PID controller

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A proportional-integral derivative controller ( PID Controller or three term controller ) is a widely used loop control feedback mechanism in the industry and various system controls other applications that require continuous modulation control. The PID controller continues to calculate the error value ${\ displaystyle e (t)}$ as the difference between the desired setpoint (SP) and the measured process variable (PV) and apply correction based on proportional, integral, and derivative terms (denoted < > P , I , and D respectively) giving the controller its name.

In practical terms, it automatically applies accurate and responsive corrections to the control function. Daily examples are the control of shipping on a road vehicle; where external influences such as gradients will cause speed changes, and the driver has the ability to change the desired speed. The PID algorithm returns the actual speed to the desired speed in an optimal way, without delay or overshoot, by controlling the engine power output of the vehicle.

The first theoretical analysis and practical application is 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 pneumatics, and then electronics, controllers. Today there is a universal use of PID concepts in applications that require accurate and optimized automatic control.

Video PID controller

Operasi fundamental

The distinguishing features of the PID controller are the ability to use three control terms from proportional, integral and derived effects on the controller output to implement accurate and optimal controls. The block diagram on the right shows the principles of how these terms are generated and applied. This shows the PID controller, which continues to calculate the error value ${\ displaystyle e (t)}$ as the difference between the desired setpoint ${\ displaystyle {\ text {SP}} = r (t)}$ and processed variables ${\ displaystyle {\ text {PV}} = y (t)}$, and apply corrections based on proportional, integral, and derivative provisions The controller tries to minimize the error over time with the adjustment of control variables ${\ displaystyle u (t)}$ , such as opening the control valve, to a new value determined by the weighted amount of the control term.

In this model:

• Term P is proportional to the current value of the SP - PV error e ( t ). For example, if the error is large and positive, the control output will be proportionally large and positive, taking into account the strengthening factor "K". Using proportional control only in the process with compensation such as temperature control, will result in an error between the setpoint and the actual process value, therefore requires an error to produce a proportional response. If there are no errors, there is no corrective response.
• The term I contributes the previous values ​​of SP-PV errors and unites them over time to generate the term I . For example, if there is a residual SP-PV error after the application of proportional control, the term integral seeks to eliminate residual errors by adding a control effect because of the historical cumulative value of the error. When errors are omitted, the term integral will stop growing. This will result in a reduced proportional effect due to falling errors, but this is compensated by the growing integral effect.
• Term D is the best estimate of future trends of SP-PV errors, based on the current rate of change. Sometimes called "anticipative control", as it effectively seeks to reduce the effects of SP-PV errors by exerting the influence of controls generated by the rate of error change. The faster the change, the greater the effect of the controller or the damper.

Tuning - This balance effect is achieved by "loop tuning" (see later) to produce optimal control functionality. The tuning constants are shown below as "K" and should be lowered for each control application, as it depends on the response characteristics of the complete external loop to the controller. This depends on the behavior of the measuring sensor, the final control element (such as the control valve), the control signal delay and the process itself. Estimated constant values ​​can usually be entered at first to know the type of application, but are usually purified, or adjusted, by "bumping" the process in practice by introducing a setpoint change and observing the system response.

Control actions - The mathematical model and the practical loop over both use "direct" control measures for all terms, which means an increase in the positive error results in a positive control output increase for the sum of terms to apply the correction. However, the result is called an "upside down" action if it is necessary to apply negative corrective actions. For example, if the valve in the flow loop is 100-0% the valve opening for the control output is 0-100% - which means that the controller action must be reversed. Some process control schemes and final control elements require this inverse action. An example is a valve to cool water, where a fail-safe mode, in case of signal loss, will open 100% of the valve; therefore the control output 0% needs to cause 100% valve opening.

Mathematical form

Fungsi kontrol keseluruhan dapat dinyatakan secara matematis sebagai

${\ displaystyle u (t) = K_ {\ text {p}} e (t) K _ {\ text {i}} \ int _ {0} ^ {t } e (t ') \, dt' K_ {\ text {d}} {\ frac {de (t)} {dt}},}  ÂÂ$

di mana ${\ displaystyle K _ {\ text {p}}}  ÂÂ$ , ${\ displaystyle K _ {\ text {i}}}  ÂÂ$ , dan ${\ displaystyle K _ {\ text {d}}}  ÂÂ$ , semua non-negatif, menunjukkan koefisien untuk masing-masing proporsional, integral, dan turunan (kadang-kadang dinotasikan P , Saya , dan D ).

Dalam bentuk standar persamaan (lihat nanti dalam artikel), ${\ displaystyle K _ {\ text {i}}}  ÂÂ$ dan ${\ displaystyle K _ {\ text {d}}}  ÂÂ$ masing-masing diganti dengan ${\ displaystyle K _ {\ text {p}}/T _ {\ text {i}}}  ÂÂ$ dan $            {\displaystyle         K_{            \text{p}          }        T_{            \text{d}          }      }        \{\backslash \; displaystyle\; K\; \_\; \{\backslash \; text\; \{p\}\}\; T\; \_\; \{\backslash \; text\; \{d\}\}\}\;  ÂÂ$ ; Keuntungan dari ini adalah bahwa ${\ displaystyle T _ {\ text {i}}}  ÂÂ$ dan ${\ displaystyle T _ {\ text {d}}}  ÂÂ$ memiliki beberapa makna fisik yang dapat dimengerti, karena mereka mewakili waktu integrasi dan waktu turunan masing-masing.

Penggunaan istilah kontrol yang selektif

Although PID controllers have three control terms, some applications only use one or two terms to provide the appropriate controls. This is achieved by setting the unused parameters to zero and is called a PI, PD, P or I controller in the absence of any other control action. PI controllers are quite common, because the action of the derivative is sensitive to noise measurement, whereas the absence of an integral term can prevent the system from reaching its target value.

Enforcement

Use of PID algorithm does not guarantee optimal control of the system or its control stability (See Ã, § PID control limitation, below.) . Situations can occur where there are excessive delays: the measurement of the pending process value, or the control action not being fast enough. In these cases, lead-lag compensation is required to be effective. The controller's response can be explained in terms of its response to errors, the extent to which the system goes beyond the setpoint, and the degree of oscillation of any system. But PID controllers are widely applicable, because they depend only on the response of process variables being measured, not on the knowledge or model of the underlying process.

Maps PID controller

History

Origins

Continuous control, before the PID controller is fully understood and implemented, has one of its origin in a centrifugal governor that uses a rotating load to control the process. It has been invented by Christian Huygens in the 17th century to set the gap between the millstones in the windmill depending on the speed of rotation, and thus compensate for the variable speed of the grain feed.

With the invention of a low-pressure stationary steam engine, there is a need for automatic speed control, and a self-designed pendulum constellation governor by James Watt, a set of rotating steel balls attached to vertical spindles with connecting sleeves, came into industry standard. It is based on the concept of factory stone crevice control.

The rotating governor's velocity control, however, is still variable under varying load conditions, where the deficiency of what is now known as proportional control alone is already evident. The error between the desired speed and the actual speed will increase as the load increases. In the 19th century the theoretical basis for the governor's operation was first described by James Clerk Maxwell in 1868 in his now famous paper On Governors. He explored the foundations of mathematics for the stability of control, and developed a good way toward solutions, but made the attraction for mathematicians to examine problems. The problem was further examined by Edward Routh in 1874, Charles Sturm and in 1895, Adolf Hurwitz, all of whom contributed to the establishment of the stability control criteria. In practice, the governor's speed was more refined, especially by American scientist Willard Gibbs, who in 1872 theoretically analyzed the governor of the Watt cone pendulum.

Around this time, the discovery of the Whitehead torpedo caused a control problem requiring accurate control of the running depth. The use of depth pressure sensors alone proved to be inadequate, and pendulums measuring the fore and aft of torpedoes combined with depth measurements to be the control pendulum and hydrostate. The pressure control only provides proportional control which, if the control gain is too high, will become unstable and go into overshoot, with considerable instability. The pendulum adds what is now known as the derived control, which damps the oscillations by detecting a dive torpedo/climbing angle and thus the rate of change of depth. This development (named Whitehead as "The Secret" to not give directions) around 1868.

Another early example of the PID type controller was developed by Elmer Sperry in 1911 for steering-boats, although his work is intuitive rather than mathematically based.

However, it was only in 1922 that the formal control law for what we now call PID or three-term control was first developed using theoretical analysis, by Russian engineer Nicolas Minorsky. Minorsky is researching and designing automatic ship steering for the US Navy and based on his analysis on the observations of a helmsman. He notes that the helmsman steers the ship based not only on current course errors, but also on past mistakes, as well as the current rate of change; This was then given mathematical treatment by Minorsky. The goal is stability, not general control, which simplifies the problem significantly. While proportional controls provide stability to small disturbances, it is not sufficient to handle stable disturbances, especially storm storms (due to steady-state errors), which required adding integral terms. Finally, derivative terms are added to improve stability and control.

Trials are performed on USS New Mexico , with controllers controlling angular velocity (not angle). The PI control produces a continuous yaw (angular error) of  ± 2  °. Adding a D element produces a yaw error  ± Ã,  ± 1/6Ã,  °, better than most a helmsman can achieve.

The Navy eventually did not adopt the system because of resistance by personnel. Similar work was done and published by several others in the 1930s.

Industrial control

The widespread use of feedback controllers does not become feasible until the development of high-gain broadband amplifiers to use the concept of negative feedback. It was developed in telephone electronics engineering by Harold Black in the late 1920s, but was not published until 1934. By independently, Clesson E Mason of Foxboro Company in 1930 created a wide-band pneumatic controller by combining nozzle and flapper high-gain pneumatic amplifiers , which was discovered in 1914, with negative feedback from the controller output. This dramatically increases the linear range of operation of the nozzle and flapper reinforcement, and integral control can also be added with the use of precision valve and bellows producing Integral terms. The result is a "Stabilog" controller that provides proportional and integral functions using feedback bellows. Then the derivative term is added by a further bellows and an adjustable orifice.

From about 1932 onwards, the use of wideband pneumatic controllers increased rapidly in a variety of control applications. Compressed air is used both to produce controller output, and to turn on process modulation devices; such as diaphragm control valves operated. They are simple low maintenance devices that operate well in harsh industrial environments, and do not present the risk of explosions in hazardous locations. They are the industry standard for decades until the emergence of discrete electronic controllers and distributed control systems.

With this controller, 3-15 psi pneumatic industrial signaling standards (0.2-1.0 bar) are established, which has zero increases to ensure the device works in linear characteristics and represents a 0-100% control range.

In the 1950s, when high-gain electronic amplifiers became cheap and reliable, electronic PID controllers became popular, and 4-20 mA current loop signals were used that mimicked pneumatic standards. However, field actuators still use a lot of pneumatic standards because of the advantages of pneumatic motive power for control valves in the process plant environment.

Most modern industrial PID controls are implemented in a distributed control system (DCS), programmable logic controller (PLC), computer-based controller or as a compact controller. Software implementation has the advantage of being relatively inexpensive and flexible with respect to the application of PID algorithms in certain control scenarios.

Electronic analog controller

Electronic analog PID control loops are often found in more complex electronic systems, for example, the head position of a disk drive, power conditioning from the power supply, or even modern seismometer motion detection circuitry. Discrete electronic analog controllers have been largely replaced by digital controllers using a microcontroller or FPGA, to implement the PID algorithm. However, discrete analog PID controllers are still used in special applications that require high bandwidth performance and low noise, such as laser diode controllers.

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Control loop example

Let's take the example of a robotic arm, which can be moved and positioned by a control loop. An electric motor can lift or lower the arm, depending on the applied forward or reverse force, but strength can not be a simple function of position due to mass inertia of the arm, force due to gravity, external forces on the arm like a load. to lift or work to be done on external objects.

• The perceived position is a process variable (PV).
• The desired position is called setpoint (SP).
• The difference between PV and SP is error (e), which quantifies whether the arm is too low or too high and how much.
• The input to the process (electric current in the motor) is the output of the PID controller. This is called either manipulated variable (MV) or control variable (CV).

By measuring the position (PV), and subtracting it from the setpoint (SP), error (e) is found, and from there the controller calculates how much electric current to supply to the motor (MV).

Proportional

The obvious method is proportional control: the motor current is adjusted proportionally with the error. However, this method fails if, for example, the arm has to lift a different load: larger weights require greater strength applied to the same error on the down side, but smaller forces if the error is on the upper side. That's where the terms integrals and derivatives play their part.

Integral

The term integral increases action in relation not only to errors but also to the time already in progress. So, if the applied force is not enough to bring the error to zero, this force will increase over time. A pure "I" controller can bring an error to zero, however, it will be a slow reaction at the beginning (because action will be small at the beginning, takes time to get significant), brutal (increased action during positive error, even if the error has started approaching zero), and slow to end (when the error switching side, this for some time will only reduce the action power of "I", not make it switch side too), push overshoot and oscillation (see below). In addition, it can even move the system from zero error: given that the system has been wrong, it could push the action when it is not needed. An alternative formulation of the integral act is to change the electric current in small persistent steps that are proportional to the current error. As time passes, the steps increase and increase depending on past mistakes; this is a discrete time equivalent to integration.

Derivatives

The term derivative does not take into account the error (meaning it can not bring it to zero: the pure D controller can not bring the system to its setpoint), but the error rate changes, trying to bring this level to zero. It aims to flatten the trajectory of error into a horizontal line, dampen the applied force, and reduce the overshoot (error on the other hand due to overly applied force). Applying too much encouragement when a small error and reducing will cause overshoot. After overshooting, if the controller makes a large correction in the opposite direction and repeatedly exceeds the desired position, the output will oscillate around the setpoint in either a constant, growing, or rotting sinusoid. If the amplitude of oscillation increases over time, the system becomes unstable. If they are down, the system is stable. If the oscillation remains at a constant magnitude, the system is slightly stable.

Damping control

In the interest of achieving controlled arrival at desired position (SP) in a timely and accurate manner, the controlled system needs to be muted critically. A well-positioned position control system will also apply the required current to the controlled motor so that the arm pushes and pulls as necessary to withstand an external force that tries to keep it away from the required position. The setpoint itself may be generated by an external system, such as a PLC or other computer system, thus continuing to vary depending on the work expected by the robot arm. A well-tuned PID control system will allow the arm to meet these changing requirements to the best of its ability.

Response to interruption

If the controller starts from a stable state with zero error (PV = SP), then further changes by the controller will respond to other measurable or measurable input changes to the process affecting the process, and hence PV. Variables that affect processes other than MV are known as interference. Generally controllers are used to deny interference and apply setpoint changes. Changes in the load on the arm is a disturbance in the robot arm control process.

Apps

Theoretically, the controller can be used to control any process that has a measurable output (PV), the known ideal value for that output (SP) and the process input (MV) that will affect the relevant PV. Controllers are used in industry to regulate temperature, pressure, force, feed rate, flow rate, chemical composition (component concentration), weight, position, velocity, and almost any other variable of measurement.

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PID controller theory

This section describes the parallel or non-interaction form of the PID controller. For other forms, please see the Nomenclature and Alternative PID forms.

The PID control scheme is named after three correction terms, whose summing is a manipulation variable (MV). The proportional, integral, and derivative terms are summed to calculate the output of the PID controller. Specify ${\ displaystyle u (t)}$ as the controller output, the final form of the PID algorithm is

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dimana

${\ displaystyle K _ {\ text {p}}}  ÂÂ$ adalah keuntungan proporsional, parameter penyetelan,

${\ displaystyle K _ {\ text {i}}}  ÂÂ$ adalah keuntungan integral, parameter penyetelan,

${\ displaystyle K _ {\ text {d}}}  ÂÂ$ adalah gain turunan, parameter penyetelan,

${\ displaystyle e (t) = \ mathrm {SP} - \ mathrm {PV} (t)}  ÂÂ$ adalah kesalahan (SP adalah setpoint, dan PV ( t ) adalah variabel proses),

${\ displaystyle t}  ÂÂ$ adalah waktu atau waktu instan (saat ini),

${\ displaystyle \ tau}  ÂÂ$ adalah variabel integrasi (mengambil nilai dari waktu 0 hingga sekarang ${\ displaystyle t}  ÂÂ$ ).

Secara ekuivalen, fungsi transfer di domain Laplace dari kontroler PID adalah

${\ displaystyle L (s) = K_ {\ text {p}} K_ {\ text {i}}/s K _ {\ text {d}} s, }  ÂÂ$

di mana ${\ displaystyle s}  ÂÂ$ adalah frekuensi kompleks.

Istilah proporsional

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 the constant K _p , called the proportional gain constant.

Istilah proporsional diberikan oleh

${\ displaystyle P _ {\ text {out}} = K_ {\ text {p}} e (t).}  ÂÂ$

A high proportional result produces a major change in output for a given change in error. If the proportional gain is too high, the system may become unstable (see section on ignition loop). In contrast, small results produce small output responses to large input errors, and less responsive or less sensitive controllers. If the proportional gain is too low, the control action may be too small in response to a system failure. The theory of tuning and industrial practice suggests that proportional terms should account for most of the output changes.

Steady state error

Since non-zero errors are required to drive it, proportional controllers generally operate with so-called fixed-state errors. Steady-state error (SSE) is proportional to the process gain and inversely proportional to the gain. SSE can be reduced by adding the term bias compensation to setpoint AND output, or dynamically corrected by adding an integral term.

The term integral

The contribution of the integral term is proportional to the magnitude of the error and the duration of the error. The integral in the PID controller is the sum of the instantaneous errors over time and gives an accumulated offset that should have been corrected earlier. The accumulated error is then multiplied by the integral gain ( K _i ) and added to the controller output.

Istilah integral diberikan oleh

${\ displaystyle I _ {\ text {out}} = K _ {\ text {i}} \ int _ {0} ^ {t} e (\ tau) \, d \ tau}  ÂÂ$ .

The term integral speeds up the movement of the process toward the setpoint and eliminates the remaining steady state errors that occur with pure proportional controllers. However, since the term integral responds to the accumulation of errors from the past, it can cause the current value to go beyond the setpoint value (see section on the tuning loop).

Derivative term

The derivative of a process error is calculated by determining the slope of the error over time and multiplying the rate of this change with the derived profit K _d . The magnitude of the derivative term contribution to the overall control action is called derived profit, K _d .

Istilah derivatif diberikan oleh

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

Derivative action predicts the behavior of the system and thereby increases the time and stability of the system. An ideal derivative is not causal, so the implementation of PID controllers includes additional low-pass filtering for derivative terms to limit high frequency acquisition and noise. Derivative actions are rarely used in practice though - with an estimate of only 25% of controllers used - due to their variable impact on system stability in real-world applications.

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Loop alignment

Tuning control loop is the adjustment of its control parameters (band/proportional gain, integral gain/reset, gain/rate derivative) to the optimal value for the desired control response. Stability (no unlimited oscillations) is a basic requirement, but beyond that, different systems have different behaviors, different applications have different requirements, and requirements may conflict with each other.

Tuning PID is a difficult problem, although there are only three parameters and in principle simple to describe, it must therefore meet complex criteria within PID control limits. There are various methods for tuning loops, and more sophisticated techniques are patent subjects; This section describes some of the traditional manual methods for loop tuning.

Designing and tuning PID controllers seems intuitively conceptual, but it can be difficult in practice, if multiple (and often conflicting) goals such as temporary short and high stability must be achieved. PID controllers often provide acceptable controls using default tuning, but performance can generally be improved with careful setup, and performance may not be acceptable with poor setup. Typically, the initial design needs to be adjusted repeatedly through computer simulations until a closed-loop system performs or compromises as desired.

Some processes have a degree of nonlinearity so that parameters that work well under full load conditions do not work when the process starts from no load; this can be corrected by gain scheduling (using different parameters in different operating areas).

Stability

If the PID controller parameters (advantages of proportional, integral and derivative provisions) are incorrectly selected, the controlled process input may become unstable, that is, its output is distorted, with or without oscillation, and limited only by saturation or mechanical damage. Instability is caused by excess excess , especially in the presence of significant lag.

In general, response stabilization is required and the process should not oscillate for any combination of process and setpoint conditions, although sometimes marginal stability (limited oscillation) is acceptable or desirable.

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

Fungsi transfer loop total adalah:

${\ displaystyle H (s) = {\ frac {K (s) G (s)} {1 K (s) G (s)}}}  ÂÂ$

dimana

${\ displaystyle K (s)}  ÂÂ$ : Fungsi transfer PID

${\ displaystyle G (s)}  ÂÂ$ : Fungsi transfer tanaman

Sistem ini disebut

Source of the article : Wikipedia