# Is this very simple system stable?

A simple feedback control loop:
input --> *(+-)* ---> 1/(s-1) ---*--> output
The feedback is simply multiplied by 2.

to *(-)* <---- X2 ------ from *
(X2 means multiplied by 2)
--------------------------------------------------------------------
Here's my question:
1. Is this system stable?
The output/input transfer function is 1/(s+1)
( H(s) = 1/(s-1), G(s) = 2, transfer func = H/(1+G*H) = 1/(s+1) )
The only pole is at -1, LHP,===> Stable
2. On the otherhand, if the input is a constant dc then 1/(s-1) becomes just "-1" (s=jw, w=0) and the feedback is -X2 (negative feedback) Therefore, the total feedback is (-1)*(-2) = 2 That is the system always multiplies the input by 2 and then feed it back to the system. A positive feedback! ===> Not stable.
Where did I make the mistake?
<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
----- Original Message -----
Newsgroups: comp.robotics.misc Sent: Monday, August 23, 2004 8:40 PM Subject: Is this very simple system stable?

H/(1+G*H)=(1/(s-1))/(1+2*(1/(s-1)))=1/((s-1)+2)=1/(s+1) Okay, I agree with you here.

This doesn't mean it is unstable. If the loop gain was one then you would know it is metastable. Actually a non inverting first or second order system with poles in the left half plain will be unstable for any negative feedback. Your example seems to show that an inverting unstable first order system will be unstable for any positive feedback or equivalently a no inverting unstable first order system will be unstable for any negative feedback. By inverting I mean a DC gain less then one.

Positive feedback is not necessarily unstable. Notice though that H(s) is unstable. Maybe this is why the result is less then intuitive.

<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>
Harry,
The system is stable. I believe the misstep is in the assumption that constant input to the system will produce the same constant input to the 1/(s-1) block. After transients have settled, the input to the 1/(s-1) block will actually be opposite in sign from the system input.
The transfer function from the system input to the input of the 1/(s-1) block is (s-1)/(s+1).
James

<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>

<% if( /^image/.test(type) ){ %>
<% } %>
<%-name%>

Hi James,
Here's another question:
The bode plot of the open loop transfer function 2/(s-1) shows that
At w=0, phase = -180 deg and gain = (20*log(2)), a negative margin
Is that an indication of "unstable"?
If not, what are the correction conditions for a stable system according to its bode plot?
(positive gain margin, positive phase margin, am I right? )
Harry