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1. The measured input to a PI controller changes stepwise (Ym(s) = 5/s) and the controller output changes initially as given below:   Calculate the values of the controller gain and integral time.   Book: Process Dynamics and Control, Third Edition or Fourth Edition, by D. E. Seborg, T. F. Edgar, D. A. Mellichamp, F. J. Doyle III, John Wiley & Sons, Inc, 2011

1. The measured input to a PI controller changes stepwise (Ym(s) = 5/s) and the controller output changes initially as given below:   Calculate the values of the controller gain and integral time.   Book: Process Dynamics and Control, Third Edition or Fourth Edition, by D. E. Seborg, T. F. Edgar, D. A. Mellichamp, F. J. Doyle III, John Wiley & Sons, Inc, 2011

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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1. The measured input to a PI controller changes stepwise (Ym(s) = 5/s) and the controller output changes initially as given below:

 

Calculate the values of the controller gain and integral time.

 

Book: Process Dynamics and Control, Third Edition or Fourth Edition, by D. E. Seborg, T. F. Edgar, D. A. Mellichamp, F. J. Doyle III, John Wiley & Sons, Inc, 2011

The image is a graphical representation often used in the context of differential equations or rate of change in an educational setting. It depicts the function \( p'(t) \) versus time \( t \). ### Description: - **Axes Representation:** - The horizontal axis (x-axis) represents time, denoted as \( t \). - The vertical axis (y-axis) represents the derivative of \( p(t) \) with respect to \( t \), denoted as \( p'(t) \). - **Graphical Elements:** - The graph consists of a piecewise linear function. - At \( t = 0 \), the function \( p'(t) \) jumps instantaneously by 4 units on the y-axis. This is depicted by a vertical segment labeled with a brace and the number "4". - For \( t > 0 \), \( p'(t) \) increases linearly with a positive slope. - **Line Segment:** - The line segment representing \( p'(t) \) for \( t > 0 \) starts from the point where \( p'(t) \) jumps. - The slope of this line segment is labeled as \( 2.5 \, \text{min}^{-1} \). ### Explanation: - **Initial Jump:** - At \( t = 0 \), there is an immediate change in \( p'(t) \) by a value of 4 units. This indicates a sudden increase in the rate of change of \( p(t) \). - **Linear Increase:** - For times greater than zero, \( p'(t) \) increases at a constant rate, which is represented by the linear segment with a slope of \( 2.5 \, \text{min}^{-1} \). This graph might represent a scenario in a physical or biological model where there is an initial sudden change followed by a steady increase in the rate of change over time. Examples could include the response of a capacitor charging in an electric circuit or a biological reaction rate after an initial stimulus. Understanding such graphs is fundamental for interpreting differential equations that model real-world phenomena.
Transcribed Image Text:The image is a graphical representation often used in the context of differential equations or rate of change in an educational setting. It depicts the function \( p'(t) \) versus time \( t \). ### Description: - **Axes Representation:** - The horizontal axis (x-axis) represents time, denoted as \( t \). - The vertical axis (y-axis) represents the derivative of \( p(t) \) with respect to \( t \), denoted as \( p'(t) \). - **Graphical Elements:** - The graph consists of a piecewise linear function. - At \( t = 0 \), the function \( p'(t) \) jumps instantaneously by 4 units on the y-axis. This is depicted by a vertical segment labeled with a brace and the number "4". - For \( t > 0 \), \( p'(t) \) increases linearly with a positive slope. - **Line Segment:** - The line segment representing \( p'(t) \) for \( t > 0 \) starts from the point where \( p'(t) \) jumps. - The slope of this line segment is labeled as \( 2.5 \, \text{min}^{-1} \). ### Explanation: - **Initial Jump:** - At \( t = 0 \), there is an immediate change in \( p'(t) \) by a value of 4 units. This indicates a sudden increase in the rate of change of \( p(t) \). - **Linear Increase:** - For times greater than zero, \( p'(t) \) increases at a constant rate, which is represented by the linear segment with a slope of \( 2.5 \, \text{min}^{-1} \). This graph might represent a scenario in a physical or biological model where there is an initial sudden change followed by a steady increase in the rate of change over time. Examples could include the response of a capacitor charging in an electric circuit or a biological reaction rate after an initial stimulus. Understanding such graphs is fundamental for interpreting differential equations that model real-world phenomena.
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