Nyquist contour

Consider a Nyquist contour for the nominal open-loop system Gm(iLu)C(iuj) with the model uncertainty given by equation (9.119). Let fa( ) be the bound of additive uncertainty and therefore be the radius of a disk superimposed upon the nominal Nyquist contour. This means that G(iuj) lies within a family of plants 7r(C(ja ) e tt) described by the disk, defined mathematically as... 

From the Nyquist stability criterion, let N k, G(iuj)) be the net number of clockwise encirclements of a point (k, 0) of the Nyquist contour. Assume that all plants in the family tt, expressed in equation (9.132) have the same number ( ) of right-hand plane (RHP) poles. 

This choice of Cg creates a dilemma—how do we evaluate H(s) on the boundary of an infinite region This problem is solved by choosing Cg to be the Nyquist contour shown in Fig. K.2. The Nyquist contour consists of the imaginary axis and a semicircle with radius,... 

Because Gol s) is strictly proper (that is, it has more poles than zeros), Gol s) 0 as / 00 and the semicircular arc of the Nyquist contour maps into the origin... 

A2. The modified Nyquist contour Q circumvents any open-loop poles that lie on the imaginary... 

The Nyquist stability criterion can be stated as A closed-loop control system is stable if, and only if, a contour in the G s)H s) plane describes a number of counterclockwise encirclements of the (—l,jO) point, the number of encirclements being equal to the number of poles of G s)H s) with positive real parts . 

Fig. 6.18 Nyquist diagram showing stable and unstable contours.

The Nyquist stability criterion that we developed in Chap. 13 can be directly applied to multivariable processes. As you should recall, the procedure is based on a complex variable theorem which says that the dilTerence between the number of zeros and poles that a function has inside a dosed contour can be found by plotting the function and looking at the number of times it endrdes the origin. 

As we will show in a minute when we have completed the rest of the contours, this means that if the controller gain is made big enough, the Nyquist plot will encircle the ( — 1, 0) point. If IV = 1, Z = 1 for this system since P = 0. Thus there will be one zero or root of the closedloop characteristic equation outside the unit circle. 

Thus, the Co contour maps into a semicircle in the GmGc plane with a radius that goes to infinity and a phase angle that goes from +7t/2 through 0 to —ttI2. See Fig. 11.4c. The Nyquist plot does not encircle the (- 1,0) point if the polar plot of GM(m)Gc iw) crosses the negative real axis inside the unit circle. The system would then be closedloop stable. 

The Nyquist stability criterion is based on two concepts from complex variable theory, contour mapping and the Principle of the Argument, We briefly review these concepts in Appendix K. More detailed descriptions are available elsewhere (Brown and Churchill, 2008 Franklin et al., 2002). We now present one of the most important results of frequency domain analysis ... 

The concept of contour encirclement plays a key role in Nyquist stability theory. A contour is said to make a clockwise encirclement of a point if the point is always to the right of the contour as the contour is traversed in the clockwise direction. Thus, a single traverse of either Ch or Cg in Fig. K.l results in a clockwise encirclement of the origin. The number of encirclements by C is related to the poles and zeroes of H s) that are located inside of Cg, by a well-known result from complex variable theory (Brown and Churchill, 2004 Franklin et al., 2005). 

Note that N is negative when P > Z. For this situation, the Ch contour encircles the origin in the counterclockwise (or negative) direction. Next, we show that the Nyquist Stability Criterion is based on a direct application of the Principle of the Argument. 

Recall that H s) was defined as H s) = 1 + Gql ) Thus, the Ch and contours have the same shape, but the Ch contour is shifted to the left by -1, relative to the contour. Consequently, encirclements of the origin by Ch are identical to encirclements of the -1 point by As a result, it is more convenient to express the Nyquist Stability Criterion in terms of Ggl(s) rather than H s). 

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