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The question has been raised whether an open-loop unstable process can be stabilized with a proportional-only controller. a) For the process and controller shown in Fig. 1, find the range of Kc values b) What is the gain of Y (s)/Ysp(s), if Kc is within the range of part (a)? c)For K = 10 and τ = 20, find the value of Kc that yields a pole at s = −0.1. Given a unit step setpoint, what is the offset for these conditions?

The question has been raised whether an open-loop unstable process can be stabilized with a proportional-only controller. a) For the process and controller shown in Fig. 1, find the range of Kc values b) What is the gain of Y (s)/Ysp(s), if Kc is within the range of part (a)? c)For K = 10 and τ = 20, find the value of Kc that yields a pole at s = −0.1. Given a unit step setpoint, what is the offset for these conditions?

Introduction to Chemical Engineering Thermodynamics
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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The question has been raised whether an open-loop unstable process can be stabilized with a proportional-only controller. a) For the process and controller shown in Fig. 1, find the range of Kc values

b) What is the gain of Y (s)/Ysp(s), if Kc is within the range of part (a)?

c)For K = 10 and τ = 20, find the value of Kc that yields a pole at s = −0.1. Given a unit step setpoint, what is the offset for these conditions?

**Figure 1: Control Block Diagram** This diagram represents a feedback control system, which is a fundamental concept in control theory used to maintain the desired output of a system. Here's a detailed explanation of each component within the block diagram: 1. **Set Point (Y_sp)**: This is the desired value or target that the system aims to achieve. It is an input to the system. 2. **Summing Junction**: This is represented by the circle with a "+" and a "−" symbol inside. It calculates the error signal (difference) by subtracting the measured process variable (feedback) from the set point (Y_sp). 3. **Controller (K_c)**: The block labeled "K_c" represents the controller gain. It processes the error signal to produce a control signal. The controller adjusts the process to reduce the error over time. 4. **Process Transfer Function (K / (1−τs))**: This block represents the dynamic behavior of the process being controlled, where "K" is the gain and "τ" is the time constant of the process. The transfer function dictates how the process responds to the control signal. 5. **Output (Y)**: This is the actual output of the system after being influenced by the controller and the process dynamics. 6. **Feedback Loop**: The line that loops back from the output (Y) to the summing junction shows the feedback mechanism. It continuously monitors the output to compare it against the set point, enabling adjustments through the controller to minimize the error. This type of diagram is useful for understanding how automatic control systems maintain the desired operating conditions by correcting deviations from set points.
Transcribed Image Text:**Figure 1: Control Block Diagram** This diagram represents a feedback control system, which is a fundamental concept in control theory used to maintain the desired output of a system. Here's a detailed explanation of each component within the block diagram: 1. **Set Point (Y_sp)**: This is the desired value or target that the system aims to achieve. It is an input to the system. 2. **Summing Junction**: This is represented by the circle with a "+" and a "−" symbol inside. It calculates the error signal (difference) by subtracting the measured process variable (feedback) from the set point (Y_sp). 3. **Controller (K_c)**: The block labeled "K_c" represents the controller gain. It processes the error signal to produce a control signal. The controller adjusts the process to reduce the error over time. 4. **Process Transfer Function (K / (1−τs))**: This block represents the dynamic behavior of the process being controlled, where "K" is the gain and "τ" is the time constant of the process. The transfer function dictates how the process responds to the control signal. 5. **Output (Y)**: This is the actual output of the system after being influenced by the controller and the process dynamics. 6. **Feedback Loop**: The line that loops back from the output (Y) to the summing junction shows the feedback mechanism. It continuously monitors the output to compare it against the set point, enabling adjustments through the controller to minimize the error. This type of diagram is useful for understanding how automatic control systems maintain the desired operating conditions by correcting deviations from set points.
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