Question

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Consider the unity feedback system depicted in Figure 1 G(s) R(s) 50K s(s + a) Figure 1 1. Determine the system's closed loop transfer function. 2. Plot the system's step response for K=10 and: • a= 2 • a=5 • a= 10 3. What happens to the system's response as a increases? Justify your answer. 4. In each case specify the system's overshoot and settling time. 5. In each case, use the final value theorem to determine the closed loop system's steady state error to a step input.

Answer 1

soln - - S T " Civen the system. ers SCSta) 2) The overall open loop system as a siven in figure is » e =lies) sosto - since aix we know that the close loop toomster en at the given fis in terms of one feedback systen: ie ces) . &cs) Where RCS) It airs rics) HES) = I - (cs) RES) srsta) 1 tk sosta) k CSI R.CS) Sesta) tk close CS loop --> TROS stastk song rct=ult) & ils 10 6 .) Kalo & a=2 (cs) RCS) s? + 2 st It 10 3t 5242141 Cessa S (52+25+10) using al fraction A3²1RASTIJA + BS?tesi A+B = 0 2Atc=0 2 C-2

put the at He obtained values C(5) = In tu (6+2) (5+1) ²0 ces Es-2 ) 52+25 +1. (5+2) (82t2stro) 842) (6+1, ²+20-2 (6+1) (6+12+ 9 et cast – e cos) - s (84) 2+ 41 - sin3t. ((t) = (1 - 4) = (1 - et (cos3t & /e sien Lost & & sin3t) wley dampe at fo-es. d = 30,-, os Bon = 2 we w10 2 - maxmoner shoot - { mp- eu-EL ? = 0.376 = 0.9934 seco Mpa 0-357 ? tp A CH 1-31_ -2 for a= 2 >t (pec) tp=0.9934 sec

a=5 - ces satsstio (es) - +55 +00) do: mp- e Elora -> [Mp = 08017 ) Cp - 2&wn - 5 28 To 5 o ę = 6.79 5 b = 1.62 sec. 7 - -- -1.939 top- stdec) tp=1-62 ree co wr= 2 a) a = 10:- (CS2 Rcs) 16 (² tlost 10 En to Ř & win =- = 1.500 hd not defined system I becz they t have no osin becomes. 5) As we have seen from overdamped abone rexpose as the value of a î ex the oscillations at the seastem rexpoure will be decreases & 8 yrstem - be conces more sleggingh in nature.

a=5 - ces satsstio (es) - +55 +00) do: mp- e Elora -> [Mp = 08017 ) Cp - 2&wn - 5 28 To 5 o ę = 6.79 5 b = 1.62 sec. 7 - -- -1.939 top- stdec) tp=1-62 ree co wr= 2 a) a = 10:- (CS2 Rcs) 16 (² tlost 10 En to Ř & win =- = 1.500 hd not defined system I becz they t have no osin becomes. 5) As we have seen from overdamped abone rexpose as the value of a î ex the oscillations at the seastem rexpoure will be decreases & 8 yrstem - be conces more sleggingh in nature.

& arba in care a=2 e for at me doesno define bo olley have Jos & wn Fe 2% ts=4 ts- 2วก 2.5 0316 X TO ž 4 sec. 1.6 SC t's ts = 60 I o s2 #2stro RCS) a=2) ces = Scs² 82.54Lo) S + ) X6- fer ess=lt sees - et Jess = z] so a=52 ces ) = et 16 SC 2+55+ 10) 840) s(2 5stio) ess -a0 (a=10 ces) ) मारना) sistlostiu) 8xlo essa e So 01 (2 losto