oscillation conditions

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houly

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Hello
I have question about an exercise, I don't know how can I solve it :

The open loop transfer function of a unity feedback control system is given by

G(s)*H(s)=K/(s(s+1)(2s+1)(3s+1))

First question :
What is the value of K which will cause sustained oscillations in the closed loop system ?


Second question :
What is the frequency of sustained oscillations ?


Do you know how can I do ? what is the method to answer them ?

regards
 

As you know: The necessary condition for oscillation is G(s)*H(s)=1

Thus, in your case: K=s*(..)(..)(..).

Step1: solve the right side to a polynom N(s) of third degree;
Step2: Set s=jw and solve for Im(N(jw))=0 because the left side of the equation is real (K);
this gives the frequency of oscillation;
Step 3: Solve for Re(N(jw))=K; so you get the required value for K.
 
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    houly

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The oscillation condition in closed loop is:
1+G(s)*H(s) = 0
--> s(s+1)(2s+1)(3s+1) + K = 0​
This can be done by hand but i prefer Derive6 :smile:

Replacing s by jw --> k + 6·w^2·(w^2 - 1) + j·(w - 11·w^3) = 0

Then, the imaginary part is 0 only when
w = 1/√11​
And the real part is 0 when
k = 6·w^2·(1-w^2) = 60/121​
 
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    houly

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I didn't check the values; they may be OK.
However, if the given function G*H is the loop gain, the starting equation must be 1-G*H=0 and therefore k must be negative.
 
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    houly

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Right: K = -60/121 ≈ -0.5
 
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    houly

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OK ! thanks a lot for explanations, it's more clear for me now do you know a good reference book which abord this ?
 

LvW said:
I didn't check the values; they may be OK.
However, if the given function G*H is the loop gain, the starting equation must be 1-G*H=0 and therefore k must be negative.
Click to expand...

Some authors prefer to write 1+H(s)G(s) = 0 and others 1-H(s)G(s) = 0

The sign depend if the feedback signal is added or subtracted at closes loop .



1+H(s)G(s) = 0





1-H(s)G(s) = 0

 

Eduardo, of course you are right.
However, when somebody tells me that a function H(s) equals the "open loop transfer function" (as houly did) I presume that the minus sign is incorporated since the loop gain (for low frequencies) always must be negative for a stable operating point.
 

So when a system use a summing block, the direct or the feedback block must be a negative term to have a stable system ?
 

houly said:
So when a system use a summing block, the direct or the feedback block must be a negative term to have a stable system ?
Click to expand...

That's right. With other words - around the loop there must be an uneven number of inverting blocks.
 
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    houly

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