0195136047.pdf

802 BASIC CONTROL SYSTEMS

c(t)=

17

3

−

17

2

e−^2 t+

17

6

e−^6 t (4)

(b) The differential equation describing the closed-loop system operation is

d^2 c

dt^2

+ 8

dc

dt

+ 12 c= 68 (r−c) (5)

The output and input variables of the closed-loop system are related by

d^2 c

dt^2

+ 8

dc

dt

+ 80 c= 68 r (6)

The two figures of merit, damping ratioξand natural frequencyωn, follow from the

characteristic equation of the closed-loop system,

s^2 + 8 s+ 80 = 0 (7)

(when the input is set equal to zero and the left side of Equation (6) is Laplace transformed

with zero initial conditions).

Identifying Equation (7) with the general form that applies to all linear second order

systems,

s^2 + 2 ξωns+ω^2 n= 0 (8)

we obtain the following by comparing coefficients:

ωn=

√

80 = 8 .95 rad/s (9)

and

ξ=

8

2 ωn

=

4

8. 95

= 0. 448 (10)

Referring to Figure 16.2.10, the maximum overshoot can be seen to be 17%. With

ξωn=4, it follows that the controlled variable reaches within 1% of its steady-state

value after the time elapse of

ts=

5

ξωn

=

5

4

= 1 .25 s

(c) The transfer function of the direct transmission path is given by

G(s)=

C(s)

E(s)

=

68

s^2 + 8 s+ 12

(11)

The closed-loop transfer function is

M(s)=

C(s)

R(s)

=

G(s)

1 +H G(s)

=

68

s^2 + 8 s+ 12

1 +

68

s^2 + 8 s+ 12

(12)