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Unformatted text preview: dy-state error to system parameters. In many cases, adjusting
t he amplifier gain K c an result in t he desired performance. I f t he r equirements 460 6 C ontinuous-Time System Analysis Using t he L aplace Transform P roblems
461 c annot b e satisfied by mere a djustment of gain, t hen some form of compensation
m ust b e used. T he loci of t he c haracteristic roots of t he s ystem are called t he r oot
locus, which proves extremely convenient in designing a feedback system.
Most of t he i nput signals a nd p ractical systems are causal. Consequently, we
a re r equired m ost of t he t ime t o deal with causal signals. W hen all signals are
restricted to t he c ausal type, t he Laplace transform analysis is g reatly simplified;
t he region of convergence of a signal becomes irrelevant t o t he a nalysis process.
T his s pecial c ase of t he Laplace transform (which is r estricted t o c ausal signals)
is called t he u nilateral Laplace transform. Much of the c hapter deals with this
variety of L aplace t ransform. Section 6.8 discusses t he general Laplace transform
( the b ilateral Laplace transform), which can handle causal a nd n oncausal signals
a nd s ystems. I n t he b ilateral transform, t he inverse transform o f F (s) is n ot u nique
b ut d epends on t he region of convergence of F (s). T hus t he region of convergence
plays a very c rucial role in t he b ilateral Laplace transform. sin t l ie
1- 6 .1-3 F md t he mverse (unilateral) Laplace transforms of t he following functions:
8 4. 2 s+5
+ 5s + 6 ( f) 5. N ahin, P .J., "Oliver Heaviside: Genius a nd Curmudgeon," I EEE S pectrum,
vol. 20, pp. 63-69, J uly 1983.
Berkey, D., Calculus, 2nd ed., Saunder's College Publishing, Philadelphia, Pa.
1988. 7. 2 ( h) Churchill, R .V., O perational M athematics, 2 nd ed, McGraw-Hill, New York,
1958. 9. Yang, J .S. a nd Levine, W.S. C hapter 10 in T he C ontrol Handbook, CRC P ress,
1996. - S- 6 + 2) ; ,ind t:.~~aplace transf~rms o f t he following functions using only Table 6.1 a nd t he
Ime-s I mg property (If needed) of t he u nilateral Laplace transform: ( e) t e-tu(t - r)
( f) sin [wort - r)] u (t - r)
( g) sin [wo(t - r)] u (t) ( d) e -tu(t - r) ( h) sin w otu(t - r) ~S!~! ~~~aYsa~~e~:. ~n6~1~~~ time-shifting property, determine t he Laplace transform
Hint: See Sec. 1.4 for discussion o f expressing such signals analytically. 6.2-3 F ind t he inverse Laplace transforms of t he following functions:
(28 + 5 )e- 2s
( a) s 2+5s+6 e -(8-1) + 3
( e)-,,-_ _
s2 - 2s + 5 3s ( b) s e- +2
s2 + 28 + 2 B y d irect i ntegration [Eq. (6.8b)] find t he Laplace transforms a nd t he region of convergence of t he following functions:
6.2-4 ( a) u (t} - u(t - 1) (e) cos Wit cos W2t u (t) ( b) t e-tu(t) ( f) cosh (at) u (t) ( e) t cos wot u (t) ( g) s inh (at) u (t) ( d) (e u - 2e- t )u(t) ( h) e - 2t cos (5t + 0) u(t) 8 + 1)2(s2 + 28 + 5) (e) e -(t-T)u(t) 6 .2-2 + 5) 3 (s ( a) u (t) - u(t - 1) Problems
1.1-1 + 1)(s + 2)4 s+1
s is + 2)2(S2 + 4s ( i) 2s + 1
(s + 1)(s2 + 2s ( b) e -(t-T)u(t - r) E...
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