R-H range of K for s⁴+20s³+15s²+2s+K

Apply the Routh array with $K$ in it. For stability, every element of the first column must be positive. Marginal stability occurs when the $s^1$ row becomes zero.

$$s^4+20s^3+15s^2+2s+K = 0$$

Routh array

(i) Range of K for stability

• From $s^1$ row: $29.8 - 20K > 0 \Rightarrow K < 1.49$
• From $s^0$ row: $K > 0$

$$0 < K < 1.49$$

(ii) Marginal stability

Yes. At $K = 1.49$ the $s^1$ row becomes all zero, and the system has a pair of roots on the $j\omega$ axis (marginally stable, sustained oscillation).

Auxiliary equation from the $s^2$ row:

$$\begin{aligned} 14.9s^2 + K &= 0 \\ 14.9s^2 + 1.49 &= 0 \\ s^2 &= -0.1 \Rightarrow s = \pm j0.316 \end{aligned}$$

So the frequency of sustained oscillation is $\omega = \sqrt{0.1} = 0.316$ rad/s.

Answer: stable for $0 < K < 1.49$; marginally stable at $K_{mar} = 1.49$, oscillating at $\omega = 0.316$ rad/s.