*1.3. PROCESS MONITORING AND CONTROL chemeng.queensu.ca the most common is known as the PID (Proportional, Integral, and Derivative) algorithm, on which this publication will focus. First we will look at the PID algorithm and its components. We will then look at the dynamics of the process being controlled. Then we will review several methods of tuning (or adjusting the parameters of) the PID control algorithm. Finally, we will look as several ways*

Improve Process Efficiency Through Regulatory Control. AN964 INTRODUCTION The purpose of this application note is to describe how a PIC16F684 can be used to implement a positional Proportional-Integral-Derivative (PID) feedback control in an inherently unstable system. An inverted pendulum is used to demonstrate this type of control. The inverted pendulum consists of three main parts: the base platform, the pendulum and the controller board, as, THREE TERM (PID) CONTROL The following is a stand alone tutorial to explain the meaning of P.I.D. or 3 Term control used in analogue and digital control systems. P.I.D. stands for Proportional, Integral and Derivative. It is difficult to understand this feature on a PLC unless you are familiar with control theory. Here is an attempt to explain it briefly using the control of the speed of an.

control theory, the Proportional-Integral-Derivative(PID) controller are still dominating in the motion control systems in the industry due to the well acquaintance of the PID control has proportional, integral, and derivative algorithms available to maintain the setpoint of the process. Steam applications use the proportional and integral part of вЂњPID;вЂќ

proportional-integral-derivative (pid) controller Due to the prevalence of pulse encoders for system state information, an all- digital proportional-integral-derivative (ADPID) is вЂ¦ вЂўExample final control elements include process valves, вЂўPopular intermediate value controller is PID (proportional-integral-derivative) controller вЂўPID computes a controller output signal based on control error: Proportional Integral Derivative term term term where: y(t) = measured process variable u(t) = controller output signal

A variation of Proportional Integral Derivative (PID) control is to use only the proportional and integral terms as PI control. The PI controller is the most popular variation, even more than full PID controllers. The value of the controller output `u(t)` is fed into the system as the manipulated variable input. The "PID" in "PID Control" stands for "Proportional, Integral, Derivative". These three terms describe the basic elements of a PID controller. Each of these elements performs a different task and has a different effect on the functioning of a system.

PID stands for вЂњproportional, integral, derivative.вЂќ These three terms describe the basic ele-ments of a PID controller. Each of these elements performs a differ-ent task and has a different effect on the functioning of a system. In a typical PID controller these elements are driven by a combi-nation of the system command and the feedback signal from the object that is being controlled The "PID" in "PID Control" stands for "Proportional, Integral, Derivative". These three terms describe the basic elements of a PID controller. Each of these elements performs a different task and has a different effect on the functioning of a system.

PID PID Controller is implemented with LABVIEW Control Design Toolkit for which the block diagram is shown in fig. PI-Control: P=0. 2: Block Diagram 5.ijetae. 3. Adjust the set-point value. D=tu/8 Fig. to a typical value for the envisaged use of the system and turn off the derivative and integral actions by setting their levels to zero. I=2/tu. then reduce this level by a factor of two or PID PID Controller is implemented with LABVIEW Control Design Toolkit for which the block diagram is shown in fig. PI-Control: P=0. 2: Block Diagram 5.ijetae. 3. Adjust the set-point value. D=tu/8 Fig. to a typical value for the envisaged use of the system and turn off the derivative and integral actions by setting their levels to zero. I=2/tu. then reduce this level by a factor of two or

The PID algorithm consists of three basic elements: proportional, integral and derivative. The setting for The setting for each of these three elements is varied in вЂ¦ PID control has proportional, integral, and derivative algorithms available to maintain the setpoint of the process. Steam applications use the proportional and integral part of вЂњPID;вЂќ

Integral control Even bias-free proportional controllers can cause steady-state errors (try the previous exercise again with Gp = 1, Gc = 2, and VB = 0). One of the first solutions to overcome this problem was the introduction of integral control. An integral controller generates a corrective effort proportional not to the present error, but to the sum of all previous errors. The level The DSP implements a digital proportional-integral-derivative feedback controller using an integrated 12-bit analog-to-digital converter to read the thermistor, and direct output of pulse-width-modulated waveforms to the H-bridge DRV592 power amplifier.

proportional-integral-derivative (pid) controller Due to the prevalence of pulse encoders for system state information, an all- digital proportional-integral-derivative (ADPID) is вЂ¦ derivative-plus-integral term in the fac- tory, the proportional band (gain) of the controller could be adjusted in the field. Field adjustment did, however, pose a problem since there was no established method of choosing the appropriate set- tings for each of the three terms of the controller. Recognizing this as a weak- ness, the Taylor Instrument Companies Extended version of a paper

PID stands for вЂњproportional, integral, derivative.вЂќ These three terms describe the basic elements of a PID controller. Each of these elements per- forms a different task and has a different effect on the functioning of a system. In a typical PID controller these elements are driven by a combination of the system command and the feedback signal from the object that is being controlled The PID controller is a simple system. Well-developed architectures exist for building complex systems from the bottom up by combining PID controllers with linear and nonlinear elements such as

The PID controller is a simple system. Well-developed architectures exist for building complex systems from the bottom up by combining PID controllers with linear and nonlinear elements such as introduce the Proportional- Integral- Derivative (PID) control algorithm. discuss the role of the three modes of the algorithm. highlight different algorithm structures. Discuss methods that have evolved over the last 50 years as aids in control loop tuning.

PID Control 123seminarsonly.com. 1.0 Introduction: In this course, the design and applications of Proportional -plus- Integral -plus- Derivative (PID) controllerвЂ™s is discussed. PID control is a technique used extensively in feedback control systems. Its origins date back to the 19th century, being used for governor speed control, and since then in numerous applications with a wide variety of actuators and sensors. The, Simple proportional-integral-derivative (PID) control almost always suffices to achieve desired closed-loop performance. The traditional approach for implementing PID control for TEC has been with analog circuitry. While PID control can be effectively implemented using a few op-amps, resistors, and capacitors, digital control offers many well known advantages over analog control. Digital.

CONTROL CHARACTERISTICS OF AN ALL-DIGITAL PROPORTIONAL. The proportional, integral and derivative components of the classical PID control law were re-envisioned in terms of frequency of occurrences or counts вЂ¦, PID Control Theory the P, I and D elements shown in Figure 3. Block diagram of the PID controller 1..2 (1 ). ID P I TT S Gs K TS = 1 P D(1 ) i KTs Ts (5) where K P is the proportional gain, TI is the integral time constant, TD is the derivative time constant, K I = K P / T I is the integral gain and KD = K P T D is the derivative gain. The three-term functionalities are highlighted below.

PID Without a PhD Tim Wescott - Embeddedrelated. Taken together these three actions constitute proportional-integral-derivative or PID control. The вЂtuningвЂ™ of a controller consists of setting the magnitudes of these three actions. Sometimes the third (derivative) action is suppressed, particularly if there are rapid fluctuating disturbances in the system which would make this action unstable. Proportional control by itself is https://en.m.wikipedia.org/wiki/Proportional_control PID Control Objectives The objective of this lab is to study basic design issues for proportional-integral-derivative control laws. Emphasis is placed on transient responses and steady-state errors. The п¬Ѓrst control problem consists in the regulation of velocity for brush DC motors and is solved using proportional-integral control. The second prob-lem consists in the regulation of position.

PID stands for вЂњproportional, integral, derivative.вЂќ These three terms describe the basic elements of a PID controller. Each of these elements per- forms a different task and has a different effect on the functioning of a system. In a typical PID controller these elements are driven by a combination of the system command and the feedback signal from the object that is being controlled PID PID Controller is implemented with LABVIEW Control Design Toolkit for which the block diagram is shown in fig. PI-Control: P=0. 2: Block Diagram 5.ijetae. 3. Adjust the set-point value. D=tu/8 Fig. to a typical value for the envisaged use of the system and turn off the derivative and integral actions by setting their levels to zero. I=2/tu. then reduce this level by a factor of two or

The study is done to determine the comparison between proportional-integral-derivative controller (PID controller) and tilt-integral-derivative (TID controller) for cardiac pacemaker systems, which can automatically control the heart rate to accurately track a desired preset profile. The controller offers good adaption of heart to the physiological needs of the patient. The parameters of the The DSP implements a digital proportional-integral-derivative feedback controller using an integrated 12-bit analog-to-digital converter to read the thermistor, and direct output of pulse-width-modulated waveforms to the H-bridge DRV592 power amplifier.

PID Control Objectives The objective of this lab is to study basic design issues for proportional-integral-derivative control laws. Emphasis is placed on transient responses and steady-state errors. The п¬Ѓrst control problem consists in the regulation of velocity for brush DC motors and is solved using proportional-integral control. The second prob-lem consists in the regulation of position The DSP implements a digital proportional-integral-derivative feedback controller using an integrated 12-bit analog-to-digital converter to read the thermistor, and direct output of pulse-width-modulated waveforms to the H-bridge DRV592 power amplifier.

AN964 INTRODUCTION The purpose of this application note is to describe how a PIC16F684 can be used to implement a positional Proportional-Integral-Derivative (PID) feedback control in an inherently unstable system. An inverted pendulum is used to demonstrate this type of control. The inverted pendulum consists of three main parts: the base platform, the pendulum and the controller board, as PID stands for вЂњproportional, integral, derivative.вЂќ These three terms describe the basic elements of a PID controller. Each of these elements per- forms a different task and has a different effect on the functioning of a system. In a typical PID controller these elements are driven by a combination of the system command and the feedback signal from the object that is being controlled

derivative-plus-integral term in the fac- tory, the proportional band (gain) of the controller could be adjusted in the field. Field adjustment did, however, pose a problem since there was no established method of choosing the appropriate set- tings for each of the three terms of the controller. Recognizing this as a weak- ness, the Taylor Instrument Companies Extended version of a paper The PID algorithm consists of three basic elements: proportional, integral and derivative. The setting for The setting for each of these three elements is varied in вЂ¦

PID control stands for proportional plus derivative plus integral control. PID control is a feedback mechanism which is used in control system. This type of control is also termed as three term control. By controlling the three parameters вЂ“ proportional, integral and derivative we can achieve different control actions for specific work. PID PID control is an important ingredient of a distributed control system. The controllers are also The controllers are also embedded in many special purpose control systems.

PID stands for "proportional, integral, derivative." These three terms describe the basic elements of a PID controller. Each of these elements performs a different task and has a different effect on the functioning of a system. PID control has proportional, integral, and derivative algorithms available to maintain the setpoint of the process. Steam applications use the proportional and integral part of вЂњPID;вЂќ

THREE TERM (PID) CONTROL The following is a stand alone tutorial to explain the meaning of P.I.D. or 3 Term control used in analogue and digital control systems. P.I.D. stands for Proportional, Integral and Derivative. It is difficult to understand this feature on a PLC unless you are familiar with control theory. Here is an attempt to explain it briefly using the control of the speed of an The proportional and derivative actions of a PID controller can also exacerbate hunting, depending on the behavior of the process. Reset windup Integral action comes into play in situations where a process has an actuator that is too small to implement an especially large control effort.

The PID controller, which consists of proportional, integral and derivative elements, is widely used in feedback control of industrial processes. In applying PID controllers, PDF The glucose-insulin system is a challenging process to model due to the feedback mechanisms present, hence the implementation of a model-based approach to the system is an on-going and

Simple proportional-integral-derivative (PID) control almost always suffices to achieve desired closed-loop performance. The traditional approach for implementing PID control for TEC has been with analog circuitry. While PID control can be effectively implemented using a few op-amps, resistors, and capacitors, digital control offers many well known advantages over analog control. Digital вЂўExample final control elements include process valves, вЂўPopular intermediate value controller is PID (proportional-integral-derivative) controller вЂўPID computes a controller output signal based on control error: Proportional Integral Derivative term term term where: y(t) = measured process variable u(t) = controller output signal

PID Control Objectives The objective of this lab is to study basic design issues for proportional-integral-derivative control laws. Emphasis is placed on transient responses and steady-state errors. The п¬Ѓrst control problem consists in the regulation of velocity for brush DC motors and is solved using proportional-integral control. The second prob-lem consists in the regulation of position The study is done to determine the comparison between proportional-integral-derivative controller (PID controller) and tilt-integral-derivative (TID controller) for cardiac pacemaker systems, which can automatically control the heart rate to accurately track a desired preset profile. The controller offers good adaption of heart to the physiological needs of the patient. The parameters of the

DESIGN OF PID CONTROLLER FOR FOPDT AND IPDT SYSTEM. A variation of Proportional Integral Derivative (PID) control is to use only the proportional and integral terms as PI control. The PI controller is the most popular variation, even more than full PID controllers. The value of the controller output `u(t)` is fed into the system as the manipulated variable input., The PID controller, which consists of proportional, integral and derivative elements, is widely used in feedback control of industrial processes. In applying PID controllers,.

(PDF) A new general glucose homeostatic model using a. control theory, the Proportional-Integral-Derivative(PID) controller are still dominating in the motion control systems in the industry due to the well acquaintance of the, PDF The glucose-insulin system is a challenging process to model due to the feedback mechanisms present, hence the implementation of a model-based approach to the system is an on-going and.

Simple proportional-integral-derivative (PID) control almost always suffices to achieve desired closed-loop performance. The traditional approach for implementing PID control for TEC has been with analog circuitry. While PID control can be effectively implemented using a few op-amps, resistors, and capacitors, digital control offers many well known advantages over analog control. Digital The proportional, integral and derivative components of the classical PID control law were re-envisioned in terms of frequency of occurrences or counts вЂ¦

The DSP implements a digital proportional-integral-derivative feedback controller using an integrated 12-bit analog-to-digital converter to read the thermistor, and direct output of pulse-width-modulated waveforms to the H-bridge DRV592 power amplifier. вЂўExample final control elements include process valves, вЂўPopular intermediate value controller is PID (proportional-integral-derivative) controller вЂўPID computes a controller output signal based on control error: Proportional Integral Derivative term term term where: y(t) = measured process variable u(t) = controller output signal

Familiar examples show how and why proportional-integral-derivative controllers behave the way they do. Keywords: Process control Control theory Controllers Loop controllers PID Vance J. VanDoren, CONTROL ENGINEERING A feedback controller is designed to generate an output that causes some corrective effort to be applied to a process so as to drive a measurable process вЂ¦ The study is done to determine the comparison between proportional-integral-derivative controller (PID controller) and tilt-integral-derivative (TID controller) for cardiac pacemaker systems, which can automatically control the heart rate to accurately track a desired preset profile. The controller offers good adaption of heart to the physiological needs of the patient. The parameters of the

1.0 Introduction: In this course, the design and applications of Proportional -plus- Integral -plus- Derivative (PID) controllerвЂ™s is discussed. PID control is a technique used extensively in feedback control systems. Its origins date back to the 19th century, being used for governor speed control, and since then in numerous applications with a wide variety of actuators and sensors. The The DSP implements a digital proportional-integral-derivative feedback controller using an integrated 12-bit analog-to-digital converter to read the thermistor, and direct output of pulse-width-modulated waveforms to the H-bridge DRV592 power amplifier.

PID control has proportional, integral, and derivative algorithms available to maintain the setpoint of the process. Steam applications use the proportional and integral part of вЂњPID;вЂќ PID stands for "proportional, integral, derivative." These three terms describe the basic elements of a PID controller. Each of these elements performs a different task and has a different effect on the functioning of a system.

Proportional-Integral-Derivative (PID) Function Block The controller block frequently shown in control system block is generally thought of as the Laplace transfer functions that act as the control compensating elements. Abstract: In process control industry, the proportional-integral-derivative (PID) controllers are one of the most important control elements. In practice, operational amplifiers are вЂ¦

control theory, the Proportional-Integral-Derivative(PID) controller are still dominating in the motion control systems in the industry due to the well acquaintance of the The "PID" in "PID Control" stands for "Proportional, Integral, Derivative". These three terms describe the basic elements of a PID controller. Each of these elements performs a different task and has a different effect on the functioning of a system. In a typical PID controller these elements are driven by a combination of the system com-mand and the feedback signal from the thing that is

Proportional-integral-derivative (PID) controllers are one of the most important control elements used in the process control industry. In practice, operational ampli ers are generally used in analog controllers. On the other hand, the operational transconductance ampli ers recently developed, which have some positive properties compared to operational ampli ers, are not used in analog The proportional and derivative actions of a PID controller can also exacerbate hunting, depending on the behavior of the process. Reset windup Integral action comes into play in situations where a process has an actuator that is too small to implement an especially large control effort.

Proportional-Integral-Derivative (PID) Function Block The controller block frequently shown in control system block is generally thought of as the Laplace transfer functions that act as the control compensating elements. вЂўExample final control elements include process valves, вЂўPopular intermediate value controller is PID (proportional-integral-derivative) controller вЂўPID computes a controller output signal based on control error: Proportional Integral Derivative term term term where: y(t) = measured process variable u(t) = controller output signal

The "PID" in "PID Control" stands for "Proportional, Integral, Derivative". These three terms describe the basic elements of a PID controller. Each of these elements performs a different task and has a different effect on the functioning of a system. In a typical PID controller these elements are driven by a combination of the system com-mand and the feedback signal from the thing that is Proportional-Integral-Derivative (PID) Function Block The controller block frequently shown in control system block is generally thought of as the Laplace transfer functions that act as the control compensating elements.

Improve Process Efficiency Through Regulatory Control. PID Control Theory the P, I and D elements shown in Figure 3. Block diagram of the PID controller 1..2 (1 ). ID P I TT S Gs K TS = 1 P D(1 ) i KTs Ts (5) where K P is the proportional gain, TI is the integral time constant, TD is the derivative time constant, K I = K P / T I is the integral gain and KD = K P T D is the derivative gain. The three-term functionalities are highlighted below, вЂўExample final control elements include process valves, вЂўPopular intermediate value controller is PID (proportional-integral-derivative) controller вЂўPID computes a controller output signal based on control error: Proportional Integral Derivative term term term where: y(t) = measured process variable u(t) = controller output signal.

CONTROL CHARACTERISTICS OF AN ALL-DIGITAL PROPORTIONAL. A variation of Proportional Integral Derivative (PID) control is to use only the proportional and integral terms as PI control. The PI controller is the most popular variation, even more than full PID controllers. The value of the controller output `u(t)` is fed into the system as the manipulated variable input., AN964 INTRODUCTION The purpose of this application note is to describe how a PIC16F684 can be used to implement a positional Proportional-Integral-Derivative (PID) feedback control in an inherently unstable system. An inverted pendulum is used to demonstrate this type of control. The inverted pendulum consists of three main parts: the base platform, the pendulum and the controller board, as.

Thermo-Electric Cooler Control Using a TMS320F2812 DSP. Simple proportional-integral-derivative (PID) control almost always suffices to achieve desired closed-loop performance. The traditional approach for implementing PID control for TEC has been with analog circuitry. While PID control can be effectively implemented using a few op-amps, resistors, and capacitors, digital control offers many well known advantages over analog control. Digital, AN964 INTRODUCTION The purpose of this application note is to describe how a PIC16F684 can be used to implement a positional Proportional-Integral-Derivative (PID) feedback control in an inherently unstable system. An inverted pendulum is used to demonstrate this type of control. The inverted pendulum consists of three main parts: the base platform, the pendulum and the controller board, as.

PID Without a PhD Tim Wescott - Embeddedrelated. 1.0 Introduction: In this course, the design and applications of Proportional -plus- Integral -plus- Derivative (PID) controllerвЂ™s is discussed. PID control is a technique used extensively in feedback control systems. Its origins date back to the 19th century, being used for governor speed control, and since then in numerous applications with a wide variety of actuators and sensors. The https://en.m.wikipedia.org/wiki/Proportional_control PID stands for вЂњproportional, integral, derivative.вЂќ These three terms describe the basic ele-ments of a PID controller. Each of these elements performs a differ-ent task and has a different effect on the functioning of a system. In a typical PID controller these elements are driven by a combi-nation of the system command and the feedback signal from the object that is being controlled.

Proportional-integral-derivative (PID) controllers are one of the most important control elements used in the process control industry. In practice, operational ampli ers are generally used in analog controllers. On the other hand, the operational transconductance ampli ers recently developed, which have some positive properties compared to operational ampli ers, are not used in analog 1.0 Introduction: In this course, the design and applications of Proportional -plus- Integral -plus- Derivative (PID) controllerвЂ™s is discussed. PID control is a technique used extensively in feedback control systems. Its origins date back to the 19th century, being used for governor speed control, and since then in numerous applications with a wide variety of actuators and sensors. The

control theory, the Proportional-Integral-Derivative(PID) controller are still dominating in the motion control systems in the industry due to the well acquaintance of the The PID controller, which consists of proportional, integral and derivative elements, is widely used in feedback control of industrial processes. In applying PID controllers,

Integral control Even bias-free proportional controllers can cause steady-state errors (try the previous exercise again with Gp = 1, Gc = 2, and VB = 0). One of the first solutions to overcome this problem was the introduction of integral control. An integral controller generates a corrective effort proportional not to the present error, but to the sum of all previous errors. The level The "PID" in "PID Control" stands for "Proportional, Integral, Derivative". These three terms describe the basic elements of a PID controller. Each of these elements performs a different task and has a different effect on the functioning of a system. In a typical PID controller these elements are driven by a combination of the system com-mand and the feedback signal from the thing that is

PID Control Theory the P, I and D elements shown in Figure 3. Block diagram of the PID controller 1..2 (1 ). ID P I TT S Gs K TS = 1 P D(1 ) i KTs Ts (5) where K P is the proportional gain, TI is the integral time constant, TD is the derivative time constant, K I = K P / T I is the integral gain and KD = K P T D is the derivative gain. The three-term functionalities are highlighted below вЂўExample final control elements include process valves, вЂўPopular intermediate value controller is PID (proportional-integral-derivative) controller вЂўPID computes a controller output signal based on control error: Proportional Integral Derivative term term term where: y(t) = measured process variable u(t) = controller output signal

AN964 INTRODUCTION The purpose of this application note is to describe how a PIC16F684 can be used to implement a positional Proportional-Integral-Derivative (PID) feedback control in an inherently unstable system. An inverted pendulum is used to demonstrate this type of control. The inverted pendulum consists of three main parts: the base platform, the pendulum and the controller board, as Familiar examples show how and why proportional-integral-derivative controllers behave the way they do. Keywords: Process control Control theory Controllers Loop controllers PID Vance J. VanDoren, CONTROL ENGINEERING A feedback controller is designed to generate an output that causes some corrective effort to be applied to a process so as to drive a measurable process вЂ¦

Abstract: In process control industry, the proportional-integral-derivative (PID) controllers are one of the most important control elements. In practice, operational amplifiers are вЂ¦ PID PID Controller is implemented with LABVIEW Control Design Toolkit for which the block diagram is shown in fig. PI-Control: P=0. 2: Block Diagram 5.ijetae. 3. Adjust the set-point value. D=tu/8 Fig. to a typical value for the envisaged use of the system and turn off the derivative and integral actions by setting their levels to zero. I=2/tu. then reduce this level by a factor of two or

The study is done to determine the comparison between proportional-integral-derivative controller (PID controller) and tilt-integral-derivative (TID controller) for cardiac pacemaker systems, which can automatically control the heart rate to accurately track a desired preset profile. The controller offers good adaption of heart to the physiological needs of the patient. The parameters of the Proportional Integral Derivative (PID) Controller - Proportional Integral Derivative (PID) Controller - Control System Video Tutorial - Control System video tutorials for GATE, IES and other PSUs exams preparation and to help Electrical Engineering Students covering Introduction, Feedback, Mathematical Models, Modelling of Mechanical Systems

PID control is an important ingredient of a distributed control system. The controllers are also The controllers are also embedded in many special purpose control systems. Proportional+Integral+Derivative (PID) algorithm can be designed from a model based perspective. The performance capabilities of PID algorithms are limited though. More sophisticated strategies, such as adaptive algorithms and predictive controllers have been proposed for improved process experiencing a revival. In particular, attempts are being made to integrate traditional SPC practice with

Proportional-integral-derivative (PID) controllers are one of the most important control elements used in the process control industry. In practice, operational ampli ers are generally used in analog controllers. On the other hand, the operational transconductance ampli ers recently developed, which have some positive properties compared to operational ampli ers, are not used in analog вЂњPID controlвЂќ is the method of feedback control that uses the PID controller as the main tool. The basic structure of conventional feedback control systems is shown in Figure 1,

The PID controller, which consists of proportional, integral and derivative elements, is widely used in feedback control of industrial processes. In applying PID controllers, PID PID Controller is implemented with LABVIEW Control Design Toolkit for which the block diagram is shown in fig. PI-Control: P=0. 2: Block Diagram 5.ijetae. 3. Adjust the set-point value. D=tu/8 Fig. to a typical value for the envisaged use of the system and turn off the derivative and integral actions by setting their levels to zero. I=2/tu. then reduce this level by a factor of two or

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