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118 MECHATRONICS
r (s) 1 y (s)
K e -t s
d
- (s+a)

FIGURE 2.53: A closed loop control system with a process which has pure time delay.

which states that the u (t)issimply t time period delayed version of u (t). Let the Laplace
2
1
d
transform of u (t)be U (s), and the Laplace transform of u (t)be U (s); it can be easily
2
1
2
1
shown from the application of Laplace transform equation that
L{u (t)} = L{u (t − t )} (2.175)
1
d
2
= e −t d s ⋅ U (s) (2.176)
1
U (s) = e −t d s (2.177)
2
U (s)
1
which shows that the transfer function of pure time delay is e −t d s where t is the magnitude
d
of the time delay.
Consider the closed loop control system shown in Figure 2.53. When t = 0.0, there
d
is no time delay in the loop. The closed loop transfer function is
Y(s) Ke −t d s
= (2.178)
R(s) (s + a) + Ke −t d s
and the closed loop system characteristic equation which determines the closed loop pole
locations is
Δ (s) = (s + a) + Ke −t d s = 0 (2.179)
cls
e −t d s
= 1 + K = 0 (2.180)
s + a
Figure 2.54 shows the root locus of the closed loop system poles for t = 0.0, and two
d
different approximations to the pure time delay: one with a first-order filter and one with a
second-order filter,
1
−t d s
e ≈ ; Approximation 1 (2.181)
t s + 1
d
1
−t d s
e ≈ ; Approximation 2 (2.182)
(t s + 1) 2
d
Clearly, the time delay approximations show that it has destabilizing effect on the closed
loop pole locations. When the time delay does not exist (t = 0.0), the closed loop system
d
is stable for all values of K :0 ⟶ ∞. When a two-pole filter approximation is made to
∗
a non-zero time delay, the closed loop system is stable only for the range K :0 ⟶ K ,
where
∗
K :1 + K ∗ 1 1 | s=jw = 0.0 (2.183)
2
(t s + 1) s + a
d
∗
and unstable for K > K .