
120 Feedback Control in Systems Biology
FIGURE 4.5: Nyquist contour in the s-plane.
The number of unstable closed-loop poles is equal to the number of uns table
open- loop poles plus the num ber of encirclements (counted as positive in the
clockwise and negative in the counter-clockwise direction) of the origin by the
Nyquist plot of D(s).
This result follows directly by applying Cauchy’s principle of the argument
to the function D(s) with the s-plane contour given in Fig. 4.5, and noting
that
1. Z and P represent the numbers of zeros and poles, respectively, of D(s)
in the right half plane, and
2. the zeros of D(s) are the closed-lo