Modern Control Engineering

Example Problems and Solutions 627

Because the designed system involves three open-loop poles on the jwaxis, the Nyquist dia-

gram will look quite complicated as we will see in what follows:

Define the open-loop transfer function of the designed system as G(s). Then

Let us choose a modified Nyquist path in the splane as shown in Figure 8–57(a). The modified

path encloses three open-loop poles (s=0, s=j1, s=j1). Now define s 1 =s+. Then,

the Nyquist path in the s 1 plane becomes as shown in Figure 8–57(b). In the s 1 plane, the open-

loop transfer function has three poles in the right-half s 1 plane.

Let us choose Since we have

Open-loop transfer function in the s 1 plane

A MATLAB program to obtain the Nyquist plot is shown in MATLAB Program 8–14. The re-

sulting Nyquist plot is shown in Figure 8–58.

=

6.104s 12 +40.48832s 1 +3.5945064

s 13 - 0.03s 12 +1.0003s 1 - 0.010001

=

6.104(s 12 - 0.02s 1 +0.0001)+40.6104(s 1 - 0.01)+ 4

(s 1 - 0.01)(s 12 - 0.02s 1 +1.0001)

G(s)=G(s 1 - 0.01)

s 0 =0.01. s=s 1 - s 0 ,

s 0

G(s)=Gc(s)

s+0.1

s^2 + 1

=

6.104s^2 +40.6104s+ 4

s(s^2 + 1 )

Figure 8–57
(a) Modified
Nyquist path in
thesplane;
(b) Nyquist path in
thes 1 plane.

jv s plane s 1 plane

(a) (b)

0 s

s 0

jv

0 s

MATLAB Program 8–14

% Nyquist Plot

num = [6.104 40.48832 3.5945064];

den = [1 -0.03 1.0003 -0.010001];

nyquist(num,den)

v = [-1500 1500 -2500 2500]; axis(v)