I am designing a control system :

with open loop gain equal to : $\frac{K}{s(s^{2}+4s+3)}$

The closed loop gain is obviously :$\frac{K}{s(s^{2}+4s+3)+K}$

I tried finding the critical frequency but found something else interesting while trying to find the critical frequency.

I applied Ruth's algorithm to the denominator of the closed loop gain and I get this result:

Obviously for a stable system k doesnt exist but this seemed super odd so I decided to set k = 2 and find all the roots using a calculator and I found that the root is in the left half plane of the Re,Im plane so the system SHOULD be stable.So what am I doing wrong?


1 Answer 1


Your setup for the Routh-Hurwitz stability criterion isn't correct. Your setup should look like:

var c1 c2
$s^3$ $1$ $3$
$s^2$ $4$ $K$
$s^1$ $ \frac{12-K}{4} $ $0$
$s^0$ $K$ $0$

So your system is stable when $0 \leq K \leq 12$.


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