Question

The transfer function relating Z(s) to the input U(s) is given below: It can be shown that the ILT of Z (s) is of the form Z(t) Find C1, C2, r1 r2 r3 , and W

Problem 1: (5Points) The transfer function relating Z(s) to the input U(s) is given below: St S It can be shown that the ILT of Z (s) is of the form rt Find C1, C2» T['T2'3' Problem 2: (20 Points) The DE for a suspension system is given by 1000z +700002+ 36000 ž-72000 + 36000 u (1) Given u(t)- 1 - cos(2t) , and L(cos(wt)- -^Tw Show that U(s)- ; Find Z(s) with zero ICs. T" (2) Using the ILT, find z(t). For t-0:0.01:7, plot(rt,z). (3) Use "impulse" command on Z(s), plot z(t) (4) Use "lsim" command on Z(s), plot z(t)

Answer 1

Given,

$Z(s)=\frac{s+5}{s^{2}+9s+20}U(s)$

and

$U(s)=\frac{9}{s^2+9}$

so,

$Z(s)=\frac{s+5}{s^{2}+9s+20}*\frac{9}{s^2+9}$

and laplace inverse of Z(s) can be calculated as below

$z(t)=L^{-1}\left\{\frac{9\left(s+5\right)}{\left(s^2+9s+20\right)\left(s^2+9\right)}\right\}$

using partial fractions and converting into simpler forms

$z(t)=L^{-1}\left\{\frac{36-9s}{25\left(s^2+9\right)}+\frac{9}{25\left(s+4\right)}\right\}$

$z(t)=L^{-1}\left\{-\frac{9s}{25\left(s^2+9\right)}+\frac{36}{25\left(s^2+9\right)}+\frac{9}{25\left(s+4\right)}\right\}$

$z(t)=\frac{12}{25}\sin \left(3t\right)-\frac{9}{25}\cos \left(3t\right)+\frac{9}{25}e^{-4t}$