PID – Basic Control System Actions

BEFORE:  Steady-State error control system

NEXT: PID – Effect of integrative and derivative control actions.

Introduction

An automatic controller compares the real value of the output of a plant with the input reference (the desired value), determines the deviation and produces a control signal that will reduce the deviation to zero or a small value. The way in which the automatic controller produces the control signal is called control action.

Classification of industrial controls

According to their control actions, industrial controllers are classified as:

  1. Two-position (On / Off)
  2. Proportional
  3. Integrals
  4. Proportional-Integrals
  5. Proportional-Derivatives
  6. Proportional-Integrals-Derivatives

Almost all industrial controllers use electricity as an energy source or a pressurized fluid, such as oil or air. The controllers can also be classified, according to the type of energy they use in their operation, like pneumatic, hydraulic or electronic. The type of controller that is used must be decided based on the nature of the plant and operational conditions, including considerations such as safety, cost, availability, reliability, precision, weight and size.

Figure 5-1 shows a typical configuration for an Industrial Control System:

The previous figure consists of a Block Diagram for an industrial control system composed of an automatic controller, an actuator, a plant and a sensor (measuring element). The controller detects the error signal, which is usually at a very low power level, and amplifies it to a sufficiently high level. The output of an automatic controller feeds an actuator that can be a pneumatic valve or an electric motor. The actuator is a power device that produces the input for The plant according to the control signal, so that the output signal approaches the reference input signal. The sensor, or measurement element, is a device that converts an output variable, such as a displacement, into another manageable variable, such as a voltage, that can be used to compare the output with the reference input signal. This element is in the feedback path of the closed-loop system. The setpoint of the controller must be converted into a reference input with the same units as the feedback signal from the sensor or from the measuring element.

Two positions control (On / Off).

In a two position control system, the acting element only has two fixed positions that, in many cases, are simply turned on and off. The On/Off control is relatively simple and cheap, which is why it is extensively used in industrial and domestic control systems.

Suppose that the output signal of the controller is u(t) and that the error signal is e(t). In the control of two positions, the signal u(t) remains at a value either maximum or minimum, depending on whether the error signal is positive or negative. In this way,

where U1 y U2 are constants. Very often, the minimum value of U2 is zero or –U1.

It is common for two-position controllers to be electrical devices, in which case an electrical valve operated by solenoids is widely used. Pneumatic proportional controllers with very high gains function as two-position controllers and are sometimes referred to as two-position pneumatic controllers.

Figures 5-3 (a) and (b) show the block diagrams for two controllers of two positions The range in which the error signal must move before the commutation is called differential gap. In Figure 5-3 (b) a differential gap is indicated. Such a gap causes the output of the controller u (t) to retain its present value until the error signal has moved slightly beyond zero. In some cases, the differential gap is the result of unintentional friction and a lost movement; however, it is often intentionally caused to avoid too frequent operation of the on and off mechanism.

Proportional control action.

For a controller with proportional control action, the relationship between the controller output u (t) and the error signal e (t) is:

or, in quantities transformed by the Laplace method:

where Kp is considered proportional gain.

Whatever the actual mechanism and the form of the operating power, the controller

proportional is, in essence, an amplifier with an adjustable gain. A block diagram of such a controller is presented in Figure 5-6.

Integral control action.

In a controller with integral control action, the value of the controller output u (t) is changed to a ratio proportional to the error signal e (t). That is to say,

O well:

where Ki is an adjustable constant. The transfer function of the integral controller is:

If the value of e (t) is doubled, the value of u (t) varies twice as fast. For an error of zero, the value of u (t) remains stationary. Sometimes, the integral control action is called adjustment control (reset). Figure 5-7 shows a block diagram of such controller.

 

Integral-proportional control action.
The control action of a proportional-integral controller (PI) is defined by:

null

or the transfer function of the controller, which is:

null

where Kp is the proportional gain and Ti is called integral time. Both Kp and Ti are adjustable. Integral time adjusts the integral control action, while a change in the value of Kp affects the integral and proportional parts of the control action.

The inverse of the integral time Ti is called the readjustment speed. The rate of readjustment is the number of times per minute that the proportional part of the control action is doubled. The rate of readjustment is measured in terms of the repetitions per minute. The Figure 5-8 (a) shows a block diagram of a PI controller. If the error signal e (t) is a unit step function, as shown in Figure 5-8 (b), the controller output u (t) becomes what is shown in Figure 5-8 (c).

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Proportional-derivative control action.
The control action of a proportional-derivative (PD) controller is defined by:

null

The transfer function is:

null

where Kp is the proportional gain and Td is a constant called derivative time. Both Kp and Td are adjustable. The derivative control action, sometimes called speed control, occurs where the magnitude of the controller output is proportional to the rate of change of the error signal. The derivative time Td is the time interval during which the action of the velocity advances the effect of the proportional control action.

Figure 5-9 (a) shows a block diagram of a PD controller. If the error signal e (t) is a unit ramp function as shown in Figure 5-9 (b), the controller output u (t) becomes that shown in Figure 5-9 (c). ). The derivative control action has a forecast nature. However, it is obvious that a derivative control action never foresees an action that has never occurred.null

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Although the derivative control action has the advantage of being forecast, it has the disadvantages that it amplifies the noise signals and can cause a saturation effect in the actuator. Note that the derivative control action is never used alone, because it is only effective during transient periods.

Proportional-Integral-derivative (PID) control action.

The combination of a proportional control action, an integral control action and a derivative control action is called proportional-integral-derivative (PID) control action.

This combined action has the advantages of each of the three individual control actions. The equation of a controller with this combined action is obtained by:

null

The transfer function is:

null

where Kp is the proportional gain, Ti is the integral time and Td is the derivative time. The block diagram of a PID controller appears in Figure 5-10 (a). If e (t) is a unit ramp function, like the one shown in Fig. 5-10 (b), the controller output u (t) becomes that of Fig. 5-10 (c).

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Effects of the sensor on the performance of the system.

Since the dynamic and static characteristics of the sensor or measuring element affects the indication of the actual value of the output variable, the sensor fulfills a function important to determine the overall performance of the control system. As usual, the sensor determines the transfer function in the feedback path. If the time constants of a sensor are negligible compared to other constants of time of the control system, the sensor transfer function simply it becomes a constant. Figures 5-11 (a), (b) and (c) show diagrams of automatic controller blocks with a first-order sensor, an overdamped second-order sensor and a second-order underdamped sensor, respectively. Often the response of a thermal sensor is of the overdamped second order type.

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BEFORE:  Steady-State error control system

NEXT: PID – Effect of integrative and derivative control actions.

Source:

  1. Ingenieria de Control Moderna, 3° ED. – Katsuhiko Ogata pp 211-232

Literature review by Larry Francis Obando – Technical Specialist – Educational Content Writer

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Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

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PID – Acciones Básicas de Sistemas de Control

ANTERIOR: Error en estado estable de un sistema de control

SIGUIENTE: PID – Efecto de las acciones de control Integral y Derivativo

Introducción

Un controlador automático compara el valor real de la salida de una planta con la entrada de referencia (el valor deseado), determina la desviación y produce una señal de control que reducirá la desviación a cero o a un valor pequeño. La manera en la cual el controlador automático produce la señal de control se denomina acción de control.

Clasificación de los controladores industriales.

De acuerdo con sus acciones de control, los controladores industriales se clasifican en:

  1. De dos posiciones o de encendido y apagado (On/Off)
  2. Proporcionales
  3. Integrales
  4. Proporcionales-Integrales
  5. Proporcionales-Derivativos
  6. Proporcionales-Integrales-Derivativos

Casi todos los controladores industriales emplean como fuente de energía la electricidad o un fluido presurizado, tal como el aceite o el aire. Los controladores también pueden clasificarse, de acuerdo con el tipo de energía que utilizan en su operación, como neumáticos, hidráulicos o electrónicos. El tipo de controlador que se use debe decidirse con base en la naturaleza de la planta y las condiciones operacionales, incluyendo consideraciones tales como seguridad, costo, disponibilidad, confiabilidad, precisión, peso y tamaño.

La Figura 5-1 muestra la configuración típica de un Sistema de Control Industrial:

La figura anterior consiste en un Diagrama de Bloques para un sistema de control industrial compuesto por un controlador automático, un actuador, una planta y un sensor (elemento de medición). El controlador detecta la señal de error, que por lo general, está en un nivel de potencia muy bajo, y la amplifica a un nivel lo suficientemente alto. La salida de un controlador automático alimenta a un actuador que puede ser una válvula neumática o un motor eléctrico. El actuador es un dispositivo de potencia que produce la entrada para La planta de acuerdo con la señal de control, a fin de que la señal de salida se aproxime a la señal de entrada de referencia. El sensor, o elemento de medición, es un dispositivo que convierte la variable de salida, tal como un desplazamiento, en otra variable manejable, tal como un voltaje, que pueda usarse para comparar la salida con la señal de entrada de referencia. Este elemento está en la trayectoria de realimentación del sistema en lazo cerrado. El punto de ajuste del controlador debe convertirse en una entrada de referencia con las mismas unidades que la señal de realimentación del sensor o del elemento de medición.

Acción de control de dos posiciones o de encendido y apagado (on/off).

En un sistema de control de dos posiciones, el elemento de actuación sólo tiene dos posiciones fijas que, en muchos casos, son simplemente encendido y apagado. El control de dos posiciones o de encendido y apagado es relativamente simple y barato, razón por la cual su uso es extendido en sistemas de control tanto industriales como domésticos.

Supongamos que la señal de salida del controlador es u(t) y que la señal de error es e(t). En el control de dos posiciones, la señal u(t) permanece en un valor ya sea máximo o mínimo, dependiendo de si la señal de error es positiva o negativa. De este modo,

donde U1 y U2 son constantes. Por lo general, el valor mínimo de U2 es cero ó –U1.

Es común que los controladores de dos posiciones sean dispositivos eléctricos, en cuyo caso se usa extensamente una válvula eléctrica operada por solenoides. Los controladores neumáticos proporcionales con ganancias muy altas funcionan como controladores de dos posiciones y, en ocasiones, se denominan controladores neumáticos de dos posiciones.

Las figuras 5-3(a) y (b) muestran los diagramas de bloques para dos controladores de dos posiciones. El rango en el que debe moverse la señal de error antes de que ocurra la conmutación se denomina brecha diferencial. En la figura 5-3(b) se señala una brecha diferencial. Tal brecha provoca que la salida del controlador u(t) conserve su valor presente hasta que la señal de error se haya desplazado ligeramente más allá de cero. En algunos casos, la brecha diferencial es el resultado de una fricción no intencionada y de un movimiento perdido; sin embargo, con frecuencia se provoca de manera intencional para evitar una operación demasiado frecuente del mecanismo de encendido y apagado.

Acción de control proporcional.

Para un controlador con acción de control proporcional, la relación entre la salida del controlador u(t) y la señal de error e(t) es:

o bien, en cantidades transformadas por el método de Laplace:

donde Kp se considera la ganancia proporcional.

Cualquiera que sea el mecanismo real y la forma de la potencia de operación, el controlador proporcional es, en esencia, un amplificador con una ganancia ajustable. En la figura 5-6 se presenta un diagrama de bloques de tal controlador.

Acción de control integral.

En un controlador con acción de control integral, el valor de la salida del controlador u(t) se cambia a una razón proporcional a la señal de error e(t). Es decir,

o bien:

en donde Ki es una constante ajustable. La función de transferencia del controlador integral es:

Si se duplica el valor de e(t), el valor de u(t) varía dos veces más rápido. Para un error de cero, el valor de u(t) permanece estacionario. En ocasiones, la acción de control integral se denomina control de reajuste (reset). La figura 5-7 muestra un diagrama de bloques de tal controlador.

Acción de control integral-proporcional.

La acción de control de un controlador proporcional-integral (PI) se define mediante:

null

o la función de transferencia del controlador, la cual es:

null

en donde Kp es la ganancia proporcional y Ti se denomina tiempo integral. Tanto Kp como Ti son ajustables. El tiempo integral ajusta la acción de control integral, mientras que un cambio en el valor de Kp afecta las partes integral y proporcional de la acción de control.

El inverso del tiempo integral Ti se denomina velocidad de reajuste. La velocidad de reajuste es la cantidad de veces por minuto que se duplica la parte proporcional de la acción de control. La velocidad de reajuste se mide en términos de las repeticiones por minuto. La Figura 5-8(a) muestra un diagrama de bloques de un controlador proporcional más integral. Si la señal de error e(t) es una función escalón unitario, como se aprecia en la Figura 5-8(b), la salida del controlador u(t) se convierte en lo que se muestra en la figura 5-8(c).

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Acción de control proporcional-derivativa.

La acción de control de un controlador proporcional-derivativa (PD) se define mediante:

null

la función de transferencia es:

null

en donde Kp es la ganancia proporcional y Td es una constante denominada tiempo derivativo. Tanto Kp como Td son ajustables. La acción de control derivativa, en ocasiones denominada control de velocidad, ocurre donde la magnitud de la salida del controlador es proporcional a la velocidad de cambio de la señal de error. El tiempo derivativo Td es el intervalo de tiempo durante el cual la acción de la velocidad hace avanzar el efecto de la acción de control proporcional.

La Figura 5-9(a) muestra un diagrama de bloques de un controlador PD. Si la señal de error e(t) es una función rampa unitaria como se aprecia en la Figura 5-9(b), la salida del controlador u(t) se convierte en la que se muestra en la figura 5-9(c). La acción de control derivativa tiene un carácter de previsión. Sin embargo, es obvio que una acción de control derivativa nunca prevé una acción que nunca ha ocurrido.

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Aunque la acción de control derivativa tiene la ventaja de ser de previsión, tiene las desventajas de que amplifica las señales de ruido y puede provocar un efecto de saturación en el actuador. Observe que la acción de control derivativa no se usa nunca sola, debido a que sólo es eficaz durante periodos transitorios.

Acción de control proporcional-Integral-derivativa (PID)

La combinación de una acción de control proporcional, una acción de control integral y una acción de control derivativa se denomina acción de control proporcional-integral-derivativa (PID). 

Esta acción combinada tiene las ventajas de cada una de las tres acciones de control individuales. La ecuación de un controlador con esta acción combinada se obtiene mediante:

null

la función de transferencia es:

null

en donde Kp es la ganancia proporcional, Ti es el tiempo integral y Td es el tiempo derivativo. El diagrama de bloques de un controlador PID aparece en la figura 5-10(a). Si e(t) es una función rampa unitaria, como la que se observa en la figura 5-10(b), la salida del controlador u(t) se convierte en la de la figura 5-10(c).

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nullnull

 

Efectos del sensor sobre el desempeño del sistema.

Dado que las características dinámica y estática del sensor o del elemento de medición afecta la indicación del valor real de la variable de salida, el sensor cumple una función importante para determinar el desempeño general del sistema de control. Por lo general, el sensor determina la función de transferencia en la trayectoria de realimentación. Si las constantes de tiempo de un sensor son insignificantes en comparación con otras constantes de tiempo del sistema de control, la función de transferencia del sensor simplemente se convierte en una constante. Las figuras 5-11(a), (b) y (c) muestran diagramas de bloques de controladores automáticos con un sensor de primer orden, un sensor de segundo orden sobreamortiguado y un sensor de segundo orden subamortiguado, respectivamente. Con frecuencia la respuesta de un sensor térmico es del tipo de segundo orden sobreamortiguado.

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ANTERIOR: Error en estado estable de un sistema de control

SIGUIENTE: PID – Efecto de las acciones de control Integral y Derivativo

Fuente:

  1. Ingenieria de Control Moderna, 3° ED. – Katsuhiko Ogata, pp 211-232

 

Revisión hecha por Larry Francis Obando – Technical Specialist – Educational Content Writer

Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, CCs.

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contact: Caracas, Quito, Guayaquil, Cuenca. telf – 0998524011

WhatsApp: +593998524011

email: dademuchconnection@gmail.com

Steady-State error – Control Systems

BEFORE: Control System Stability

NEXT: PID -Basic control system actions

Introduction

Errors in a control system can be attributed to many factors. Changes in the reference input will cause unavoidable errors during transient periods and may also cause steady-state errors. Imperfections in the system components, such as static friction, backlash, and amplifier drift, as well as aging or deterioration, will cause errors at steady state. In this section, however, we shall not discuss errors due to imperfections in the system components. Rather, we shall investigate a type of steady-state error that is caused by the incapability of a system to follow particular types of inputs.

Steady-state error is the difference between the input and the output for a prescribed test input as time tends to infinity. Test inputs used for steady-state error analysis and design are summarized in Table 7.1. In order to explain how these test signals are used, let us assume a position control system, where the output position follows the input commanded position.

Step inputs represent constant position and thus are useful in determining the ability of the control system to position itself with respect to a stationary target. An antenna position control is an example of a system that can be tested for accuracy using step inputs.

Ramp inputs represent constant-velocity inputs to a position control system by their linearly increasing amplitude. These waveforms can be used to test a system’s ability to follow a linearly increasing input or, equivalently, to track a constant velocity target. For example, a position control system that tracks a satellite that moves across the sky at a constant angular velocity.

Parabolas inputs, whose second derivatives are constant, represent constant acceleration inputs to position control systems and can be used to represent accelerating targets, such as a missile.

Any physical control system inherently suffers steady-state error in response to certain types of inputs. A system may have no steady-state error to a step input, but the same system may exhibit nonzero steady-state error to a ramp input. (The only way we may be able to eliminate this error is to modify the system structure.) Whether a given system will exhibit steady-state error for a given type of input depends on the type of open-loop transfer function of the system.

Definition of the error in steady state depending on the configuration of the system.

The steady-state errors of linear control systems depend on the type of the reference signal and the type of system. Before undertaking the error in steady state, it must be clarified what is the meaning of the system error.

The error can be seen as a signal that should quickly be reduced to zero, if this is possible. Consider the system of Figure 7-5:

Where r (t) is the input signal, u (t) is the acting signal, b (t) is the feedback signal and y (t) is the output signal. The error e (t) of the system can be defined as:

We must remember that r (t) and y (t) do not necessarily have the same dimensions. On the other hand, when the system has unit feedback, H (s) = 1, the input r (t) is the reference signal and the error is simply:

That is, the error is the acting signal, u (t). When H (s) is not equal to 1, u (t) may or may not be the error, depending on the form and purpose of H (s). Therefore, the reference signal must be defined when H (s) is not equal to 1.

The error in steady state is defined as:

To establish a systematic study of the error in steady state for linear systems, we will classify the control systems as follows:

  1. Unit feedback systems,
  2. Non-unit feedback systems.

Steady-State Error in Unity-feedback control systems

Consider the system shown in Figure 5-49:

The closed-loop transfer function for this can be obtained as:

The transfer function between the error signal e(t) and the input signal r(t) is:

Where the error e(t) is the difference between the input signal and the output signal. The final-value theorem provides a convenient way to find the steady-state performance of a stable system. Since:

The steady-state error is:

This last equation allows us to calculate the steady-state error Ess, given the input R(s) and the transfer function G(s). We then substitute several inputs for R(s) and then draw conclusions about the relationship that exists between the open-loop system G(s)  and the nature of the steady-state error Ess.

  • Step Input: Using R(s) =1/S, we obtain:

Where:

Is the gain of the forward transfer function. In order to have zero steady-state error,

To satisfy the last condition, G(s) must have the following form:

And for the limit to be infinite the denominator must be equal to zero a S goes to zero. So n>=1, that is, at least one pole must be at the origin, equal to say that at least one pure integration must be present in the forward path. The steady-state respond for this case of zero steady-state  error is similar to that shown in Figure 7-2a, ouput 1.

If there are no integrations, the n=0, and it yields a finite error. This is the case shown in Figure 7-2a, output 2.

In summary, for a step input to a unity feedback system, the steady-state error will be zero if there is at least one pure integration in the forward path.  

  • Ramp Input:  Using R(s) =1/Sˆ2, we obtain:

To have zero steady-state error to a ramp input we must have:

To satisfy this G(s) must take the form where n>=2. In other words, there must be at least two integrations in the forward path. An example of a steady-state error for a ramp input is shown in Figure 7.2b, output 1:

If only one integrator exist in the forward path then lim sG(s) is finite rather tan infinite and this lead to a constant error, as shown in Figure 7.2b, output 2. If there is only one integrator in the forward path then lim sG(s) =0, and the steady-state error will be infinite and lead to diverging ramp, as shown in Figure 7.2b, output 3.

  • Parabolic Input: Using R(s) =1/Sˆ3, we obtain:

In order to have zero steady-state error for a parabolic input, we must have:

To satisfy this G(s) n must be n>=3. In other words, there must be at least three integrations in the forward path. If there only two integrators in the forward path then lim sˆ2G(s) is finite rather tan infinite and this lead to a constant error. If there one or zero integrators in the forward path then e() is infinite.

Classification of Control Systems (System Types) and Static Errors Constant.

System Type.  Control system may be classified according to their ability to follow step inputs, ramp inputs or parabolic inputs and so on. This is a reasonable classification scheme because most of the actual inputs can be considered a combination of such inputs. Consider the unity-feedback control system with the following open-loop transfer function G(s):

It involves the term SˆN in the denominator, representing a pole of multiplicity N at the origin. A system is called type 0, type 1, type 2,…if N=0, 1, 2…respectively. As the type increases, accuracy is improved. However, this agraves the stability problem. If  G(s) is written so that each term in the numerator and denominator, except the term SˆN, approaches unity as s approaches zero, then the open-loop gain K is directly related to the steady-state error.  

Static Error Constant. The Static Error Constants defined in the following are figures of merit of control systems. The higher the constants, the smaller the steady-state error.

  • Static Position Error Constant Kp. The steady-state error of a system for a unit-step input is:

The Static Position Error Constant Kp is defined by:

Thus the steady-state error in terms of the Static Position Error Constant Kp is given by:

For a type 0 system:

For a type 1 or higher system:

  • Static Velocity Error Constant Kv. The steady-state error of a system for a unit-ramp input is given by:

The Static Velocity Error Constant Kv is defined by:

Thus the steady-state error in terms of the Static Velocity Error Constant Kv is given by:

For a type 0 system:

For a type 1 system:

For For a type 2 system or higher:

  • Static Acceleration Error Constant Ka. The steady-state error of a system for a unit-parabolic input is given by:

The Static Acceleration Error Constant Kv is defined by:

Thus the steady-state error in terms of the Static Acceleration Error Constant Ka is given by:

For a type 0 system:

For a type 1 system:

For a type 2 system:

For a type 3 system or higher:

Table 7.2 ties together the concepts of steady-state error, static error constants and system type. The table shows the static error constants and the steady-state error as a functions of the input waveform and the system type.

Steady-State Error for Non-unity Feedback Systems.

Control systems often do not have unity feedback because of the compensation used to improve performance or because of the physical model of the system. In these cases the most practical way to analyze the steady-state error is to take the system and form a unity feedback system by adding and subtracting unity feedback paths as shown in Figure 7.15:

Donde G(s)=G1(s)G2(s) y H(s)=H1(s)/G1(s). Notice that these steps require that input and output signals have the same units.

BEFORE: Control System Stability

NEXT: PID -Basic control system actions

Sources:

  1. Control Systems Engineering, Nise pp 340, 353
  2. Sistemas de Control Automatico Benjamin C Kuo pp 390, 395
  3. Modern_Control_Engineering, Ogata 4t pp 301,305

Written by: Larry Francis Obando – Technical Specialist – Educational Content Writer.

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, Caracas.

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contact: Caracas, Quito, Guayaquil, Cuenca – Telf. 00593998524011

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Related:

The Block Diagram – Control Engineering

Dinámica de un Sistema Masa-Resorte-Amortiguador

Block Diagram of Electromechanical Systems – DC Motor

Transient-response Specifications

Control System Stability

Control System Stability

BEFORE: Transient-response specifications

NEXT: Steady-state error in Control Systems

Introduction

The most important problem in linear control systems concerns stability. It is the most important system specification among the three requirements enter into the design of a control system: transient response, stability and steady-state error. That is, under what conditions will a system become unstable? If it is unstable, how should we stabilize the system?

The total response of a system is the sum of the forced and natural responses:

null

Using this concept, we present the following definition of stability, instability and marginal stability:

  • A linear time-invariant system is stable if the natural response approaches to zero as time approaches infinity.
  • A linear time-invariant system is unstable if the natural responses grows without bound as time approaches infinity.
  • A linear time-invariant system is marginally stable if the natural response neither decays nor grows but remains constant or oscillates as time approaches infinity.

Thus the definition of stability implies that only the forced response remains as the natural response approaches zero. An alternate definition of stability is regards the total response and implies the first definition based upon the natural response:

  • A system is stable if every bounded input yields a bounded output. We call this statement a bounded-input, bounded-output (BIBO) definition of stability.

We now realize that if the input is bounded but the output is unbounded, the system is unstable. If the input is unbounded we will see an unbounded total response and we cannot draw any about conclusion about stability.

Physically, an unstable system whose natural response grows without bound can cause damage to the system, to adjacent property, or to human life. Many times systems are designed with limited stops to prevent total runaway.

Negative feedback tends to improve stability. From the study of the system poles we must recall that poles in the left half-plane (lhp) yield either pure exponential decay or damped sinusoidal natural responses. These natural responses decay to zero when time approaches infinity. Thus, if the closed-loop system poles are in the left half of the plane and hence have a negative real part, the system is stable. That is:

  • Stable systems have closed-loop transfer functions with poles only in the left half-plane.

Poles in the right half-plane (rhp) yield either pure exponentially increasing or exponentially increasing sinusoidal natural responses, which approche infinity when time approaches infinity. Thus:

  • Unstable systems have closed-loop transfer functions with at least one pole in the right half-plane and /or poles of multiplicity greater than 1 on the imaginary axis.

Finally, the system that has imaginary axis poles of multiplicity 1 yields pure sinusoidal oscillations as a natural response. These responses neither increase nor decrease in amplitude. Thus,

  • Marginally stable systems have closed-loop transfer functions with only imaginary axis poles of multiplicity one and poles in the left half-plane.

Figure 6.1a shows a unit step response of a stable system, while Figure 6.1b shows an unstable system.

null

[1]

Routh’s Stability Criterion

Routh’s Stability criterion tell us whether or not there are unstable roots in a polynomial equation without actually solving for them. This stability criterion applies to polynomials with only a finite number of terms. When the criterion is applied to a control system, information about absolute stability can be obtained directly from the coefficients of the characteristic equation.

The method requires two steps: 1) Generate a data table called a Routh Table and 2) Interpret the to tell how many closed-loop system poles are in the left half-plane, the right half-plane, and on the jw-axis. The power of the method lies in design rather than analysis. For example, if you have an unknown parameter in the denominator of a transfer function, it is difficult to determine via a calculator the range of this parameter to yield stability. We shall see that The Routh-Hurwitz criterion can yield a closed-form expression for the range of the unknown parameter.

  • Generating a basic Routh Table. Considering the equivalent closed-loop transfer function in Figure 6.3. Since we are interested in the system poles, we focus our attention on the denominator. We first create the Routh Table shown in Table 6.1:

null

Begin by labeling the rows with powers of s from the highest power of the denominator of the closed-loop transfer function to s^0. Next start with the coefficient of the highest power of s in the denominator and list, horizontally in the first row every other coefficient.

null

In the second row, list horizontally, starting with the next higher power of s, every coefficient that was skipped in the first row. The remain entries are filled as follows:

null

Each entry is a negative determinant of entries in the previous two rows divided by the entry in the first column directly above the calculated row. The left-hand column of the determinant is always the first column of the previous two rows, and the right-hand column is the elements of the column above and to the right.

Figures 6.4 shows an example of building the Routh Table :

null

null

null

The complete array of coefficients is triangular. Note that in developing the array an entire row maybe divided or multiplied by a positive number in order to simplify the subsequent numerical calculation without altering the stability conclusion.

Consider the following characteristic equation, example 5-13:

null

The first two rows can be obtained directly from the given polynomial. The second is divided by two but we arrive to the same conclusion:

null

  • Interpreting the basic Routh Table. Simply stated, the Routh-Hurwitz criterion declares that the number of roots of the polynomial that are in the right-half plane is equal to the number of sign changes in the first column.

If the closed-loop transfer function has all poles in the left-hand plane the system is stable. Thus, the system is stable if there are no sign changes in the first column of the Routh Table.

The last case, Example 5-13, is one of an unstable system. In that example the number of sign changes in the first column is equal to two. This means that there are two roots with positive real parts. Table 6.3 is also an unstable system. There, the first change occurs from 1 in the s^2 row to -72 in the s^1 row. The second occurs from -72 in the  s^1 row to 103 in the  s^0 row. Thus, the system has two poles in the right-half plane.

Routh’s stability criterion is of limited usefulness when applying to control system analysis because it does not suggest how to improve relative stability or how to stabilize a unstable system. It is possible, however, to determine the effects of changing one or two parameters of the system by examining the values that causes instability. In the following we consider the problem of determining the stability range of a parameter value. Consider the system of the Figure 5-38. Let us determine the range of K for stability.

nullnull

The characteristic equation is:

null

And the Routh Table:

null

For stability, K must be positive and all coefficient in the first column must be positive. Therefore:

null

When K=14/9 the system becomes oscillatory and, mathematically, the oscillation is sustained at constant amplitude.

Routh-Hurwitz Criterion Special Cases

Two special cases can occur: (1) The Routh table sometimes will have a zero only in the first column of a row, (2) The Routh table sometimes will have an entire row that consists of zeros

  • Zero only in the first column. If the first elemento of a row is zero, division by zero will be required to form the next row. To avoid this phenomenon, an epsilon ε is assigned to replace the zero in the first column. The value is then allowed to approach zero from either the positive or the negative side, after which the signs of the entries in the first column can be determined. To see the application of this, let us look the follow example: determine the stability of the closed-loop transfer function T(s):

null

The solution is shown in table 6.4:

null

We must begin by assembling the Routh table down to the row where a zero appears only in the first column (the s^3 row). Next, replace the zero by a small number ε complete the table. To begin the interpretation we must first assume a sign, positive or negative for the quantity  ε. Table 6.5  shows the first column of table 6.4 along with the resulting signs for choices of  ε positive and ε negative.

null

If is chosen ε positive Table 6.5 shows a sign change from the s^3 row to the s^2 row, and there will be another sign change from the s^2 row to the s^1 row. Hence the system is unstable and has two poles in the right half-plane. Alternatively, we could chose ε negative.  Table 6.5 then shows a sign change from the s^4 row to the s^3 row. Another sign change would occur from the s^3 row to the s^2 row. Our result would be exactly the same as that for a positive choice for ε. Thus, the system is unstable.

  • Entire Row is zero. We now look at the second special case. Sometimes while making a Routh table, we can find that an entire row consists of zeros because there is a even polynomial that is a factor of the original polynomial. This case must be handled differently from the previous case. Next example shows how to construct and interpret the Routh table wne an entire row of zeros is present.

Determine the number of right half-plane poles in the closed-loop transfer function T(s):

null

Start by forming the Routh table for the denominator. We get Table 6.7:

null

At the second we multiply by 1/7 for convenience. We stop at the third row since the entire row consist of zeros and use the following procedure. First we return to the row immediately above the row of zeros and form an auxiliary polynomial using the entries in that row as coefficients. The polynomial will start with the power of s in the label column and continue by skipping every other power of s. Thus, the polynomial formed for this example is:

null

Next we differentiate the polynomial with respect to s and we obtain:  

null

Finally we use the coefficients of this last equation to replace the row of zeros. Again, for convenience the third row is multiplied by ¼ after replacing the zeros. The remainder of the table is formed in a straightforward manner by following the standard form shown in Table 6.2

We get Table 6.7. It shows that all entries in the first column are positive. Hence, there are no right half-plane poles and the system is stable.

BEFORE: Transient-response specifications

NEXT: Steady-state error in Control Systems

Source:

  1. Control Systems Engineering, Nise pp 301-320
  2. Sistemas de Control Automatico Benjamin C Kuo pp
  3. Modern_Control_Engineering, Ogata 4t pp 288,

Written by: Larry Francis Obando – Technical Specialist – Educational Content Writer.

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, Caracas.

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contact: Caracas, Quito, Guayaquil, Cuenca – Telf. 00593998524011

WhatsApp: +593984950376

email: dademuchconnection@gmail.com

Attention:

If what you need is to solve urgently a problem of a “Mass-Spring-Damper System ” (find the output X (t), graphs in Matlab of the 2nd Order system and relevant parameters, etc.), or a “System of Electromechanical Control “… to deliver to your teacher in two or three days, or with greater urgency … or simply need an advisor to solve the problem and study for the next exam … send me the problem…I Will Write The Solution To any Control System Problem…

, …I will give you the answer in digital and I give you a video-conference to explain the solution … it also includes simulation in Matlab. In the link above, you will find the description of the service and its cost.

Related:

The Block Diagram – Control Engineering

Dinámica de un Sistema Masa-Resorte-Amortiguador

Block Diagram of Electromechanical Systems – DC Motor

Transient-response Specifications

Steady-state error in Control Systems

Transient-response Specifications

NEXT: Control System Stability

Introduction

The response in time of a control system is usually divided into two parts: the transient response and the steady-state response. Let y (t) be the response of a system in continuous time, then:

where yt (t) is the transient response, while yss (t) is the steady state response.

The transient response of a control system is important since both its amplitude and its duration must be kept within tolerable or prescribed limits. It is defined as the part of the response in time that tends to zero when the time becomes very large. Thus,

All real stable control systems present a transient phenomenon before reaching the steady state response. For analysis and design purposes it is necessary to assume some basic types of test inputs to evaluate the performance of a system. The proper selection of these test signals allows the prediction of system performance with other more complex inputs. The following signals are used: Step function, which represents an instantaneous change in the reference input; Ramp function, which represents a linear change over time; Parabolic function, which represents a faster order than the ramp. These signals have the common characteristic that they are simple to write in mathematical form, it is rarely necessary or feasible to use faster functions. In Figure 7-1 you can see these functions:

[2]

For a linear control system, the analysis and characterization of the transient response is performed frequently using the unit step function Us (t), shown in Figure 7-1a with R = 1. A typical response of a control system to a unit step input is shown in Figure 7-11:

[2]

The transient response of a practical control system often exhibits damped oscillations before reaching the steady state. That’s happens because systems have energy storage and cannot responds immediately. The transient-response to a unit step input depends on the initial conditions. That’s why it is a common practice to use the standard initial conditions that the system is at rest initially with the output an all time derivatives thereof zero.

Second-order systems and Transient-response specifications.

Figure 5-5a shows a Servo System as an example of a second-order system. It consists of a proportional controller and load elements (inertia and viscous friction elements):

null

[3]

The closed-loop Transfer Function of the system shown in Figure 5-5c is:

In the transient-response analysis it is convenient to write:

Where σ is called the attenuation; ωn is the undamped natural frequency; and ζ  the damping ratio of the system.  ζ  is the ratio of the actual damping B to the critical damping Bc equal to two times the square-root of JK:

In terms of ωn y σ, the system shown in Figure 5-5c can be expressed as Figure 5-6:

[3]

Now, the Transfer Function C(s)/R(s) can be written as:

This form is called The Standard Form. The dynamic behavior of a second-order system can be now described in terms of the two parameters ωn and σ. In short, the cases of second-order response as a function of σ are summarized in Figure 4.11 (for a better review see FIRST and SECOND ORDER SYSTEMS):

[1]

In specifying the transient-response characteristics of a control system to a unit-step input, it is common to specify the following parameters associated with the underdamped response:

 

  1. Delay time, Td
  2. Rise time, Tr
  3. Peak time, Tp
  4. Percent overshoot (%OS) or Maximum overshoot (Mp)
  5. Settling time, Ts

These specifications are defined as follows:

Delay time (Td): it is the time required for the response to reach half the final value the very first time.

Rise time (Tr): it is the time required for the response to rise from 10% to 90%. In other words, to go from 0.1 of the final value to 0.9 of the final value.

Peak time (Tp): it is the time required for the response to reach the first peak of the overshoot.

Maximum overshoot (Mp): it is the maximum peak value of the response curve measured from unity. It is also the amount that the waveform overshoots the final value, expressed as a percentage of the steady-state value.

Settling time (Ts):  it is the time required for the transient damping oscillations to reach and stay within ±2% or ±5% of the final or steady-state value.

These specifications are graphically shown in Figure 5-8:

[3]

It is important to remark that these specifications don’t necessarily apply to any given case. For example, the terms peak time and maximum overshoot do not apply to overdamped systems.

Except for certain applications where oscillations can’t be tolerated, it is desirable that the transient-response be sufficiently fast and sufficiently damped. Thus, for a desirable transient response of a second-order system, the damping ratio must be between 0.4 and 0.8. Small values of σ (σ<0.4) yields excessive overshoot in the transient response, and systems with a large value of σ (σ>0.8) responds sluggishly. We will also see that the maximum overshoot and the rise time conflict with each other. In other words, they cannot be made smaller simultaneously.

Analytically:
Rise time (Tr):

where ωd is the damped natural frequency:

and ß is defined by the Figure 5-9:

[3]

Peak time (Tp):

Settling time (Ts):

Transient-response of Higher-order systems.

It could be seen that the transient response of a system higher than a second-order is the sum of the responses of first-order and second order systems.  

Transient-response of a First-order system.

We briefly discuss the transient response of a first-order system. A first-order system without zeros can be described by the transfer function shown in Figure 4.4(a).

[1]

If the input is a unit step, where R(s)=1, the Laplace transform of the step response is C(s), where:

Taking the inverse transform:

Figure 4-5 shows a typical response of this system to a unit step input:

[1]

We call 1/a the time constant of the response. The parameter a is the only one needed to describe the transient response for a first-order system. Thus, the time constant can be considered a transient response specification for a first order system, since it is related to the speed at which the system responds to a step input. Since the pole of the transfer function is at a, we can say the pole is located at the reciprocal of the time constant, and the farther the pole from the imaginary axis, the faster the transient response.

The other specifications for a first-order system are:

Rise time (Tr):

Settling time (Ts):

NEXT: Control System Stability

Source:

  1. Control Systems Engineering, Nise pp 177-181
  2. Sistemas de Control Automatico Benjamin C Kuo p 385,
  3. Modern_Control_Engineering, Ogata 4t pp 224, 232

Written by: Larry Francis Obando – Technical Specialist – Educational Content Writer.

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, Caracas.

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contact: Caracas, Quito, Guayaquil, Cuenca – Telf. 00593998524011

WhatsApp: +593984950376

email: dademuchconnection@gmail.com

Attention:

If what you need is to solve urgently a problem of a “Mass-Spring-Damper System ” (find the output X (t), graphs in Matlab of the 2nd Order system and relevant parameters, etc.), or a “System of Electromechanical Control “… to deliver to your teacher in two or three days, or with greater urgency … or simply need an advisor to solve the problem and study for the next exam … send me the problem…I Will Write The Solution To any Control System Problem…

, …I will give you the answer in digital and I give you a video-conference to explain the solution … it also includes simulation in Matlab. In the link above, you will find the description of the service and its cost.

Related:

The Block Diagram – Control Engineering

Dinámica de un Sistema Masa-Resorte-Amortiguador

Block Diagram of Electromechanical Systems – DC Motor

Control System Stability

Steady-state error in Control Systems

Block Diagram of Electromechanical Systems – DC Motor

Introduction

Electromechanical Systems are hybrids of electrical and mechanical variables. Applications for electromechanical components range from robot control, sun and star trackers, disk-drive position control, DC machines control and central Air-Conditioning systems for residences.

The structure of an Electromechanical Drive System is given in Figure 2.1. It consist of energy/power source, reference values for the quantities to be controlled, electronic controller, gating circuit for converter, electronic converter (rectifier, inverter, power electronic controller), current sensors (shunts, current transformer, Hall sensor), voltage sensor (voltage divider, potential transformer), speed sensors (tachometers) and displacement sensors (encoders), rotating three-phase machines, mechanical gearbox, and the application-specific load (pump, fan, automobile). In Figure 2-1 all but the mechanical gear are represented by a Transfer Functions (output variables as a function of time). Meanwhile, the mechanical gear is represented by the Transfer Characteristic (output variable Xout as a function of the input variable Xin).

[1]

The armature-controlled dc servomotor is perhaps the most important component found in robotic applications. But in general, in electromechanical systems, we can use induction motors and synchronous motors as well, the two most common three-phase AC Machines. Here, we just derive the Dynamic of the DC Motor in order to represent it through block diagrams. So we will have to pass from schematic to block diagram as shown in Figure 2.35:

[2]

The Block Diagram for a DC Motor

To derive the block diagram representation for a separately excited DC machine, firstly we must derive the Dynamic of the System, the differential equations that govern the DC machine. After that, we use the Laplace Transform to build the block diagram. The motor can be controlled by field or by armature. In this case, let’s suppose that a stationary permanent magnet or a stationary electromagnet generates a constant magnetic flux Ф called the Fixed Field according to Figure 2.35. As a result, the motor is controlled by a voltage ea applied to the armature terminals. The armature is a rotating circuit through which a current ia, flows. When the armature passes through Ф at right angles, it feels a force F=BLia where B is the magnetic field strength and L is the length of the conductor. The resulting torque Tm turns the rotor, the rotating member of the motor. For linear analysis it is supposed that this torque or par is proportional to the flux Ф and the armature current ia, and from here, we obtain the first important relation:

[3]

As Ф is constant, KmФ = Ki, so the previous equation is as follows,

Where Ki is The Constant of Proportionality, also called the motor torque constant (or par constant) and is one the parameter given by the manufacturers when selling a motor. Ki also frequently called KT comes in N-m/A.

(Note: when the motor is controlled by the current in the field, in order to have a linear system the current of the armature must be considered to be constant and the motor torque is given by Tm= KmiF)

There is another phenomenon that occurs in the motor: A conductor moving at right angles to a magnetic field generates a voltage Vb at the terminals of the conductor. Since the current-carrying armature is rotating in a magnetic field, its voltage is proportional to speed. In this way we obtain the second important equation:

[2]

We call Vb the Back Electromotive Force (or back emf); Kb is a constant of proportionality called the back emf constant, another given parameter. It is also important to notice that θm is the angular displacement and the angular velocity of the motor is ωm:

Although the motor is itself an open-loop system, later we will see that The Back EFM Vb, generates a feedback-loop inside the motor, acting as an “electric friction” that tends to improve the stability of the motor.

After this and applying the Kirchhoff’s law for voltages, we find the relation between the armature current ia, the applied armature voltage ea, and the back efm Vb (or eb). Clearing conveniently we obtain:

Where La and Ra represent the inductance and the resistance of the armature respectively. Vb = eb. (La and Ra are given parameters of the motor)

Now, applying Newton’s Law for Rotational Mechanical Systems and clearing conveniently:

Where TL represents the load, JM is the inertia of the rotor, and Bm viscous friction coefficient (JM and Bm are given parameters of the motor)

Actually, the completed Dynamic of the DC Motor in open-loop operation is the following set of differential equations:

[3]

To get a block diagram of this dynamic we have to apply Laplace Transform to this set of equations while focused on the value we are concerned as the output: θm. So, we now that

After applying Laplace:

θm(s) = Ωm(s)/S =(Ωm(s))*(1/S)

We can represent this simple relationship between θm(s) and Ωm(s) using a Block Diagram by recalling the fact that every block has a Transfer Function inside. In this case, the transfer function is:

θm(s)/Ωm(s) = 1/S

The block diagram to represent this system is:

We see now that we need to know what is Ωm(s). To know that we use

We clear for dθm/dt and apply Laplace. We obtain:

Ωm(s)=(Tm(s) – TL(s))/(Jm(s) + Bm(s))

As a dog following its tail, we continue this procedure using each of the rest of the equations just one time, till all the variables are cleared, except the inputs TL(s) and Ea(s).

To get Tm(s) we use:

And we get:

Tm(s)= KiIa(S)

Finally, from:

We get:

Ia(S)= (Ea(s)-Eb(s))/(SLa+Ra)

Eb(s)=KbΩm(s)

Now, using, the interconnecting, θm(s), Ωm(s), Tm(s), Ia(S), Ea(s), Eb(s) and TL(s), with functional blocks and its transfer functions, summing points and pickoff points, we get The Block Diagram for a DC Motor operating as an open-loop:

[3]

Here we can corroborate what we pointed out before, that the back electromotive force, proportional to Ωm (s), represented in the diagram as Eb (s), generates a feedback loop that tends to stabilize the system.

The most commonly used electromechanical system is shown in Figure 2.15., operating at about constant speed without feedback control. Some of the more commonly occurring drive systems are presented using linear transfer function within each and every block of these diagrams as in Figure 2.15. Consequently, we usually need the transfer function of the motor and its load to represent it just through one block.

 The Transfer Function for a DC Motor and Load

Let’s consider the configuration shown in Figure 2.37:

[2]

The relation between the input Ea(s) and the output θm(s) is given by:

(Note: To see how this equation is derived, please see page 81 of bibliography resource [2])

If we assume that the armature inductance La is small compared to the armature resistance Ra, which is usual for the motor DC, we obtain:

The desired Transfer Function θm(s)/Ea(s) is found to be:

[2]

Where Jm is the equivalent inertia after load inertia JL is reflected back to the armature, and Dm is the equivalent damping after load damping DL is reflected back to the armature, and:

Usually, to find the factors Kt/Ra and Kb, the user counts with the following relations and a Torque-Speed Curve of the motor provided by the manufacturer, as it is shown in the following example:

[2]

As a consequence, we can now represent the motor and its load through one block in the diagram:

Closed-loop operation.

The open-loop operation is acceptable for a lot of drive application where a constant speed (a chainsaw) or position (an elevator) is sufficient. However, if a variable speed is needed (conveyor belt) or the position must be accurate (an Antenna), a closed-loop control, with negative feedback, must be chosen. But, in a closed-loop operation, others devices are needed to assess or transform signals: sensors, transducers and amplifiers. A typical closed-loop operation is the position control schema for an Antenna Azimuth shown as follows:

Which equivalent block diagram is as follows:

[2]

The potentiometer. 

A potentiometer is an electromechanical transducer that converts mechanical energy into electrical energy. The input of the device is a form of mechanical displacement that can be translational or rotational. When a voltage is applied across the fixed terminals of the potentiometer, the output voltage, which is measured between the variable terminal and the ground, is proportional to the input shift. Figure 4-29 shows the schema for a rotational potentiometer.

[3]

When the potentiometer housing is connected to the reference, the output voltage e(t) will be proportional to the position of the axis θc(t) in the case of a rotatory movement. So,

where Ks is the constant of proportionality. So E(s)/θc(s) = Ks will be the transfer function for a block representing this relationship

Potentiometers are often used to feedback the output position in motor control systems. Here it works as a mechanical sensor. Doing so, they allow comparing the actual position of the load with the reference position. The comparison generates an error signal that is then amplified to move the motor until it reaches the correct position. Hence its name, servomotor, slaves of the reference position.

The Tachometer. 

Like potentiometers, tachometers are electromechanical devices that convert mechanical energy into electrical energy. It works essentially as a voltage generator, with the voltage output proportional to the angular velocity of the input shaft. Figure 4-33 reflects the common use of a tachometer in a speed control system:

[3]

The Dynamic of the tachometer can be represented by:

Where et(t) is the output voltage, θ(t) is the displacement of the motor in radians, ω(t) is the speed of the rotor in rad/s, and Kt is the constant of the tachometer. Later, Et(s)/Ω(s) = Kt. In terms of: the displacement of the motor:

The gear train. 

Gear trains are used very frequently in electromechanical systems in order to reduce speed, amplify torque or to achieve the more efficient power transfer by matching the driving member with a given load. Consider the gear train of Figure 2-30 (for a more exhaustive analysis please refer to page 66 of the bibliography [5]):

[5]

Generally, in practice, the most commonly used procedure is to reflect the inertia and the damping of the load arrow to the motor shaft. The result is as follows:

Where J1eq is the equivalent inertia seen by the motor arrow and b1eq is the equivalent damping coefficient seen by the motor arrow. In the block diagrams, it is customary to represent the gear train by a block with a proportional transfer function, a constant Kg (which stands for Kgears) equivalent to the n1/n2.

Example:
  1. Before continuing, let’s see an example of how to apply the theory studied to a fairly common case: Obtain mathematical model of the position control system of the Figure. Obtain your block diagram and the transfer function between the angle of the load and the reference angle θc(s)/θc(s).

null

Solution:

null

To see the whole answer see:

Ejemplo 1 – Función de Transferencia de Sistema Electromecánico

DC Motor/Amplifier System Block Diagram

The CD Motor is always driven by a power amplifier that acts as an energy source. For this reason, it is more practical to present the Torque-Speed Curve of the combinación DC Motor/Amplifier. The Figure 4-51 shows a block diagram for the DC motor-Amplifier arrangement. And Figure 4-52 the Torque-Speed Curve:

[3]

The Proportional amplifier

A good example of a proportional amplifier is an Operational Amplifier with negative resistive feedback and inverting configuration such as shown in Figure 2.7a.

[1]

Operational amplifiers, often called Op Amps, are often used to amplify signals in sensor circuits. The Transfer Function of the Operational Amplifier is shown in Figure 2.7b under the name of G2:

Other configurations of this class are shown in Table 3-1 and Table 4-1 with their respective transfer functions:

Table 3-1

[4]

Table 4-1

[3]

Power Electronic

Performance of servomotors used for robotics applications highly depends on electric power amplifiers and control electronics, broadly termed Power Electronic. The actuators in robotic applications, mostly DC motors, must be controlled precisely so that desired motions of arms and legs may be attained. This requires a power amplifier to drive de desired level of voltage (or current indirectly) to the motor armature. The use of a linear amplifier, as the operational amplifier discussed in the previous section, is power-inefficient and impractical since it entails a large amount of power loss. An alternative is to control the voltage via ON-OFF switching. Pulse Width Modulation or PWM for short is the most commonly used method for varying the average voltage to the DC motor (For a detailed review of PWM see Motor Drives, p 663, bibliography [7])

Briefly, a typical motor drive system is expected to have some of the systems block diagram indicated in Figure 27.1. The load may be a conveyor system, a traction system, the rolls of a mill drive, the cutting tool of a numerically controlled machine tool, the compressor of an air conditioner, a ship propulsion system, a control valve for a boiler, a robotic arm, and so on.

[7]

The Power Electronic converter block in the previous diagram, for PWM control with inner current loop, may use diodes, MOSFETs, GTOs or IGBTs. Servo drive systems normally use the full four-quadrant converter of Figure 27.9, which allows bidirectional drives and regenerative braking capabilities.

[7]

The PWM is a technique to control an effective armature voltage by using the ON-OFF switching alone. Figure 2.3.3 illustrates the PWM signal:

[6]

PWM varies the ratio of the time length of the complete ON state to the complete OFF state. A single cycle of ON and OFF states is called the PWM period, whereas the percentage of the ON stage in a single period is called duty rate. The first PWM signal of Figure 2.3.3, is of 60% duty, and the second one is 25%. If the voltage supply is V=10 volts, the average voltage actually transmitted to the DC motor is 6 volts and 2.5 volts respectively. The PWM period is set to be shorter than the time constant associated with the mechanical motion. The PWM frequency is usually between 2 and 20 KHz, whereas the bandwidth of a motion control system is at most 100 Hz. Therefore, the discrete switching does not influence the mechanical motion in most cases.

If the electric time constant Te is much larger than PWM period, the actual current flowing to the motor armature is a smooth curve, as illustrated in Figure 2.3.4:

[6]

Design of Control Systems - Next Topic

The fundamental objective of analyzing a control system is to facilitate its design. The Block Diagram is the first step because it represents the mathematical model of the system that we want to analyze and improve. The dynamics of a controlled linear process can be represented by the block diagram of Figure 10-1:

[3]

Most control systems are built so that the output vector y (t) meets certain specifications that define what the system should do and how to do it in the desired way. These specifications are unique for each individual application. Relative stability, steady-state precision (error) and transient response are the most commonly used specifications in the market. The essential problem involves determining the control signal U (t) within a prescribed range so that the specifications required by the customer are met.

After determining the above, the engineer must design a fixed configuration of the system and the place where the controller will be placed in relation to the controlled process, which also involves designing the elements that make up the controller, that is, determining the controller parameters. Because most control efforts involve modifying or compensating the performance characteristics displayed by the system during the transient response analysis and steady state response, the design of a fixed configuration is also called compensation.

In short, the art and science of designing control systems can be summarized in three steps, which leads to our following issues:

  1. Determine what should be done and how to do it
    1. Transient Response Specifications
    2. Stability – Routh Criterion
    3. Stead-State Error
  2. Determine the driver configuration
    1. Proportional, Integral and Derivative Control Actions
    2. PD Controller
    3. PI Controller
    4. PID controller
  3. Determine the controller parameters

Source

  1. Chapter 2, Block Diagram of EM Systems, pp 21, 43(23) (Fuchs E.F., Masoum M.A.S. (2011) Block Diagrams of Electromechanical Systems. In: Power Conversion of Renewable Energy Systems. Springer, Boston, MA)
  2. Control Systems Engineering, Nise pp 79, 81
  3. Sistemas de Control Automatico Benjamin C Kuo pp 159, 203
  4. Modern_Control_Engineering, Ogata 4t pp 103,
  5. dinamica_de_sistemas p 66
  6. Actuators and Drive System – Robótica
  7. Libro Rashid – Power Electronic Handbook p 663-666

Written by: Larry Francis Obando – Technical Specialist – Educational Content Writer.

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, Caracas.

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contact: Caracas, Quito, Guayaquil, Cuenca – Telf. 00593998524011

WhatsApp: +593998524011

email: dademuchconnection@gmail.com

Attention:

If what you need is to solve urgently a problem of a “Mass-Spring-Damper System ” (find the output X (t), graphs in Matlab of the 2nd Order system and relevant parameters, etc.), or a “System of Electromechanical Control “… to deliver to your teacher in two or three days, or with greater urgency … or simply need an advisor to solve the problem and study for the next exam … send me the problem…I Will Write The Solution To any Control System Problem…

, …I will give you the answer in digital and I give you a video-conference to explain the solution … it also includes simulation in Matlab. In the link above, you will find the description of the service and its cost.

Related:

Ejemplo 1 – Función de Transferencia de Sistema Electromecánico

The Block Diagram – Control Engineering

Dinámica de un Sistema Masa-Resorte-Amortiguador

Ejemplo 1 – Función Transferencia de Sistema masa-resorte-amortiguador

Transient Response Specifications

Control System Stability

PID – Basic Control System Actions

The Block Diagram – Control Engineering

Foundations

A control system may consist of a number of components. To show the function performed by each component, control engineers commonly use a diagram called the block diagram.They are used in various control actions at automated control systems. The first application is in representing physical systems.

A block diagram of a system is a pictorial representation of the functions performed by each component and of the flow of signals. Such a diagram depicts the interrelationship that exists among the various components. Differing from a purely mathematical representation, a block diagram has the advantage of indicating more realistically the signals flow of the actual system.The Figure 1.9b and c show a schematic for an Antenna azimuth position control system, and Figure 1.9d shows its functional block diagram:

[2]

The functional block is a symbol for the mathematical operation on the input signal to the block that produces the output. The Transfer Functions of the components are usually entered in the corresponding blocks, which are connected by arrows to indicate the direction of the signal flow.

Figure 3-2 shows an element of the block diagram (In order to guide the readers to the source, I preferred to use the same reference from the book where I found the information). The dimension of the output signal of the block is the dimension of the input signal multiplied by the dimension of the transfer function in the block.

[1]

The main contribution of the block diagrams lies in the fact that the functional operation of the entire system can be visualized more readily by examining its block diagram than by examining the physical system itself. The block diagram contains information concerning dynamic behavior but it does not include any information about the physical construction of the system. Consequently, many dissimilar or unrelated systems can be represented by the same block diagram.

So, each block can be considered as a subsystem. When multiple subsystems are interconnected it is necessary to add new elements to the block diagram: summing points and pickoff points. The characteristics of each element are shown in Figure 5-2:

[2]

One of the most important components of a control system is that which acts as a union point to the comparison of signals. That is what a summing point do. Examples of the devices involved in these operations are the potentiometer and the differential amplifier. Figure 3-3 shows the diversity of this operations:

[3]

A pickoff point, as shown in Figure 5-2, distributes the input signal to several output points.

We will now examine some common topologies for interconnecting subsystems and derive the single Transfer Function representation. These common topologies will form the basis for reducing more complicated systems to a single block.

Cascade Form

Figure 5-3 shows an example of cascade configuration. Intermediate signals values are shown at the output of each subsystem. Each signal is derived from the product of the imput times the Transfer Function.

[2]

The Equivalent Transfer Function Ge(s) is shown in Figure 5-3-b and it is the output Laplace Transform divided by the input Laplace Transform as follows:

Parallel Form

Figure 5-5 shows an example of parallel subsystems.

[2]

Again, by writing the output of each system we can find the equivalent Transfer Function. Parallel subsystems have a common input and the output form by the algebraic sum of the output from all of the subsystems.The equivalent Transfer Function Ge(s) is the output transform divided by the input transform as follows:

Feedback Form

Figure 3-4 shows an example of a block diagram of a Feedback system, also known as Closed-Loop System. The output C(s) is fed back to the summing point to compare it with the reference input signal R(s).

[1]

Generally, when the output signal C(s) is fed back to the summing point for comparison with the input, it is necessary to transform its form so that it gets the same form of the input signal. For example, the output could be an assessment of the temperature, thus it has the dimensions of the temperature, but the input could be a level of voltage, so it has a typical dimension of electricity. The control system must implement a transducer to set both signals in the same dimension. The conversion is accomplished by the feedback element whose Transfer Function is H(s):

[1]

For the system shown in Figure 3-5 the output C(s) an input R(s) are related as follows:

The Transfer Function relating C(s) and R(s) is called The Closed-Loop Transfer Function and it relates the closed-loop system Dynamic to the Dynamic of the feedforward elements and feedback elements.

To finish, Table 3-1 shows the algebraic rules for Block Diagrams:

[1]

The next example shows how to obtain the Transfer Function of a closed-loop model by using the rules from Table 3-1 at simplifying Block Diagrams:

BD Reductionn

[1]

Sources:

  1. Modern_Control_Engineering, Ogata 4t pp 71,116
  2. Control Systems Engineering, Nise p236
  3. Sistemas de Control Automatico, Kuo p108

Posted by: Larry Francis Obando – Technical Specialist –  Educational Content Writer

Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, Caracas.

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contact: Caracas, Quito, Guayaquil, Cuenca.

Telephone number: 00593-998524011

WhatsApp: +593984950376

dademuchconnection@gmail.com

Attention:

If what you need is to reduce a complex Block Diagram, to solve urgently a problem of a “Mass-Spring-Damper System ” (find the output X (t), graphs in Matlab of the 2nd Order system and relevant parameters, etc.), or to solve a “System of Electromechanical Control “… to deliver to your teacher in two or three days, or with greater urgency … or simply need an advisor to solve the problem and study for the next exam … send me the problem...I Will Write The Solution To any Control System Problem…

, …I will give you the answer in digital and I give you a video-conference to explain the solution … it also includes simulation in Matlab. In the link above, you will find the description of the service and its cost.

Related:

Block Diagram of Electromechanical Systems

Dinámica de un Sistema Masa-Resorte-Amortiguador