# Determine the transfer function (H(s)) of the following system using Laplace transform properties. ddot{{y}}+{4}dot{{y}}+{4}{y}=-{x}+{2}dot{{x}} Note: Assume that all initial conditions are zero

Determine the transfer function (H(s)) of the following system using Laplace transform properties.
$\stackrel{¨}{y}+4\stackrel{˙}{y}+4y=-x+2\stackrel{˙}{x}$
Note: Assume that all initial conditions are zero
You can still ask an expert for help

## Want to know more about Laplace transform?

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Brittany Patton
Step 1
According to the given information, it is required to determine the transfer function H(s) using Laplace transform properties.$\stackrel{¨}{y}+4\stackrel{˙}{y}+4y=-x+2\stackrel{˙}{x}$
Step 2
Now represent the above differential equation in transfer function form. It is also given that initial conditions are zero.
$\stackrel{¨}{y}+4\stackrel{˙}{y}+4y=-x+2\stackrel{˙}{x}$
$L\left(\stackrel{¨}{y}+4\stackrel{˙}{y}+4y\right)=L\left(-x+2\stackrel{˙}{x}\right)$
${s}^{2}Y\left(s\right)-sY\left(0\right)-{y}^{\prime }\left(0\right)+4\left[sY\left(s\right)-y\left(0\right)\right]+4Y\left(s\right)=-X\left(s\right)+2\left[sX\left(s\right)-x\left(0\right)\right]$
since initial conditions are all 0:
${s}^{2}Y\left(s\right)+4sY\left(s\right)+4Y\left(s\right)=-X\left(s\right)+2sX\left(s\right)$
$\left({s}^{2}+4s+4\right)Y\left(s\right)=\left(-1+2s\right)X\left(s\right)$
$H\left(s\right)=\frac{Y\left(s\right)}{H\left(s\right)}=\frac{-1+2s}{{s}^{2}+4s+4}$