Question

Given a system with the transfer function

p(S)= (s+1)/(s(2s^2+4s+3)(2s+1))

Each section must specify the way of solution / explanation / reasoning
A. 8 points (Is the system in an open circle asymptomatic or BIBO stable or unstable?
B. (8 pts) Closes a control circle with a proportional controller. What is the range of K values for which
The closed circle is stable?
third. 4 points (what is the constant state error of the system in the open circle for step entry
Unit?
D. ) 4 points (what is the error of the constant state of the system in the closed circle with a controller held inside
The domain you found in section b) Select a value as you wish (for a single entry level?)

Answer 1

Question A

The system transfer function is given as

The poles are at

0

We can see that the poles have negative real part. One pole is at the origin. So the system is asymptotically stable

The system is not BIBO stable because there is a pole at the origin. So if we give a bounded input, we might not get a bounded output.

Question B

The proportional controller

So the characteristic equation becomes

Lets form the Routh array as follows

For the system to be stable, the elements in the first column of the Routh array has to be of the same sign

So

So

So combing all conditions, for the system to be stable

Question C

When the system is open circle

Let be the output and be the input . Then

The input is a constant unit step. So

Taking Laplace transform we get

So

The final value of the output will be

Using Final Value Theorem we can write

So the steady state error is infinity

Question D

The open loop transfer function is

The steady state error for a constant unit step input is given as

Where is the position error constant given by

So the steady state error is

So as long as K is between 0 and 5.660254, the steady state error will be zero.