454 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response MethodNote that a similar analysis can be made if G(s)H(s)involves poles and/or zeros on
thejvaxis. The Nyquist stability criterion can now be generalized as follows:
Nyquist stability criterion[for a general case whenG(s)H(s)has poles and/or zeros
on thejvaxis]: In the system shown in Figure 7–44, if the open-loop transfer function
G(s)H(s)haskpoles in the right-half splane, then for stability the G(s)H(s)locus,
as a representative point straces on the modified Nyquist path in the clockwise di-
rection, must encircle the –1+j0pointktimes in the counterclockwise direction.
7–6 Stability Analysis
In this section, we shall present several illustrative examples of the stability analysis of
control systems using the Nyquist stability criterion.
If the Nyquist path in the splane encircles Zzeros and Ppoles of 1+G(s)H(s)and
does not pass through any poles or zeros of 1+G(s)H(s)as a representative point s
moves in the clockwise direction along the Nyquist path, then the corresponding con-
tour in the G(s)H(s)plane encircles the –1+j0pointN=Z-Ptimes in the clock-
wise direction. (Negative values of Nimply counterclockwise encirclements.)
In examining the stability of linear control systems using the Nyquist stability crite-
rion, we see that three possibilities can occur:
1.There is no encirclement of the –1+j0point. This implies that the system is sta-
ble if there are no poles of G(s)H(s)in the right-half splane; otherwise, the sys-
tem is unstable.
2.There are one or more counterclockwise encirclements of the –1+j0point. In this
case the system is stable if the number of counterclockwise encirclements is the
same as the number of poles of G(s)H(s)in the right-half splane; otherwise, the
system is unstable.
3.There are one or more clockwise encirclements of the –1+j0point. In this case
the system is unstable.
In the following examples, we assume that the values of the gain Kand the time con-
stants (such as T,T 1 ,and ) are all positive.T 2
jv
s PlanesGH PlaneRej 0 +j 0 –`+j`- j`
e 1v = 0 +v = 0 – –^1`Imv =–`v =`Figure 7–52
s-Plane contour and the
G(s)H(s)locus in the GH
plane, where
G(s)H(s)=KCs^2 (Ts+1)D.Openmirrors.com