Signal Processing and Linear Systems-B.P.Lathi copy

T he c ommand conv multiplies d s w ith d2s a nd gives

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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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