A First Course in FUZZY and NEURAL CONTROL

2.7. PROPORTIONAL-INTEGRAL-DERIVATIVE CONTROL 55

Performing partial fractions expansion, we obtain

Y(s)=

10

s

−

10

s+0. 05

The inverse Laplace transform yields

y(t)=10u(t)− 10 e−^0.^05 tu(t)

whereu(t)is a unit step. The graph in Figure 2.19 shows that the vehicle takes
more than 100 seconds to reach the steady-state speed of 10 meters/second.
Clearly, this does not satisfy our rise time criterion of less than 5 seconds.

0

2

4

6

8

10

(^2040) t 60 80 100
Figure 2.19.y(t)=10u(t)− 10 e−^0.^05 tu(t)
From the above analysis, we have determined that a controller is needed to
improve the performance. The performance of this system can be improved by
providing a unity feedback controller. Figure 2.20 is the block diagram of a
typical unity feedback system.
Figure 2.20. Unity feedback system
We choose the standard PID transfer function for the controller, namely,
Gc(s)=KP+

KI

s

+KDs=

KDs^2 +KPs+KI

s

The plant transfer function is as derived above, namely,

Gp(s)=

1

ms+b