Consider the cascaded system as shown in the figure. Neglecting the faster component of the transient response, which one of the following options is a first-order pole-only approximation such that the steady-state values of the unit step responses of the original and the approximated systems are same? 
To solve this problem, we need to find the first-order pole-only approximation of the given cascaded transfer function and ensure that it has the same steady-state value for the unit step response as the original system.
The given cascaded system has two transfer functions:
- \(\frac{1}{s+1}\)
- \(\frac{s+40}{s+20}\)
The overall transfer function of the cascaded system is:
\(G(s) = \frac{1}{s+1} \cdot \frac{s+40}{s+20} = \frac{s+40}{(s+1)(s+20)}\)
Neglecting the faster component of the transient response, the pole at the higher frequency should be neglected since its transient effects decay faster. The system is dominated by the slower response, which corresponds to a lower frequency pole.
To find the steady-state response to a unit step input, evaluate the zero-frequency (s=0) gain of the original system:
\(G(s=0) = \frac{0+40}{(0+1)(0+20)} = \frac{40}{20} = 2\)
We need a first-order system with the same steady-state value, \(2\). This can be achieved by a first-order system:
\(G_{\text{approx}}(s) = \frac{K}{s+a}\)
Given options are:
- \(\frac{1}{s+1}\)
- \(\frac{2}{s+1}\)
- \(\frac{1}{s+20}\)
- \(\frac{2}{s+20}\)
To have the same steady-state gain, \(\frac{2}{s+1}\) is the correct approximation. This gives a steady-state value of:
\(G_{\text{approx}}(s=0) = \frac{2}{0+1} = 2\)
Thus, the approximation \(\frac{2}{s+1}\) yields the same steady-state gain.
Therefore, the correct option is:
\(\frac{2}{s+1}\)
Important Questions from Time Response Analysis
Match List I with List II:
List I
(Effeet of ξ)
List II
(Condition of System)
(A) 0 < ξ < 1 (I) Over damped (B) ξ > 1 (II) Undamped (C) ξ = 0 (III) Unstable (D) ξ = −1 (IV) Under damped Choose the correct answer from the options given below:
What is the value of ωn in the given transfer function?
\(G\left( s \right) = \frac{{36}}{{{s^2} + 4.2s + 36}}\)
Which of the following is correct for over-damped and under-damped system, respectively?
What will be the time response expression for a standard first order system having unit step function \(\frac{1}{s}\) as the input
A second order control system is NOT required to satisfy the following specification:
