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lecture_29

# lecture_29 - MIT OpenCourseWare http/ocw.mit.edu 2.004...

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MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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X X o o a " b r e a k a w a y " p o i n t a " b r e a k - i n " p o i n t s j w K = 0 K = 0 K i n c r e a s i n g K i n c r e a s i n g K = ¥ K = ¥ Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 29 1 Reading: Nise: Chapter 8 1 Root Locus Refinement The complete set of sketching rules contains additional methods to make a sketched plot more accurate. While these were useful in the days before ubiquitous computation, today with the existence of tools such as MATLAB makes these graphical refinements somewhat unnecessary. We therefore just mention them here and refer you to Nise, Section 8.5, for more detail. 1.1 Real-Axis Breakaway and Break-In Points A breakaway point is the point on a real axis segment of the root locus between two real poles where the two real closed-loop poles meet and diverge to become complex conjugates. Similarly, a break-in point is the point on a real axis segment of the root locus between two real zeros where two real closed-loop complex conjugate zeros meet and diverge to become real. Because the closed-loop poles originate from open-loop poles (when K = 0), a breakaway point will correspond to the point of maximum K along the real-axis segment. Similarly, a break-in point will correspond to the point of minimum K on the real axis segment between the two zeros. The closed-loop characteristic equations is 1 + KG ( s ) = 0, so that along the root locus segments on the real axis ( s = σ ) 1 D ( σ ) K = G ( σ ) = N ( σ ) 1 copyright c D.Rowell 2008 29–1
X X o o " b r e a k a w a y " p o i n t a " b r e a k - i n " p o i n t I j M - 1 - 2 - 4 - 6 X X o o s K = 0 K = 0 K = ¥ K = ¥ K K s s 0 0 s s b b The breakaway/break-in points (maximum/minimum points) will therefore occur where dK d D ( σ ) = dK N ( σ ) = 0 or when N ( σ ) D ( σ ) N ( σ ) D ( σ ) = 0 .

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