KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec38

# KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec38

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1 Chapter 14 Nyquist Stability Criterion The Nyquist stability criterion is similar to the Bode criterion in that it determines closed-loop stability from the open-loop frequency response characteristics. The Nyquist stability criterion is based on two concepts from complex variable theory, contour mapping and the Principle of the Argument . Nyquist Stability Criterion . Consider an open-loop transfer function G OL ( s ) that is proper and has no unstable pole-zero cancellations. Let N be the number of times that the Nyquist plot for G OL (s) encircles the -1 point in the clockwise direction. Also let P denote the number of poles of G OL (s) that lie to the right of the imaginary axis. Then, Z = N + P where Z is the number of roots of the characteristic equation that lie to the right of the imaginary axis ( that is, its number of “zeros”). The closed-loop system is stable if and only if Z = 0.

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2 Chapter 14 Some important properties of the Nyquist stability criterion are: 1. It provides a necessary and sufficient condition for closed- loop stability based on the open-loop transfer function. 2. The reason the -1 point is so important can be deduced from the characteristic equation, 1 + G OL ( s ) = 0. This equation can also be written as G OL ( s ) = -1, which implies that AR OL = 1 and , as noted earlier. The -1 point is referred to as the critical point . 3. Most process control problems are open-loop stable. For
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KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec38

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