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Unformatted text preview: (Continued on next page) 18.2 Roots of Stability Nyquist Criterion 87 e ( s ) 1 S ( s ) = = , r ( s ) 1 + P ( s ) C ( s ) where P ( s ) represents the plant transfer function, and C ( s ) the compensator. The closed loop characteristic equation , whose roots are the poles of the closedloop system, is 1 + P ( s ) C ( s ) = 0, equivalent to P ( s ) C ( s ) + P ( s ) C ( s ) = 0, where the underline and overline denote the denominator and numerator, respectively. The Nyquist criterion allows us to assess the stability properties of a system based on P ( s ) C ( s ) only. This method for design involves plotting the complex loci of P ( s ) C ( s ) for the range = [ , ]. There is no explicit calculation of the closedloop poles, and in this sense the design approach is quite different from the rootlocus method (see Ogata). 18.2.1 Mapping Theorem We impose a reasonable assumption from the outset: The number of poles in P ( s ) C ( s ) exceeds the number of zeros. It is a reasonable constraint because otherwise the loop transfer function could pass signals with infinitely high frequency. In the case of a PID controller (two zeros) and a secondorder zeroless plant, this constraint can be easily met by adding a highfrequency rolloff to the compensator, the equivalent of lowpass filtering the error signal. Let F ( s ) = 1 + P ( s ) C ( s ). The heart of the Nyquist analysis is the mapping theorem, which answers the following question: How do paths in the splane map into paths in the Fplane? We limit ourselves to closed, clockwise (CW) paths in the splane, and the remarkable result of the mapping theorem is Every zero of F ( s ) enclosed in the splane generates exactly one CW encirclement of the origin in the F ( s )plane. Conversely, every pole of F ( s ) enclosed in the splane generates exactly one CCW encirclement of the origin in the F ( s )plane. Since CW and CCW encir clements of the origin may cancel, the relation is often written Z P = CW . The trick now is to make the trajectory in the splane enclose all unstable poles, i.e., the path encloses the entire righthalf plane, moving up the imaginary axis, and then proceeding to the right at an arbitrarily large radius, back to the negative imaginary axis. Since the zeros of F ( s ) are in fact the poles of the closedloop transfer function, e.g., S ( s ), stability requires that there are no zeros of F ( s ) in the righthalf splane. This leads to a slightly shorter form of the above relation: P = CCW. (200) In words, stability requires that the number of unstable poles in F ( s ) is equal to the number of CCW encirclements of the origin, as s sweeps around the entire righthalf splane....
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.154 taught by Professor Michaeltriantafyllou during the Fall '04 term at MIT.
 Fall '04
 MichaelTriantafyllou

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