Question

Im{s} x Х 1 2 Re{s} -2 -1 1 Х -1 Assuming a zero-state system at reference time, determine the behavior of the output as time approaches infinity for the input x(t). x(t) = (5te-2t + 2t)u(t) 0.1 0 0.2 10 0.8

Answer 1

plot in s-domain, A Im (s) - 1 * -2 -- 2 Rics) - 1 -1 x Х o represents Zero (Numerator of transfer function) →x represents pole C denominator of transfer function) > from the plot, Zerors are S=0 S =-Y → poles are, S =-2 S=-2+j SE-2-3 Sa -1+j S=-1-j . → Transfer function : ycs) S (5+1) XCS) (5+2)(5+2-1) (3+2+3) CS+1-3) (5+1+j)

VCS) X(S) = SCS+1) (5+2)(8+23+1) (CS+1)71) S (S+1) (5+2) (5445+4+1) Cs421+1+1) ycs) = S(5+1) (S+2)(52+45+5)(8+26+2) XCS) -2t given input act) = (ste +2t ) uct) 5 X (5) - (87272 * 2 / 2 55?+ 2C5+2) S²(S+2) 2 55+265745+4) 35+2)2 552+28+85+8 52C5+2)2 7585+8 S²(S+2)² XCS) = XCS) = 2 from Yos) SCS+1) •XC5) (5+2) (s+45+5)(s+25+2)

7 Substitute @ in the above equation, $CS+1) 784485+8 Y(s) - (5+2) ($14545) (s 425 +2) SHCS+2)? (5+1)(75485+8) S (5+233 (s?+45+5)(8+26+2) ycs) -2t → The output from the given system as time approachur infinity for 24) =(5test) uc) is given by gcao) = it ttoo yet) Lt in Laplace domain, 9Cz) = sys) sto (S+1) (75485+8) $(5+233 (s=45+5)(s+as+2) (0+1) (76024860)+8) (6+2) (0914601+5)(o 42(0)+2) 1x8 Lt sto & y looy 8x5X2 9.00) = yoo) = 0.1 Answer 5 so output at too = 0.1