R-H range of K for K(s+1)/(s³+bs²+3s+1)

Result: stable for $K>-1$ and $b(3+K)>1+K$. For $K>0$ this means any $K$ when $b\ge1$, and $0<K<\frac{3b-1}{1-b}$ when $\frac13<b<1$. It just oscillates at $K=\frac{3b-1}{1-b}$, with $\omega=\sqrt{3+K}=\sqrt{\frac{2}{1-b}}$ rad/s.

The characteristic equation is $1+G(s)=0$. The Routh array gives the stability conditions in terms of $K$ and $b$. Sustained oscillation occurs when the $s^1$ row becomes zero.

Characteristic equation

$$\begin{aligned} s^3+bs^2+3s+1 + K(s+1) &= 0 \\ s^3 + bs^2 + (3+K)s + (1+K) &= 0 \end{aligned}$$

Routh array

Conditions for stability

1. $b > 0$
2. $1+K>0 \Rightarrow K>-1$
3. $b(3+K) - (1+K) > 0 \Rightarrow K(b-1) > 1-3b$

Interpreting condition 3 (for $K>0$, $b>0$):

• If $b \ge 1$: condition 3 holds for every $K>0$, so the system is stable for all $K>0$.
• If $b < 1$: $K < \dfrac{3b-1}{1-b}$. A positive range exists only if $b > \frac{1}{3}$.

$$\text{Range: } 0 < K < \frac{3b-1}{1-b}\quad\left(\tfrac13<b<1\right)$$

Sustained oscillation

The system just oscillates when the $s^1$ row is zero:

$$b(3+K) = 1+K \quad\Rightarrow\quad K_{mar} = \frac{3b-1}{1-b}$$

Auxiliary equation from the $s^2$ row:

$$\begin{aligned} bs^2 + (1+K) &= 0 \\ \omega^2 &= \frac{1+K}{b} = 3+K = \frac{2}{1-b} \\ \omega &= \sqrt{\frac{2}{1-b}}\ \text{rad/s} \end{aligned}$$

Example: for $b = 0.5$: $K_{mar} = \frac{0.5}{0.5}=1$ and $\omega = \sqrt{4} = 2$ rad/s. Check: $s^3+0.5s^2+4s+2 = (s^2+4)(s+0.5)$, which has roots $\pm j2$.