Page 134 - Mechatronics with Experiments

120 MECHATRONICS
where
−t d s −t d Re(s) −jt d Im(s)
e = e ⋅ e (2.185)
−jt d w
= (Real Number) ⋅ e (2.186)
e −jt d w = cos(t w) − j sin(t w) (2.187)
d d
where the cos(t w) and sin(t w) terms have an infinite number of roots for w. Each solution
d
d
for w corresponds to a point on a separate branch of the root locus. The most significant
branch of the infinite number of root locus branches is the one closest to the origin. It is
that branch that goes unstable before the other branches, and hence dominates the transient
response and stability of the closed loop system.
While accurate analysis of a pure time delay in the root locus is rather difficult due
to the infinite number of branches, the analysis in the frequency domain is rather easy.
The Bode plot of a pure time delay is simply a unit magnitude and linear phase angle as a
function of frequency. However, when we plot the phase angle in logarithmic scale of the
frequency, it looks nonlinear (Figure 2.55a).

G (jw) = e −t d s | s=jw (2.188)
1
= e −t d jw (2.189)
= 1.0 ⋅ e −j(t d w)) (2.190)
|G (jw)| = 1.0 (2.191)
1
[rad] =−t s ⋅ w [rad∕s] (2.192)
d
[deg] =−57.3deg∕s t s ⋅ w [rad∕s] (2.193)
d
The Bode plot of the pure time delay is shown in Figure 2.55a.
s=tf(’s’);
G1 = exp(-1.0*s) ; % Time delay
G2 = 1/(s+1) ;
G3 = G1 * G2;

figure(1) ; grid on;
subplot(2,1,1) ; bode(G1,’b’); grid on; % Bode plot of time delay only: eˆ(-td s)
subplot(2,1,2) ; bode(G1,’b’,G2,’g’,G3,’r’); grid on;
% Bode plot of, eˆ(-td s), 1/(s+1) and eˆ(-td s) / (s+1)

Notice that the effect of the time delay on the stability (through phase margin and
gain margin measures) of the closed loop system is obvious: it adds phase lag to the loop
transfer function and reduces its stability margins. Eventually, it will make the closed loop
system unstable as the closed loop gain gets larger. Figure 2.55b shows the Bode plot of a
closed loop system (Figure 2.53), with and without time delay in the loop transfer function,
for the following parameters a = 1.0, t = 1.0 s. Clearly, the closed loop system is stable
d
for all values of the gain K when there is no time delay, as was confirmed by the root
locus as well. However, when the time delay is taken into account, the closed loop system
eventually goes unstable due to added phase lag by the time delay (the same result was also
confirmed by the root locus analysis above). The closed loop system becomes unstable at a
value of gain K where the phase margin becomes zero. This gain value can be determined
from the Bode plot as follows:

1. Read the value of frequency when the total phase of the loop transfer function
◦
∗
(including the time delay) is −180 : w = w .
∗
2. At that frequency, read the value of the loop transfer function gain, |G(jw )|.
∗
3. The closed loop gain which defines the stability margin is: K = 1 ∗ .
|G(jw )|