09-10Chapter 7-5 Nyquist’s Stability Criterion(2)

09-10Chapter 7-5 Nyquist’s Stability Criterion(2)

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Nyquist’s stability criterion_Part 2 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 2 Outline of Chapter 7 9 troduction 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 ……… Winter sweet
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Nyquist’s stability criterion_Part 2 requency Response Nyquist’s stability criterion Frequency Response 4. Nyquist s stability criterion Examples of Nyquist’s Criterion Using Direct Polar Plot The number of rotation N can be determined by drawing a radial line from the –1+j0 point . Then, at each -270 ° ω =0 1 2 intersection of this radial line with the curve of G(s)H(s), the direction f increasing frequency is noted. -1+j0 3 Radial line of increasing frequency is noted. -180 ° ω =- (s)H(s) ω =+ 4 B(s) As the curve crosses the radial line in a ccw or cw direction, this crossing ω =0 + G(s)H(s) ω 5 6 is positive or negative. The sum of the crossings, including e sign is equal to N 2 = N 0 = P 2 = = N P Z R R The closed-loop is unstable. 7 the sign, is equal to N . 3 R
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Nyquist’s stability criterion_Part 2 requency Response Nyquist’s stability criterion Frequency Response 4. Nyquist s stability criterion Examples of Nyquist’s Criterion Using Direct Polar Plot(P278- Example 7-15- 3. Type 2 system. Consider the transfer function ) 1 )( 1 )( 1 ( ) 1 ( ) ( ) ( 3 2 1 2 4 2 s T s T s T s s T K s H s G + + + + = Where T 4 >T 1 +T 2 +T 3 . The following figure shows the mapping of G(s)H(s) for the contour Q of the s-plane The mapping means that for a given oint in the s lane there corresponds a G(s)H(s)] 23 point in the s-plane there corresponds a given value of G(s)H(s) or B(s) The presence of the s 2 term in the [( )( ) ] p denominator of G(s)H(s) results in a net rotation of 360 ° in the vicinity of ω =0. 14 7 ω =+ ω =- -1+j0 B(s) For the complete contour, the net rotation is zero. Since P R =0, the system is stable. 4 5 6
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Nyquist’s stability criterion_Part 2 requency Response Nyquist’s stability criterion ) 1 )( 1 )( 1 ( ) 1 ( ) ( ) ( 3 2 1 2 4 2 s T s T s T s s T K s H s G + + + + = Frequency Response 4. Nyquist s stability criterion Examples of Nyquist’s Criterion Using Direct Polar Plot(P278- ne nd one w ossing The value of N can be determined by drawing the line radiating from –1+j0 point. (we can choose negative real axis as the line) . One cw and one ccw crossing 23 0 1 1 = = N = -1+j0 ω =+ 14 0 = = N P Z R R 0 = R P ω - G(s)H(s) B(s) This system can be made unstable by increasing the gain sufficiently for the (s)H(s) lot oss e egative al 7 5 6 Radial line G(s)H(s) plot to cross the negative real axis to the left of the –1+j0 point.
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09-10Chapter 7-5 Nyquist’s Stability Criterion(2)

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