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David J

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1. A process can be represented by the first order equation

##4\frac{dy(t)}{dt}+y(t)=3u(t)##

Assume the initial steady state is steady (y=0 at t=-0)

(a) Determine the transfer function of this process in the s domain

(b) If the input is a ramp change in u(t)=4t, determine the value of y(t) when t=10s

For part (b) I used ##y(t) = 4[-\tau+t+\tau e^-\frac{t}{\tau}##]

##4\frac{dy(t)}{dt}+y(t)=3u(t)##

##4[sy(s)-y(s)]+y(s)=3u(s)##

So ##4[sy(s)+y(s)=3u(s)##

or ##y(s)[4s+1]=3u(s)##

So the transfer function (output / input) or gain for this process in the `s` domain would be:-

##G(s)=\frac{y(s)}{3u(s)}=\frac{1}{4s+1}##

The attachment (part A example) is from my notes which I have basically tried to adapt to this solution and follow

I am still working on part b and will post my solution in a couple of days but I was wondering if someone could look at this so far and advise.

##4\frac{dy(t)}{dt}+y(t)=3u(t)##

Assume the initial steady state is steady (y=0 at t=-0)

(a) Determine the transfer function of this process in the s domain

(b) If the input is a ramp change in u(t)=4t, determine the value of y(t) when t=10s

**2.**For part (a) i had to use laplace transform on ##4\frac{dy(t)}{dt}+y(t)=3u(t)## term by termFor part (b) I used ##y(t) = 4[-\tau+t+\tau e^-\frac{t}{\tau}##]

**3. First of all for part a**##4\frac{dy(t)}{dt}+y(t)=3u(t)##

##4[sy(s)-y(s)]+y(s)=3u(s)##

So ##4[sy(s)+y(s)=3u(s)##

or ##y(s)[4s+1]=3u(s)##

So the transfer function (output / input) or gain for this process in the `s` domain would be:-

##G(s)=\frac{y(s)}{3u(s)}=\frac{1}{4s+1}##

The attachment (part A example) is from my notes which I have basically tried to adapt to this solution and follow

I am still working on part b and will post my solution in a couple of days but I was wondering if someone could look at this so far and advise.

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