# Steady state error in non-unity feedback

#### Karanbir Singh Mann

Joined Jul 27, 2017
3
Refer the attached image.

My solution:-
Error--> E(s)=R(s)-C(s)
=R(s)-[G(s)/1+G(s)H(s)]*R(s)
where H(s)=2
Steady state error--> e=lim s.E(s) ....where s approaches zero(final value theorem)
So my solution gives value e=0.5

But the actual answer is e=0.
And the sample solution takes
E(s)=R(s)/Kh - C(s)
where Kh is a constant defined as H(0)=2

I don't get it . I had read that even in non unity feedback system we take E(s) as difference between R(s) and C(s). Am I conceptually wrong here?

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#### WBahn

Joined Mar 31, 2012
29,844
Your process includes an integrator. If you have ANY non-zero steady state error, then your output will increase without bound.

#### MrAl

Joined Jun 17, 2014
11,256
Refer the attached image.

My solution:-
Error--> E(s)=R(s)-C(s)
=R(s)-[G(s)/1+G(s)H(s)]*R(s)
where H(s)=2
Steady state error--> e=lim s.E(s) ....where s approaches zero(final value theorem)
So my solution gives value e=0.5

But the actual answer is e=0.
And the sample solution takes
E(s)=R(s)/Kh - C(s)
where Kh is a constant defined as H(0)=2

I don't get it . I had read that even in non unity feedback system we take E(s) as difference between R(s) and C(s). Am I conceptually wrong here?
Hello there,

If we take the diagram at face value then the system settles at Vout=1/2 and there's no way around that.

However, if they also ASSERT that the output steady state error is zero and provide you with a factor that multiplies the input signal, then we have to include that factor to get the right result. Granted though i dont think they presented this question very concisely.

A time when this may happen is when we design the system to provide an output that is NOT the same as the input, but the output we get is the output we want anyway. In other words, we want the output to be a RATIO of the input, not the same as the input. This is actually very common with things like voltage regulators.
So if we wanted an output of 1/2 but we only had a reference of 1, we might do a system like that. To find the steady state error then we would have to divide that 1 by 2 before we subtracted.

As i said though i dont think they presented the problem very concisely, unless they gave you the answer of zero at the same time that they gave you the problem diagram itself.

So the question is, did they give you the actual final result and explanation of how they got that result at the very same time that they gave you the problem diagram?

It could also be that they wanted you to see that the non unity feedback system must be handled differently simply because the design goal is different.

Last edited:

#### Karanbir Singh Mann

Joined Jul 27, 2017
3
Your process includes an integrator. If you have ANY non-zero steady state error, then your output will increase without bound.
could you please elaborate?

#### Karanbir Singh Mann

Joined Jul 27, 2017
3
Hello there,

If we take the diagram at face value then the system settles at Vout=1/2 and there's no way around that.

However, if they also ASSERT that the output steady state error is zero and provide you with a factor that multiplies the input signal, then we have to include that factor to get the right result. Granted though i dont think they presented this question very concisely.

A time when this may happen is when we design the system to provide an output that is NOT the same as the input, but the output we get is the output we want anyway. In other words, we want the output to be a RATIO of the input, not the same as the input. This is actually very common with things like voltage regulators.
So if we wanted an output of 1/2 but we only had a reference of 1, we might do a system like that. To find the steady state error then we would have to divide that 1 by 2 before we subtracted.

As i said though i dont think they presented the problem very concisely, unless they gave you the answer of zero at the same time that they gave you the problem diagram itself.

So the question is, did they give you the actual final result and explanation of how they got that result at the very same time that they gave you the problem diagram?

It could also be that they wanted you to see that the non unity feedback system must be handled differently simply because the design goal is different.
no the final answer and explanation was given later...
I get what u r saying...if that's the case then they didn't frame the question correctly maybe
thanks man

#### WBahn

Joined Mar 31, 2012
29,844
could you please elaborate?
There's a few issues.

First, the signal after the summing point is NOT the error. The error is the difference between the desired output and the actual output. But the input signal is NOT the desired output. The desired output is half of the input signal.

Imagine using a mechanical lever system to adjust the height of an object. But your object is halfway between the pivot and the end of the board which is where you take your measurements. If you want the height of the object to be, say, 10 cm above the ground, then the input you have to put on the stick you are making the measurements at has to go at 20 cm. If you just subtract the actual output from the input, you'll get an "error" of 10 cm even when the output is exactly where you want it to be. When your input is 20 cm, your DESIRED output is 10 cm.

Switching over to the effect of the integrator.

Let's say that you have a steady state error that is 0.000000001. The integrator is going integrate that error over time and the resulting drive single is going to grow and grow and grow until it is large enough to force the system to reduce the error. If the system is stable, the error will be driven to zero. Not something small, but to zero.

#### WBahn

Joined Mar 31, 2012
29,844
I had read that even in non unity feedback system we take E(s) as difference between R(s) and C(s). Am I conceptually wrong here?
Yes, you are conceptually wrong. R(s) may be the input, but it is NOT the desired output.

In order for E(s) to have any meaning as an error, it must be the difference between two things measured using equivalent scales.

#### MrAl

Joined Jun 17, 2014
11,256
could you please elaborate?

Hello again,

I think WBahn summed it up pretty well in post #7. I will provide an exemplary case.

Case 1:
We have a voltage regulator that outputs 999 volts DC.
We use a reference voltage (from say a zener) that is 10 volts in the hopes of obtaining 1000 volts output which means a gain of 100.
The question now is:
Is the error 999-10=989 volts or
is the error 999-10*100=-1 volts?

So you see the error was not Vout-Vin it was Vout-Vin*K where K=100, so the error is -1 volts.

Since they did not give you the result with the question, that means they want you to recognize that systems that do not have unity feedback have to be handled slightly different than those that do. Granted, you may still have to be on the lookout for systems that are not designed correctly as this could have been a huge mistake on the designers part.