Let's break down the problem **step by step**: --- ## **Given System** Plant transfer function: \[ G(s) = \frac{100}{s^3 + 24s^2 + 156s + 224} \] **Disturbance:** A sinusoidal input with frequencies between 10 and 12 \(\text{rad/s}\). **Controllers to consider:** - \(K_1 = 2\) (Proportional controller) - \(K_2 = s + 5\) (Lead controller) --- ## **Block Diagram Analysis** The disturbance \(d_y\) enters after the plant, affecting the output. --- ## **Step 1: Transfer Function from Disturbance to Output** The **sensitivity function** \(S(s)\) relates the output to the disturbance: \[ C(s) = S(s) D_y(s) \] where \[ S(s) = \frac{1}{1 + K(s) G(s)} \] --- ## **Step 2: Evaluate Sensitivity for Both Controllers** ### **(a) For \(K_1 = 2\):** \[ S_1(s) = \frac{1}{1 + 2G(s)} \] ### **(b) For \(K_2 = s+5\):** \[ S_2(s) = \frac{1}{1 + (s+5)G(s)} \] --- ## **Step 3: Robustness to Sinusoidal Disturbance** **Key Point:** A controller is **more robust** to disturbances at a certain frequency if \(|S(j\omega)|\) is **small** at that frequency. Lower \(|S(j\omega)|\) means better disturbance rejection. We care about frequencies \(\omega\) between 10 and 12 rad/s. --- ### **Evaluate \(|S_1(j\omega)|\) and \(|S_2(j\omega)|\) at \(\omega = 10-12\) rad/s** #### **Estimate the Magnitude:** #### For \(K_1 = 2\): \[ S_1(j\omega) = \frac{1}{1 + 2G(j\omega)} \] #### For \(K_2 = s+5\): \[ S_2(j\omega) = \frac{1}{1 + (j\omega+5)G(j\omega)} \] --- ### **Magnitude of \(G(j\omega)\) at High Frequency** At high frequencies (\(\omega\) large): \[ G(j\omega) \approx \frac{100}{(j\omega)^3} \] The magnitude: \[ |G(j\omega)| \approx \frac{100}{\omega^3} \] At \(\omega = 10\): \[ |G(j10)| \approx \frac{100}{100} = 1 \] At \(\omega = 12\): \[ |G(j12)| \approx \frac{100}{1728} \approx .58 \] --- ### **Plug Values into Sensitivity Functions** #### For \(K_1 = 2\): \[ |S_1(j10)| = \frac{1}{|1 + 2 \cdot G(j10)|} \] With \(2 \cdot G(j10) \approx 2\): \[ |S_1(j10)| \approx \frac{1}{|1 + 2|} = \frac{1}{3} \approx .33 \] #### For \(K_2 = (j\omega + 5)\): \[ K_2(j10) = j10 + 5 \] Magnitude: \(\sqrt{10^2 + 5^2} = \sqrt{125} \approx 11.18\) So, \[ K_2(j10)G(j10) \approx 11.18 \cdot 1 = 11.18 \] \[ |S_2(j10)| = \frac{1}{|1 + 11.18|} \approx \frac{1}{12.18} \approx .082 \] #### At \(\omega = 12\): - \(K_2(j12) = j12 + 5 \rightarrow \sqrt{12^2 + 5^2} = \sqrt{169} = 13\) - \(K_2(j12)G(j12) \approx 13 \cdot .58 \approx 7.54\) - \(|S_2(j12)| \approx \frac{1}{1 + 7.54} = \frac{1}{8.54} \approx .117\) --- ## **Step 4: Conclusion** - **\(K_2 = s + 5\) results in a significantly lower sensitivity (better disturbance rejection) in the frequency range of interest (10-12 rad/s) compared to \(K_1 = 2\).** - Therefore, **\(K_2\) is the better controller for robustness to the specified disturbance.** --- ## **Final Answer** > **The better controller is \(K_2 = s + 5\). It provides much lower sensitivity to disturbances at the problematic frequencies (10-12 rad/s), thus making the system more robust to those sinusoidal disturbances.**