1. **Problem Statement:** Given a control system with parameter $b=2$, determine the range of $K$ for system stability using the Routh-Hurwitz criterion. 2. **Simplify the block diagram:** The system has blocks: 4, feedback with $1+b=3$, block $\frac{1}{s+2}$, block $\frac{1}{s^2+3s}$, and feedback with gain $K$. The open-loop transfer function is: $$G(s)H(s) = \frac{4 \cdot \frac{1}{s+2} \cdot \frac{1}{s^2+3s}}{1+3} = \frac{4}{(s+2)(s^2+3s)} \cdot \frac{1}{3} = \frac{4}{3(s+2)(s^2+3s)}$$ Including feedback $K$ at the first summing junction, the characteristic equation denominator is: $$1 + K \cdot \frac{4}{3(s+2)(s^2+3s)} = 0$$ Multiply both sides by denominator: $$3(s+2)(s^2+3s) + 4K = 0$$ 3. **Form characteristic polynomial:** Expand: $$(s+2)(s^2+3s) = s^3 + 3s^2 + 2s^2 + 6s = s^3 + 5s^2 + 6s$$ Multiply by 3: $$3s^3 + 15s^2 + 18s + 4K = 0$$ Characteristic polynomial: $$3s^3 + 15s^2 + 18s + 4K = 0$$ 4. **Routh array setup:** Coefficients: $a_3=3$, $a_2=15$, $a_1=18$, $a_0=4K$ Routh array: $$\begin{array}{c|cc} s^3 & 3 & 18 \\ s^2 & 15 & 4K \\ s^1 & b_1 & 0 \\ s^0 & 4K & \\\end{array}$$ Calculate $b_1$: $$b_1 = \frac{15 \times 18 - 3 \times 4K}{15} = \frac{270 - 12K}{15}$$ 5. **Stability conditions:** All first column elements must be positive: - $3 > 0$ (always true) - $15 > 0$ (always true) - $b_1 = \frac{270 - 12K}{15} > 0 \Rightarrow 270 - 12K > 0 \Rightarrow K < 22.5$ - $4K > 0 \Rightarrow K > 0$ 6. **Final range for stability:** $$0 < K < 22.5$$ 7. **Summary:** Using the Routh-Hurwitz criterion, the system is stable if and only if $K$ lies between 0 and 22.5.