K range for roots left of s = −2

"Roots more negative than $s=-2$" means all closed-loop poles must lie to the left of the line $s=-2$. Shift the imaginary axis by putting $s = z - 2$, then apply the Routh criterion in $z$.

Characteristic equation

$$\begin{aligned} 1 + \frac{K(s+10)}{s(s+3)} &= 0 \\ s^2 + (3+K)s + 10K &= 0 \end{aligned}$$

Shift the axis: $s = z - 2$

$$\begin{aligned} (z-2)^2 + (3+K)(z-2) + 10K &= 0 \\ z^2 - 4z + 4 + (3+K)z - 6 - 2K + 10K &= 0 \\ z^2 + (K-1)z + (8K-2) &= 0 \end{aligned}$$

Routh array in z

Conditions

• $z^1$ row: $K - 1 > 0 \Rightarrow K > 1$
• $z^0$ row: $8K - 2 > 0 \Rightarrow K > 0.25$

Both must hold, so

$$K > 1$$

Check: at $K = 1$, $s^2 + 4s + 10 = 0$ gives $s = -2 \pm j2.45$, exactly on the line $s=-2$. For $K>1$ the real part of the roots (or both real roots) is more negative than $-2$.

Answer: all roots are more negative than $s = -2$ for $K > 1$.