The open-loop transfer function of the system shown in the figure is: \[ G(s) = \frac{K s (s + 2)}{(s + 5)(s + 7)} \] For \( K \geq 0 \), which of the following real axis point(s) is/are on the root locus?
Show Hint
To determine if a point on the real axis lies on the root locus, count the number of real poles and real zeros to the right of the point. If the count is odd, the point lies on the root locus.
The open-loop transfer function is:
\[
G(s)H(s) = \frac{K s (s + 2)}{(s + 5)(s + 7)}
\]
Poles: \( s = -5, -7 \)
Zeros: \( s = 0, -2 \)
According to the root locus rule, on the real axis, a point lies on the root locus if the total number of real poles and real zeros to the right of that point is odd.
Check each option:
For \( s = -1 \): Right of \( -1 \) are zeros at \( 0 \) and \( -2 \) (only \( 0 \) is to the right), count = 1 (odd) \( \Rightarrow \) on root locus
For \( s = -4 \): Right of \( -4 \) are zeros at \( 0, -2 \), no poles. Count = 2 (even) \( \Rightarrow \) not on root locus
For \( s = -6 \): Right of \( -6 \) are \( -5, -2, 0 \), count = 3 (odd) \( \Rightarrow \) on root locus
For \( s = -10 \): All poles and zeros are to the right, count = 4 (even) \( \Rightarrow \) not on root locus
