09-10Chapter 7-4 Nyquist’s stability criterion(1)

09-10Chapter 7-4 Nyquist’s stability criterion(1)

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Nyquist’s stability criterion_Part 1 Principle of Automatic Control CHAPTER 7 requency Response (P244 Ch 8) By Hui Hui Wang Wang Frequency Response (P.244, Ch.8) 浙江大学控制科学与工程学系
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Nyquist’s stability criterion_Part 1 Outline of Chapter 7 9 Introduction 9 Bode Plots (Logarithmic plots) 9 Direct Polar Plots 9 Nyquist’s Stability Criterion_Part 1 9 Nyquist’s Stability Criterion_Part 2 9 Phase Margin and Gain Margin and Their Relation to Stability 9 ……… Wintersweet
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Nyquist’s stability criterion_Part 1 Frequency Response: 4. Nyquist’s stability criterion qy p yq y The Nyquist stability criterion provides a simple graphical procedure for determining closed-loop stability from the frequency-response curves of the open-loop transfer function G(j ω )H(j ω ). or a stable system the roots of the characteristic equation nnot be permitted to lie in the RH s- lane or on the j axis. If ) ( ) ( 1 ) ( s H s G s B + = For a stable system the roots of the characteristic equation cannot be permitted to lie in the RH s plane or on the j ω 1 1 ) ( D N s G = ) ( D N s H = 2 1 2 1 2 1 1 ) ( D D N N D D D D N N s B + = + = Zeros of B(s) 2 2 2 1 2 1 The closed-loop transfer function of a system is l f ) 2 1 2 1 2 1 ) ( ) ( 1 ) ( ) ( ) ( ) ( N N D D D N s H s G s G s R s C s + = + = = φ Poles of Φ (s) 3
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Nyquist’s stability criterion_Part 1 Frequency Response: 4. Nyquist’s stability criterion The condition for stability may therefore be restated as follows: qy p yq y For a stable system none of the zeros of B(s) can lie in the RH s-plane or on the imaginary axis. yquist’s stability criterion relates e number f zeros and the poles i it ti Nyquist’s stability criterion relates the number of zeros and the poles of B(s) that lie in the RH s-plane to the polar plot of G(s)H(s) 1 1 ) ( D N s G = Limitations: Assumption is that all the control systems are inherently linear or that eir limits of operation are confined to give a linear operation 2 2 ) ( D N s H = 2 1 2 1 ) ( ) ( D D N N s H s G = their limits of operation are confined to give a linear operation. The degree of the denominator D 1 D 2 is equal to or greater than the egree of the numerator N f the open op transfer function degree of the numerator N 1 N 2 of the open-loop transfer function G(s)H(s). That means lim s →∞ G(s)H(s) 0 or a constant. 4
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Nyquist’s stability criterion_Part 1 Frequency Response: 4. Nyquist’s stability criterion Mathematical Basis for Nyquist’s Stability Criterion th i l th t B( ) i ti l f ti th h t i ti qy p yq y Poles of Φ (s) For the special case that B(s) is a rational fraction, the characteristic function B(s) is rationalized, factored and then written in the form L ---- oted the marks Z nd p Poles of GH ) ( ) )( ( ) ( ) )( ( ) ( 2 1 2 1 n w p s p s p s Z s Z s Z s s B = L Some of the poles and zeros of a generalized function B(s) are arbitrarily j ω ------Noted the marks Z i and p j .
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09-10Chapter 7-4 Nyquist’s stability criterion(1)

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