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THE NYQUIST STABILITY CRITERION In a system with the open loop transfer function
of KZ(s)/P(s) the characteristic
equation in 1+ KZ(s)/P(s). |
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To develop the Nyquist criterion, the
characteristic function itself is written as a function as a ratio of
polynomials so that
where –r1, -r2, . . . are the roots of the characteristic equation (or closed loop poles) and that –p1, -p2, . . . are the poles of both the characteristic function and open loop transfer function (open loop poles). Poles and roots at the origin have been omitted in the interests of simplicity. |
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Figure 1 Encirclement of one root |
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Locate some roots and poles on the s plane, arbitrarily, as indicated in Fig. 1 a. In addition, constrain s to take on the values along the arbitrary closed contour that encloses only the root r1 and is made to travel in a clockwise direction. For each value of s on the contour there will be a corresponding value of the characteristic function 1+KZ(s)/P(s). The closed contour in the s plane will map into a closed contour in the 1+KZ(s)/P(s) plane. In Fig. 1a, draw the vectors from all the
poles and roots to a point s on the contour (only one of these
vectors are shown in Figure 1a). When s
has made one complete clockwise revolution about r1, the vector (s
+ r1) will have also
made one clockwise revolution for a change in angle of 360o. Since
the other roots and poles are outside the contour, their vectors will not
make complete revolutions but will merely nod up and down as s travels around
the contour. The clockwise revolution of the (s + r1)
vector will produce a clockwise revolution about the origin in the 1 + KZ(s)/P(s) plane as shown in Fig.
1b, since the root r1 is
in the numerator of the function. The characteristic function, unfortunately, is
rarely in the factored form of Equation
1, if it were, the location of the roots would be
immediately known and further investigation would be unnecessary. Mapping of
the contour in the KZ(s)/P(s) plane
is more appropriate and convenient and is shown in Fig. 1c. The shape of the
contour is unchanged, but now the clockwise encirclement of the origin in the
1 + KZ(s)/P(s) plane corresponds to
a clockwise encirclement of the (–1,0)
point in the KZ(s)/P(s)
plane. If the contour in the s plane is redrawn so as to encircle both a root and a pole, as
shown in Fig. 2, the vectors from both r1
and p3 will each make
one complete revolution as s
travels the contour. Since the (s +
r1) vector is in the
numerator, it will contribute +360 to
the change in angle of the 1+KZ(s)/P(s),
whereas the (s + p3) vector, being in the denominator,
will contribute –360 to the change.
Consequently, there will be no net change in the angle of the 1+KZ(s)/P(s) vector; it will not
encircle the origin in the 1+KZ(s)/P(s)
plane or the (–1,0) point in the KZ(s)/P(s)
plane as shown in Fig 2b and c.
Figure 2 Encirclement of
one pole and one root |
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The closed contour in the s plane may be drawn to include any region desired. For the sake
of consistency, this contour will always be traveled in a clockwise
direction. Each root within the contour will produce a clockwise encirclement
of the (–1,0) point in the KZ(s)/P(s)
plane; each pole within the contour will produce a counterclockwise
encirclement of the (–1,0) point. The results of the development to this
point can be summarized by the relationship Nn=Nr-Np where Nn
is the number of encirclements of the –1 point, Nr is the number of roots within the contour in the s plane, and Np is the number of poles within the s contour. If Nn is positive, the (–1,0) point is encircled in a
clockwise manner and Nr>Np.
If Nn is positive, the
(–1,0) point is encircled in a counter-clockwise manner Nr<Np. Equation
2 can be used to detect the presence of roots in the
right half of the s plane by
selecting a contour that completely encloses the right-half plane. Such a
contour is shown if Fig. 3 and is divided into three regions. In Region I, s=jw and w takes on values from w=
0+ to w = ¥ along the positive
imaginary axis. In Region II, s=Rejq , where R is infinite and q changes
from +90 to -90 . Finally, in Region III, s=jw again and w
goes from -¥ to 0 along the negative imaginary axis. Since complex factors
always occur as conjugate pairs, the map of Region III will be the mirror image
about the real axis of Region I. The map of Region II is obtained by setting s equal to Rejq in KZ(s)/P(s)
and taking the limit as R goes to
infinity. Poles at the origin and on the imaginary axis of the s plane produce singularities in KZ(s)/P(s) and are bypassed by the
small semicircles around them. |
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Figure 3 The closed
contour of the right half of the s
plane |
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The map of the s-plane
contour of Figure 3 into the KZ(s)/P(s)
plane is known as a Nyquist diagram. Figure
4 shows a sample of such mappings for a stable (a) and an unstable (b)
systems. he application of Equation
2 to the Nyquist diagram is called the Nyquist stability criterion and is
written as Nr=Nn+Np If we know KZ(s)/P(s),
we know the number of poles in the right-half plane. If we count the number
of times the Nyquist diagram encircles the –1 point, we know the number of
roots in the right-half plane. If the encirclements are clockwise, Nn is positive; if
counterclockwise, Nn is
negative. Any root in right half plane denotes an unstable system. One of the several expressions for the Nyquist
stability criterion states that the Nyqust diagram of a stable system (a system with no root in
the right half plane) must encircle the –1 point in the counterclockwise
direction as many times as there are poles in the right half of the s plane. Most closed-loop systems
are open-loop stable, do not have any pole (open-loop pole) in the right half
of the s plane, i.e. Np =0. Such closed-loop systems are stable (will
not have any root in the right half plane, i.e, Nr =0) if Nn=0,
i.e., the
Nyquist diagram of an open-loop stable system does not encircle the (–1,0)
point. |
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Figure 4 |
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A simplification in the plotting of KZ(s)/P(s), results from the fact that
any KZ(s)/P(s), representing a
physical system will have more poles than zeros and thus zero response at
infinite frequency. This mean that
the map of Region II corresponding to s
at infinity (see Figure 3) will be a point of infinitesimally small value
near the origin of KZ(s)/P(s).
Therefore, one can accomplish a complete evaluation of a physical system KZ(s)/P(s), by letting s traverse the imaginary axis from –j¥ to +j¥. Note that the evaluation
of KZ(s)/P(s) from s=0 to s=+j¥ is the same as finding the frequency response of the open loop
system. Because KZ(-jw)/P(-jw)
is the complex conjugate of KZ(jw)/P(jw), one can easily obtain the entire plot of KZ(s)/P(s) by reflecting the 0£s£+ jw portion (the solid lines in Figure 4) about the real axis to get
the - jw £s£0 portion (the dashed lines in Figure 4). Hence, one sees that the closed loop
stability can be determined in all cases by examining the frequency response
of the open loop system on a polar plot. In
practice, for most systems, one need not carry out a complete evaluation of
KZ(s)/P(s) with subsequent inspection of the –1 encirclement; a simple look
at the frequency response may suffice to determine the stability. If the open loop frequency
response encircles the (-1,0) point, the closed loop system is unstable and
if it does not encircle the (-1,0) point, the system is stable. Encirclement
of the (-1,0) point is the basis for defining stability
margins (gain and phase margins) in the context of frequency response. |
