Assertion (A) : In control systems, steady state response in the final requirement for calculating the efficiency of the system. Reason (R) : The transient response is also critical for the determination of the steady state response.
(A) is true, but (R) is false.
The assertion is acceptable, but the reason states a dependence that does not exist — option 3.
The two parts of a response are computed independently. The total response of a linear system separates as
\(c(t)=c_{tr}(t)+c_{ss}(t)\)
and the steady-state part is found by the final-value theorem, which needs nothing but the transfer function and the input:
\(e_{ss}=\lim_{s\to0}\dfrac{sR(s)}{1+G(s)H(s)}\)
The limit as \(s\to0\) is a statement about the system's behaviour at zero frequency — the DC gain and the number of integrators in the forward path. The transient, by contrast, is governed by the pole locations, which control how fast the terms \(e^{-\sigma t}\) die away. The transient determines how long the system takes to reach the steady state, never what that steady state is. A sluggish system and a fast one with the same DC gain settle at exactly the same final value.
| Set by | Measured as | |
|---|---|---|
| Transient | Pole locations | Rise time, overshoot, settling time |
| Steady state | DC gain, system type | Kp, Kv, Ka |
The assertion, read fairly, is sound. Steady-state error is what finally decides whether a system does its job — whether the lift stops level with the floor, whether the furnace holds the set temperature. Overshoot and settling time matter, but a system that settles beautifully at the wrong value has failed. In that sense the steady-state response is the final requirement, and the word "efficiency" is doing loose duty for accuracy.
The genuine connection between the two runs the opposite way from what R claims, and is worth stating because it is the central tension of control design: adding integrators or raising the gain improves the steady-state error but degrades the transient, pushing poles towards the imaginary axis and increasing overshoot. They trade against each other — which is precisely why PID controllers exist, the integral term for the steady state and the derivative term to restore the transient. But trading against each other is not the same as one determining the other.
Flagged because A's wording is imprecise; on the substantive point, however, R is false.
Hence, (A) is true, but (R) is false.
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