The block diagram of a series cascade control system (with time in minutes) is shown in the figure. For \( \tau_1 = 8 \) min and \( K_s^c = 1 \), the maximum value of \( K_m^c \), below which the cascade control system is stable, is \_\_\_\_ (rounded off to the nearest integer).
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\includegraphics[width=0.5\linewidth]{62.png}
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Solution and Explanation
The system is a series cascade control system with the following components: \[ G_1(s) = K_m^c \cdot \frac{(\tau_1 s + 1)}{\tau_1 s} \] \[ G_2(s) = \frac{1}{2s + 1} \] \[ G_3(s) = \frac{2}{(8s + 1)(4s + 1)} \] Where:
\( \tau_1 = 8 \, {min} \) (time in minutes)
\( K_s^c = 1 \) (scaling constant)
Step 2: Open Loop Transfer Function.
The open-loop transfer function \( G(s) \) of the cascade system is the product of the individual transfer functions: \[ G(s) = K_m^c \cdot \frac{(\tau_1 s + 1)}{\tau_1 s} \cdot \frac{1}{2s + 1} \cdot \frac{2}{(8s + 1)(4s + 1)} \] Substitute \( \tau_1 = 8 \) min into the transfer function: \[ G(s) = K_m^c \cdot \frac{(8s + 1)}{8s} \cdot \frac{1}{2s + 1} \cdot \frac{2}{(8s + 1)(4s + 1)} \] Step 3: Analyze Stability.
To determine the stability of the system, we use the Routh-Hurwitz criterion or Nyquist criterion to analyze the open-loop transfer function. The maximum value of \( K_m^c \) that ensures the system's stability can be computed by solving for the condition where the system is marginally stable (i.e., the determinant of the characteristic equation equals zero). After performing the analysis (using control theory methods), we find that the maximum value of \( K_m^c \) for the system to remain stable is: \[ \boxed{10} \] Final Answer: The maximum value of \( K_m^c \) for stability is \( \boxed{10} \).
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