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# lec18 - (Continued on next page 18.2 Roots of Stability...

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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 closed-loop 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 closed-loop poles, and in this sense the design approach is quite different from the root-locus 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 second-order zero-less plant, this constraint can be easily met by adding a high-frequency rolloff to the compensator, the equivalent of low-pass 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 s -plane map into paths in the F -plane? We limit ourselves to closed, clockwise (CW) paths in the s -plane, and the remarkable result of the mapping theorem is Every zero of F ( s ) enclosed in the s -plane generates exactly one CW encirclement of the origin in the F ( s ) -plane. Conversely, every pole of F ( s ) enclosed in the s -plane 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 s -plane enclose all unstable poles, i.e., the path encloses the entire right-half 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 closed-loop transfer function, e.g., S ( s ), stability requires that there are no zeros of F ( s ) in the right-half s -plane. 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 right-half s -plane. 18.2.2 Nyquist Criterion The Nyquist criterion now follows from one translation. Namely, encirclements of the origin by F ( s ) are equivalent to encirclements of the point ( 1 + 0 j ) by F ( s ) 1, or P ( s ) C ( s ).
88 18 CONTROL SYSTEMS LOOPSHAPING Then the stability criterion can be cast in terms of the unstable poles of P ( s ) C ( s ) , instead of those of F ( s ): P = CCW ⇒∀ closed-loop stability (201) This is in fact the complete Nyquist criterion for stability. It is a necessary and suﬃcient condition that the number of

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