Chapter 5 Classical Feedback Analysis
Xmath Control Design Module 5-16 ni.com
plant is open-loop stable, then there should be no encirclements.
If the plant has one open-loop unstable pole, there should be one negative
(counter-clockwise) encirclement.
The stability criterion is most easily derived from the SISO
transfer-function representation of a system. The Nyquist plot for a MIMO
system consists of a set of plots, one for each output, each containing as
many input frequency response curves as there are system inputs. You can
derive any plot from a context menu. If you close a feedback loop around a
SISO system in transfer function format, you obtain a closed-loop system
as shown in Figure 5-6.
Figure 5-6. Closed-Loop System Containing a Variable Gain K
You obtain the following closed-loop transfer function from Y(s) to U(s):
Thus, the closed-loop roots are the roots of the equation 1 + KH(s) = 0.
The complex frequency response of KH(s), evaluated for s=jω in
continuous time and ejωT for discrete systems, will encircle (–1,0) in the
complex plane if 1 + KH(s) encircles (0,0). If you are examining the
Nyquist plot of H(s), you will notice that an encirclement of (–1/K,0) by
H(s) is the same as an encirclement of (–1,0) by KH(s). This fact allows you
to use one Nyquist plot to determine the stability of a system for any and
all values of K.
nyquist( )H = nyquist(Sys,{F,keywords})
The nyquist( ) function is structured very similarly to bode( ) and
nichols( ) in that it is largely a wrapper on the freq( ) function to
obtain the system’s frequency response. The output H is just the output from
the call to freq( ). The main difference from the other two functions is
that nyquist( ) does not calculate the decibel gain and the phase of the
system’s response. It generates the Nyquist plot by plotting the real part of
each point of the response against the imaginary part.
U(s) Y(s)
K
+
–H(s) = num(s)
den(s)
Ys()
Us()
----------- KH s()
1KH s()+
-------------------------=