Obtain the response ykT of the following system YsXS 1s 1s2

Obtain the response y(kT) of the following system: Y(s)/X*(S) = 1/(s +1)(s+2) where x(t) is the unit-step function and x*(t) is its impulse-sampled version. Assume that the sampling period T is 0.1 sec.

Solution

laplace transform of unit step function=1/s

y(s)=1/[s(s+1)(s+2)]=a/s+b/(s+1)+c/(s+2)

1=a(s+1)(s+2)+bs(s+2)+cs(s+1)

1=a(s^2+3s+2)+b(s^2+2s)+c(s^2+s)

1=s^2(a+b+c)+s(3a+2b+c)+2a

compare

2a=1

a=0.5

a+b+c=0

3a+2b+c=0

from the above equations

b=-1

c=0.5

y(s)=0.5/s-1/(s+1)+0.5/(s+2)

apply inverse laplace transform

y(t)=0.5-1*e^-t+0.5*e^(-2t)