Let's break down the problem and solve it step by step. --- ## **Step 1: Understanding Gain Margin** - **Gain Margin (GM)** in dB tells us how much we can multiply the open-loop gain before the system reaches instability (i.e., before the gain crossover frequency makes the phase -180°). - The gain margin of **20.4 dB** means the system gain can be increased by 20.4 dB before reaching instability. --- ## **Step 2: Converting Gain Margin from dB to Absolute Units** The decibel gain margin is related to the absolute gain by: \[ GM_{dB} = 20 \log_{10}(K_{max}) \] where \( K_{max} \) is the maximum proportional gain factor. So, \[ K_{max} = 10^{\frac{GM_{dB}}{20}} \] Given: \( GM_{dB} = 20.4 \) dB \[ K_{max} = 10^{\frac{20.4}{20}} \] \[ K_{max} = 10^{1.02} \] \[ K_{max} \approx 10.47 \] --- ## **Step 3: Answer** **The highest proportional control gain value (in absolute units) that can be used is:** \[ \boxed{10.47} \] --- ### **Summary Table** | Parameter | Value | |----------------------------|------------| | Gain Margin (dB) | 20.4 dB | | Gain Margin (absolute) | 10.47 | | Highest Proportional Gain | 10.47 | --- **Note:** The phase margin and frequency information is not needed to answer this question, since the limiting factor for proportional gain is the gain margin. If you need further explanation on phase margin or why it’s not used here, let me know!