4. 2. Solved Problems

Example 2: Maximum Controller Gain for Stability The characteristic equation of a closed loop system using a proportional controller with gain \(K_c\) is \[12 s^3+19 s^2 + 8s + 1 + K_c = 0\] At the onset of instability, the value of \(K_c\) is            (G - 2009-42)

(a) 35/3 (b) 10 (c) 25/3 (d) 20/3

Solution: 

The Routh array of the given characteristic equations is given below:

Row 1 \(\quad 12\) 8
2 \(\quad 19\) \(1+K_c\)
3 \(\quad \dfrac{19\times8-12\times(1+K_c)}{19}\)
4 \(\quad 1+K_c\)

For stability all the elements in the first column of Routh array should be positive. Therefore, \[\begin{aligned} \frac{19\times8-12\times(1+K_c)}{19} &> 0 \\ \text{i.e.,} \qquad 152 &> 12+12K_c \\ \Longrightarrow \qquad K_c &< 35/3 \end{aligned}\] At values of \(K_c>35/3\), the closed loop system is unstable.