Question:
Consider a unity negative feedback control system with forward path gain:
\[
G(s) = \frac{K}{(s+1)(s+2)(s+3)}.
\]
\vspace{0.5cm}
\begin{center}
\includegraphics[width=8cm]{37.png}
\end{center}
The impulse response of the closed-loop system decays faster than \(e^{-t}\) if \(\_\_\_\_\).
Consider a unity negative feedback control system with forward path gain:
\[
G(s) = \frac{K}{(s+1)(s+2)(s+3)}.
\]
\vspace{0.5cm}
\begin{center}
\includegraphics[width=8cm]{37.png}
\end{center}
The impulse response of the closed-loop system decays faster than \(e^{-t}\) if \(\_\_\_\_\).
Show Hint
To ensure a desired decay rate, analyze the real parts of the closed-loop poles using the characteristic equation and adjust \(K\) accordingly.
Updated On: Jan 31, 2025
- \(1 \leq K \leq 5\)
- \(7 \leq K \leq 21\)
- \(-4 \leq K \leq -1\)
- \(-24 \leq K \leq -6\)
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The Correct Option is A
Solution and Explanation
Step 1: Derive the characteristic equation for the closed-loop system.
The closed-loop transfer function is given by: \[ T(s) = \frac{G(s)}{1 + G(s)} = \frac{\frac{K}{(s+1)(s+2)(s+3)}}{1 + \frac{K}{(s+1)(s+2)(s+3)}}. \] The characteristic equation of the system is: \[ (s+1)(s+2)(s+3) + K = 0. \] Step 2: Condition for faster decay than \(e^{-t}\).
For the impulse response to decay faster than \(e^{-t}\), all poles of the closed-loop system must have real parts less than \(-1\). Step 3: Determine the range of \(K\).
Solving the characteristic equation and analyzing the root locations, the condition \(1 \leq K \leq 5\) ensures that all poles of the system have real parts less than \(-1\). Final Answer: \[\boxed{{(1) } 1 \leq K \leq 5}\]
The closed-loop transfer function is given by: \[ T(s) = \frac{G(s)}{1 + G(s)} = \frac{\frac{K}{(s+1)(s+2)(s+3)}}{1 + \frac{K}{(s+1)(s+2)(s+3)}}. \] The characteristic equation of the system is: \[ (s+1)(s+2)(s+3) + K = 0. \] Step 2: Condition for faster decay than \(e^{-t}\).
For the impulse response to decay faster than \(e^{-t}\), all poles of the closed-loop system must have real parts less than \(-1\). Step 3: Determine the range of \(K\).
Solving the characteristic equation and analyzing the root locations, the condition \(1 \leq K \leq 5\) ensures that all poles of the system have real parts less than \(-1\). Final Answer: \[\boxed{{(1) } 1 \leq K \leq 5}\]
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