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counterclockwise movement along the contour B in the F -plane because the phase
angle in the F-plane is negaLive of that in the s-plane.
Suppose the zero of F(s) - s - zi lies inside the contour A as shown in
Fig. 5.17c. Then the vector V' makes one complete rotation of 360 deg in the
F-plane so that the contour B encloses the origin. Similarly, we have a pole of
the mapping function F(s) ~ l/(s - pl) that lies inside the contour A; then the
image contour B in the F -plane also encloses the origin as shown in Fig. 5.17d.
If the contour A encloses an equal number of poles and zeros of the mapping
function F(s), then clockwise encirclement of the origin due to the zeros cancels
the counterclockwise encirclement due to poles, and thFe image contour B does not.
enclose the origin as shown in Fig. 5.17e.
Nyqt.ust p/ot. Suppose the contour A in the s-plane is a semicircle of infi-
nite radius covering the entire right half of the s-plane, then the corresponding
image contour in tl~e F -plane is said to be the Nyquist plot of right half of the
s-plane through the given mapping function F (s).lf we have zeros and/or poles in
the right half of the s-plane, then the image contour B in the F -plane will encircle
the origin n times where n = nz - nP and nz and np are, respectively, the number
of zeros and poles of the mapping function F (s) located in the right half of the
s-plane. If n > 0, then we will have n clockwise encirclements and, if n < 0, we
will have that many counterclockwise encirclements of- the origin.
476 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Nyquist criterion of stability. The transfer function of a closed-loop system
is given by
G(s)
T(s) : ~+ (s~H(s)
Ng Dh
= Ds Dh + Ng Nh
(5.138)
(5.139)
where G(s) = Ng]Dg and H(s) : Nh/Dh.
The poles of the closed-loop transfer function T(s) are generally not known and
have to be determined actually by solving the closed-loop characteristic equation
1 + G(s)H (s):O. Note that the poles of T(s) are the zeros of 1 + G(s)H (s). The
closed-loop system will be unstable if any of the poles of T(S) are located in the
right half of the s-plane. The usefulness of the Nyquist stability criterion is that
it enables us to know whether any of the poles of T(s) are located in the right
half of the s-plane without actually soMng the closed-loop characteristic equation
l+G(s)H (s) :O.In this way,it gives us an idea whether the given closed-lotop sys-
tem is stable or not without actually knowing the location ofthe closed-loop poles.
This information is very useful in evaluating the stability of a closed-loop system
as a certain system parameter, say the gain k, is varied.
Suppose we make a Nyquist plot of the function F(s) = l+G(s)H (s). Note that
the zeros of this function F(s) are the poles of the closed-loop transfer function
T(s) and the poles of F(s) are the combined poles of the open-loop transfer
function G(s)H(s), which are known. Then, the Nyquist criterion fo:stability
centers around the determination of the parameter, N = P - Z, where N is th~.
number of encirclements of the origin in the F -plane, P is the number of poles of
F (s) located in the right half of the s-plane, and Z is the number of zeros of F(s)
that are located in the right half of the s-plane. Note that a positive value of N
corresponds to counterclockwise encirclement of the origin and a negative value to
clockwise encirclement. Here, P is known but Z is not known. Therefore, unless
we have a method to determine Z, we cannot sketch a Nyquist plot and determine
the system stability. As said before, we do not have an easy method of finding Z.
Suppose we use the function G(s)H(s) as the mapping function instead of I +
G(s)H (s) because all the poles and zeros ofthe function G (s)H (s) are known. The
resulting Nyquist plot is the same as that ofl+G(s)H (s) except that it is displaced
by one unit to the left of the origin. Then, instead of counting the encirclement of
the origin, we can count the encirclement of the point -l. Everything else remains
the same, and we can now use the Nyquist plot to determine the system stability,
With this modification, the Nyquist criterion for the stability of a closed-loop
system can be restated as follows.
If a contour A in the s-plane that covers the entire right half of the s-plane is
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