Unformatted text preview: ty a nd makes t he s ystem sensitive t o p arameter v ariations. Hence i t is
generally d esirable t o have a larger value for ( ( small P O). For a fast response i t is
desirable t o h ave s mall values for t r , t s , t p a nd t d·
T his d iscussion shows t hat for a second-order system in Eq. 6.81, all t he t ransient p arameters ( PO, t r , t s , t p a nd t d ) a re related t o t he pole location of T (s).
F rom t he p oint o f view of t he s ystem design, i t will be convenient t o d raw t he
c ontours r epresenting different values of t ransient p arameters i n t he s -plane. For
instance, each r adial line d rawn from t he origin in t he s -plane represents a c onstant
( line (see Fig. 6.37). Since t he P O is directly related t o ( (Fig. 6.39), each radial
line also r epresents a line of c onstant P O, a s depicted in Fig. 6.40. Similarly, each
vertical line r epresents c onstant ( wn (see Fig. 6.37). Because t s = 4 /(wn, t he lines
representing c onstant s ettling t ime a re vertical lines, as shown in Fig. 6.40. T his figure also shows t he c ontours for c onstant t r . T hese c ontours allows us t o d etermine
by inspection, t he i mportant t ransient c haracteristics ( PO, t r, ts) o f a second-order
s ystem from t he knowledge of i ts pole locations. Moreover, if we a re r equired t o
s ynthesize a second-order system t o m eet some given t ransient specifications, we
c an find t he d esired T(8) w ith t he h elp of t his figure.
As an e xample c onsider t he p osition control system in Fig. 6.36a. L et t he - 10
- 12 ~~
," Ii; ~
I - 14 - 16 F ig. 6 .40 Contours of second-order system pole location for constant PO, constant t .,
and constant tr I II 8 plane.
t ransient s pecifications for t his s ystem be given as
P O S 16%, t r S 0.5 s econds t s S 2 s econds (6.86) We delineate a ppropriate c ontours in Fig. 6.40 t o m eet t he above specifications. T he
s haded region defined by these contours in Fig. 6.41 meets all t he t hree r equirements.
Hence T (s) m ust b e chosen so t hat b oth o f its poles lie in t he s haded region. T he
t ransfer function T (s) for t he closed-loop system in Fig. 6.36a is given by T (s) = K K G(s) 1 + K G(s) S2 + 88 + K (6.87a) T his shows t hat l ocations of t he p oles of T (s) c an b e a djusted by changing t he gain
K . We m ust choose t he gain K so t hat t he poles lie in t he s haded region in Fig.
6.41. T he poles o f T (s) a re t he r oots of t he c haracteristic e quation 438 6 C ontinuous-Time S ystem Analysis Using t he L aplace Transform 439 6.7 Application to Feedback a nd C ontrol S tate E quations
Higher-order Systems t O ur discussion, so far, has been limited to second-order T (s) only. I fT(s) h as
a dditional poles which are far away to t he left of jw-axis, t hey have only a negligible
effect on t he t ransient behavior of t he system. T he reason is t hat t he t ime constants
of such poles are considerably smaller when compared t o t he t ime c onstant of the
complex conjugate poles n ear t he j w-axis. Consequently, t he e xponentials arising
because of poles far away f...
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