Un controlador proporcional-integral-derivado ( controlador PID o controlador de tres términos ) es un mecanismo de circuito de control que emplea retroalimentación que se usa ampliamente en sistemas de control industrial y una variedad de otras aplicaciones que requieren control modulado continuamente. Un controlador PID calcula continuamente un valor de error como la diferencia entre un punto de ajuste deseado (SP) y una variable de proceso medida (PV) y aplica una corrección basada en términos proporcionales , integrales y derivados (indicados P , I y D respectivamente), de ahí el nombre.
En términos prácticos, aplica automáticamente una corrección precisa y sensible a una función de control. Un ejemplo cotidiano es el control de crucero de un automóvil, en el que subir una colina reduciría la velocidad si solo se aplicara una potencia constante del motor. El algoritmo PID del controlador restaura la velocidad medida a la velocidad deseada con un retardo mínimo y un exceso al aumentar la potencia de salida del motor.
El primer análisis teórico y aplicación práctica se realizó en el campo de los sistemas de gobierno automático para barcos, desarrollado desde principios de la década de 1920 en adelante. Luego se utilizó para el control automático de procesos en la industria manufacturera, donde se implementó ampliamente, primero en controladores neumáticos y luego electrónicos . Hoy en día, el concepto PID se utiliza universalmente en aplicaciones que requieren un control automático preciso y optimizado.
Operación fundamental
La característica distintiva del controlador PID es la capacidad de utilizar los tres términos de control de influencia proporcional, integral y derivada en la salida del controlador para aplicar un control preciso y óptimo. El diagrama de bloques de la derecha muestra los principios de cómo se generan y aplican estos términos. Muestra un controlador PID, que calcula continuamente un valor de error. como la diferencia entre un punto de ajuste deseado y una variable de proceso medida : y aplica una corrección basada en términos proporcionales , integrales y derivados . El controlador intenta minimizar el error a lo largo del tiempo mediante el ajuste de una variable de control. , como la apertura de una válvula de control , a un nuevo valor determinado por una suma ponderada de los términos de control.
En este modelo:
- El término P es proporcional al valor actual del error SP - PV. Por ejemplo, si el error es grande y positivo, la salida de control será proporcionalmente grande y positiva, teniendo en cuenta el factor de ganancia "K". El uso del control proporcional solo resultará en un error entre el punto de ajuste y el valor real del proceso porque requiere un error para generar la respuesta proporcional. Si no hay error, no hay respuesta correctiva.
- El término I tiene en cuenta los valores pasados del error SP - PV y los integra a lo largo del tiempo para producir el término I. Por ejemplo, si hay un error residual SP - PV luego de la aplicación del control proporcional, el término integral busca eliminar el error residual agregando un efecto de control debido al valor histórico acumulado del error. Cuando se elimina el error, el término integral dejará de crecer. Esto dará como resultado que el efecto proporcional disminuya a medida que disminuye el error, pero esto se compensa con el efecto integral creciente.
- El término D es una mejor estimación de la tendencia futura del error SP - PV, basada en su tasa de cambio actual. A veces se le llama "control anticipatorio", ya que busca efectivamente reducir el efecto del error SP - PV ejerciendo una influencia de control generada por la tasa de cambio de error. Cuanto más rápido sea el cambio, mayor será el efecto de control o amortiguación. [1]
Afinación : el equilibrio de estos efectos se logra mediante la afinación de bucle para producir la función de control óptima. Las constantes de ajuste se muestran a continuación como "K" y deben derivarse para cada aplicación de control, ya que dependen de las características de respuesta del lazo completo externo al controlador. Estos dependen del comportamiento del sensor de medición, el elemento de control final (como una válvula de control), cualquier retraso en la señal de control y el proceso en sí. Por lo general, los valores aproximados de las constantes se pueden ingresar inicialmente conociendo el tipo de aplicación, pero normalmente se refinan o ajustan "golpeando" el proceso en la práctica al introducir un cambio de punto de ajuste y observar la respuesta del sistema.
Acción de control : el modelo matemático y el bucle práctico anterior utilizan una acción de control directa para todos los términos, lo que significa que un error positivo en aumento da como resultado una corrección de salida de control positivo en aumento. El sistema se denomina acción inversa si es necesario aplicar una acción correctiva negativa. Por ejemplo, si la válvula en el circuito de flujo tenía una apertura de válvula del 100–0% para una salida de control del 0–100%, lo que significa que la acción del controlador debe invertirse. Algunos esquemas de control de procesos y elementos de control final requieren esta acción inversa. Un ejemplo sería una válvula para agua de refrigeración, donde el modo a prueba de fallas , en caso de pérdida de señal, sería 100% de apertura de la válvula; por lo tanto, la salida del controlador al 0% debe causar la apertura de la válvula al 100%.
Forma matemática
La función de control general
dónde , , y , todos no negativos, denotan los coeficientes de los términos proporcional , integral y derivado respectivamente (a veces se denotan como P , I y D ).
En la forma estándar de la ecuación (ver más adelante en el artículo), y son reemplazados respectivamente por y ; la ventaja de esto es que y tienen un significado físico comprensible, ya que representan el tiempo de integración y el tiempo derivado respectivamente.
Uso selectivo de términos de control
Aunque un controlador PID tiene tres términos de control, algunas aplicaciones solo necesitan uno o dos términos para proporcionar un control adecuado. Esto se logra estableciendo los parámetros no utilizados en cero y se denomina controlador PI, PD, P o I en ausencia de las otras acciones de control. Los controladores PI son bastante comunes en aplicaciones donde la acción derivada sería sensible al ruido de medición, pero el término integral a menudo es necesario para que el sistema alcance su valor objetivo.
Aplicabilidad
El uso del algoritmo PID no garantiza un control óptimo del sistema o su estabilidad de control . Pueden ocurrir situaciones en las que haya retrasos excesivos: la medición del valor del proceso se retrasa o la acción de control no se aplica con la suficiente rapidez. En estos casos , se requiere que la compensación de adelanto-retraso sea efectiva. La respuesta del controlador se puede describir en términos de su capacidad de respuesta a un error, el grado en que el sistema sobrepasa un punto de ajuste y el grado de cualquier oscilación del sistema . Pero el controlador PID es ampliamente aplicable ya que se basa solo en la respuesta de la variable de proceso medida, no en el conocimiento o un modelo del proceso subyacente.
Historia
Orígenes
El control continuo, antes de que los controladores PID fueran completamente comprendidos e implementados, tiene uno de sus orígenes en el gobernador centrífugo , que utiliza pesos giratorios para controlar un proceso. Esto había sido inventado por Christiaan Huygens en el siglo XVII para regular el espacio entre las muelas en los molinos de viento en función de la velocidad de rotación y, por lo tanto, compensar la velocidad variable de la alimentación del grano. [2] [3]
Con la invención de la máquina de vapor estacionaria de baja presión , surgió la necesidad de un control automático de velocidad, y apareció el gobernador de "péndulo cónico" diseñado por James Watt , un conjunto de bolas de acero giratorias unidas a un eje vertical mediante brazos articulados. para ser un estándar de la industria. Esto se basó en el concepto de control del espacio entre las piedras de molino. [4]
Sin embargo, el control de velocidad del gobernador giratorio seguía siendo variable en condiciones de carga variable, donde era evidente la deficiencia de lo que ahora se conoce como control proporcional solo. El error entre la velocidad deseada y la velocidad real aumentaría al aumentar la carga. En el siglo XIX, la base teórica del funcionamiento de los gobernadores fue descrita por primera vez por James Clerk Maxwell en 1868 en su ahora famoso artículo On Governors . Exploró la base matemática para la estabilidad del control y avanzó bastante hacia una solución, pero hizo un llamamiento a los matemáticos para que examinaran el problema. [5] [4] El problema fue examinado más a fondo en 1874 por Edward Routh , Charles Sturm , y en 1895, Adolf Hurwitz , todos los cuales contribuyeron al establecimiento de criterios de estabilidad de control. [4] En aplicaciones posteriores, los reguladores de velocidad se refinaron aún más, en particular por el científico estadounidense Willard Gibbs , quien en 1872 analizó teóricamente el regulador de péndulo cónico de Watt.
Aproximadamente en este momento, la invención del torpedo Whitehead planteó un problema de control que requería un control preciso de la profundidad de carrera. El uso de un sensor de presión de profundidad por sí solo resultó inadecuado, y un péndulo que medía el cabeceo de proa y popa del torpedo se combinó con la medición de profundidad para convertirse en el control de péndulo e hidrostato . El control de presión proporcionó solo un control proporcional que, si la ganancia de control era demasiado alta, se volvería inestable y se sobrepasaría con una inestabilidad considerable de retención de profundidad. El péndulo agregó lo que ahora se conoce como control derivado, que amortiguó las oscilaciones al detectar el ángulo de picado / ascenso del torpedo y, por lo tanto, la tasa de cambio de profundidad. [6] Este desarrollo (nombrado por Whitehead como "El Secreto" para no dar pistas sobre su acción) fue alrededor de 1868. [7]
Otro ejemplo temprano de un controlador de tipo PID fue desarrollado por Elmer Sperry en 1911 para la dirección de barcos, aunque su trabajo fue más intuitivo que matemático. [8]
Sin embargo, no fue hasta 1922 que el ingeniero ruso-estadounidense Nicolas Minorsky desarrolló por primera vez una ley de control formal para lo que ahora llamamos PID o control de tres términos utilizando un análisis teórico . [9] Minorsky estaba investigando y diseñando la dirección automática de un barco para la Marina de los Estados Unidos y basó su análisis en las observaciones de un timonel . Señaló que el timonel dirigía el barco basándose no solo en el error de rumbo actual, sino también en el error pasado, así como en la tasa de cambio actual; [10] Minorsky le dio un tratamiento matemático a esto. [4] Su objetivo era la estabilidad, no el control general, lo que simplificó significativamente el problema. Si bien el control proporcional proporcionó estabilidad frente a pequeñas perturbaciones, fue insuficiente para hacer frente a una perturbación constante, en particular un vendaval rígido (debido a un error de estado estable ), que requirió agregar el término integral. Finalmente, se agregó el término derivado para mejorar la estabilidad y el control.
Las pruebas se llevaron a cabo en el USS New Mexico , con los controladores controlando la velocidad angular (no el ángulo) del timón. El control PI produjo una guiñada sostenida (error angular) de ± 2 °. La adición del elemento D produjo un error de guiñada de ± 1/6 °, mejor de lo que podrían lograr la mayoría de los timonel. [11]
La Marina finalmente no adoptó el sistema debido a la resistencia del personal. Varios otros llevaron a cabo un trabajo similar y lo publicaron en la década de 1930.
Control industrial
El amplio uso de controladores de retroalimentación no fue factible hasta el desarrollo de amplificadores de banda ancha de alta ganancia para utilizar el concepto de retroalimentación negativa. Esto había sido desarrollado en electrónica de ingeniería telefónica por Harold Black a fines de la década de 1920, pero no se publicó hasta 1934. [4] Independientemente, Clesson E Mason de Foxboro Company en 1930 inventó un controlador neumático de banda ancha combinando la boquilla y la aleta alta - Amplificador neumático de ganancia, que se había inventado en 1914, con retroalimentación negativa de la salida del controlador. Esto aumentó dramáticamente el rango de operación lineal de la boquilla y el amplificador de aleta, y el control integral también podría agregarse mediante el uso de una válvula de purga de precisión y un fuelle que genera el término integral. El resultado fue el controlador "Stabilog" que proporcionó funciones tanto proporcionales como integrales mediante fuelles de retroalimentación. [4] El término integral se llamó Reset . [12] Más tarde, el término derivado se añadió mediante un fuelle adicional y un orificio ajustable.
Desde aproximadamente 1932 en adelante, el uso de controladores neumáticos de banda ancha aumentó rápidamente en una variedad de aplicaciones de control. La presión de aire se utilizó para generar la salida del controlador y también para alimentar dispositivos de modulación del proceso, como válvulas de control operadas por diafragma . Eran dispositivos simples de bajo mantenimiento que funcionaban bien en entornos industriales hostiles y no presentaban riesgos de explosión en ubicaciones peligrosas . Fueron el estándar de la industria durante muchas décadas hasta la llegada de los controladores electrónicos discretos y los sistemas de control distribuido .
Con estos controladores, se estableció un estándar de señalización de la industria neumática de 3 a 15 psi (0,2 a 1,0 bar), que tenía un cero elevado para garantizar que los dispositivos funcionaran dentro de su característica lineal y representaba el rango de control de 0 a 100%.
En la década de 1950, cuando los amplificadores electrónicos de alta ganancia se volvieron baratos y confiables, los controladores PID electrónicos se hicieron populares y el estándar neumático fue emulado por señales de bucle de corriente de 10-50 mA y 4-20 mA (este último se convirtió en el estándar de la industria). Los actuadores de campo neumáticos todavía se utilizan ampliamente debido a las ventajas de la energía neumática para válvulas de control en entornos de plantas de proceso.
La mayoría de los controles PID modernos en la industria se implementan como software de computadora en sistemas de control distribuido (DCS), controladores lógicos programables (PLC) o controladores compactos discretos .
Controladores analógicos electrónicos
Los lazos de control PID analógicos electrónicos se encontraban a menudo dentro de sistemas electrónicos más complejos, por ejemplo, el posicionamiento de la cabeza de una unidad de disco , el acondicionamiento de energía de una fuente de alimentación o incluso el circuito de detección de movimiento de un sismómetro moderno . Los controladores analógicos electrónicos discretos han sido reemplazados en gran medida por controladores digitales que utilizan microcontroladores o FPGA para implementar algoritmos PID. Sin embargo, los controladores PID analógicos discretos todavía se utilizan en aplicaciones de nicho que requieren un alto ancho de banda y un rendimiento de bajo ruido, como los controladores de diodo láser. [13]
Ejemplo de lazo de control
Considere un brazo robótico [14] que se puede mover y colocar mediante un bucle de control. Un motor eléctrico puede levantar o bajar el brazo, dependiendo de la potencia de avance o retroceso aplicada, pero la potencia no puede ser una función simple de la posición debido a la masa inercial del brazo, las fuerzas debidas a la gravedad, las fuerzas externas en el brazo, como una carga. levantar o trabajar en un objeto externo.
- La posición detectada es la variable de proceso (PV).
- La posición deseada se llama punto de ajuste (SP).
- La diferencia entre PV y SP es el error (e), que cuantifica si el brazo es demasiado bajo o demasiado alto y por cuánto.
- La entrada al proceso (la corriente eléctrica en el motor) es la salida del controlador PID. Se llama variable manipulada (MV) o variable de control (CV).
Al medir la posición (PV) y restarla del punto de ajuste (SP), se encuentra el error (e) y, a partir de él, el controlador calcula la cantidad de corriente eléctrica que debe suministrar al motor (MV).
Proporcional
El método obvio es el control proporcional : la corriente del motor se establece en proporción al error existente. Sin embargo, este método falla si, por ejemplo, el brazo tiene que levantar diferentes pesos: un peso mayor necesita una fuerza mayor aplicada para el mismo error en el lado inferior, pero una fuerza menor si el error es al revés. Ahí es donde los términos integral y derivado juegan su papel.
Integral
Un término integral aumenta la acción en relación no solo con el error sino también con el tiempo durante el cual ha persistido. Entonces, si la fuerza aplicada no es suficiente para llevar el error a cero, esta fuerza aumentará a medida que pase el tiempo. Un controlador "I" puro podría llevar el error a cero, pero sería tanto una reacción lenta al principio (porque la acción sería pequeña al principio, necesitando tiempo para volverse significativa) como brutal (la acción aumenta mientras el el error es positivo, incluso si el error ha comenzado a acercarse a cero).
Derivado
Un término derivado no considera el error (lo que significa que no puede llevarlo a cero: un controlador D puro no puede llevar el sistema a su punto de ajuste), sino la tasa de cambio de error, tratando de llevar esta tasa a cero. Su objetivo es aplanar la trayectoria del error en una línea horizontal, amortiguando la fuerza aplicada y, por lo tanto, reduce el sobreimpulso (error en el otro lado debido a una fuerza aplicada demasiado grande). Aplicar demasiada integral cuando el error es pequeño y la disminución conducirá a un sobreimpulso. Después de sobrepasar, si el controlador aplicara una gran corrección en la dirección opuesta y repetidamente sobrepasara la posición deseada, la salida oscilaría alrededor del punto de ajuste en una sinusoide constante, creciente o decreciente . Si la amplitud de las oscilaciones aumenta con el tiempo, el sistema es inestable. Si disminuyen, el sistema es estable. Si las oscilaciones permanecen en una magnitud constante, el sistema es marginalmente estable .
Control de amortiguación
Con el fin de lograr una llegada controlada a la posición deseada (SP) de manera oportuna y precisa, el sistema controlado debe amortiguarse críticamente . Un sistema de control de posición bien ajustado también aplicará las corrientes necesarias al motor controlado para que el brazo empuje y tire según sea necesario para resistir las fuerzas externas que intentan alejarlo de la posición requerida. El punto de ajuste en sí puede ser generado por un sistema externo, como un PLC u otro sistema informático, de modo que varíe continuamente según el trabajo que se espera que realice el brazo robótico. Un sistema de control PID bien ajustado permitirá que el brazo cumpla con estos requisitos cambiantes de la mejor manera posible.
Respuesta a disturbios
Si un controlador comienza desde un estado estable con error cero (PV = SP), los cambios adicionales del controlador serán en respuesta a cambios en otras entradas medidas o no medidas al proceso que afectan el proceso y, por lo tanto, el PV. Las variables que afectan el proceso distintas del MV se conocen como perturbaciones. Generalmente, los controladores se utilizan para rechazar perturbaciones e implementar cambios en los puntos de ajuste. Un cambio de carga en el brazo constituye una perturbación del proceso de control del brazo del robot.
Aplicaciones
En teoría, un controlador puede usarse para controlar cualquier proceso que tenga una salida medible (PV), un valor ideal conocido para esa salida (SP) y una entrada al proceso (MV) que afectará la PV relevante. Los controladores se utilizan en la industria para regular temperatura , presión , fuerza , velocidad de alimentación , [15] velocidad de flujo , composición química ( concentraciones de componentes ), peso , posición , velocidad y prácticamente todas las demás variables para las que existe una medición.
Teoría del controlador
- Esta sección describe la forma paralela o no interactiva del controlador PID. Para otros formularios, consulte la sección Nomenclatura alternativa y formularios PID .
El esquema de control PID recibe el nombre de sus tres términos correctores, cuya suma constituye la variable manipulada (MV). Los términos proporcional, integral y derivado se suman para calcular la salida del controlador PID. Definiendo como salida del controlador, la forma final del algoritmo PID es
dónde
- es la ganancia proporcional, un parámetro de ajuste,
- es la ganancia integral, un parámetro de ajuste,
- es la ganancia derivada, un parámetro de ajuste,
- is the error (SP is the setpoint, and PV( t) is the process variable),
- is the time or instantaneous time (the present),
- is the variable of integration (takes on values from time 0 to the present ).
Equivalently, the transfer function in the Laplace domain of the PID controller is
where is the complex frequency.
Proportional term
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 Kp, called the proportional gain constant.
The proportional term is given by
A high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable (see the section on loop tuning). In contrast, a small gain results in a small output response to a large input error, and a less responsive or less sensitive controller. If the proportional gain is too low, the control action may be too small when responding to system disturbances. Tuning theory and industrial practice indicate that the proportional term should contribute the bulk of the output change.[citation needed]
Steady-state error
The steady-state error is the difference between the desired final output and the actual one.[16] Because a non-zero error is required to drive it, a proportional controller generally operates with a steady-state error.[a] Steady-state error (SSE) is proportional to the process gain and inversely proportional to proportional gain. SSE may be mitigated by adding a compensating bias term to the setpoint AND output or corrected dynamically by adding an integral term.
Integral term
The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error. The integral 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 (Ki) and added to the controller output.
The integral term is given by
The integral term accelerates the movement of the process towards setpoint and eliminates the residual steady-state error that occurs with a pure proportional controller. However, since the integral term responds to accumulated errors from the past, it can cause the present value to overshoot the setpoint value (see the section on loop tuning).
Derivative term
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 Kd. The magnitude of the contribution of the derivative term to the overall control action is termed the derivative gain, Kd.
The derivative term is given by
Derivative action predicts system behavior and thus improves settling time and stability of the system.[17][18] An ideal derivative is not causal, 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[citation needed] – because of its variable impact on system stability in real-world applications.
Sintonización de bucle
Tuning 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 behavior, different applications have different requirements, and requirements may conflict with one another.
PID tuning is a difficult problem, even though there are only three parameters and in principle is simple to describe, because it must satisfy complex criteria within the limitations of PID control. There are accordingly various methods for loop tuning, and more sophisticated techniques are the subject of patents; this section describes some traditional manual methods for loop tuning.
Designing and tuning a PID controller appears to be conceptually intuitive, but can be hard in practice, if multiple (and often conflicting) objectives such as short transient and high stability are to be achieved. PID controllers often provide acceptable control using default tunings, but performance can generally be improved by careful tuning, and performance may be unacceptable with poor tuning. Usually, initial designs need to be adjusted repeatedly through computer simulations until the closed-loop system performs or compromises as desired.
Some processes have a degree of nonlinearity and so parameters that work well at full-load conditions don't work when the process is starting up from no-load; this can be corrected by gain scheduling (using different parameters in different operating regions).
Stability
If the PID controller parameters (the gains of the proportional, integral and derivative terms) are chosen incorrectly, the controlled process input can be unstable, i.e., its output diverges, with or without oscillation, and is limited only by saturation or mechanical breakage. Instability is caused by excess gain, particularly in the presence of significant lag.
Generally, stabilization of response is required and the process must not oscillate for any combination of process conditions and setpoints, though sometimes marginal stability (bounded oscillation) is acceptable or desired.[citation needed]
Mathematically, the origins of instability can be seen in the Laplace domain.[19]
The total loop transfer function is:
where
- is the PID transfer function and
- is the plant transfer function
The system is called unstable where the closed loop transfer function diverges for some .[19] This happens for situations where . Typically, this happens when with a 180 degree phase shift. Stability is guaranteed when for frequencies that suffer high phase shifts. A more general formalism of this effect is known as the Nyquist stability criterion.
Optimal behavior
The optimal behavior on a process change or setpoint change varies depending on the application.
Two basic requirements are regulation (disturbance rejection – staying at a given setpoint) and command tracking (implementing setpoint changes) – these refer to how well the controlled variable tracks the desired value. Specific criteria for command tracking include rise time and settling time. 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.
Overview of tuning methods
There are several methods for tuning a PID loop. The most effective methods generally involve the development of some form of process model, then choosing P, I, and D based on the dynamic model parameters. Manual tuning methods can be relatively time-consuming, particularly for systems with long loop times.
The choice of method will depend largely on whether or not 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 using this response to determine the control parameters.[citation needed]
Method | Advantages | Disadvantages |
---|---|---|
Manual tuning | No math required; online. | Requires experienced personnel.[citation needed] |
Ziegler–Nichols [b] | Proven method; online. | Process upset, some trial-and-error, very aggressive tuning.[citation needed] |
Tyreus Luyben | Proven method; online. | Process upset, some trial-and-error, very aggressive tuning.[citation needed] |
Software tools | Consistent tuning; online or offline - can employ computer-automated control system design (CAutoD) techniques; may include valve and sensor analysis; allows simulation before downloading; can support non-steady-state (NSS) tuning. | Some cost or training involved.[21] |
Cohen–Coon | Good process models. | Some math; offline; only good for first-order processes.[citation needed] |
Åström-Hägglund | Can be used for auto tuning; amplitude is minimum so this method has lowest process upset | The process itself is inherently oscillatory.[citation needed] |
Manual tuning
If the system must remain online, one tuning method is to first set and values to zero. Increase the until the output of the loop oscillates, then the should be set to approximately half of that value for a "quarter amplitude decay" type response. Then increase until any offset is corrected in sufficient time for the process. However, too much will cause instability. Finally, increase , if required, until the loop is acceptably quick to reach its reference after a load disturbance. However, too much will cause excessive response and overshoot. A fast PID loop tuning usually overshoots slightly to reach the setpoint more quickly; however, some systems cannot accept overshoot, in which case an overdamped closed-loop system is required, which will require a setting significantly less than half that of the setting that was causing oscillation.[citation needed]
Parameter | Rise time | Overshoot | Settling time | Steady-state error | Stability |
---|---|---|---|---|---|
Decrease | Increase | Small change | Decrease | Degrade | |
Decrease | Increase | Increase | Eliminate | Degrade | |
Minor change | Decrease | Decrease | No effect in theory | Improve if small |
Ziegler–Nichols method
Another heuristic tuning method is known as the Ziegler–Nichols method, introduced by John G. Ziegler and Nathaniel B. Nichols in the 1940s. As in the method above, the and gains are first set to zero. The proportional gain is increased until it reaches the ultimate gain, , at which the output of the loop starts to oscillate constantly. and the oscillation period are used to set the gains as follows:
Control Type | |||
---|---|---|---|
P | — | — | |
PI | — | ||
PID |
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 and are dependent on the oscillation period .
Cohen–Coon parameters
This method was developed in 1953 and is based on a first-order + time delay model. Similar to the Ziegler–Nichols method, a set of tuning parameters were developed to yield a closed-loop response with a decay ratio of 1/4. 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.
Relay (Åström–Hägglund) method
Published in 1984 by Karl Johan Åström and Tore Hägglund,[24] the relay method temporarily operates the process using bang-bang control and measures the resultant oscillations. The output is switched (as if by a relay, hence the name) between two values of the control variable. The values must be chosen so the process will cross the setpoint, but need not be 0% and 100%; by choosing suitable values, dangerous oscillations can be avoided.
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 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.
Specifically, the ultimate period is assumed to be equal to the observed period, and the ultimate gain is computed as where a is the amplitude of the process variable oscillation, and b is the amplitude of the control output change which caused it.
There are numerous variants on the relay method.[25]
First order with dead time model
The transfer function for a first-order process, with dead time, is:
where kp is the process gain, τp is the time constant, θ is the dead time, and u(s) is a step change input. Converting this transfer function to the time domain results in:
using the same parameters found above.
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 will ensure that the output is not changing due to a disturbance (for best results, try to minimize disturbances when performing the step test).
One way to determine the parameters for the first-order process is using the 63.2% method. In this method, the process gain (kp) is equal to the change in output divided by the change in input. The dead time (θ) is the amount of time between when the step change occurred and when the output first changed. The time constant (τp) 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 the time to reach a new steady-state value can take a while if the process has large time constants.[26]
Tuning software
Most modern industrial facilities no longer tune loops using the manual calculation methods shown above. Instead, PID tuning and loop optimization software are used to ensure consistent results. These software packages will gather the data, develop process models, and suggest optimal tuning. Some software packages can even develop tuning by gathering data from reference changes.
Mathematical PID loop tuning induces an impulse in the system and then uses the controlled system's frequency response to design the PID loop values. In loops with response times of several minutes, mathematical loop tuning is recommended, because trial and error can take days just to find a stable set of loop values. Optimal values are harder to find. Some digital loop controllers offer a self-tuning feature in which very small setpoint changes are sent to the process, allowing the controller itself to calculate optimal tuning values.
Another approach calculates initial values via the Ziegler–Nichols method, and uses a numerical optimization technique to find better PID coefficients.[27]
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.[28]
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 will model the dynamics of a process, through a disturbance, and calculate PID control parameters in response.[29]
Limitaciones
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 optimal control. The fundamental difficulty with PID control is that it is a feedback control system, with constant parameters, and no direct knowledge of the process, and thus overall performance is reactive and a compromise. While PID control is the best controller in an observer without a model of the process, better performance can be obtained by overtly modeling the actor of the process without resorting to an observer.
PID controllers, when used alone, can give poor performance when the PID loop gains must be reduced so that the control system does not overshoot, oscillate or hunt about the control setpoint value. They also have difficulties in the presence of non-linearities, may trade-off regulation versus response time, do not react to changing process behavior (say, the process changes after it has warmed up), and have lag in responding to large disturbances.
The most significant improvement is to incorporate feed-forward control with knowledge about the system, and using the PID only to control error. Alternatively, PIDs can be modified in more minor ways, such as by changing the parameters (either gain scheduling in different use cases or adaptively modifying them based on performance), improving measurement (higher sampling rate, precision, and accuracy, and low-pass filtering if necessary), or cascading multiple PID controllers.
Linearity
Another problem faced with PID controllers is that they are linear and symmetric. Thus, performance of PID controllers in non-linear systems (such as HVAC systems) is variable. For example, in temperature control, a common use case is active heating (via a heating element) but passive cooling (heating off, but no cooling), so overshoot can only be corrected slowly – it cannot be forced downward. In this case the PID should be tuned to be overdamped, to prevent or reduce overshoot, though this reduces performance (it increases settling time).
Noise in derivative
A problem with the derivative term is that it amplifies higher frequency measurement or process noise that can cause large amounts of change in the output. It is often helpful to filter the measurements with a low-pass filter 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 median filter may be used, which improves the filtering efficiency and practical performance.[30] 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 PI controller.
Modificaciones al algoritmo
The basic PID algorithm presents some challenges in control applications that have been addressed by minor modifications to the PID form.
Integral windup
One common problem resulting from the ideal PID implementations is integral windup. 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:
- Disabling the integration until the PV has entered the controllable region
- Preventing the integral term from accumulating above or below pre-determined bounds
- Back-calculating the integral term to constrain the regulator output within feasible bounds.[31]
Overshooting from known disturbances
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 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.
PI controller
A PI controller (proportional-integral controller) is a special case of the PID controller in which the derivative (D) of the error is not used.
The controller output is given by
where is the error or deviation of actual measured value (PV) from the setpoint (SP).
A PI controller can be modelled easily in software such as Simulink or Xcos using a "flow chart" box involving Laplace operators:
where
- = proportional gain
- = integral gain
Setting a value for is often a trade off between decreasing overshoot and increasing settling time.
The lack of derivative action may make the system more steady in the steady state in the case of noisy data. This is because derivative action is more sensitive to higher-frequency terms in the inputs.
Without derivative action, a PI-controlled system is less responsive to real (non-noise) and relatively fast alterations in state and so the system will be slower to reach setpoint and slower to respond to perturbations than a well-tuned PID system may be.
Deadband
Many PID loops control a mechanical device (for example, a valve). Mechanical maintenance can be a major cost and wear leads to control degradation in the form of either stiction or backlash in the mechanical response to an input signal. The rate of mechanical wear is mainly a function of how often a device is activated to make a change. Where wear is a significant concern, the PID loop may have an output deadband to reduce the frequency of activation of the output (valve). This is accomplished by modifying the controller to hold its output steady if the change would be small (within the defined deadband range). The calculated output must leave the deadband before the actual output will change.
Setpoint step change
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 the case of an instantaneous step change. As a result, some PID algorithms incorporate some of the following modifications:
- Setpoint ramping
- In this modification, the setpoint is gradually moved from its old value to a newly specified value using a linear or first-order differential ramp function. This avoids the discontinuity present in a simple step change.
- Derivative of the process variable
- In this case the PID controller measures the derivative of the measured process variable (PV), rather than the derivative of the error. This quantity is always continuous (i.e., never has a step change as a result of changed setpoint). This modification is a simple case of setpoint weighting.
- Setpoint weighting
- Setpoint weighting adds adjustable factors (usually between 0 and 1) to the setpoint in the error in the proportional and derivative element of the controller. The error in the integral term must be the true control error to avoid steady-state control errors. These two extra parameters do not affect the response to load disturbances and measurement noise and can be tuned to improve the controller's setpoint response.
Feed-forward
The control system performance can be improved by combining the feedback (or closed-loop) control of a PID controller with feed-forward (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 error 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 the setpoint and on extra measured disturbances. Setpoint weighting is a simple form of feed forward.
For example, in most motion control systems, in order to accelerate a mechanical load under control, more force is required from the actuator. If a velocity loop PID controller is being used to control the speed of the load and command the force being applied by the actuator, then it is beneficial to take the desired instantaneous acceleration, scale that value appropriately and add it to the output of the PID velocity loop controller. This means that whenever the load is being accelerated or decelerated, a proportional amount of force is commanded from the actuator regardless of the feedback value. The PID loop in this situation uses the feedback information to change the combined output to reduce the remaining difference between the process setpoint and the feedback value. Working together, the combined open-loop feed-forward controller and closed-loop PID controller can provide a more responsive control system.
Bumpless operation
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.[32] A partial implementation is to store the integral gain times the error rather than storing the error and postmultiplying by the integral gain, which prevents discontinuous output when the I gain is changed, but not the P or D gains.
Other improvements
In addition to feed-forward, PID controllers are often enhanced through methods such as PID gain scheduling (changing parameters in different operating conditions), fuzzy logic, or computational verb logic.[33][34] 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 fractional order. The order of the integrator and differentiator add increased flexibility to the controller.[35]
Control en cascada
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 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[citation needed] that the working frequency of the controller is increased and the time constant of the object is reduced by using cascaded PID controllers.[vague].
For example, a temperature-controlled circulating bath has two PID controllers in cascade, each with its own thermocouple temperature sensor. The outer controller controls the temperature of the water using a thermocouple located far from the heater, where it accurately reads the temperature of the bulk of the water. The error term of this PID controller is the difference between the desired bath temperature and measured temperature. Instead of controlling the heater directly, the outer PID controller sets a heater temperature goal for the inner PID controller. The inner PID controller controls the temperature of the heater using a thermocouple attached to the heater. The inner controller's error term is the difference between this heater temperature setpoint and the measured temperature of the heater. Its output controls the actual heater to stay near this setpoint.
The proportional, integral, and differential terms of the two controllers will be very different. The outer PID controller has a long time constant – all the water in the tank needs to heat up or cool down. The inner loop responds much more quickly. Each controller can be tuned to match the physics of the system it controls – heat transfer and thermal mass of the whole tank or of just the heater – giving better total response.[36][37]
Nomenclatura y formas alternativas
Standard versus parallel (ideal) form
The form of the PID controller most often encountered in industry, and the one most relevant to tuning algorithms is the standard form. In this form the gain is applied to the , and terms, yielding:
where
- is the integral time
- is the derivative time
In this standard form, the parameters have a clear physical meaning. In particular, the inner summation produces a new single error value which is compensated for future and past errors. The proportional error term is the current error. The derivative components term attempts to predict the error value at 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 seconds (or samples). The resulting compensated single error value is then scaled by the single gain to compute the control variable.
In the parallel form, shown in the controller theory section
the gain parameters are related to the parameters of the standard form through and . 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.
Reciprocal gain, a.k.a. proportional band
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 dimensioned units such as temperature. It is common in this case to express the gain not as "output per degree", but rather in the reciprocal form of a proportional band , 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.
Basing derivative action on PV
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 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.
Basing proportional action on PV
Most commercial control systems offer the option 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 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.
King[38] describes an effective chart-based method.
Laplace form
Sometimes it is useful to write the PID regulator in Laplace transform form:
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.
Series/interacting form
Another representation of the PID controller is the series, or interacting form
where the parameters are related to the parameters of the standard form through
- , , and
with
- .
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.
Discrete implementation
The analysis for designing a digital implementation of a PID controller in a microcontroller (MCU) or FPGA device requires the standard form of the PID controller to be discretized.[39] Approximations for first-order derivatives are made by backward finite differences. The integral term is discretized, with a sampling time , as follows,
The derivative term is approximated as,
Thus, a velocity algorithm for implementation of the discretized PID controller in a MCU is obtained by differentiating , using the numerical definitions of the first and second derivative and solving for and finally obtaining:
s.t.
Pseudocódigo
Here is a simple software loop that implements a PID algorithm:[40]
- Kp - proportional gain
- Ki - integral gain
- Kd - derivative gain
- dt - loop interval time
previous_error := 0integral := 0loop: error := setpoint − measured_value proportional := error; integral := integral + error × dt derivative := (error − previous_error) / dt output := Kp × proportional + Ki × integral + Kd × derivative previous_error := error wait(dt) goto loop
In this example, two variables that will be maintained within the loop are initialized to zero, then the loop begins. The current error is calculated by subtracting the measured_value (the process variable, or PV) from the current setpoint (SP). Then, integral and derivative values are calculated, and these and the error are combined with three preset gain terms – the proportional gain, the integral gain and the derivative gain – to derive an output value.
In the real world, this is D-to-A converted and passed into the process under control as the manipulated variable (MV). The current error is stored elsewhere for re-use in the next differentiation, the program then waits until dt seconds have passed since start, and the loop begins again, reading in new values for the PV and the setpoint and calculating a new value for the error.[40]
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 preemption delaying the algorithm.
Ver también
- Control theory
Notas
- ^ The only exception is where the target value is the same as the value obtained when the controller output is zero.
- ^ A common assumption often made for Proportional-Integral-Derivative (PID) control design, as done by Ziegler and Nichols, is to take the integral time constant to be four times the derivative time constant. Although this choice is reasonable, selecting the integral time constant to have this value may have had something to do with the fact that, for the ideal case with a derivative term with no filter, the PID transfer function consists of two real and equal zeros in the numerator.[20]
Referencias
- ^ Araki, M. "PID Control" (PDF).
- ^ Hills, Richard L (1996), Power From the Wind, Cambridge University Press
- ^ Richard E. Bellman (December 8, 2015). Adaptive Control Processes: A Guided Tour. Princeton University Press. ISBN 9781400874668.
- ^ a b c d e f Bennett, Stuart (1996). "A brief history of automatic control" (PDF). IEEE Control Systems Magazine. 16 (3): 17–25. doi:10.1109/37.506394. Archived from the original (PDF) on 2016-08-09. Retrieved 2014-08-21.
- ^ Maxwell, J. C. (1868). "On Governors" (PDF). Proceedings of the Royal Society. 100.
- ^ Newpower, Anthony (2006). Iron Men and Tin Fish: The Race to Build a Better Torpedo during World War II. Praeger Security International. ISBN 978-0-275-99032-9. p. citing Gray, Edwyn (1991), The Devil's Device: Robert Whitehead and the History of the Torpedo, Annapolis, MD: U.S. Naval Institute, p. 33.
- ^ Sleeman, C. W. (1880), Torpedoes and Torpedo Warfare, Portsmouth: Griffin & Co., pp. 137–138,
which constitutes what is termed as the secret of the fish torpedo.
- ^ "A Brief Building Automation History". Archived from the original on 2011-07-08. Retrieved 2011-04-04.
- ^ Minorsky, Nicolas (1922). "Directional stability of automatically steered bodies". J. Amer. Soc. Naval Eng. 34 (2): 280–309. doi:10.1111/j.1559-3584.1922.tb04958.x.
- ^ Bennett 1993, p. 67
- ^ Bennett, Stuart (June 1986). A history of control engineering, 1800-1930. IET. pp. 142–148. ISBN 978-0-86341-047-5.
- ^ Shinskey, F Greg (2004), The power of external-reset feedback (PDF), Control Global
- ^ Neuhaus, Rudolf. "Diode Laser Locking and Linewidth Narrowing" (PDF). Retrieved June 8, 2015.
- ^ "Position control system" (PDF). Hacettepe University Department of Electrical and Electronics Engineering.
- ^ 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". Material Processing Technology. 213 (1): 2015–2032. doi:10.1016/j.jmatprotec.2013.05.023.
- ^ Lipták, Béla G. (2003). Instrument Engineers' Handbook: Process control and optimization (4th ed.). CRC Press. p. 108. ISBN 0-8493-1081-4.
- ^ "Introduction: PID Controller Design". University of Michigan.
- ^ Tim Wescott (October 2000). "PID without a PhD" (PDF). EE Times-India. Cite journal requires
|journal=
(help) - ^ a b Bechhoefer, John (2005). "Feedback for Physicists: A Tutorial Essay On Control". Reviews of Modern Physics. 77 (3): 783–835. Bibcode:2005RvMP...77..783B. CiteSeerX 10.1.1.124.7043. doi:10.1103/revmodphys.77.783.
- ^ Atherton, Drek P (December 2014). "Almost Six Decades in Control Engineering". IEEE Control Systems Magazine. 34 (6): 103–110. doi:10.1109/MCS.2014.2359588. S2CID 20233207.
- ^ Li, Y., et al. (2004) CAutoCSD - Evolutionary search and optimisation enabled computer automated control system design, Int J Automation and Computing, vol. 1, No. 1, pp. 76-88. ISSN 1751-8520.
- ^ Kiam Heong Ang; Chong, G.; Yun Li (2005). "PID control system analysis, design, and technology" (PDF). IEEE Transactions on Control Systems Technology. 13 (4): 559–576. doi:10.1109/TCST.2005.847331. S2CID 921620.
- ^ Jinghua Zhong (Spring 2006). "PID Controller Tuning: A Short Tutorial" (PDF). Archived from the original (PDF) on 2015-04-21. Retrieved 2011-04-04. Cite journal requires
|journal=
(help) - ^ Åström, K.J.; Hägglund, T. (July 1984). "Automatic Tuning of Simple Regulators". IFAC Proceedings Volumes. 17 (2): 1867–1872. doi:10.1016/S1474-6670(17)61248-5.
- ^ Hornsey, Stephen (29 October 2012). "A Review of Relay Auto-tuning Methods for the Tuning of PID-type Controllers". Reinvention. 5 (2).
- ^ Bequette, B. Wayne (2003). Process Control: Modeling, Design, and Simulation. Upper Saddle River, New Jersey: Prentice Hall. p. 129. ISBN 978-0-13-353640-9.
- ^ Heinänen, Eero (October 2018). A Method for automatic tuning of PID controller following Luus-Jaakola optimization (PDF) (Master's Thesis ed.). Tampere, Finland: Tampere University of Technology. Retrieved Feb 1, 2019.
- ^ Li, Yun; Ang, Kiam Heong; Chong, Gregory C.Y. (February 2006). "Patents, software, and hardware for PID control: An overview and analysis of the current art" (PDF). IEEE Control Systems Magazine. 26 (1): 42–54. doi:10.1109/MCS.2006.1580153. S2CID 18461921.
- ^ Soltesz, Kristian (January 2012). On Automation of the PID Tuning Procedure (PDF) (Licentiate theis). Lund university. 847ca38e-93e8-4188-b3d5-8ec6c23f2132.
- ^ Li, Y. and Ang, K.H. and Chong, G.C.Y. (2006) PID control system analysis and design - Problems, remedies, and future directions. IEEE Control Systems Magazine, 26 (1). pp. 32-41. ISSN 0272-1708
- ^ Cooper, Douglas. "Integral (Reset) Windup, Jacketing Logic and the Velocity PI Form". Retrieved 2014-02-18.
- ^ Cooper, Douglas. "PI Control of the Heat Exchanger". Practical Process Control by Control Guru. Retrieved 2014-02-27.
- ^ Yang, T. (June 2005). "Architectures of Computational Verb Controllers: Towards a New Paradigm of Intelligent Control". International Journal of Computational Cognition. 3 (2): 74–101. CiteSeerX 10.1.1.152.9564.
- ^ Liang, Yilong; Yang, Tao (2009). "Controlling fuel annealer using computational verb PID controllers". Proceedings of the 3rd International Conference on Anti-Counterfeiting, Security, and Identification in Communication: 417–420.
- ^ Tenreiro Machado JA, et al. (2009). "Some Applications of Fractional Calculus in Engineering". Mathematical Problems in Engineering. 2010: 1–34. doi:10.1155/2010/639801. hdl:10400.22/4306.
- ^ [1] 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
- ^ [2] | The Benefits of Cascade Control | September 22, 2020 | Watlow
- ^ King, Myke (2011). Process Control: A Practical Approach. Wiley. pp. 52–78. ISBN 978-0-470-97587-9.
- ^ "Discrete PI and PID Controller Design and Analysis for Digital Implementation". Scribd.com. Retrieved 2011-04-04.
- ^ a b "PID process control, a "Cruise Control" example". CodeProject. 2009. Retrieved 4 November 2012.
- Bequette, B. Wayne (2006). Process Control: Modeling, Design, and Simulation. Prentice Hall PTR. ISBN 9789861544779.
Otras lecturas
- Liptak, Bela (1995). Instrument Engineers' Handbook: Process Control. Radnor, Pennsylvania: Chilton Book Company. pp. 20–29. ISBN 978-0-8019-8242-2.
- Tan, Kok Kiong; Wang Qing-Guo; Hang Chang Chieh (1999). Advances in PID Control. London, UK: Springer-Verlag. ISBN 978-1-85233-138-2.
- King, Myke (2010). Process Control: A Practical Approach. Chichester, UK: John Wiley & Sons Ltd. ISBN 978-0-470-97587-9.
- Van Doren, Vance J. (July 1, 2003). "Loop Tuning Fundamentals". Control Engineering.
- Sellers, David. "An Overview of Proportional plus Integral plus Derivative Control and Suggestions for Its Successful Application and Implementation" (PDF). Archived from the original (PDF) on March 7, 2007. Retrieved 2007-05-05.
- Graham, Ron; Mike McHugh (2005-10-03). "FAQ on PID controller tuning". Mike McHugh. Archived from the original on February 6, 2005. Retrieved 2009-01-05.
- Aidan O'Dwyer (2009). Handbook of PI and PID Controller Tuning Rules (PDF) (3rd ed.). Imperial College Press. ISBN 978-1-84816-242-6.
enlaces externos
- PID tuning using Mathematica
- PID tuning using Python
- Principles of PID Control and Tuning
- Introduction to the key terms associated with PID Temperature Control
PID tutorials
- PID Control in MATLAB/Simulink and Python with TCLab
- What's All This P-I-D Stuff, Anyhow? Article in Electronic Design
- Shows how to build a PID controller with basic electronic components (pg. 22)
- PID Without a PhD
- PID Control with MATLAB and Simulink
- PID with single Operational Amplifier
- Proven Methods and Best Practices for PID Control
- Principles of PID Control and Tuning
- PID Tuning Guide: A Best-Practices Approach to Understanding and Tuning PID Controllers
- Michael Barr (2002-07-30), Introduction to Closed-Loop Control, Embedded Systems Programming, archived from the original on 2010-02-09
- Jinghua Zhong, Mechanical Engineering, Purdue University (Spring 2006). "PID Controller Tuning: A Short Tutorial" (PDF). Archived from the original (PDF) on 2015-04-21. Retrieved 2013-12-04.CS1 maint: multiple names: authors list (link)
- Introduction to P,PI,PD & PID Controller with MATLAB
Online calculators
- PID tutorial, free PID tuning tools, advanced PID control schemes, on-line PID simulators
- Online PID Tuning applet from University of Texas Control Group
- Online PID tuning application