Let's solve the problem step by step using the **Routh-Hurwitz criterion** to determine the stability of the system. --- ## **Step 1: Write the given transfer functions** - **Controller:** \( G_{\text{kontrolcü}} = 11 + \frac{1}{3s} \) - **System:** \( G_{\text{sistem}} = \frac{10}{s(s+4)(s+7)} \) - **Sensor Transfer Function:** Last 3 digits of student number = **037**, so \( H(s) = .037 \) --- ## **Step 2: Find the open-loop transfer function** \[ G_{\text{open-loop}}(s) = G_{\text{kontrolcü}} \cdot G_{\text{sistem}} = \left(11 + \frac{1}{3s}\right) \cdot \frac{10}{s(s+4)(s+7)} \] Rewrite the controller: \[ 11 + \frac{1}{3s} = \frac{33s + 1}{3s} \] So, \[ G_{\text{open-loop}}(s) = \frac{33s + 1}{3s} \cdot \frac{10}{s(s+4)(s+7)} = \frac{10(33s + 1)}{3s^2(s+4)(s+7)} \] --- ## **Step 3: Find the closed-loop transfer function** The closed-loop transfer function is: \[ T(s) = \frac{G_{\text{open-loop}}(s)}{1 + G_{\text{open-loop}}(s)H(s)} \] The denominator (characteristic equation) is: \[ 1 + G_{\text{open-loop}}(s)H(s) = \] Plug in \( H(s) = .037 \): \[ 1 + \frac{10(33s + 1)}{3s^2(s+4)(s+7)} \cdot .037 = \] \[ 1 + \frac{.37(33s + 1)}{s^2(s+4)(s+7)} = \] \[ s^2(s+4)(s+7) + .37(33s+1) = \] Expand \( s^2(s+4)(s+7) \): \[ (s+4)(s+7) = s^2 + 11s + 28 \] \[ s^2(s^2 + 11s + 28) = s^4 + 11s^3 + 28s^2 \] So the equation becomes: \[ s^4 + 11s^3 + 28s^2 + .37 \times 33s + .37 \times 1 = \] \[ s^4 + 11s^3 + 28s^2 + 12.21s + .37 = \] --- ## **Step 4: Apply the Routh-Hurwitz Criterion** Write the coefficients: | \( s^4 \) | \( s^3 \) | \( s^2 \) | \( s^1 \) | \( s^ \) | |-----------|-----------|-----------|-----------|-----------| | 1 | 11 | 28 | 12.21 | .37 | Set up the Routh array: \[ \begin{array}{c|cc} s^4 & 1 & 28 \\ s^3 & 11 & 12.21 \\ s^2 & A & .37 \\ s^1 & B & \\ s^ & .37 & \\ \end{array} \] Calculate \( A \) and \( B \): ### Calculate \( A \): \[ A = \frac{11 \times 28 - 1 \times 12.21}{11} = \frac{308 - 12.21}{11} = \frac{295.79}{11} \approx 26.89 \] ### Calculate \( B \): \[ B = \frac{26.89 \times 12.21 - 11 \times .37}{26.89} = \frac{328.333 - 4.07}{26.89} = \frac{324.263}{26.89} \approx 12.06 \] --- ## **Step 5: Check the first column sign** First column is: - \( 1 \) - \( 11 \) - \( 26.89 \) - \( 12.06 \) - \( .37 \) **All positive!** --- ## **Final Answer** ### **The system is stable** according to the Routh-Hurwitz criterion. --- ### **Summary Table** | Row | Coefficient | Value | |-----|-------------|------------| | s⁴ | 1 | | | s³ | 11 | | | s² | 26.89 | | | s¹ | 12.06 | | | s⁰ | .37 | | **All coefficients are positive, so the system is stable.**