Gp4_Chapter 9 - EE2010 Systems and Control (Chapter 9)...

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EE2010 Systems and Control (Chapter 9) 9-1 How do we assess stability of the following feedback system? Closed loop stability can be obtained by calculating the poles of the CL system. The calculation of poles requires the solving of roots of the polynomial characteristic equation of the form: When n > 2, the roots of the above equation are difficult to obtain. A tool such as Matlab is needed. Are there other ways of assessing closed loop stability, apart from RL? Nyquist Stability Criterion (NSC): A graphical method that makes use of the OL TF { G ( s ) K ( s )} to determine CL stability Assessment of Stability - Revisit G ( s ) + K ( s ) Y ( s ) R ( s ) E ( s ) 0 ... ) ( ) ( 1 0 2 2 1 1 a s a s a s s K s G n n n n n
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EE2010 Systems and Control (Chapter 9) 9-2 We define closed contours and consider their values. The values along the C -contour are complex variables Some simple facts to consider before developing NSC Re C j e R s R Im Examples of C contours C may be a circular contour: Values of s around the C contour are given by , where R = radius of the contour and 0    2 . C may be a semi-circular contour: This is called D -contour because of the shape, where s = j on the imaginary axis and on the semi-circle. j e R s D R j e R s Re Im j e R s X j s j s X s -plane s -plane
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EE2010 Systems and Control (Chapter 9) 9-3 j j Re Im -3 Plot of F ( s ) encloses the origin once in the clockwise direction Consider the values of s around this D - contour Suppose we have a polynomial function: F ( s ) = s 3. Is the zero of F ( s ) inside the D -contour? YES j j Re Im What does the plot of F ( s ) look like if we plot for all values of s in the D - contour? 3 What happens if a complex function is evaluated for all values around the C -contour? F ( s )-plane s -plane
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EE2010 Systems and Control (Chapter 9) 9-4 j j Re Im Consider the values of s around this D - contour Suppose we have a polynomial function: . Is the pole of F ( s ) inside the D -contour? NO Plot of F ( s ) does not enclose the origin but is in the clockwise direction What does the plot of F ( s ) look like if we plot for all values of s in the D - contour? 3 1 ) ( s s F Re Im 0  3 1 -3 F ( s )-plane s -plane
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Systems and Control (Chapter 9) 9-5 Consider the values of s around this D - contour Suppose we have a polynomial function: . Is the zero/pole of F ( s ) inside the D -contour? NO for zero; YES for pole Plot of F ( s ) encloses the origin once in the anticlockwise direction What does the plot of F ( s ) look like if we plot for all values of s in the D - contour? 3 -plane
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This note was uploaded on 07/11/2011 for the course ECE 2010 taught by Professor Tankaychen during the Spring '11 term at National University of Singapore.

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Gp4_Chapter 9 - EE2010 Systems and Control (Chapter 9)...

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