R-H range of K for K/[(s+2)(s+4)(s²+6s+25)]

Write the closed-loop characteristic equation $1+G(s)=0$ and apply the Routh criterion. Sustained oscillation occurs when the $s^1$ row becomes zero, and the auxiliary equation from the $s^2$ row gives the frequency.

Characteristic equation

$$\begin{aligned} (s+2)(s+4)(s^2+6s+25) + K &= 0 \\ (s^2+6s+8)(s^2+6s+25) + K &= 0 \\ s^4 + 12s^3 + 69s^2 + 198s + (200+K) &= 0 \end{aligned}$$

Routh array

Range of K

• $s^1$ row: $7995 - 12K > 0 \Rightarrow K < 666.25$
• $s^0$ row: $200 + K > 0 \Rightarrow K > -200$

$$-200 < K < 666.25 \qquad (\text{for positive gain: } 0 < K < 666.25)$$

Value of K for sustained oscillation

$$K_{mar} = \frac{7995}{12} = 666.25$$

Frequency of oscillation

Auxiliary equation from the $s^2$ row:

$$\begin{aligned} 52.5s^2 + (200 + 666.25) &= 0 \\ s^2 &= -\frac{866.25}{52.5} = -16.5 \\ \omega &= \sqrt{16.5} = 4.062\ \text{rad/s} \end{aligned}$$

Check: with $K = 666.25$ the roots are $\pm j4.062$ and $-6 \pm j4.062$.

Answer: stable for $0 < K < 666.25$; sustained oscillation at $K = 666.25$ with $\omega = 4.06$ rad/s.