Consider a unity negative feedback control system with forward path gain: \[ G(s) = \frac{K}{(s+1)(s+2)(s+3)}. \] The impulse response of the closed-loop system decays faster than \(e^{-t}\) if \(\_\_\_\_\).

Question

A source transmits a symbol \(s\), taken from \(\{-4, 0, 4\}\) with equal probability, over an additive white Gaussian noise channel. The received noisy symbol \(r\) is given by \(r = s + w\), where the noise \(w\) is zero mean with variance 4 and is independent of \(s\). Using: \[ Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty e^{-\frac{t^2}{2}} dt, \] the optimum symbol error probability is \(\_\_\_\_\).

Answer 1

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: \[\boxed{{(1) } 1 \leq K \leq 5}\]

Consider a unity negative feedback control system with forward path gain: \[ G(s) = \frac{K}{(s+1)(s+2)(s+3)}. \] The impulse response of the closed-loop system decays faster than \(e^{-t}\) if \(\_\_\_\_\).

Show Hint

The Correct Option is A

Solution and Explanation

Top Questions on Linear 2-port network parameters

Questions Asked in GATE EC exam