Question

2. Consider the causal LTI system in which the input x(t) and output y(t) are related through the following block diagram respresentation: x(t) y(t) + 1/s # 7 -5 1/s 10 6 a. Find the system function. (2.5 points). b. Determine the differential equation relating y(t) and x(t). (2.5 points). c. Show that this system can be realized as a cascade interconnection of four first order subsystems. Give the system function for each subsystem, and sketch the block diagram of interconnected subsystems. (2.5 points). d. Show that this system can be realized as a paralel interconnection of lower order subsystems. Give the system function for each subsystem, and sketch the block diagram of interconnected subsystems. (2.5 points).

Answer 1

| 6 w(s) 2 W(S). (F) 6 X(+)- X(5) Vs W(S) (SMS S S2 low(s) S2 Ys 6 W(S) S2 6 101 WIS) at summer (1) x(S) + lowls) + 7 W(S) S W(S) 52 X (5) = W(S) (s) ( -5 -10 ها - S2 at Summer (2) W(S) + 6 W(S) Sw(s) = Y(5) S2 S

from I each it = Y(S) - W(S) ( w(s) (-5-102) Ol X(S) D/3 1 + 2-5/ S Y(S) X (5) 1-7-1952 Y(S) S²+6-55 X15) 5275-10 Y(S) s? 55+6 x(S) S? 75-10 System Function H(S) = S255+6 S²_75-10 = (4.

b) differential ean relating 4(+) % *(+) Y(S) X(S) S²55+6 5275-10 Y(s) (5275-10) = X(5) ( 52 55+6) s?y(s) - 75418) -10 Y15) = 8?x(S)- 5 5x4) + 6x(5) d? yl) 7 dyla) Lloy(+) dtan dt d2XLt) 50x(+) +60(+) Id2ylt) -7 dy (+) _loy(+)= d?n[+). dt2 dt 50 m(t) dt 2 dt +6 m(t) c) H(S) = s? 55+6 327S-10 (S-3) (S-2) (S-8.21) (S+1.216) xls) sobstemi Sobslem 2 306 Syst 3 S-2 S-3 subsystema 1 S+1.216 1905) S-8.21 block diagram of cascade interconnection

d) H(S) = S²55 +6 S27S-0 s2-75-lo) 3255+6 82_75-10 + + 2 st 16 H(S) = It It 25+16 5275-10 2S+16 (S-S) (5-52) doing partial fraction H(S) = 1 + A + (S-SI) S-S2 Si = 7+J09 S2=7 - våg Si 8.216 S2 = 1.216 A = 4+ (S-s) (2 +16) Susi 2 (S-S) (S-S2) 2 S1+16 3.437 ( S.-S2) A =

SAS2 B = Lt B = 1+ (S-S2) (25+16) (5-Si) (5-S2) 2 S2+16 Sasa (52-51). 21 7-J69) +16 -2 Jog B B = -1.437 H(S) = 1+ 3.437 1.437 (S-8.21) (5+1.216) Subsystem Irans fer function TEI=1 TF2 = 3.437 S- 8:21 TB=-11437 SH.216 1 + 815) YES) 7 3:437 S-8.211 + block diagram +1.437 S+1.216)