:

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

1235:. 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. 343: 3053: 2344:

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

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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

1183:{\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)} 7147: 6191: 1199:

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.

6997: 4602: 4341:. 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. 5156: 4985: 3127:

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

9570: 6418: 9617: 8814: 6153: 5862: 5835: 5807: 5772: 5740: 5476: 4393: 4298: 4271: 4244: 3931: 3903: 3875: 3837: 3810: 3783: 3756: 3729: 3673: 3632: 3587: 3542: 3463: 3436: 3405: 3378: 3351: 3324: 3297: 3270: 3243: 2094: 6974: 6945: 2875: 2846: 1724: 435: 387: 204: 75: 5873: 5222: 4777:

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

4365:

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.

3078:(implementing setpoint changes). These terms refer to how well the controlled variable tracks the desired value. Specific criteria for command tracking include 2899: 2234: 2114: 2072: 4770:

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

7929:

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:

2702:

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

1335:. Pressure control provided only a proportional control that, if the control gain was too high, would become unstable and go into overshoot with considerable 10033: 2546:

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

4798:

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.

1350:

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

1456:

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

4743:

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

4470: 1385:

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.

1227:

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

8782:{\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}} 10891: 7011: 11254: 9903: 4219:

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

4774:

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

10160: 646:{\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}},} 453:

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

3196:

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

2423: 1614:

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

6581: 6390:{\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)} 1522:

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

10914: 4844:

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

2675:, 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 1401:

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

11056: 10140: 3353:

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

493:

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

11223: 10445: 9865: 3167:) techniques; may include valve and sensor analysis; allows simulation before downloading; can support non-steady-state (NSS) tuning. 11057:"An Overview of Proportional plus Integral plus Derivative Control and Suggestions for Its Successful Application and Implementation" 2569: 2134: 1547:

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

1480:

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

10184: 2641:

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

4902:(proportional-integral controller) is a special case of the PID controller in which the derivative (D) of the error is not used. 1888:{\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}},} 1423: 736:, all non-negative, denote the coefficients for the proportional, integral, and derivative terms respectively (sometimes denoted 10677:

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

5323:

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

3193:

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

1540:

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.

1526:

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.

6839: 2402:

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

11170: 11115: 11032: 11011: 10992: 10874: 10562: 10123: 10096: 9988: 4709:

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

3140:

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

1358:. Minorsky was researching and designing automatic ship steering for the US Navy and based his analysis on observations of a 11102: 10368: 5263:

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

9724: 1259:

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

11082: 2722: 45:

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

10933: 10261: 8532: 5447:

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

1991: 11208: 10742:

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

10531: 6786: 4911: 4216:

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

2553:. The magnitude of the contribution of the derivative term to the overall control action is termed the derivative gain, 10014: 9573:(integration time). The above implementation allows to perform an I-only controller which may be useful in some cases. 5288:

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

10388: 2289: 11276: 11261: 10964: 10784: 10777:

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

10245: 9875: 5429: 5064: 4790:

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

1366:), which required adding the integral term. Finally, the derivative term was added to improve stability and control. 942:

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

112:

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

11196: 11142: 8228: 5702:{\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)} 5411: 3151:

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

10037: 9743:

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

6738: 5368:

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

4720:

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

3207:

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

437:. For example, if the error is large, the control output will be proportionately large by using the gain factor "K 1287:

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

1268:

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

11291: 5407: 1422:

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

1332: 982: 812: 10579: 6692: 5163: 2657:

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

1687:

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

945: 854: 11043: 9539:

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

5355:

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

2688: 1619: 1465: 1021:

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

4822:

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

3380:, if required, until the loop is acceptably quick to reach its reference after a load disturbance. Too much 11286: 1426:. They were the industry standard for many decades until the advent of discrete electronic controllers and 125:. The PID concept has been used widely in applications requiring accurate and optimized automatic control. 11137: 9582: 9577: 5340: 5271: 3056: 1212: 122: 9926: 9621: 9008:{\displaystyle C(z)=K_{p}+K_{i}\Delta _{t}{\frac {z}{z-1}}+{\frac {K_{d}}{\Delta _{t}}}{\frac {z-1}{z}}} 5516: 5481: 4162: 10842: 6017: 4309: 4023: 3693: 3687: 3055:

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

1427: 6072: 5310:

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

4114: 252: 212: 11147: 9703: 8847: 5837:

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

4398: 4315: 207: 154: 78: 42: 11159: 11022: 10756: 10483: 10339: 10300: 10276: 9807: 9665: 8825:

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

5005: 289: 11281: 9696: 7289:{\displaystyle {\dot {f}}(t_{k})={\dfrac {df(t_{k})}{dt}}={\dfrac {f(t_{k})-f(t_{k-1})}{\Delta t}}} 5392: 4308:

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

3758:

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

3011: 1224: 11164: 4312:, a set of tuning parameters were developed to yield a closed-loop response with a decay ratio of 2954: 1962: 1933: 1904: 918: 891: 785: 758: 712: 685: 658: 9630: 9627:

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

8860: 5396: 4692:{\displaystyle y(t)=k_{\text{p}}\Delta u\left(1-e^{\frac {-t-\theta }{\tau _{\text{p}}}}\right),} 4081: 3990: 3944: 3170:"Black box tuning" that requires specification of an objective describing the optimal behaviour. 3067:

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

2904: 1492:. Discrete electronic analog controllers have been largely replaced by digital controllers using 1284:

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

11176: 10853: 10772: 10684: 9755: 4818:

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

1606:

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

1448:

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

10751: 10334: 6553:{\displaystyle G(s)=K_{p}+{\frac {K_{i}}{s}}+K_{d}{s}={\frac {K_{d}{s^{2}}+K_{p}{s}+K_{i}}{s}}} 6161: 5302: 5245: 4350: 1370: 10608:"Patents, software, and hardware for PID control: An overview and analysis of the current art" 10115: 9619:

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

6979: 5343:. The order of the integrator and differentiator add increased flexibility to the controller. 3326:

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

3148:

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

11063: 10470: 10287: 10091: 5128: 4970: 1393: 11132: 9545: 5331:

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

4872:

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

4766:. The fundamental difficulty with PID control is that it is a feedback control system, with 3161:

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

11230: 10676: 10452: 10326: 10190:. Hacettepe University Department of Electrical and Electronics Engineering. Archived from 9595: 8792: 7003: 6131: 5840: 5813: 5785: 5750: 5718: 5454: 4786: 4371: 4276: 4249: 4222: 3909: 3881: 3853: 3815: 3788: 3761: 3734: 3707: 3701: 3651: 3610: 3565: 3520: 3441: 3414: 3383: 3356: 3329: 3302: 3275: 3248: 3221: 2079: 1658: 1406: 1347:

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

1269: 1251:

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

448:

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

346: 86: 6950: 6921: 6004:{\displaystyle u(t)=K_{p}e(t)+K_{i}\int _{0}^{t}e(\tau )\,d\tau +K_{d}{\frac {d}{dt}}e(t)} 4596:) is a step change input. Converting this transfer function to the time domain results in 2851: 2822: 1700: 411: 363: 180: 51: 8: 10655: 9950: 9018:

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

4806: 4778: 1300: 11221: 10330: 10191: 9954: 979:

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

11202:

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

10630: 10426: 10074: 5207: 2884: 2703: 2502: 2219: 2099: 2057: 1590: 1469: 1410: 1397:

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

1273: 1228: 11186: 11181: 10515: 11248: 11111: 11028: 11007: 10988: 10960: 10870: 10780: 10680: 10649: 10558: 10241: 10222: 10207: 10119: 9984: 9871: 6915: 6409: 6128:

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

5055: 4354: 2695:, and is limited only by saturation or mechanical breakage. Instability is caused by 2121: 1670: 1615: 1461: 1402: 1327: 10634: 4395:

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

1418:. Later the derivative term was added by a further bellows and adjustable orifice. 10823: 10813: 10622: 10581:

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

10510: 10418: 10344: 10218: 10070: 10058: 9918: 4869:

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

1544: 1531: 1493: 1378: 1355: 1351: 1292: 247: 150: 82: 11214: 10430: 9585:

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

4560:{\displaystyle y(s)={\frac {k_{\text{p}}e^{-\theta s}}{\tau _{\text{p}}s+1}}u(s),} 1444: 357:

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

10659: 10317:

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

6903: 5332: 4859: 4853: 4823: 4760: 4716:) is equal to the change in output divided by the change in input. The dead time 3697: 2676: 1208: 169:

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

8846:

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

6686:

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

5224:

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

2365:

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

1452: 11086: 10892:"Discrete PI and PID Controller Design and Analysis for Digital Implementation" 10607: 9719: 6014:

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

4771: 2710: 2125: 1519: 1509: 113: 11191: 10697: 10587:(Master's Thesis ed.). Tampere, Finland: Tampere University of Technology 10532:"A Review of Relay Auto-tuning Methods for the Tuning of PID-type Controllers" 10389:"A Review of Relay Auto-tuning Methods for the Tuning of PID-type Controllers" 10348: 9270:

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

7132:{\displaystyle {\dot {u}}(t)=K_{p}{\dot {e}}(t)+K_{i}e(t)+K_{d}{\ddot {e}}(t)} 11270: 10626: 10422: 9981:

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

6902:

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

4828: 4357:

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

3083: 2642: 1662: 1320: 1316: 1312: 390: 138: 10403: 3468: 5356: 4736:

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

4353:

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

2672: 2096:

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

1485: 1438: 1344: 1308: 394: 34: 10818: 10801: 4866:

Disabling the integration until the PV has entered the controllable region

10828: 10718: 8854: 5336: 2692: 2490:{\displaystyle I_{\text{out}}=K_{\text{i}}\int _{0}^{t}e(\tau )\,d\tau .} 1642: 1523: 1515: 1489: 1336: 1272:, which uses rotating weights to control a process. This was invented by 1232: 11222:

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

1299:

theoretical basis for the operation of governors was first described by

1255: 10498: 6676:{\displaystyle G(s)=K_{c}({\frac {1}{\tau _{i}{s}}}+1)(\tau _{d}{s}+1)} 3408: 1677:, and practically every other variable for which a measurement exists. 1481: 1323:, who in 1872 theoretically analyzed Watt's conical pendulum governor. 1288: 354: 94: 10663: 9922: 11148:

Introduction to the key terms associated with PID Temperature Control

10369:"Simple analytic rules for model reduction and PID controller tuning" 9808:"Control Systems, Robotics and Automation – Volume VII - PID Control" 3079: 1654: 1582: 1277: 490: 11171:

Shows how to build a PID controller with basic electronic components

10773:"Controlling fuel annealer using computational verb PID controllers" 5381: 3218:

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

10698:"Integral (Reset) Windup, Jacketing Logic and the Velocity PI Form" 10021:

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

9999:

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

6155:

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

5298: 5250: 5241: 5047: 4858:

One common problem resulting from the ideal PID implementations is

2403: 1646: 1586: 1359: 1281: 1248: 350: 90: 38: 10916:

Hardware Implimentation [sic] of FPGA based PID Controller

6914:. Approximations for first-order derivatives are made by backward 4805:

An asymmetric application, for example, is temperature control in

4464:

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

3137:

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

1405:. This had been developed in telephone engineering electronics by 1243: 10061:(1922). "Directional stability of automatically steered bodies". 2521:

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

2517: 2377:

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

2373: 2362: 2248:

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

1726:

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

11001: 10238:

Instrument Engineers' Handbook: Process control and optimization

5297:

The control system performance can be improved by combining the

4890: 3785:

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

1343:

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

10855:| The Benefits of Cascade Control | September 22, 2020 | Watlow 10606:

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

6571:

Another representation of the PID controller is the series, or

4453:

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

3163: 3074:(disturbance rejection – staying at a given setpoint) and 2631:{\displaystyle D_{\text{out}}=K_{\text{d}}{\frac {de(t)}{dt}}.} 2244: 2206:{\displaystyle L(s)=K_{\text{p}}+K_{\text{i}}/s+K_{\text{d}}s,} 1666: 1585:

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

10987:. Radnor, Pennsylvania: Chilton Book Company. pp. 20–29. 10557:. Upper Saddle River, New Jersey: Prentice Hall. p. 129. 6910:

device requires the standard form of the PID controller to be

2709:

Mathematically, the origins of instability can be seen in the

5867:

In the parallel form, shown in the controller theory section

4819: 4706:

try to minimize disturbances when performing the step test).

4358: 2679:(using different parameters in different operating regions). 1674: 1650: 479:– The mathematical model and practical loop above both use a 6122: 6907: 5051: 5046:

A PI controller can be modelled easily in software such as

1497: 1247:

Early PID theory was developed by observing the actions of

4449:

is the amplitude of the process variable oscillation, and

2881:

where the closed-loop transfer function diverges for some

461:

the change, the greater the controlling or damping effect.

408:

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

10802:"Some Applications of Fractional Calculus in Engineering" 10401: 5335:(changing parameters in different operating conditions), 3411:

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

3099:

using this response to determine the control parameters.

1564:

where the integral and derivative terms play their part.

489:

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

133: 11262:

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

11002:

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

6882:{\displaystyle \alpha =1+{\frac {\tau _{d}}{\tau _{i}}}} 5442: 1460:

Most modern PID controls in industry are implemented as

9542:

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

4876: 2699:

gain, particularly in the presence of significant lag.

1688: 1518:

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

9758: 8843:

previous_error := error wait(dt) goto loop

4320: 3008:

with a 180° phase shift. Stability is guaranteed when

10404:"PID control system analysis, design, and technology" 9668: 9633: 9598: 9548: 9482: 9425: 9358: 9279: 9195: 9143: 9088: 9027: 8893: 8863: 8795: 8633: 8535: 8463: 8401: 8336: 8309: 8231: 8161: 8099: 8027: 7938: 7590: 7308: 7224: 7183: 7150: 7014: 6982: 6953: 6924: 6842: 6789: 6741: 6695: 6584: 6421: 6408:

Sometimes it is useful to write the PID regulator in

6224: 6164: 6134: 6075: 6020: 5876: 5843: 5816: 5788: 5753: 5721: 5557: 5519: 5484: 5457: 5371: 5210: 5166: 5131: 5067: 5008: 4973: 4914: 4605: 4473: 4401: 4374: 4318: 4279: 4252: 4225: 4165: 4117: 4084: 4026: 3993: 3947: 3912: 3884: 3856: 3818: 3791: 3764: 3737: 3710: 3654: 3613: 3568: 3523: 3444: 3417: 3386: 3359: 3332: 3305: 3278: 3251: 3224: 3014: 2957: 2907: 2887: 2854: 2825: 2725: 2572: 2426: 2292: 2222: 2137: 2102: 2082: 2060: 1994: 1965: 1936: 1907: 1735: 1703: 1488:, or even the movement-detection circuit of a modern 1030: 985: 948: 921: 894: 857: 815: 788: 761: 715: 688: 661: 507: 414: 366: 292: 255: 215: 183: 54: 11160:

PID Control in MATLAB/Simulink and Python with TCLab

10679:. IEEE Control Systems Magazine, 26 (1). pp. 32-41. 10201: 4987:

is the error or deviation of actual measured value (

2809:{\displaystyle H(s)={\frac {K(s)G(s)}{1+K(s)G(s)}},} 2361:

gain. SSE may be mitigated by adding a compensating

10063:

Journal of the American Society for Naval Engineers

10019:, Portsmouth: Griffin & Co., pp. 137–138, 8618:{\displaystyle T_{i}=K_{p}/K_{i},T_{d}=K_{d}/K_{p}} 6205: 4785:The most significant improvement is to incorporate 4459: 3188: 2041:{\displaystyle e(t)=\mathrm {SP} -\mathrm {PV} (t)} 10890: 9863: 9776: 9686: 9654: 9611: 9564: 9527: 9259: 9007: 8876: 8838:dt - loop interval time (assumes reasonable scale) 8808: 8781: 8617: 8518: 8211: 7918: 7570: 7288: 7131: 6991: 6968: 6939: 6881: 6823:{\displaystyle T_{d}={\frac {\tau _{d}}{\alpha }}} 6822: 6773: 6727: 6675: 6552: 6389: 6185: 6147: 6111: 6061: 6003: 5856: 5829: 5801: 5766: 5734: 5701: 5540: 5505: 5470: 5301:(or closed-loop) control of a PID controller with 5216: 5192: 5150: 5111: 5035: 4979: 4957:{\displaystyle K_{P}\Delta +K_{I}\int \Delta \,dt} 4956: 4691: 4559: 4437: 4387: 4333: 4292: 4265: 4238: 4203: 4150: 4102: 4059: 4011: 3965: 3925: 3897: 3869: 3831: 3804: 3777: 3750: 3723: 3667: 3626: 3581: 3536: 3457: 3430: 3399: 3372: 3345: 3318: 3299:until the output of the loop oscillates; then set 3291: 3264: 3237: 3047: 3000: 2943: 2893: 2869: 2840: 2808: 2630: 2489: 2333: 2228: 2205: 2108: 2088: 2066: 2040: 1978: 1949: 1920: 1887: 1718: 1182: 1013: 971: 934: 907: 880: 843: 801: 774: 728: 701: 674: 645: 429: 381: 337: 278: 238: 198: 69: 11192:Proven Methods and Best Practices for PID Control 11100: 11080: 10957:Process Control: Modeling, Design, and Simulation 10934:"PID process control, a "Cruise Control" example" 10555:Process Control: Modeling, Design, and Simulation 10001:, Annapolis, MD: U.S. Naval Institute, p. 33 7002:Differentiating both sides of PID equation using 6196: 4839: 4456:There are numerous variants on the relay method. 4344: 3438:setting significantly less than half that of the 1537:The desired position is called the setpoint (SP). 1193: 11268: 10799: 10744:International Journal of Computational Cognition 10383: 10381: 8817:null, it prevents from sending the output to 0. 6400:King describes an effective chart-based method. 3692:Another heuristic tuning method is known as the 2706:(bounded oscillation) is acceptable or desired. 2334:{\displaystyle P_{\text{out}}=K_{\text{p}}e(t).} 2074:is the time or instantaneous time (the present), 1475: 1276:in the 17th century to regulate the gap between 1207:The use of the PID algorithm does not guarantee 10985:Instrument Engineers' Handbook: Process Control 10411:IEEE Transactions on Control Systems Technology 5112:{\displaystyle C={\frac {G(1+\tau s)}{\tau s}}} 11213:, Embedded Systems Programming, archived from 11104:Handbook of PI and PID Controller Tuning Rules 10496: 9752:Note that for very small intervals (e.g. 60Hz/ 8519:{\displaystyle u(t_{k})=u(t_{k-1})+K_{p}\left} 3183:Offline; only good for first-order processes. 468:– The balance of these effects is achieved by 11027:. Chichester, UK: John Wiley & Sons Ltd. 10443: 10378: 10312: 10310: 10161:"Diode Laser Locking and Linewidth Narrowing" 10095:harvnb error: no target: CITEREFBennett1993 ( 9785:holds true for all pseudocode presented here. 3089: 1928:is the proportional gain, a tuning parameter, 11253:: CS1 maint: multiple names: authors list ( 11206: 6774:{\displaystyle T_{i}=\tau _{i}\cdot \alpha } 2877:is the plant transfer function. A system is 10926: 10274: 10112:A history of control engineering, 1800-1930 5410:. Unsourced material may be challenged and 4813: 3132: 1986:is the derivative gain, a tuning parameter, 1626: 23:proportional–integral–derivative controller 10605: 10402:Kiam Heong Ang; Chong, G.; Yun Li (2005). 10316: 10307: 9702:A variant of the above algorithm using an 7581:Applying backward difference again gives, 6897: 6566: 6210:Most commercial control systems offer the 3704:in the 1940s. As in the method above, the 3175: 11224:"PID Controller Tuning: A Short Tutorial" 11041: 10827: 10817: 10800:Tenreiro Machado JA, et al. (2009). 10770: 10755: 10723:Practical Process Control by Control Guru 10651:On Automation of the PID Tuning Procedure 10514: 10446:"PID Controller Tuning: A Short Tutorial" 10366: 10338: 9867:Adaptive Control Processes: A Guided Tour 8711: 6330: 6260: 6123:Reciprocal gain, a.k.a. proportional band 5954: 5647: 5430:Learn how and when to remove this message 4947: 4793: 3681: 2477: 2280:, called the proportional gain constant. 2048:is the error (SP is the setpoint, and PV( 1957:is the integral gain, a tuning parameter, 1833: 1689:§ Alternative nomenclature and forms 1231:a setpoint, and the degree of any system 1120: 1014:{\displaystyle K_{\text{p}}/T_{\text{i}}} 844:{\displaystyle K_{\text{p}}/T_{\text{i}}} 585: 11110:(3rd ed.). Imperial College Press. 10954: 10552: 10497:Åström, K.J.; Hägglund, T. (July 1984). 10240:(4th ed.). CRC Press. p. 108. 10138: 10057: 9978: 6728:{\displaystyle K_{p}=K_{c}\cdot \alpha } 5193:{\displaystyle {\frac {G}{\tau }}=K_{I}} 4889: 4303: 4273:are dependent on the oscillation period 3467: 2516: 2372: 2243: 1451: 1443: 1392: 1254: 1242: 972:{\displaystyle K_{\text{p}}T_{\text{d}}} 881:{\displaystyle K_{\text{p}}T_{\text{d}}} 755:of the equation (see later in article), 141:of a PID controller in a feedback loop. 132: 128: 11081:Graham, Ron; Mike McHugh (2005-10-03). 10647: 10529: 10499:"Automatic Tuning of Simple Regulators" 10362: 10360: 10358: 10152: 10109: 10087: 9949: 9901: 9897: 9895: 9893: 9891: 9889: 9887: 9864:Richard E. Bellman (December 8, 2015). 9825:"9.3: PID Tuning via Classical Methods" 6976:are discretized with a sampling period 5257: 4702:using the same parameters found above. 1503: 1376:, with the controllers controlling the 1363: 85:(PV) and applies a correction based on 11269: 10982: 10235: 9904:"A brief history of automatic control" 7141:Derivative terms are approximated as, 3839:are used to set the gains as follows: 3465:setting that was causing oscillation. 2414:) and added to the controller output. 1326:About this time, the invention of the 11187:PID with single Operational Amplifier 11024:Process Control: A Practical Approach 10867:Process Control: A Practical Approach 10262:"Introduction: PID Controller Design" 10034:"A Brief Building Automation History" 9848: 9805: 5443:Standard versus parallel (ideal) form 5326: 5317: 2716:The closed-loop transfer function is 2351: 1388: 11197:Principles of PID Control and Tuning 11182:PID Control with MATLAB and Simulink 11165:What's All This P-I-D Stuff, Anyhow? 11143:Principles of PID Control and Tuning 11042:Van Doren, Vance J. (July 1, 2003). 11020: 10864: 10806:Mathematical Problems in Engineering 10741: 10664:847ca38e-93e8-4188-b3d5-8ec6c23f2132 10571: 10530:Hornsey, Stephen (29 October 2012). 10355: 10142:The power of external-reset feedback 9884: 9725:Active disturbance rejection control 5408:adding citations to reliable sources 5375: 4877:Overshooting from known disturbances 3472:Effects of varying PID parameters (K 3156: 2951:. In other words, this happens when 2239: 1680: 496: 345:, and applies a correction based on 206:as the difference between a desired 77:as the difference between a desired 11210:Introduction to Closed-Loop Control 11054: 10158: 5054:using a "flow chart" box involving 3062: 2901:. This happens in situations where 888:; the advantage of this being that 13: 10975: 10912: 10719:"PI Control of the Heat Exchanger" 10716: 10695: 10648:Soltesz, Kristian (January 2012). 10075:10.1111/j.1559-3584.1922.tb04958.x 9983:. Praeger Security International. 9495: 9443: 9371: 9345: 9208: 9161: 9101: 9075: 8976: 8933: 8865: 8756: 8737: 8713: 8476: 8419: 8349: 8312: 8174: 8117: 8040: 8017: 7907: 7896: 7826: 7717: 7643: 7559: 7435: 7361: 7276: 6983: 6235: 6229: 6226: 5541:{\displaystyle D_{\mathrm {out} }} 5532: 5529: 5526: 5506:{\displaystyle I_{\mathrm {out} }} 5497: 5494: 5491: 5372:Alternative nomenclature and forms 5346: 5277:Derivative of the process variable 5009: 4974: 4944: 4925: 4905:The controller output is given by 4847: 4833: 4730: 4631: 4368:Specifically, the ultimate period 4204:{\displaystyle 3{K_{u}}{T_{u}}/40} 3484:) on the step response of a system 3119: 2848:is the PID transfer function, and 2512: 2283:The proportional term is given by 2025: 2022: 2014: 2011: 1755: 1752: 1661:, chemical composition (component 1609: 1165: 1146: 1122: 630: 611: 587: 14: 11303: 11126: 10771:Liang, Yilong; Yang, Tao (2009). 9706:(IIR) filter for the derivative: 8627:Note: This method solves in fact 6062:{\displaystyle K_{i}=K_{p}/T_{i}} 5864:to compute the control variable. 4060:{\displaystyle 0.54{K_{u}}/T_{u}} 3124:No mathematics required; online. 393:, to a new value determined by a 11153: 10223:10.1016/j.jmatprotec.2013.05.023 9962:Proceedings of the Royal Society 9695: 9620: 6206:Basing proportional action on PV 6112:{\displaystyle K_{d}=K_{p}T_{d}} 5380: 5274:present in a simple step change. 4885: 4460:First-order model with dead time 4151:{\displaystyle 1.2{K_{u}}/T_{u}} 3213: 2671:Some processes have a degree of 2661: 2563:The derivative term is given by 2506: 2368: 2345: 1303:in 1868 in his now-famous paper 1218: 1202: 469: 279:{\displaystyle {\text{PV}}=y(t)} 239:{\displaystyle {\text{SP}}=r(t)} 11006:. London, UK: Springer-Verlag. 10906: 10883: 10858: 10847: 10836: 10793: 10764: 10735: 10710: 10689: 10670: 10641: 10599: 10578:Heinänen, Eero (October 2018). 10546: 10523: 10490: 10437: 10395: 10268: 10254: 10229: 10177: 10132: 10103: 10081: 10051: 9746: 9737: 6403: 5292: 4438:{\displaystyle K_{u}=4b/\pi a,} 4334:{\displaystyle {\tfrac {1}{4}}} 1635: 1554: 1369:Trials were carried out on the 109:respectively), hence the name. 16:Control loop feedback mechanism 11083:"FAQ on PID controller tuning" 10211:Material Processing Technology 10026: 10006: 9972: 9943: 9870:. Princeton University Press. 9857: 9842: 9817: 9799: 9768: 9687:{\displaystyle 3<=N<=10} 9522: 9510: 9475: 9463: 9397: 9391: 9310: 9298: 9289: 9283: 9037: 9031: 8903: 8897: 8750: 8744: 8708: 8702: 8668: 8662: 8643: 8637: 8508: 8489: 8456: 8437: 8380: 8367: 8276: 8257: 8248: 8235: 8206: 8187: 8154: 8135: 8071: 8058: 7983: 7964: 7955: 7942: 7891: 7872: 7863: 7844: 7821: 7802: 7793: 7780: 7755: 7742: 7712: 7693: 7684: 7671: 7638: 7619: 7610: 7597: 7554: 7535: 7517: 7504: 7473: 7460: 7430: 7411: 7402: 7389: 7356: 7337: 7328: 7315: 7271: 7252: 7243: 7230: 7205: 7192: 7176: 7163: 7126: 7120: 7092: 7086: 7067: 7061: 7033: 7027: 6963: 6957: 6934: 6928: 6670: 6646: 6643: 6610: 6594: 6588: 6431: 6425: 6379: 6373: 6326: 6320: 6277: 6271: 6238: 6232: 6197:Basing derivative action on PV 5998: 5992: 5951: 5945: 5911: 5905: 5886: 5880: 5691: 5685: 5644: 5638: 5597: 5591: 5567: 5561: 5095: 5080: 5036:{\displaystyle \Delta =SP-PV.} 4894:Basic block of a PI controller 4840:Modifications to the algorithm 4753: 4615: 4609: 4551: 4545: 4483: 4477: 4345:Relay (Åström–Hägglund) method 3036: 3030: 3024: 3018: 2988: 2984: 2978: 2972: 2966: 2959: 2929: 2923: 2917: 2911: 2864: 2858: 2835: 2829: 2797: 2791: 2785: 2779: 2765: 2759: 2753: 2747: 2735: 2729: 2648: 2611: 2605: 2474: 2468: 2417:The integral term is given by 2325: 2319: 2147: 2141: 2035: 2029: 2004: 1998: 1868: 1862: 1830: 1824: 1790: 1784: 1765: 1759: 1745: 1739: 1713: 1707: 1543:The input to the process (the 1484:, the power conditioning of a 1466:programmable logic controllers 1333:pendulum-and-hydrostat control 1194:Selective use of control terms 1159: 1153: 1117: 1111: 1070: 1064: 1040: 1034: 624: 618: 582: 576: 542: 536: 517: 511: 424: 418: 376: 370: 338:{\displaystyle e(t)=r(t)-y(t)} 332: 326: 317: 311: 302: 296: 273: 267: 233: 227: 193: 187: 64: 58: 1: 11085:. Mike McHugh. Archived from 10615:IEEE Control Systems Magazine 10516:10.1016/S1474-6670(17)61248-5 10444:Jinghua Zhong (Spring 2006). 10110:Bennett, Stuart (June 1986). 10016:Torpedoes and Torpedo Warfare 9911:IEEE Control Systems Magazine 9792: 8820: 3272:values to zero. Increase the 3048:{\displaystyle K(s)G(s)<1} 1596: 1476:Electronic analog controllers 809:are respectively replaced by 501:The overall control function 11167:Article in Electronic Design 11133:PID tuning using Mathematica 10275:Tim Wescott (October 2000). 9853:, Cambridge University Press 8857:of a PID can be written as ( 3001:{\displaystyle |K(s)G(s)|=1} 2682: 1979:{\displaystyle K_{\text{d}}} 1950:{\displaystyle K_{\text{i}}} 1921:{\displaystyle K_{\text{p}}} 935:{\displaystyle T_{\text{d}}} 908:{\displaystyle T_{\text{i}}} 802:{\displaystyle K_{\text{d}}} 775:{\displaystyle K_{\text{i}}} 729:{\displaystyle K_{\text{d}}} 702:{\displaystyle K_{\text{i}}} 675:{\displaystyle K_{\text{p}}} 7: 11207:Michael Barr (2002-07-30), 10955:Bequette, B. Wayne (2006). 10553:Bequette, B. Wayne (2003). 9777:{\textstyle .016{\bar {6}}} 9713: 9655:{\displaystyle \tau _{d}/N} 9576:In the real world, this is 8877:{\displaystyle \Delta _{t}} 5235: 4103:{\displaystyle 0.60{K_{u}}} 4012:{\displaystyle 0.45{K_{u}}} 3966:{\displaystyle 0.50{K_{u}}} 3812:and the oscillation period 3201:Simple control rule (SIMC) 3070:Two basic requirements are 3057:Nyquist stability criterion 2944:{\displaystyle K(s)G(s)=-1} 2052:) is the process variable), 1567: 1530:The sensed position is the 1428:distributed control systems 1216: 389:, such as the opening of a 10: 11310: 11044:"Loop Tuning Fundamentals" 10367:Skogestad, Sigurd (2003). 9979:Newpower, Anthony (2006). 9813:. Japan: Kyoto University. 9592:A common issue when using 4851: 3685: 3493:a parameter independently 3090:Overview of tuning methods 2662:limitations of PID control 2507:the section on loop tuning 2346:the section on loop tuning 2236:is the complex frequency. 1507: 1263: 1238: 43:industrial control systems 10869:. Wiley. pp. 52–78. 10577: 10349:10.1103/revmodphys.77.783 10319:Reviews of Modern Physics 10264:. University of Michigan. 10185:"Position control system" 10139:Shinskey, F Greg (2004), 9849:Hills, Richard L (1996), 9704:infinite impulse response 6999:, k is the sample index. 6186:{\displaystyle 100/K_{p}} 3103:Choosing a tuning method 2128:of the PID controller is 11277:Classical control theory 10627:10.1109/MCS.2006.1580153 10503:IFAC Proceedings Volumes 10423:10.1109/TCST.2005.847331 10236:Lipták, Béla G. (2003). 9902:Bennett, Stuart (1996). 9730: 6992:{\displaystyle \Delta t} 4814:Noise in derivative term 2505:the setpoint value (see 1627:Response to disturbances 11138:PID tuning using Python 11004:Advances in PID Control 10013:Sleeman, C. W. (1880), 8884:is the sampling time): 6898:Discrete implementation 6567:Series/interacting form 5478:gain is applied to the 5151:{\displaystyle G=K_{P}} 4980:{\displaystyle \Delta } 3843:Ziegler–Nichols method 41:that is widely used in 11101:Aidan O'Dwyer (2009). 10478:Cite journal requires 10295:Cite journal requires 9829:Engineering LibreTexts 9778: 9688: 9656: 9613: 9566: 9565:{\displaystyle s^{-1}} 9529: 9261: 9009: 8878: 8829:Kp - proportional gain 8810: 8783: 8619: 8520: 8213: 7920: 7572: 7290: 7133: 6993: 6970: 6941: 6883: 6824: 6775: 6729: 6677: 6554: 6391: 6187: 6149: 6113: 6063: 6005: 5858: 5831: 5803: 5768: 5736: 5703: 5542: 5507: 5472: 5218: 5194: 5152: 5113: 5037: 4981: 4958: 4895: 4794:Linearity and symmetry 4693: 4588:is the dead time, and 4584:is the time constant, 4561: 4439: 4389: 4335: 4310:Ziegler–Nichols method 4294: 4267: 4240: 4205: 4152: 4104: 4061: 4013: 3967: 3927: 3899: 3871: 3833: 3806: 3779: 3752: 3725: 3694:Ziegler–Nichols method 3688:Ziegler–Nichols method 3682:Ziegler–Nichols method 3669: 3628: 3583: 3538: 3485: 3459: 3432: 3401: 3374: 3347: 3320: 3293: 3266: 3239: 3049: 3002: 2945: 2895: 2871: 2842: 2810: 2632: 2543: 2491: 2399: 2335: 2270: 2230: 2207: 2110: 2090: 2068: 2042: 1980: 1951: 1922: 1889: 1720: 1559:The obvious method is 1457: 1449: 1398: 1260: 1252: 1184: 1015: 973: 936: 909: 882: 845: 803: 776: 730: 703: 676: 647: 431: 397:of the control terms. 383: 339: 280: 240: 200: 166: 71: 11292:Industrial automation 10983:Liptak, Bela (1995). 10959:. Prentice Hall PTR. 9779: 9689: 9657: 9614: 9612:{\displaystyle K_{d}} 9567: 9530: 9262: 9010: 8879: 8811: 8809:{\displaystyle u_{0}} 8784: 8620: 8521: 8214: 7921: 7573: 7291: 7134: 6994: 6971: 6942: 6884: 6825: 6776: 6730: 6678: 6555: 6392: 6188: 6150: 6148:{\displaystyle K_{p}} 6114: 6064: 6006: 5859: 5857:{\displaystyle K_{p}} 5832: 5830:{\displaystyle T_{i}} 5804: 5802:{\displaystyle T_{d}} 5769: 5767:{\displaystyle T_{d}} 5737: 5735:{\displaystyle T_{i}} 5704: 5543: 5508: 5473: 5471:{\displaystyle K_{p}} 5219: 5195: 5153: 5114: 5038: 4993:) from the setpoint ( 4982: 4959: 4893: 4852:Further information: 4694: 4577:is the process gain, 4562: 4440: 4390: 4388:{\displaystyle T_{u}} 4349:Published in 1984 by 4336: 4304:Cohen–Coon parameters 4295: 4293:{\displaystyle T_{u}} 4268: 4266:{\displaystyle K_{d}} 4241: 4239:{\displaystyle K_{i}} 4206: 4153: 4105: 4062: 4014: 3968: 3928: 3926:{\displaystyle K_{d}} 3900: 3898:{\displaystyle K_{i}} 3872: 3870:{\displaystyle K_{p}} 3834: 3832:{\displaystyle T_{u}} 3807: 3805:{\displaystyle K_{u}} 3780: 3778:{\displaystyle K_{u}} 3753: 3751:{\displaystyle K_{d}} 3726: 3724:{\displaystyle K_{i}} 3686:Further information: 3670: 3668:{\displaystyle K_{d}} 3629: 3627:{\displaystyle K_{d}} 3584: 3582:{\displaystyle K_{i}} 3539: 3537:{\displaystyle K_{p}} 3471: 3460: 3458:{\displaystyle K_{p}} 3433: 3431:{\displaystyle K_{p}} 3402: 3400:{\displaystyle K_{p}} 3375: 3373:{\displaystyle K_{d}} 3348: 3346:{\displaystyle K_{i}} 3321: 3319:{\displaystyle K_{p}} 3294: 3292:{\displaystyle K_{p}} 3267: 3265:{\displaystyle K_{d}} 3240: 3238:{\displaystyle K_{i}} 3180:Good process models. 3050: 3003: 2946: 2896: 2872: 2843: 2811: 2633: 2520: 2492: 2376: 2336: 2247: 2231: 2208: 2111: 2091: 2089:{\displaystyle \tau } 2069: 2043: 1981: 1952: 1923: 1890: 1721: 1508:Further information: 1455: 1447: 1396: 1258: 1246: 1225:lead–lag compensation 1211:of the system or its 1185: 1016: 974: 937: 910: 883: 846: 804: 777: 731: 704: 677: 648: 432: 384: 340: 281: 241: 201: 165:) is the measured PV. 136: 129:Fundamental operation 72: 31:three-term controller 10779:. Asid'09: 417–420. 9997:Gray, Edwyn (1991), 9756: 9666: 9631: 9596: 9546: 9277: 9025: 8891: 8861: 8835:Kd - derivative gain 8793: 8631: 8533: 8229: 7936: 7588: 7306: 7148: 7012: 6980: 6969:{\displaystyle e(t)} 6951: 6940:{\displaystyle u(t)} 6922: 6840: 6787: 6739: 6693: 6582: 6419: 6222: 6162: 6132: 6073: 6018: 5874: 5841: 5814: 5786: 5751: 5719: 5555: 5517: 5482: 5455: 5404:improve this section 5258:Setpoint step change 5208: 5204:Setting a value for 5164: 5129: 5065: 5006: 4971: 4912: 4787:feed-forward control 4603: 4471: 4399: 4372: 4316: 4277: 4250: 4223: 4163: 4115: 4082: 4024: 3991: 3945: 3910: 3882: 3854: 3816: 3789: 3762: 3735: 3708: 3702:Nathaniel B. Nichols 3652: 3645:No effect in theory 3611: 3566: 3521: 3442: 3415: 3384: 3357: 3330: 3303: 3276: 3249: 3222: 3012: 2955: 2905: 2885: 2870:{\displaystyle G(s)} 2852: 2841:{\displaystyle K(s)} 2823: 2723: 2570: 2424: 2290: 2220: 2135: 2100: 2080: 2058: 1992: 1963: 1934: 1905: 1733: 1719:{\displaystyle u(t)} 1701: 1504:Control loop example 1468:(PLCs), or discrete 1270:centrifugal governor 1028: 983: 946: 919: 892: 855: 813: 786: 759: 713: 686: 659: 505: 430:{\displaystyle e(t)} 412: 382:{\displaystyle u(t)} 364: 290: 253: 213: 199:{\displaystyle e(t)} 181: 81:(SP) and a measured 70:{\displaystyle e(t)} 52: 37:mechanism employing 11287:Control engineering 11089:on February 6, 2005 11048:Control Engineering 11021:King, Myke (2010). 10936:. CodeProject. 2009 10865:King, Myke (2011). 10819:10.1155/2010/639801 10331:2005RvMP...77..783B 10277:"PID without a PhD" 9851:Power From the Wind 8698: 6315: 5941: 5634: 5451:. In this form the 5158:= proportional gain 3844: 3510:Steady-state error 3494: 3104: 2464: 1820: 1470:compact controllers 1424:hazardous locations 1301:James Clerk Maxwell 1107: 572: 10913:Thakur, Bhushana. 9806:Araki, M. (2009). 9774: 9684: 9652: 9609: 9562: 9525: 9505: 9453: 9381: 9257: 9218: 9171: 9111: 9005: 8874: 8832:Ki - integral gain 8806: 8779: 8684: 8615: 8516: 8484: 8427: 8357: 8330: 8209: 8182: 8125: 8048: 7916: 7568: 7286: 7284: 7218: 7129: 6989: 6966: 6937: 6916:finite differences 6879: 6820: 6771: 6725: 6673: 6550: 6387: 6301: 6183: 6145: 6109: 6059: 6001: 5927: 5854: 5827: 5799: 5764: 5732: 5699: 5620: 5538: 5503: 5468: 5327:Other improvements 5318:Bumpless operation 5284:Setpoint weighting 5214: 5190: 5148: 5109: 5033: 4977: 4954: 4896: 4689: 4557: 4435: 4385: 4331: 4329: 4290: 4263: 4236: 4201: 4148: 4100: 4057: 4009: 3963: 3923: 3895: 3867: 3842: 3829: 3802: 3775: 3748: 3721: 3665: 3624: 3579: 3534: 3488: 3486: 3455: 3428: 3397: 3370: 3343: 3316: 3289: 3262: 3235: 3102: 3045: 2998: 2941: 2891: 2867: 2838: 2806: 2704:marginal stability 2691:, with or without 2628: 2544: 2487: 2450: 2400: 2358:steady-state error 2352:Steady-state error 2331: 2271: 2226: 2203: 2120:Equivalently, the 2106: 2086: 2064: 2038: 1976: 1947: 1918: 1885: 1806: 1716: 1458: 1450: 1411:nozzle and flapper 1399: 1389:Industrial control 1364:steady-state error 1291:'s self-designed " 1274:Christiaan Huygens 1261: 1253: 1219:§ Limitations 1180: 1093: 1011: 969: 932: 905: 878: 841: 799: 772: 726: 699: 672: 643: 558: 427: 379: 335: 276: 236: 196: 167: 67: 11177:PID Without a PhD 11117:978-1-84816-242-6 11034:978-0-470-97587-9 11013:978-1-85233-138-2 10994:978-0-8019-8242-2 10876:978-0-470-97587-9 10717:Cooper, Douglas. 10696:Cooper, Douglas. 10564:978-0-13-353640-9 10282:. EE Times-India. 10159:Neuhaus, Rudolf. 10125:978-0-86341-047-5 10059:Minorsky, Nicolas 9990:978-0-275-99032-9 9923:10.1109/37.506394 9771: 9504: 9452: 9380: 9255: 9217: 9170: 9110: 9003: 8985: 8958: 8764: 8730: 8681: 8656: 8483: 8426: 8356: 8329: 8181: 8124: 8047: 7914: 7903: 7833: 7724: 7650: 7566: 7532: 7501: 7442: 7368: 7283: 7217: 7160: 7117: 7058: 7024: 7004:Newton's notation 6877: 6818: 6635: 6548: 6465: 6410:Laplace transform 6365: 6299: 6157:proportional band 5987: 5680: 5618: 5548:terms, yielding: 5440: 5439: 5432: 5217:{\displaystyle G} 5175: 5107: 4678: 4675: 4628: 4540: 4527: 4499: 4355:bang-bang control 4351:Karl Johan Åström 4328: 4214: 4213: 3679: 3678: 3211: 3210: 2894:{\displaystyle s} 2801: 2623: 2593: 2580: 2447: 2434: 2313: 2300: 2240:Proportional term 2229:{\displaystyle s} 2194: 2173: 2160: 2122:transfer function 2109:{\displaystyle t} 2067:{\displaystyle t} 1973: 1944: 1915: 1880: 1850: 1803: 1778: 1681:Controller theory 1616:critically damped 1591:marginally stable 1462:computer software 1403:negative feedback 1328:Whitehead torpedo 1213:control stability 1173: 1139: 1091: 1088: 1053: 1008: 993: 966: 956: 929: 902: 875: 865: 838: 823: 796: 769: 723: 696: 669: 638: 604: 555: 530: 497:Mathematical form 259: 219: 149:) is the desired 11299: 11258: 11252: 11244: 11242: 11241: 11235: 11229:. Archived from 11228: 11218: 11121: 11109: 11097: 11095: 11094: 11077: 11075: 11074: 11069:on March 7, 2007 11068: 11062:. Archived from 11061: 11055:Sellers, David. 11051: 11038: 11017: 10998: 10970: 10946: 10945: 10943: 10941: 10930: 10924: 10923: 10921: 10910: 10904: 10903: 10901: 10900: 10894: 10887: 10881: 10880: 10862: 10856: 10851: 10845: 10840: 10834: 10833: 10831: 10821: 10797: 10791: 10790: 10768: 10762: 10761: 10759: 10739: 10733: 10732: 10730: 10729: 10714: 10708: 10707: 10705: 10704: 10693: 10687: 10674: 10668: 10667: 10645: 10639: 10638: 10612: 10603: 10597: 10596: 10594: 10592: 10586: 10575: 10569: 10568: 10550: 10544: 10543: 10527: 10521: 10520: 10518: 10509:(2): 1867–1872. 10494: 10488: 10487: 10481: 10476: 10474: 10466: 10464: 10463: 10457: 10451:. Archived from 10450: 10441: 10435: 10434: 10408: 10399: 10393: 10392: 10385: 10376: 10375: 10373: 10364: 10353: 10352: 10342: 10314: 10305: 10304: 10298: 10293: 10291: 10283: 10281: 10272: 10266: 10265: 10258: 10252: 10251: 10233: 10227: 10226: 10217:(1): 2015–2032. 10205: 10199: 10198: 10196: 10189: 10181: 10175: 10174: 10172: 10170: 10165: 10156: 10150: 10149: 10148:, Control Global 10147: 10136: 10130: 10129: 10114:. IET. pp. 10107: 10101: 10100: 10085: 10079: 10078: 10055: 10049: 10048: 10046: 10045: 10036:. Archived from 10030: 10024: 10023: 10010: 10004: 10002: 9995:p. citing 9994: 9976: 9970: 9969: 9959: 9947: 9941: 9940: 9938: 9937: 9931: 9925:. Archived from 9908: 9899: 9882: 9881: 9861: 9855: 9854: 9846: 9840: 9839: 9837: 9836: 9821: 9815: 9814: 9812: 9803: 9786: 9783: 9781: 9780: 9775: 9773: 9772: 9764: 9750: 9744: 9741: 9699: 9693: 9691: 9690: 9685: 9661: 9659: 9658: 9653: 9648: 9643: 9642: 9624: 9618: 9616: 9615: 9610: 9608: 9607: 9578:D-to-A converted 9571: 9569: 9568: 9563: 9561: 9560: 9534: 9532: 9531: 9526: 9506: 9503: 9502: 9493: 9492: 9483: 9459: 9455: 9454: 9451: 9450: 9441: 9440: 9439: 9426: 9420: 9419: 9387: 9383: 9382: 9379: 9378: 9369: 9368: 9359: 9353: 9352: 9343: 9342: 9330: 9329: 9266: 9264: 9263: 9258: 9256: 9254: 9253: 9252: 9233: 9232: 9231: 9219: 9216: 9215: 9206: 9205: 9196: 9190: 9189: 9177: 9173: 9172: 9169: 9168: 9159: 9158: 9157: 9144: 9138: 9137: 9117: 9113: 9112: 9109: 9108: 9099: 9098: 9089: 9083: 9082: 9073: 9072: 9060: 9059: 9044: 9014: 9012: 9011: 9006: 9004: 8999: 8988: 8986: 8984: 8983: 8974: 8973: 8964: 8959: 8957: 8943: 8941: 8940: 8931: 8930: 8918: 8917: 8883: 8881: 8880: 8875: 8873: 8872: 8815: 8813: 8812: 8807: 8805: 8804: 8788: 8786: 8785: 8780: 8778: 8777: 8765: 8763: 8759: 8753: 8740: 8734: 8732: 8731: 8728: 8716: 8697: 8692: 8683: 8682: 8679: 8658: 8657: 8654: 8624: 8622: 8621: 8616: 8614: 8613: 8604: 8599: 8598: 8586: 8585: 8573: 8572: 8563: 8558: 8557: 8545: 8544: 8525: 8523: 8522: 8517: 8515: 8511: 8507: 8506: 8485: 8482: 8474: 8473: 8464: 8455: 8454: 8433: 8429: 8428: 8425: 8417: 8416: 8415: 8402: 8379: 8378: 8363: 8359: 8358: 8355: 8347: 8346: 8337: 8331: 8328: 8327: 8318: 8310: 8291: 8290: 8275: 8274: 8247: 8246: 8218: 8216: 8215: 8210: 8205: 8204: 8183: 8180: 8172: 8171: 8162: 8153: 8152: 8131: 8127: 8126: 8123: 8115: 8114: 8113: 8100: 8094: 8093: 8070: 8069: 8054: 8050: 8049: 8046: 8038: 8037: 8028: 8016: 8015: 8003: 8002: 7982: 7981: 7954: 7953: 7925: 7923: 7922: 7917: 7915: 7913: 7905: 7904: 7902: 7894: 7890: 7889: 7862: 7861: 7839: 7834: 7832: 7824: 7820: 7819: 7792: 7791: 7775: 7772: 7770: 7769: 7754: 7753: 7738: 7737: 7725: 7723: 7715: 7711: 7710: 7683: 7682: 7666: 7664: 7663: 7651: 7649: 7641: 7637: 7636: 7609: 7608: 7592: 7577: 7575: 7574: 7569: 7567: 7565: 7557: 7553: 7552: 7534: 7533: 7525: 7516: 7515: 7503: 7502: 7494: 7490: 7488: 7487: 7472: 7471: 7456: 7455: 7443: 7441: 7433: 7429: 7428: 7401: 7400: 7384: 7382: 7381: 7369: 7367: 7359: 7355: 7354: 7327: 7326: 7310: 7295: 7293: 7292: 7287: 7285: 7282: 7274: 7270: 7269: 7242: 7241: 7225: 7219: 7216: 7208: 7204: 7203: 7184: 7175: 7174: 7162: 7161: 7153: 7138: 7136: 7135: 7130: 7119: 7118: 7110: 7107: 7106: 7082: 7081: 7060: 7059: 7051: 7048: 7047: 7026: 7025: 7017: 6998: 6996: 6995: 6990: 6975: 6973: 6972: 6967: 6946: 6944: 6943: 6938: 6888: 6886: 6885: 6880: 6878: 6876: 6875: 6866: 6865: 6856: 6829: 6827: 6826: 6821: 6819: 6814: 6813: 6804: 6799: 6798: 6780: 6778: 6777: 6772: 6764: 6763: 6751: 6750: 6734: 6732: 6731: 6726: 6718: 6717: 6705: 6704: 6682: 6680: 6679: 6674: 6663: 6658: 6657: 6636: 6634: 6633: 6628: 6627: 6614: 6609: 6608: 6559: 6557: 6556: 6551: 6549: 6544: 6543: 6542: 6530: 6525: 6524: 6512: 6511: 6510: 6500: 6499: 6489: 6484: 6479: 6478: 6466: 6461: 6460: 6451: 6446: 6445: 6396: 6394: 6393: 6388: 6386: 6382: 6366: 6364: 6353: 6351: 6350: 6338: 6329: 6314: 6309: 6300: 6298: 6297: 6285: 6280: 6254: 6253: 6241: 6192: 6190: 6189: 6184: 6182: 6181: 6172: 6154: 6152: 6151: 6146: 6144: 6143: 6118: 6116: 6115: 6110: 6108: 6107: 6098: 6097: 6085: 6084: 6068: 6066: 6065: 6060: 6058: 6057: 6048: 6043: 6042: 6030: 6029: 6010: 6008: 6007: 6002: 5988: 5986: 5975: 5973: 5972: 5940: 5935: 5926: 5925: 5901: 5900: 5863: 5861: 5860: 5855: 5853: 5852: 5836: 5834: 5833: 5828: 5826: 5825: 5808: 5806: 5805: 5800: 5798: 5797: 5773: 5771: 5770: 5765: 5763: 5762: 5741: 5739: 5738: 5733: 5731: 5730: 5708: 5706: 5705: 5700: 5698: 5694: 5681: 5679: 5668: 5666: 5665: 5633: 5628: 5619: 5617: 5616: 5604: 5582: 5581: 5547: 5545: 5544: 5539: 5537: 5536: 5535: 5512: 5510: 5509: 5504: 5502: 5501: 5500: 5477: 5475: 5474: 5469: 5467: 5466: 5435: 5428: 5424: 5421: 5415: 5384: 5376: 5341:fractional order 5267:Setpoint ramping 5223: 5221: 5220: 5215: 5199: 5197: 5196: 5191: 5189: 5188: 5176: 5168: 5157: 5155: 5154: 5149: 5147: 5146: 5118: 5116: 5115: 5110: 5108: 5106: 5098: 5075: 5042: 5040: 5039: 5034: 4986: 4984: 4983: 4978: 4963: 4961: 4960: 4955: 4940: 4939: 4924: 4923: 4698: 4696: 4695: 4690: 4685: 4681: 4680: 4679: 4677: 4676: 4673: 4667: 4653: 4630: 4629: 4626: 4566: 4564: 4563: 4558: 4541: 4539: 4529: 4528: 4525: 4518: 4517: 4516: 4501: 4500: 4497: 4490: 4452: 4448: 4444: 4442: 4441: 4436: 4425: 4411: 4410: 4394: 4392: 4391: 4386: 4384: 4383: 4340: 4338: 4337: 4332: 4330: 4321: 4299: 4297: 4296: 4291: 4289: 4288: 4272: 4270: 4269: 4264: 4262: 4261: 4245: 4243: 4242: 4237: 4235: 4234: 4210: 4208: 4207: 4202: 4197: 4192: 4191: 4190: 4180: 4179: 4178: 4157: 4155: 4154: 4149: 4147: 4146: 4137: 4132: 4131: 4130: 4109: 4107: 4106: 4101: 4099: 4098: 4097: 4066: 4064: 4063: 4058: 4056: 4055: 4046: 4041: 4040: 4039: 4018: 4016: 4015: 4010: 4008: 4007: 4006: 3972: 3970: 3969: 3964: 3962: 3961: 3960: 3932: 3930: 3929: 3924: 3922: 3921: 3904: 3902: 3901: 3896: 3894: 3893: 3876: 3874: 3873: 3868: 3866: 3865: 3845: 3841: 3838: 3836: 3835: 3830: 3828: 3827: 3811: 3809: 3808: 3803: 3801: 3800: 3784: 3782: 3781: 3776: 3774: 3773: 3757: 3755: 3754: 3749: 3747: 3746: 3730: 3728: 3727: 3722: 3720: 3719: 3696:, introduced by 3674: 3672: 3671: 3666: 3664: 3663: 3633: 3631: 3630: 3625: 3623: 3622: 3588: 3586: 3585: 3580: 3578: 3577: 3543: 3541: 3540: 3535: 3533: 3532: 3495: 3487: 3464: 3462: 3461: 3456: 3454: 3453: 3437: 3435: 3434: 3429: 3427: 3426: 3406: 3404: 3403: 3398: 3396: 3395: 3379: 3377: 3376: 3371: 3369: 3368: 3352: 3350: 3349: 3344: 3342: 3341: 3325: 3323: 3322: 3317: 3315: 3314: 3298: 3296: 3295: 3290: 3288: 3287: 3271: 3269: 3268: 3263: 3261: 3260: 3244: 3242: 3241: 3236: 3234: 3233: 3105: 3101: 3076:command tracking 3063:Optimal behavior 3054: 3052: 3051: 3046: 3007: 3005: 3004: 2999: 2991: 2962: 2950: 2948: 2947: 2942: 2900: 2898: 2897: 2892: 2876: 2874: 2873: 2868: 2847: 2845: 2844: 2839: 2815: 2813: 2812: 2807: 2802: 2800: 2768: 2742: 2637: 2635: 2634: 2629: 2624: 2622: 2614: 2597: 2595: 2594: 2591: 2582: 2581: 2578: 2496: 2494: 2493: 2488: 2463: 2458: 2449: 2448: 2445: 2436: 2435: 2432: 2340: 2338: 2337: 2332: 2315: 2314: 2311: 2302: 2301: 2298: 2235: 2233: 2232: 2227: 2212: 2210: 2209: 2204: 2196: 2195: 2192: 2180: 2175: 2174: 2171: 2162: 2161: 2158: 2115: 2113: 2112: 2107: 2095: 2093: 2092: 2087: 2073: 2071: 2070: 2065: 2047: 2045: 2044: 2039: 2028: 2017: 1985: 1983: 1982: 1977: 1975: 1974: 1971: 1956: 1954: 1953: 1948: 1946: 1945: 1942: 1927: 1925: 1924: 1919: 1917: 1916: 1913: 1894: 1892: 1891: 1886: 1881: 1879: 1871: 1854: 1852: 1851: 1848: 1819: 1814: 1805: 1804: 1801: 1780: 1779: 1776: 1758: 1725: 1723: 1722: 1717: 1545:electric current 1532:process variable 1494:microcontrollers 1379:angular velocity 1356:Nicolas Minorsky 1352:Russian American 1293:conical pendulum 1222: 1189: 1187: 1186: 1181: 1179: 1175: 1174: 1172: 1168: 1162: 1149: 1143: 1141: 1140: 1137: 1125: 1106: 1101: 1092: 1090: 1089: 1086: 1077: 1055: 1054: 1051: 1020: 1018: 1017: 1012: 1010: 1009: 1006: 1000: 995: 994: 991: 978: 976: 975: 970: 968: 967: 964: 958: 957: 954: 941: 939: 938: 933: 931: 930: 927: 914: 912: 911: 906: 904: 903: 900: 887: 885: 884: 879: 877: 876: 873: 867: 866: 863: 850: 848: 847: 842: 840: 839: 836: 830: 825: 824: 821: 808: 806: 805: 800: 798: 797: 794: 781: 779: 778: 773: 771: 770: 767: 735: 733: 732: 727: 725: 724: 721: 708: 706: 705: 700: 698: 697: 694: 681: 679: 678: 673: 671: 670: 667: 652: 650: 649: 644: 639: 637: 633: 627: 614: 608: 606: 605: 602: 590: 571: 566: 557: 556: 553: 532: 531: 528: 436: 434: 433: 428: 388: 386: 385: 380: 359:control variable 344: 342: 341: 336: 285: 283: 282: 277: 260: 257: 248:process variable 245: 243: 242: 237: 220: 217: 205: 203: 202: 197: 151:process variable 83:process variable 76: 74: 73: 68: 11309: 11308: 11302: 11301: 11300: 11298: 11297: 11296: 11282:Control devices 11267: 11266: 11246: 11245: 11239: 11237: 11233: 11226: 11156: 11129: 11124: 11118: 11107: 11092: 11090: 11072: 11070: 11066: 11059: 11035: 11014: 10995: 10978: 10976:Further reading 10973: 10967: 10950: 10949: 10939: 10937: 10932: 10931: 10927: 10919: 10911: 10907: 10898: 10896: 10889: 10888: 10884: 10877: 10863: 10859: 10852: 10848: 10841: 10837: 10798: 10794: 10787: 10769: 10765: 10757:10.1.1.152.9564 10740: 10736: 10727: 10725: 10715: 10711: 10702: 10700: 10694: 10690: 10675: 10671: 10660:Lund university 10646: 10642: 10610: 10604: 10600: 10590: 10588: 10584: 10576: 10572: 10565: 10551: 10547: 10528: 10524: 10495: 10491: 10479: 10477: 10468: 10467: 10461: 10459: 10455: 10448: 10442: 10438: 10406: 10400: 10396: 10387: 10386: 10379: 10371: 10365: 10356: 10340:10.1.1.124.7043 10315: 10308: 10296: 10294: 10285: 10284: 10279: 10273: 10269: 10260: 10259: 10255: 10248: 10234: 10230: 10206: 10202: 10194: 10187: 10183: 10182: 10178: 10168: 10166: 10163: 10157: 10153: 10145: 10137: 10133: 10126: 10108: 10104: 10094: 10086: 10082: 10056: 10052: 10043: 10041: 10032: 10031: 10027: 10012: 10011: 10007: 9996: 9991: 9977: 9973: 9957: 9948: 9944: 9935: 9933: 9929: 9906: 9900: 9885: 9878: 9862: 9858: 9847: 9843: 9834: 9832: 9823: 9822: 9818: 9810: 9804: 9800: 9795: 9790: 9789: 9763: 9762: 9757: 9754: 9753: 9751: 9747: 9742: 9738: 9733: 9716: 9711: 9667: 9664: 9663: 9644: 9638: 9634: 9632: 9629: 9628: 9603: 9599: 9597: 9594: 9593: 9553: 9549: 9547: 9544: 9543: 9540: 9498: 9494: 9488: 9484: 9481: 9446: 9442: 9435: 9431: 9427: 9424: 9415: 9411: 9407: 9403: 9374: 9370: 9364: 9360: 9357: 9348: 9344: 9338: 9334: 9325: 9321: 9320: 9316: 9278: 9275: 9274: 9245: 9241: 9234: 9224: 9220: 9211: 9207: 9201: 9197: 9194: 9182: 9178: 9164: 9160: 9153: 9149: 9145: 9142: 9133: 9129: 9125: 9121: 9104: 9100: 9094: 9090: 9087: 9078: 9074: 9068: 9064: 9055: 9051: 9050: 9046: 9045: 9043: 9026: 9023: 9022: 8989: 8987: 8979: 8975: 8969: 8965: 8963: 8947: 8942: 8936: 8932: 8926: 8922: 8913: 8909: 8892: 8889: 8888: 8868: 8864: 8862: 8859: 8858: 8844: 8823: 8800: 8796: 8794: 8791: 8790: 8773: 8769: 8755: 8754: 8736: 8735: 8733: 8727: 8723: 8712: 8693: 8688: 8678: 8674: 8653: 8649: 8632: 8629: 8628: 8609: 8605: 8600: 8594: 8590: 8581: 8577: 8568: 8564: 8559: 8553: 8549: 8540: 8536: 8534: 8531: 8530: 8496: 8492: 8475: 8469: 8465: 8462: 8444: 8440: 8418: 8411: 8407: 8403: 8400: 8390: 8386: 8374: 8370: 8348: 8342: 8338: 8335: 8323: 8319: 8311: 8308: 8301: 8297: 8296: 8292: 8286: 8282: 8264: 8260: 8242: 8238: 8230: 8227: 8226: 8194: 8190: 8173: 8167: 8163: 8160: 8142: 8138: 8116: 8109: 8105: 8101: 8098: 8089: 8085: 8081: 8077: 8065: 8061: 8039: 8033: 8029: 8026: 8011: 8007: 7998: 7994: 7993: 7989: 7971: 7967: 7949: 7945: 7937: 7934: 7933: 7906: 7895: 7879: 7875: 7851: 7847: 7840: 7838: 7825: 7809: 7805: 7787: 7783: 7776: 7774: 7773: 7771: 7765: 7761: 7749: 7745: 7733: 7729: 7716: 7700: 7696: 7678: 7674: 7667: 7665: 7659: 7655: 7642: 7626: 7622: 7604: 7600: 7593: 7591: 7589: 7586: 7585: 7558: 7542: 7538: 7524: 7523: 7511: 7507: 7493: 7492: 7491: 7489: 7483: 7479: 7467: 7463: 7451: 7447: 7434: 7418: 7414: 7396: 7392: 7385: 7383: 7377: 7373: 7360: 7344: 7340: 7322: 7318: 7311: 7309: 7307: 7304: 7303: 7275: 7259: 7255: 7237: 7233: 7226: 7223: 7209: 7199: 7195: 7185: 7182: 7170: 7166: 7152: 7151: 7149: 7146: 7145: 7109: 7108: 7102: 7098: 7077: 7073: 7050: 7049: 7043: 7039: 7016: 7015: 7013: 7010: 7009: 6981: 6978: 6977: 6952: 6949: 6948: 6923: 6920: 6919: 6904:microcontroller 6900: 6871: 6867: 6861: 6857: 6855: 6841: 6838: 6837: 6809: 6805: 6803: 6794: 6790: 6788: 6785: 6784: 6759: 6755: 6746: 6742: 6740: 6737: 6736: 6713: 6709: 6700: 6696: 6694: 6691: 6690: 6659: 6653: 6649: 6629: 6623: 6619: 6618: 6613: 6604: 6600: 6583: 6580: 6579: 6569: 6538: 6534: 6526: 6520: 6516: 6506: 6502: 6501: 6495: 6491: 6490: 6488: 6480: 6474: 6470: 6456: 6452: 6450: 6441: 6437: 6420: 6417: 6416: 6406: 6357: 6352: 6346: 6342: 6331: 6316: 6310: 6305: 6293: 6289: 6284: 6261: 6259: 6255: 6249: 6245: 6225: 6223: 6220: 6219: 6208: 6199: 6177: 6173: 6168: 6163: 6160: 6159: 6139: 6135: 6133: 6130: 6129: 6125: 6103: 6099: 6093: 6089: 6080: 6076: 6074: 6071: 6070: 6053: 6049: 6044: 6038: 6034: 6025: 6021: 6019: 6016: 6015: 5979: 5974: 5968: 5964: 5936: 5931: 5921: 5917: 5896: 5892: 5875: 5872: 5871: 5848: 5844: 5842: 5839: 5838: 5821: 5817: 5815: 5812: 5811: 5793: 5789: 5787: 5784: 5783: 5776:derivative time 5758: 5754: 5752: 5749: 5748: 5726: 5722: 5720: 5717: 5716: 5672: 5667: 5661: 5657: 5629: 5624: 5612: 5608: 5603: 5587: 5583: 5577: 5573: 5556: 5553: 5552: 5525: 5524: 5520: 5518: 5515: 5514: 5490: 5489: 5485: 5483: 5480: 5479: 5462: 5458: 5456: 5453: 5452: 5445: 5436: 5425: 5419: 5416: 5401: 5385: 5374: 5349: 5347:Cascade control 5333:gain scheduling 5329: 5320: 5295: 5260: 5238: 5209: 5206: 5205: 5200:= integral gain 5184: 5180: 5167: 5165: 5162: 5161: 5142: 5138: 5130: 5127: 5126: 5099: 5076: 5074: 5066: 5063: 5062: 5007: 5004: 5003: 4972: 4969: 4968: 4935: 4931: 4919: 4915: 4913: 4910: 4909: 4888: 4879: 4860:integral windup 4856: 4854:Integral windup 4850: 4848:Integral windup 4842: 4824:low-pass filter 4816: 4796: 4756: 4733: 4731:Tuning software 4726: 4715: 4672: 4668: 4654: 4652: 4648: 4641: 4637: 4625: 4621: 4604: 4601: 4600: 4583: 4576: 4524: 4520: 4519: 4506: 4502: 4496: 4492: 4491: 4489: 4472: 4469: 4468: 4462: 4450: 4446: 4421: 4406: 4402: 4400: 4397: 4396: 4379: 4375: 4373: 4370: 4369: 4347: 4319: 4317: 4314: 4313: 4306: 4284: 4280: 4278: 4275: 4274: 4257: 4253: 4251: 4248: 4247: 4230: 4226: 4224: 4221: 4220: 4193: 4186: 4182: 4181: 4174: 4170: 4169: 4164: 4161: 4160: 4142: 4138: 4133: 4126: 4122: 4121: 4116: 4113: 4112: 4093: 4089: 4088: 4083: 4080: 4079: 4051: 4047: 4042: 4035: 4031: 4030: 4025: 4022: 4021: 4002: 3998: 3997: 3992: 3989: 3988: 3956: 3952: 3951: 3946: 3943: 3942: 3917: 3913: 3911: 3908: 3907: 3889: 3885: 3883: 3880: 3879: 3861: 3857: 3855: 3852: 3851: 3823: 3819: 3817: 3814: 3813: 3796: 3792: 3790: 3787: 3786: 3769: 3765: 3763: 3760: 3759: 3742: 3738: 3736: 3733: 3732: 3715: 3711: 3709: 3706: 3705: 3698:John G. Ziegler 3690: 3684: 3659: 3655: 3653: 3650: 3649: 3618: 3614: 3612: 3609: 3608: 3573: 3569: 3567: 3564: 3563: 3528: 3524: 3522: 3519: 3518: 3483: 3479: 3475: 3449: 3445: 3443: 3440: 3439: 3422: 3418: 3416: 3413: 3412: 3391: 3387: 3385: 3382: 3381: 3364: 3360: 3358: 3355: 3354: 3337: 3333: 3331: 3328: 3327: 3310: 3306: 3304: 3301: 3300: 3283: 3279: 3277: 3274: 3273: 3256: 3252: 3250: 3247: 3246: 3229: 3225: 3223: 3220: 3219: 3216: 3189:Åström-Hägglund 3133:Ziegler–Nichols 3092: 3065: 3013: 3010: 3009: 2987: 2958: 2956: 2953: 2952: 2906: 2903: 2902: 2886: 2883: 2882: 2853: 2850: 2849: 2824: 2821: 2820: 2769: 2743: 2741: 2724: 2721: 2720: 2685: 2677:gain scheduling 2651: 2615: 2598: 2596: 2590: 2586: 2577: 2573: 2571: 2568: 2567: 2559: 2552: 2541: 2534: 2527: 2515: 2513:Derivative term 2459: 2454: 2444: 2440: 2431: 2427: 2425: 2422: 2421: 2413: 2397: 2390: 2383: 2371: 2354: 2310: 2306: 2297: 2293: 2291: 2288: 2287: 2279: 2268: 2261: 2254: 2242: 2221: 2218: 2217: 2191: 2187: 2176: 2170: 2166: 2157: 2153: 2136: 2133: 2132: 2101: 2098: 2097: 2081: 2078: 2077: 2059: 2056: 2055: 2021: 2010: 1993: 1990: 1989: 1970: 1966: 1964: 1961: 1960: 1941: 1937: 1935: 1932: 1931: 1912: 1908: 1906: 1903: 1902: 1872: 1855: 1853: 1847: 1843: 1815: 1810: 1800: 1796: 1775: 1771: 1751: 1734: 1731: 1730: 1702: 1699: 1698: 1683: 1638: 1629: 1612: 1610:Control damping 1599: 1570: 1557: 1512: 1506: 1478: 1391: 1315:, and in 1895, 1266: 1241: 1209:optimal control 1205: 1196: 1164: 1163: 1145: 1144: 1142: 1136: 1132: 1121: 1102: 1097: 1085: 1081: 1076: 1060: 1056: 1050: 1046: 1029: 1026: 1025: 1005: 1001: 996: 990: 986: 984: 981: 980: 963: 959: 953: 949: 947: 944: 943: 926: 922: 920: 917: 916: 899: 895: 893: 890: 889: 872: 868: 862: 858: 856: 853: 852: 835: 831: 826: 820: 816: 814: 811: 810: 793: 789: 787: 784: 783: 766: 762: 760: 757: 756: 720: 716: 714: 711: 710: 693: 689: 687: 684: 683: 666: 662: 660: 657: 656: 629: 628: 610: 609: 607: 601: 597: 586: 567: 562: 552: 548: 527: 523: 506: 503: 502: 499: 440: 413: 410: 409: 400:In this model: 365: 362: 361: 291: 288: 287: 256: 254: 251: 250: 246:and a measured 216: 214: 211: 210: 182: 179: 178: 131: 97:terms (denoted 53: 50: 49: 17: 12: 11: 5: 11307: 11306: 11295: 11294: 11289: 11284: 11279: 11265: 11264: 11259: 11219: 11204: 11199: 11194: 11189: 11184: 11179: 11174: 11168: 11162: 11155: 11152: 11151: 11150: 11145: 11140: 11135: 11128: 11127:External links 11125: 11123: 11122: 11116: 11098: 11078: 11052: 11039: 11033: 11018: 11012: 10999: 10993: 10979: 10977: 10974: 10972: 10971: 10965: 10951: 10948: 10947: 10925: 10905: 10882: 10875: 10857: 10846: 10835: 10792: 10785: 10763: 10734: 10709: 10688: 10669: 10640: 10598: 10570: 10563: 10545: 10522: 10489: 10480:|journal= 10436: 10417:(4): 559–576. 10394: 10377: 10354: 10325:(3): 783–835. 10306: 10297:|journal= 10267: 10253: 10246: 10228: 10200: 10197:on 2014-05-13. 10176: 10151: 10131: 10124: 10102: 10080: 10069:(2): 280–309. 10050: 10025: 10005: 9989: 9971: 9955:"On Governors" 9951:Maxwell, J. C. 9942: 9883: 9876: 9856: 9841: 9816: 9797: 9796: 9794: 9791: 9788: 9787: 9770: 9767: 9761: 9745: 9735: 9734: 9732: 9729: 9728: 9727: 9722: 9720:Control theory 9715: 9712: 9708: 9683: 9680: 9677: 9674: 9671: 9651: 9647: 9641: 9637: 9606: 9602: 9559: 9556: 9552: 9537: 9536: 9535: 9524: 9521: 9518: 9515: 9512: 9509: 9501: 9497: 9491: 9487: 9480: 9477: 9474: 9471: 9468: 9465: 9462: 9458: 9449: 9445: 9438: 9434: 9430: 9423: 9418: 9414: 9410: 9406: 9402: 9399: 9396: 9393: 9390: 9386: 9377: 9373: 9367: 9363: 9356: 9351: 9347: 9341: 9337: 9333: 9328: 9324: 9319: 9315: 9312: 9309: 9306: 9303: 9300: 9297: 9294: 9291: 9288: 9285: 9282: 9268: 9267: 9251: 9248: 9244: 9240: 9237: 9230: 9227: 9223: 9214: 9210: 9204: 9200: 9193: 9188: 9185: 9181: 9176: 9167: 9163: 9156: 9152: 9148: 9141: 9136: 9132: 9128: 9124: 9120: 9116: 9107: 9103: 9097: 9093: 9086: 9081: 9077: 9071: 9067: 9063: 9058: 9054: 9049: 9042: 9039: 9036: 9033: 9030: 9016: 9015: 9002: 8998: 8995: 8992: 8982: 8978: 8972: 8968: 8962: 8956: 8953: 8950: 8946: 8939: 8935: 8929: 8925: 8921: 8916: 8912: 8908: 8905: 8902: 8899: 8896: 8871: 8867: 8841: 8840: 8839: 8836: 8833: 8830: 8822: 8819: 8803: 8799: 8776: 8772: 8768: 8762: 8758: 8752: 8749: 8746: 8743: 8739: 8726: 8722: 8719: 8715: 8710: 8707: 8704: 8701: 8696: 8691: 8687: 8677: 8673: 8670: 8667: 8664: 8661: 8652: 8648: 8645: 8642: 8639: 8636: 8612: 8608: 8603: 8597: 8593: 8589: 8584: 8580: 8576: 8571: 8567: 8562: 8556: 8552: 8548: 8543: 8539: 8527: 8526: 8514: 8510: 8505: 8502: 8499: 8495: 8491: 8488: 8481: 8478: 8472: 8468: 8461: 8458: 8453: 8450: 8447: 8443: 8439: 8436: 8432: 8424: 8421: 8414: 8410: 8406: 8399: 8396: 8393: 8389: 8385: 8382: 8377: 8373: 8369: 8366: 8362: 8354: 8351: 8345: 8341: 8334: 8326: 8322: 8317: 8314: 8307: 8304: 8300: 8295: 8289: 8285: 8281: 8278: 8273: 8270: 8267: 8263: 8259: 8256: 8253: 8250: 8245: 8241: 8237: 8234: 8220: 8219: 8208: 8203: 8200: 8197: 8193: 8189: 8186: 8179: 8176: 8170: 8166: 8159: 8156: 8151: 8148: 8145: 8141: 8137: 8134: 8130: 8122: 8119: 8112: 8108: 8104: 8097: 8092: 8088: 8084: 8080: 8076: 8073: 8068: 8064: 8060: 8057: 8053: 8045: 8042: 8036: 8032: 8025: 8022: 8019: 8014: 8010: 8006: 8001: 7997: 7992: 7988: 7985: 7980: 7977: 7974: 7970: 7966: 7963: 7960: 7957: 7952: 7948: 7944: 7941: 7927: 7926: 7912: 7909: 7901: 7898: 7893: 7888: 7885: 7882: 7878: 7874: 7871: 7868: 7865: 7860: 7857: 7854: 7850: 7846: 7843: 7837: 7831: 7828: 7823: 7818: 7815: 7812: 7808: 7804: 7801: 7798: 7795: 7790: 7786: 7782: 7779: 7768: 7764: 7760: 7757: 7752: 7748: 7744: 7741: 7736: 7732: 7728: 7722: 7719: 7714: 7709: 7706: 7703: 7699: 7695: 7692: 7689: 7686: 7681: 7677: 7673: 7670: 7662: 7658: 7654: 7648: 7645: 7640: 7635: 7632: 7629: 7625: 7621: 7618: 7615: 7612: 7607: 7603: 7599: 7596: 7579: 7578: 7564: 7561: 7556: 7551: 7548: 7545: 7541: 7537: 7531: 7528: 7522: 7519: 7514: 7510: 7506: 7500: 7497: 7486: 7482: 7478: 7475: 7470: 7466: 7462: 7459: 7454: 7450: 7446: 7440: 7437: 7432: 7427: 7424: 7421: 7417: 7413: 7410: 7407: 7404: 7399: 7395: 7391: 7388: 7380: 7376: 7372: 7366: 7363: 7358: 7353: 7350: 7347: 7343: 7339: 7336: 7333: 7330: 7325: 7321: 7317: 7314: 7297: 7296: 7281: 7278: 7273: 7268: 7265: 7262: 7258: 7254: 7251: 7248: 7245: 7240: 7236: 7232: 7229: 7222: 7215: 7212: 7207: 7202: 7198: 7194: 7191: 7188: 7181: 7178: 7173: 7169: 7165: 7159: 7156: 7128: 7125: 7122: 7116: 7113: 7105: 7101: 7097: 7094: 7091: 7088: 7085: 7080: 7076: 7072: 7069: 7066: 7063: 7057: 7054: 7046: 7042: 7038: 7035: 7032: 7029: 7023: 7020: 6988: 6985: 6965: 6962: 6959: 6956: 6936: 6933: 6930: 6927: 6899: 6896: 6891: 6890: 6874: 6870: 6864: 6860: 6854: 6851: 6848: 6845: 6831: 6830: 6817: 6812: 6808: 6802: 6797: 6793: 6782: 6770: 6767: 6762: 6758: 6754: 6749: 6745: 6724: 6721: 6716: 6712: 6708: 6703: 6699: 6684: 6683: 6672: 6669: 6666: 6662: 6656: 6652: 6648: 6645: 6642: 6639: 6632: 6626: 6622: 6617: 6612: 6607: 6603: 6599: 6596: 6593: 6590: 6587: 6568: 6565: 6561: 6560: 6547: 6541: 6537: 6533: 6529: 6523: 6519: 6515: 6509: 6505: 6498: 6494: 6487: 6483: 6477: 6473: 6469: 6464: 6459: 6455: 6449: 6444: 6440: 6436: 6433: 6430: 6427: 6424: 6405: 6402: 6398: 6397: 6385: 6381: 6378: 6375: 6372: 6369: 6363: 6360: 6356: 6349: 6345: 6341: 6337: 6334: 6328: 6325: 6322: 6319: 6313: 6308: 6304: 6296: 6292: 6288: 6283: 6279: 6276: 6273: 6270: 6267: 6264: 6258: 6252: 6248: 6244: 6240: 6237: 6234: 6231: 6228: 6207: 6204: 6198: 6195: 6180: 6176: 6171: 6167: 6142: 6138: 6124: 6121: 6106: 6102: 6096: 6092: 6088: 6083: 6079: 6056: 6052: 6047: 6041: 6037: 6033: 6028: 6024: 6012: 6011: 6000: 5997: 5994: 5991: 5985: 5982: 5978: 5971: 5967: 5963: 5960: 5957: 5953: 5950: 5947: 5944: 5939: 5934: 5930: 5924: 5920: 5916: 5913: 5910: 5907: 5904: 5899: 5895: 5891: 5888: 5885: 5882: 5879: 5851: 5847: 5824: 5820: 5796: 5792: 5779: 5778: 5761: 5757: 5746: 5729: 5725: 5710: 5709: 5697: 5693: 5690: 5687: 5684: 5678: 5675: 5671: 5664: 5660: 5656: 5653: 5650: 5646: 5643: 5640: 5637: 5632: 5627: 5623: 5615: 5611: 5607: 5602: 5599: 5596: 5593: 5590: 5586: 5580: 5576: 5572: 5569: 5566: 5563: 5560: 5534: 5531: 5528: 5523: 5499: 5496: 5493: 5488: 5465: 5461: 5444: 5441: 5438: 5437: 5388: 5386: 5379: 5373: 5370: 5364:of the system 5348: 5345: 5328: 5325: 5319: 5316: 5294: 5291: 5290: 5289: 5285: 5282: 5278: 5275: 5268: 5259: 5256: 5237: 5234: 5213: 5202: 5201: 5187: 5183: 5179: 5174: 5171: 5159: 5145: 5141: 5137: 5134: 5120: 5119: 5105: 5102: 5097: 5094: 5091: 5088: 5085: 5082: 5079: 5073: 5070: 5044: 5043: 5032: 5029: 5026: 5023: 5020: 5017: 5014: 5011: 4976: 4965: 4964: 4953: 4950: 4946: 4943: 4938: 4934: 4930: 4927: 4922: 4918: 4887: 4884: 4878: 4875: 4874: 4873: 4870: 4867: 4849: 4846: 4841: 4838: 4815: 4812: 4795: 4792: 4755: 4752: 4732: 4729: 4724: 4713: 4700: 4699: 4688: 4684: 4671: 4666: 4663: 4660: 4657: 4651: 4647: 4644: 4640: 4636: 4633: 4624: 4620: 4617: 4614: 4611: 4608: 4581: 4574: 4568: 4567: 4556: 4553: 4550: 4547: 4544: 4538: 4535: 4532: 4523: 4515: 4512: 4509: 4505: 4495: 4488: 4485: 4482: 4479: 4476: 4461: 4458: 4434: 4431: 4428: 4424: 4420: 4417: 4414: 4409: 4405: 4382: 4378: 4346: 4343: 4327: 4324: 4305: 4302: 4287: 4283: 4260: 4256: 4233: 4229: 4212: 4211: 4200: 4196: 4189: 4185: 4177: 4173: 4168: 4158: 4145: 4141: 4136: 4129: 4125: 4120: 4110: 4096: 4092: 4087: 4077: 4071: 4070: 4067: 4054: 4050: 4045: 4038: 4034: 4029: 4019: 4005: 4001: 3996: 3986: 3980: 3979: 3976: 3973: 3959: 3955: 3950: 3940: 3934: 3933: 3920: 3916: 3905: 3892: 3888: 3877: 3864: 3860: 3849: 3826: 3822: 3799: 3795: 3772: 3768: 3745: 3741: 3718: 3714: 3683: 3680: 3677: 3676: 3662: 3658: 3646: 3643: 3640: 3637: 3634: 3621: 3617: 3605: 3604: 3601: 3598: 3595: 3592: 3589: 3576: 3572: 3560: 3559: 3556: 3553: 3550: 3547: 3544: 3531: 3527: 3515: 3514: 3511: 3508: 3507:Settling time 3505: 3502: 3499: 3481: 3477: 3473: 3452: 3448: 3425: 3421: 3394: 3390: 3367: 3363: 3340: 3336: 3313: 3309: 3286: 3282: 3259: 3255: 3232: 3228: 3215: 3212: 3209: 3208: 3205: 3202: 3198: 3197: 3194: 3191: 3185: 3184: 3181: 3178: 3172: 3171: 3168: 3159: 3157:Software tools 3153: 3152: 3149: 3146: 3145:Tyreus Luyben 3142: 3141: 3138: 3135: 3129: 3128: 3125: 3122: 3116: 3115: 3114:Disadvantages 3112: 3109: 3091: 3088: 3064: 3061: 3044: 3041: 3038: 3035: 3032: 3029: 3026: 3023: 3020: 3017: 2997: 2994: 2990: 2986: 2983: 2980: 2977: 2974: 2971: 2968: 2965: 2961: 2940: 2937: 2934: 2931: 2928: 2925: 2922: 2919: 2916: 2913: 2910: 2890: 2866: 2863: 2860: 2857: 2837: 2834: 2831: 2828: 2817: 2816: 2805: 2799: 2796: 2793: 2790: 2787: 2784: 2781: 2778: 2775: 2772: 2767: 2764: 2761: 2758: 2755: 2752: 2749: 2746: 2740: 2737: 2734: 2731: 2728: 2711:Laplace domain 2684: 2681: 2650: 2647: 2639: 2638: 2627: 2621: 2618: 2613: 2610: 2607: 2604: 2601: 2589: 2585: 2576: 2557: 2550: 2542:held constant) 2539: 2532: 2525: 2514: 2511: 2498: 2497: 2486: 2483: 2480: 2476: 2473: 2470: 2467: 2462: 2457: 2453: 2443: 2439: 2430: 2411: 2398:held constant) 2395: 2388: 2381: 2370: 2367: 2353: 2350: 2342: 2341: 2330: 2327: 2324: 2321: 2318: 2309: 2305: 2296: 2277: 2269:held constant) 2266: 2259: 2252: 2241: 2238: 2225: 2214: 2213: 2202: 2199: 2190: 2186: 2183: 2179: 2169: 2165: 2156: 2152: 2149: 2146: 2143: 2140: 2126:Laplace domain 2118: 2117: 2105: 2085: 2075: 2063: 2053: 2037: 2034: 2031: 2027: 2024: 2020: 2016: 2013: 2009: 2006: 2003: 2000: 1997: 1987: 1969: 1958: 1940: 1929: 1911: 1896: 1895: 1884: 1878: 1875: 1870: 1867: 1864: 1861: 1858: 1846: 1842: 1839: 1836: 1832: 1829: 1826: 1823: 1818: 1813: 1809: 1799: 1795: 1792: 1789: 1786: 1783: 1774: 1770: 1767: 1764: 1761: 1757: 1754: 1750: 1747: 1744: 1741: 1738: 1715: 1712: 1709: 1706: 1694: 1693: 1682: 1679: 1663:concentrations 1637: 1634: 1628: 1625: 1611: 1608: 1598: 1595: 1569: 1566: 1556: 1553: 1549: 1548: 1541: 1538: 1535: 1520:electric motor 1510:Control system 1505: 1502: 1477: 1474: 1390: 1387: 1265: 1262: 1240: 1237: 1204: 1201: 1195: 1192: 1191: 1190: 1178: 1171: 1167: 1161: 1158: 1155: 1152: 1148: 1135: 1131: 1128: 1124: 1119: 1116: 1113: 1110: 1105: 1100: 1096: 1084: 1080: 1075: 1072: 1069: 1066: 1063: 1059: 1049: 1045: 1042: 1039: 1036: 1033: 1004: 999: 989: 962: 952: 925: 898: 871: 861: 834: 829: 819: 792: 765: 719: 692: 665: 642: 636: 632: 626: 623: 620: 617: 613: 600: 596: 593: 589: 584: 581: 578: 575: 570: 565: 561: 551: 547: 544: 541: 538: 535: 526: 522: 519: 516: 513: 510: 498: 495: 477:Control action 463: 462: 454: 442: 438: 426: 423: 420: 417: 378: 375: 372: 369: 334: 331: 328: 325: 322: 319: 316: 313: 310: 307: 304: 301: 298: 295: 275: 272: 269: 266: 263: 235: 232: 229: 226: 223: 195: 192: 189: 186: 130: 127: 114:cruise control 66: 63: 60: 57: 27:PID controller 15: 9: 6: 4: 3: 2: 11305: 11304: 11293: 11290: 11288: 11285: 11283: 11280: 11278: 11275: 11274: 11272: 11263: 11260: 11256: 11250: 11236:on 2015-04-21 11232: 11225: 11220: 11217:on 2010-02-09 11216: 11212: 11211: 11205: 11203: 11200: 11198: 11195: 11193: 11190: 11188: 11185: 11183: 11180: 11178: 11175: 11172: 11169: 11166: 11163: 11161: 11158: 11157: 11154:PID tutorials 11149: 11146: 11144: 11141: 11139: 11136: 11134: 11131: 11130: 11119: 11113: 11106: 11105: 11099: 11088: 11084: 11079: 11065: 11058: 11053: 11049: 11045: 11040: 11036: 11030: 11026: 11025: 11019: 11015: 11009: 11005: 11000: 10996: 10990: 10986: 10981: 10980: 10968: 10966:9789861544779 10962: 10958: 10953: 10952: 10935: 10929: 10918: 10917: 10909: 10893: 10886: 10878: 10872: 10868: 10861: 10854: 10850: 10843: 10839: 10830: 10829:10400.22/4306 10825: 10820: 10815: 10811: 10807: 10803: 10796: 10788: 10786:9781424438839 10782: 10778: 10774: 10767: 10758: 10753: 10750:(2): 74–101. 10749: 10745: 10738: 10724: 10720: 10713: 10699: 10692: 10686: 10682: 10678: 10673: 10665: 10661: 10657: 10653: 10652: 10644: 10636: 10632: 10628: 10624: 10620: 10616: 10609: 10602: 10583: 10582: 10574: 10566: 10560: 10556: 10549: 10541: 10537: 10533: 10526: 10517: 10512: 10508: 10504: 10500: 10493: 10485: 10472: 10458:on 2015-04-21 10454: 10447: 10440: 10432: 10428: 10424: 10420: 10416: 10412: 10405: 10398: 10390: 10384: 10382: 10370: 10363: 10361: 10359: 10350: 10346: 10341: 10336: 10332: 10328: 10324: 10320: 10313: 10311: 10302: 10289: 10278: 10271: 10263: 10257: 10249: 10247:0-8493-1081-4 10243: 10239: 10232: 10224: 10220: 10216: 10212: 10204: 10193: 10186: 10180: 10162: 10155: 10144: 10143: 10135: 10127: 10121: 10117: 10113: 10106: 10098: 10093: 10089: 10084: 10076: 10072: 10068: 10064: 10060: 10054: 10040:on 2011-07-08 10039: 10035: 10029: 10022: 10018: 10017: 10009: 10000: 9992: 9986: 9982: 9975: 9967: 9963: 9956: 9952: 9946: 9932:on 2016-08-09 9928: 9924: 9920: 9916: 9912: 9905: 9898: 9896: 9894: 9892: 9890: 9888: 9879: 9877:9781400874668 9873: 9869: 9868: 9860: 9852: 9845: 9830: 9826: 9820: 9809: 9802: 9798: 9765: 9759: 9749: 9740: 9736: 9726: 9723: 9721: 9718: 9717: 9707: 9705: 9700: 9698: 9681: 9678: 9675: 9672: 9669: 9649: 9645: 9639: 9635: 9625: 9623: 9604: 9600: 9590: 9586: 9584: 9579: 9574: 9557: 9554: 9550: 9519: 9516: 9513: 9507: 9499: 9489: 9485: 9478: 9472: 9469: 9466: 9460: 9456: 9447: 9436: 9432: 9428: 9421: 9416: 9412: 9408: 9404: 9400: 9394: 9388: 9384: 9375: 9365: 9361: 9354: 9349: 9339: 9335: 9331: 9326: 9322: 9317: 9313: 9307: 9304: 9301: 9295: 9292: 9286: 9280: 9273: 9272: 9271: 9249: 9246: 9242: 9238: 9235: 9228: 9225: 9221: 9212: 9202: 9198: 9191: 9186: 9183: 9179: 9174: 9165: 9154: 9150: 9146: 9139: 9134: 9130: 9126: 9122: 9118: 9114: 9105: 9095: 9091: 9084: 9079: 9069: 9065: 9061: 9056: 9052: 9047: 9040: 9034: 9028: 9021: 9020: 9019: 9000: 8996: 8993: 8990: 8980: 8970: 8966: 8960: 8954: 8951: 8948: 8944: 8937: 8927: 8923: 8919: 8914: 8910: 8906: 8900: 8894: 8887: 8886: 8885: 8869: 8856: 8851: 8849: 8837: 8834: 8831: 8828: 8827: 8826: 8818: 8801: 8797: 8774: 8770: 8766: 8760: 8747: 8741: 8724: 8720: 8717: 8705: 8699: 8694: 8689: 8685: 8675: 8671: 8665: 8659: 8650: 8646: 8640: 8634: 8625: 8610: 8606: 8601: 8595: 8591: 8587: 8582: 8578: 8574: 8569: 8565: 8560: 8554: 8550: 8546: 8541: 8537: 8512: 8503: 8500: 8497: 8493: 8486: 8479: 8470: 8466: 8459: 8451: 8448: 8445: 8441: 8434: 8430: 8422: 8412: 8408: 8404: 8397: 8394: 8391: 8387: 8383: 8375: 8371: 8364: 8360: 8352: 8343: 8339: 8332: 8324: 8320: 8315: 8305: 8302: 8298: 8293: 8287: 8283: 8279: 8271: 8268: 8265: 8261: 8254: 8251: 8243: 8239: 8232: 8225: 8224: 8223: 8201: 8198: 8195: 8191: 8184: 8177: 8168: 8164: 8157: 8149: 8146: 8143: 8139: 8132: 8128: 8120: 8110: 8106: 8102: 8095: 8090: 8086: 8082: 8078: 8074: 8066: 8062: 8055: 8051: 8043: 8034: 8030: 8023: 8020: 8012: 8008: 8004: 7999: 7995: 7990: 7986: 7978: 7975: 7972: 7968: 7961: 7958: 7950: 7946: 7939: 7932: 7931: 7930: 7910: 7899: 7886: 7883: 7880: 7876: 7869: 7866: 7858: 7855: 7852: 7848: 7841: 7835: 7829: 7816: 7813: 7810: 7806: 7799: 7796: 7788: 7784: 7777: 7766: 7762: 7758: 7750: 7746: 7739: 7734: 7730: 7726: 7720: 7707: 7704: 7701: 7697: 7690: 7687: 7679: 7675: 7668: 7660: 7656: 7652: 7646: 7633: 7630: 7627: 7623: 7616: 7613: 7605: 7601: 7594: 7584: 7583: 7582: 7562: 7549: 7546: 7543: 7539: 7529: 7526: 7520: 7512: 7508: 7498: 7495: 7484: 7480: 7476: 7468: 7464: 7457: 7452: 7448: 7444: 7438: 7425: 7422: 7419: 7415: 7408: 7405: 7397: 7393: 7386: 7378: 7374: 7370: 7364: 7351: 7348: 7345: 7341: 7334: 7331: 7323: 7319: 7312: 7302: 7301: 7300: 7279: 7266: 7263: 7260: 7256: 7249: 7246: 7238: 7234: 7227: 7220: 7213: 7210: 7200: 7196: 7189: 7186: 7179: 7171: 7167: 7157: 7154: 7144: 7143: 7142: 7139: 7123: 7114: 7111: 7103: 7099: 7095: 7089: 7083: 7078: 7074: 7070: 7064: 7055: 7052: 7044: 7040: 7036: 7030: 7021: 7018: 7007: 7005: 7000: 6986: 6960: 6954: 6931: 6925: 6917: 6913: 6909: 6905: 6895: 6872: 6868: 6862: 6858: 6852: 6849: 6846: 6843: 6836: 6835: 6834: 6815: 6810: 6806: 6800: 6795: 6791: 6783: 6768: 6765: 6760: 6756: 6752: 6747: 6743: 6722: 6719: 6714: 6710: 6706: 6701: 6697: 6689: 6688: 6687: 6667: 6664: 6660: 6654: 6650: 6640: 6637: 6630: 6624: 6620: 6615: 6605: 6601: 6597: 6591: 6585: 6578: 6577: 6576: 6574: 6564: 6545: 6539: 6535: 6531: 6527: 6521: 6517: 6513: 6507: 6503: 6496: 6492: 6485: 6481: 6475: 6471: 6467: 6462: 6457: 6453: 6447: 6442: 6438: 6434: 6428: 6422: 6415: 6414: 6413: 6411: 6401: 6383: 6376: 6370: 6367: 6361: 6358: 6354: 6347: 6343: 6339: 6335: 6332: 6323: 6317: 6311: 6306: 6302: 6294: 6290: 6286: 6281: 6274: 6268: 6265: 6262: 6256: 6250: 6246: 6242: 6218: 6217: 6216: 6213: 6203: 6194: 6178: 6174: 6169: 6165: 6158: 6140: 6136: 6120: 6104: 6100: 6094: 6090: 6086: 6081: 6077: 6054: 6050: 6045: 6039: 6035: 6031: 6026: 6022: 5995: 5989: 5983: 5980: 5976: 5969: 5965: 5961: 5958: 5955: 5948: 5942: 5937: 5932: 5928: 5922: 5918: 5914: 5908: 5902: 5897: 5893: 5889: 5883: 5877: 5870: 5869: 5868: 5865: 5849: 5845: 5822: 5818: 5794: 5790: 5777: 5759: 5755: 5747: 5745: 5744:integral time 5727: 5723: 5715: 5714: 5713: 5695: 5688: 5682: 5676: 5673: 5669: 5662: 5658: 5654: 5651: 5648: 5641: 5635: 5630: 5625: 5621: 5613: 5609: 5605: 5600: 5594: 5588: 5584: 5578: 5574: 5570: 5564: 5558: 5551: 5550: 5549: 5521: 5486: 5463: 5459: 5450: 5449:standard form 5434: 5431: 5423: 5420:December 2012 5413: 5409: 5405: 5399: 5398: 5394: 5389:This section 5387: 5383: 5378: 5377: 5369: 5367: 5361: 5358: 5353: 5344: 5342: 5338: 5334: 5324: 5315: 5311: 5308: 5304: 5300: 5286: 5283: 5279: 5276: 5273: 5272:discontinuity 5269: 5266: 5265: 5264: 5255: 5252: 5247: 5243: 5233: 5229: 5225: 5211: 5185: 5181: 5177: 5172: 5169: 5160: 5143: 5139: 5135: 5132: 5125: 5124: 5123: 5103: 5100: 5092: 5089: 5086: 5083: 5077: 5071: 5068: 5061: 5060: 5059: 5057: 5053: 5049: 5030: 5027: 5024: 5021: 5018: 5015: 5012: 5002: 5001: 5000: 4998: 4997: 4992: 4991: 4951: 4948: 4941: 4936: 4932: 4928: 4920: 4916: 4908: 4907: 4906: 4903: 4901: 4900:PI controller 4892: 4886:PI controller 4883: 4871: 4868: 4865: 4864: 4863: 4861: 4855: 4845: 4837: 4835: 4834:PI controller 4830: 4829:median filter 4825: 4821: 4811: 4808: 4803: 4799: 4791: 4788: 4783: 4780: 4775: 4773: 4769: 4765: 4763: 4751: 4747: 4744: 4741: 4737: 4728: 4723: 4719: 4712: 4707: 4703: 4686: 4682: 4669: 4664: 4661: 4658: 4655: 4649: 4645: 4642: 4638: 4634: 4622: 4618: 4612: 4606: 4599: 4598: 4597: 4595: 4591: 4587: 4580: 4573: 4554: 4548: 4542: 4536: 4533: 4530: 4521: 4513: 4510: 4507: 4503: 4493: 4486: 4480: 4474: 4467: 4466: 4465: 4457: 4454: 4432: 4429: 4426: 4422: 4418: 4415: 4412: 4407: 4403: 4380: 4376: 4366: 4362: 4360: 4356: 4352: 4342: 4325: 4322: 4311: 4301: 4285: 4281: 4258: 4254: 4231: 4227: 4217: 4198: 4194: 4187: 4183: 4175: 4171: 4166: 4159: 4143: 4139: 4134: 4127: 4123: 4118: 4111: 4094: 4090: 4085: 4078: 4076: 4073: 4072: 4068: 4052: 4048: 4043: 4036: 4032: 4027: 4020: 4003: 3999: 3994: 3987: 3985: 3982: 3981: 3977: 3974: 3957: 3953: 3948: 3941: 3939: 3936: 3935: 3918: 3914: 3906: 3890: 3886: 3878: 3862: 3858: 3850: 3848:Control type 3847: 3846: 3840: 3824: 3820: 3797: 3793: 3770: 3766: 3743: 3739: 3716: 3712: 3703: 3699: 3695: 3689: 3660: 3656: 3647: 3644: 3641: 3638: 3636:Minor change 3635: 3619: 3615: 3607: 3606: 3602: 3599: 3596: 3593: 3590: 3574: 3570: 3562: 3561: 3557: 3554: 3552:Small change 3551: 3548: 3545: 3529: 3525: 3517: 3516: 3512: 3509: 3506: 3503: 3500: 3497: 3496: 3492: 3470: 3466: 3450: 3446: 3423: 3419: 3410: 3392: 3388: 3365: 3361: 3338: 3334: 3311: 3307: 3284: 3280: 3257: 3253: 3230: 3226: 3214:Manual tuning 3206: 3203: 3200: 3199: 3195: 3192: 3190: 3187: 3186: 3182: 3179: 3177: 3174: 3173: 3169: 3166: 3165: 3160: 3158: 3155: 3154: 3150: 3147: 3144: 3143: 3139: 3136: 3134: 3131: 3130: 3126: 3123: 3121: 3120:Manual tuning 3118: 3117: 3113: 3110: 3107: 3106: 3100: 3096: 3087: 3085: 3084:settling time 3081: 3077: 3073: 3068: 3060: 3058: 3042: 3039: 3033: 3027: 3021: 3015: 2995: 2992: 2981: 2975: 2969: 2963: 2938: 2935: 2932: 2926: 2920: 2914: 2908: 2888: 2880: 2861: 2855: 2832: 2826: 2803: 2794: 2788: 2782: 2776: 2773: 2770: 2762: 2756: 2750: 2744: 2738: 2732: 2726: 2719: 2718: 2717: 2714: 2712: 2707: 2705: 2700: 2698: 2694: 2690: 2680: 2678: 2674: 2669: 2665: 2663: 2658: 2655: 2646: 2644: 2625: 2619: 2616: 2608: 2602: 2599: 2587: 2583: 2574: 2566: 2565: 2564: 2561: 2556: 2549: 2538: 2531: 2524: 2519: 2510: 2508: 2504: 2484: 2481: 2478: 2471: 2465: 2460: 2455: 2451: 2441: 2437: 2428: 2420: 2419: 2418: 2415: 2410: 2405: 2394: 2387: 2380: 2375: 2369:Integral term 2366: 2364: 2359: 2349: 2347: 2328: 2322: 2316: 2307: 2303: 2294: 2286: 2285: 2284: 2281: 2276: 2265: 2258: 2251: 2246: 2237: 2223: 2200: 2197: 2188: 2184: 2181: 2177: 2167: 2163: 2154: 2150: 2144: 2138: 2131: 2130: 2129: 2127: 2123: 2103: 2083: 2076: 2061: 2054: 2051: 2032: 2018: 2007: 2001: 1995: 1988: 1967: 1959: 1938: 1930: 1909: 1901: 1900: 1899: 1882: 1876: 1873: 1865: 1859: 1856: 1844: 1840: 1837: 1834: 1827: 1821: 1816: 1811: 1807: 1797: 1793: 1787: 1781: 1772: 1768: 1762: 1748: 1742: 1736: 1729: 1728: 1727: 1710: 1704: 1692: 1690: 1685: 1684: 1678: 1676: 1672: 1668: 1664: 1660: 1656: 1652: 1648: 1644: 1633: 1624: 1621: 1617: 1607: 1604: 1594: 1592: 1588: 1584: 1578: 1575: 1565: 1562: 1552: 1546: 1542: 1539: 1536: 1533: 1529: 1528: 1527: 1525: 1524:inertial mass 1521: 1517: 1511: 1501: 1499: 1495: 1491: 1487: 1483: 1473: 1471: 1467: 1463: 1454: 1446: 1442: 1440: 1435: 1431: 1429: 1425: 1419: 1417: 1412: 1408: 1404: 1395: 1386: 1383: 1381: 1380: 1375: 1374: 1367: 1365: 1361: 1357: 1353: 1348: 1346: 1341: 1338: 1334: 1329: 1324: 1322: 1321:Willard Gibbs 1318: 1317:Adolf Hurwitz 1314: 1313:Charles Sturm 1310: 1306: 1302: 1296: 1294: 1290: 1285: 1283: 1279: 1275: 1271: 1257: 1250: 1245: 1236: 1234: 1230: 1226: 1220: 1214: 1210: 1203:Applicability 1200: 1176: 1169: 1156: 1150: 1133: 1129: 1126: 1114: 1108: 1103: 1098: 1094: 1082: 1078: 1073: 1067: 1061: 1057: 1047: 1043: 1037: 1031: 1024: 1023: 1022: 1002: 997: 987: 960: 950: 923: 896: 869: 859: 832: 827: 817: 790: 763: 754: 753:standard form 749: 747: 743: 739: 717: 690: 663: 653: 640: 634: 621: 615: 598: 594: 591: 579: 573: 568: 563: 559: 549: 545: 539: 533: 524: 520: 514: 508: 494: 492: 487: 482: 478: 474: 471: 467: 459: 455: 451: 447: 443: 421: 415: 407: 403: 402: 401: 398: 396: 392: 391:control valve 373: 367: 360: 356: 352: 348: 329: 323: 320: 314: 308: 305: 299: 293: 270: 264: 261: 249: 230: 224: 221: 209: 190: 184: 177: 172: 171:control terms 164: 160: 156: 152: 148: 144: 140: 139:block diagram 135: 126: 124: 118: 115: 110: 108: 104: 100: 96: 92: 88: 84: 80: 61: 55: 48: 44: 40: 36: 32: 28: 24: 19: 11238:. Retrieved 11231:the original 11215:the original 11209: 11103: 11091:. Retrieved 11087:the original 11071:. Retrieved 11064:the original 11047: 11023: 11003: 10984: 10956: 10938:. Retrieved 10928: 10915: 10908: 10897:. Retrieved 10895:. Scribd.com 10885: 10866: 10860: 10849: 10838: 10809: 10805: 10795: 10776: 10766: 10747: 10743: 10737: 10726:. Retrieved 10722: 10712: 10701:. Retrieved 10691: 10672: 10650: 10643: 10621:(1): 42–54. 10618: 10614: 10601: 10589:. Retrieved 10580: 10573: 10554: 10548: 10539: 10535: 10525: 10506: 10502: 10492: 10471:cite journal 10460:. Retrieved 10453:the original 10439: 10414: 10410: 10397: 10322: 10318: 10288:cite journal 10270: 10256: 10237: 10231: 10214: 10210: 10203: 10192:the original 10179: 10167:. Retrieved 10154: 10141: 10134: 10111: 10105: 10088:Bennett 1993 10083: 10066: 10062: 10053: 10042:. Retrieved 10038:the original 10028: 10020: 10015: 10008: 9998: 9980: 9974: 9965: 9961: 9945: 9934:. Retrieved 9927:the original 9917:(3): 17–25. 9914: 9910: 9866: 9859: 9850: 9844: 9833:. Retrieved 9831:. 2020-05-19 9828: 9819: 9801: 9748: 9739: 9701: 9626: 9591: 9587: 9575: 9541: 9269: 9017: 8852: 8845: 8824: 8626: 8528: 8221: 7928: 7580: 7298: 7140: 7008: 7001: 6911: 6901: 6892: 6832: 6685: 6572: 6570: 6562: 6407: 6404:Laplace form 6399: 6211: 6209: 6200: 6156: 6126: 6013: 5866: 5780: 5775: 5743: 5711: 5448: 5446: 5426: 5417: 5402:Please help 5390: 5365: 5362: 5357:thermocouple 5354: 5350: 5330: 5321: 5312: 5306: 5303:feed-forward 5296: 5293:Feed-forward 5261: 5239: 5230: 5226: 5203: 5121: 5045: 4995: 4994: 4989: 4988: 4966: 4904: 4899: 4897: 4880: 4857: 4843: 4817: 4807:HVAC systems 4804: 4800: 4797: 4784: 4776: 4767: 4761: 4757: 4748: 4745: 4742: 4738: 4734: 4721: 4717: 4710: 4708: 4704: 4701: 4593: 4589: 4585: 4578: 4571: 4569: 4463: 4455: 4367: 4363: 4348: 4307: 4218: 4215: 4074: 3983: 3937: 3691: 3490: 3217: 3162: 3097: 3093: 3075: 3071: 3069: 3066: 2878: 2818: 2715: 2708: 2701: 2696: 2686: 2673:nonlinearity 2670: 2666: 2659: 2653: 2652: 2640: 2562: 2554: 2547: 2545: 2536: 2529: 2522: 2499: 2416: 2408: 2401: 2392: 2385: 2378: 2357: 2355: 2343: 2282: 2274: 2272: 2263: 2256: 2249: 2215: 2119: 2049: 1897: 1695: 1686: 1639: 1636:Applications 1630: 1613: 1602: 1600: 1579: 1573: 1571: 1561:proportional 1560: 1558: 1555:Proportional 1550: 1513: 1486:power supply 1479: 1459: 1439:current loop 1436: 1432: 1420: 1415: 1407:Harold Black 1400: 1384: 1377: 1372: 1368: 1349: 1345:Elmer Sperry 1342: 1325: 1309:Edward Routh 1305:On Governors 1304: 1297: 1286: 1267: 1206: 1197: 752: 750: 745: 741: 737: 654: 500: 485: 480: 476: 475: 465: 464: 457: 449: 445: 405: 399: 395:weighted sum 358: 347:proportional 175: 170: 168: 162: 158: 146: 142: 119: 111: 106: 102: 98: 87:proportional 46: 35:control loop 30: 26: 22: 20: 18: 10536:Reinvention 8855:Z-transform 6912:discretized 6573:interacting 5337:fuzzy logic 5058:operators: 4754:Limitations 3648:Improve if 3489:Effects of 3111:Advantages 2693:oscillation 2649:Loop tuning 1643:temperature 1516:robotic arm 1514:Consider a 1490:seismometer 1337:instability 1233:oscillation 470:loop tuning 176:error value 123:controllers 121:electronic 47:error value 11271:Categories 11240:2013-12-04 11093:2009-01-05 11073:2007-05-05 10940:4 November 10899:2011-04-04 10728:2014-02-27 10703:2014-02-18 10656:Licentiate 10462:2011-04-04 10044:2011-04-04 9936:2014-08-21 9835:2024-05-31 9793:References 9583:reading in 8821:Pseudocode 5281:weighting. 3600:Eliminate 3513:Stability 3504:Overshoot 3501:Rise time 3498:Parameter 3491:increasing 3409:overdamped 3176:Cohen–Coon 3072:regulation 1603:derivative 1597:Derivative 1482:disk drive 1373:New Mexico 1289:James Watt 1278:millstones 1229:overshoots 355:derivative 157:(SP), and 95:derivative 10752:CiteSeerX 10685:0272-1708 10335:CiteSeerX 9769:¯ 9636:τ 9555:− 9517:− 9508:ϵ 9496:Δ 9470:− 9461:ϵ 9444:Δ 9422:− 9409:− 9389:ϵ 9372:Δ 9346:Δ 9305:− 9247:− 9239:− 9226:− 9209:Δ 9184:− 9162:Δ 9140:− 9127:− 9102:Δ 9076:Δ 8994:− 8977:Δ 8952:− 8934:Δ 8866:Δ 8718:τ 8706:τ 8686:∫ 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1354:engineer 1282:windmills 1127:τ 1115:τ 1095:∫ 592:τ 580:τ 560:∫ 491:fail-safe 321:− 11249:cite web 11173:(pg. 22) 10812:: 1–34. 10658:theis). 10635:18461921 9953:(1868). 9714:See also 8850:filter: 5299:feedback 5251:deadband 5246:backlash 5242:stiction 5236:Deadband 5048:Simulink 4772:observer 4768:constant 3603:Degrade 3558:Degrade 2879:unstable 2689:diverges 2404:integral 1671:position 1647:pressure 1587:sinusoid 1574:integral 1568:Integral 1430:(DCSs). 1360:helmsman 1249:helmsmen 351:integral 208:setpoint 155:setpoint 153:(PV) or 91:integral 79:setpoint 39:feedback 10327:Bibcode 10169:June 8, 10116:142–148 7006:gives: 5774:is the 5742:is the 5412:removed 5397:sources 5056:Laplace 4764:control 4762:optimal 3108:Method 3095:times. 2124:in the 1264:Origins 1239:History 1221:, below 751:In the 486:reverse 33:) is a 11114: 11031: 11010: 10991: 10963: 10873: 10783: 10754: 10683: 10633: 10591:Feb 1, 10561: 10431:921620 10429: 10337: 10244: 10122: 9987: 9874: 9662:where 8789:where 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5689:t 5686:( 5683:e 5677:t 5674:d 5670:d 5663:d 5659:T 5655:+ 5649:d 5645:) 5639:( 5636:e 5631:t 5626:0 5614:i 5610:T 5606:1 5601:+ 5598:) 5595:t 5592:( 5589:e 5585:( 5579:p 5575:K 5571:= 5568:) 5565:t 5562:( 5559:u 5533:t 5530:u 5527:o 5522:D 5498:t 5495:u 5492:o 5487:I 5464:p 5460:K 5433:) 5427:( 5422:) 5418:( 5414:. 5400:. 5212:G 5186:I 5182:K 5178:= 5170:G 5144:P 5140:K 5136:= 5133:G 5104:s 5096:) 5093:s 5087:+ 5084:1 5081:( 5078:G 5072:= 5069:C 5031:. 5028:V 5025:P 5019:P 5016:S 5013:= 4952:t 4949:d 4937:I 4933:K 4929:+ 4921:P 4917:K 4725:p 4722:τ 4718:θ 4714:p 4711:k 4687:, 4683:) 4674:p 4659:t 4650:e 4643:1 4639:( 4635:u 4627:p 4623:k 4619:= 4616:) 4613:t 4610:( 4607:y 4594:s 4592:( 4590:u 4586:θ 4582:p 4579:τ 4575:p 4572:k 4555:, 4552:) 4549:s 4546:( 4543:u 4537:1 4534:+ 4531:s 4526:p 4514:s 4504:e 4498:p 4494:k 4487:= 4484:) 4481:s 4478:( 4475:y 4451:b 4447:a 4433:, 4430:a 4423:/ 4419:b 4416:4 4413:= 4408:u 4404:K 4381:u 4377:T 4326:4 4323:1 4286:u 4282:T 4259:d 4255:K 4232:i 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2584:= 2575:D 2558:d 2555:K 2551:d 2548:K 2540:i 2537:K 2533:p 2530:K 2528:( 2526:d 2523:K 2485:. 2479:d 2475:) 2469:( 2466:e 2461:t 2456:0 2446:i 2442:K 2438:= 2429:I 2412:i 2409:K 2407:( 2396:d 2393:K 2389:p 2386:K 2384:( 2382:i 2379:K 2329:. 2326:) 2323:t 2320:( 2317:e 2312:p 2308:K 2304:= 2295:P 2278:p 2275:K 2267:d 2264:K 2260:i 2257:K 2255:( 2253:p 2250:K 2224:s 2201:, 2198:s 2193:d 2189:K 2185:+ 2182:s 2178:/ 2172:i 2168:K 2164:+ 2159:p 2155:K 2151:= 2148:) 2145:s 2142:( 2139:L 2104:t 2062:t 2050:t 2036:) 2033:t 2030:( 2026:V 2023:P 2015:P 2012:S 2008:= 2005:) 2002:t 1999:( 1996:e 1972:d 1968:K 1943:i 1939:K 1914:p 1910:K 1883:, 1877:t 1874:d 1869:) 1866:t 1863:( 1860:e 1857:d 1849:d 1845:K 1841:+ 1835:d 1831:) 1825:( 1822:e 1817:t 1812:0 1802:i 1798:K 1794:+ 1791:) 1788:t 1785:( 1782:e 1777:p 1773:K 1769:= 1766:) 1763:t 1760:( 1756:V 1753:M 1749:= 1746:) 1743:t 1740:( 1737:u 1714:) 1711:t 1708:( 1705:u 1691:. 1215:( 1177:) 1170:t 1166:d 1160:) 1157:t 1154:( 1151:e 1147:d 1138:d 1134:T 1130:+ 1123:d 1118:) 1112:( 1109:e 1104:t 1099:0 1087:i 1083:T 1079:1 1074:+ 1071:) 1068:t 1065:( 1062:e 1058:( 1052:p 1048:K 1044:= 1041:) 1038:t 1035:( 1032:u 1007:i 1003:T 998:/ 992:p 988:K 965:d 961:T 955:p 951:K 928:d 924:T 901:i 897:T 874:d 870:T 864:p 860:K 837:i 833:T 828:/ 822:p 818:K 795:d 791:K 768:i 764:K 746:D 742:I 738:P 722:d 718:K 695:i 691:K 668:p 664:K 641:, 635:t 631:d 625:) 622:t 619:( 616:e 612:d 603:d 599:K 595:+ 588:d 583:) 577:( 574:e 569:t 564:0 554:i 550:K 546:+ 543:) 540:t 537:( 534:e 529:p 525:K 521:= 518:) 515:t 512:( 509:u 458:D 450:I 446:I 439:p 425:) 422:t 419:( 416:e 406:P 377:) 374:t 371:( 368:u 333:) 330:t 327:( 324:y 318:) 315:t 312:( 309:r 306:= 303:) 300:t 297:( 294:e 274:) 271:t 268:( 265:y 262:= 234:) 231:t 228:( 225:r 222:= 194:) 191:t 188:( 185:e 163:t 161:( 159:y 147:t 145:( 143:r 107:D 103:I 99:P 65:) 62:t 59:( 56:e 25:(

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