Given the following information about a continuoustime LTI s

Given the following information about a continuous-time LTI system: (1) it is causal, (2) its system function H(s) is rational and has three poles: -1, -2, and -3, respectively, (3) If x(t) = 5, then y(t) = 0, (4) h(t = 0^+) = 8, and (5) H(s) has a zero at -1/8, determine the exact form of H(s).

Solution

Ploes are at s=-1,-2,-3

Zero is at s=-1/8

Let H(s)= k*s*(s+1/8)/[(s+1)(s+2)(s+3)]

When x(t)=5, then X(s)=5/s

Y(s)= H(s)*X(s)= k(s+1/8)/[(s+1)(s+2)(s+3)]

= (-k7/16)/(s+1) +(15k/8)/(s+2) -(23k/16)/(s+3)

  

y(t) = 0,

Now. From initial value theorem

h(0+)= s--> infinity. SH(S)

8 = k

Hence. H(s)= 8*s*(s+1/8)/[(s+1)(s+2)(s+3)]