Assertion–Reason (Control Systems): For a stable linear time-invariant system driven by a sinusoidal input, the steady-state response depends on initial conditions; moreover, the steady-state frequency response is obtained by substituting s = jω into the transfer function.

Introduction / Context:
In classical control systems, we frequently compare time-domain intuition (initial conditions, transients, steady state) with frequency-domain tools (Bode, Nyquist). This assertion–reason item tests whether steady-state sinusoidal response depends on initial conditions and recalls how frequency response is constructed from a transfer function.

Given Data / Assumptions:

• System is linear, time-invariant (LTI) and stable.
• Input is a pure sinusoid of fixed frequency ω.
• Transfer function G(s) exists and poles lie in the open left-half plane.

Concept / Approach:
For a stable LTI system, the total response = transient response + forced (steady-state) response. Transients depend on initial conditions but decay to zero as t → ∞. The steady-state sinusoidal output has the same frequency as the input, scaled by |G(jω)| and phase-shifted by ∠G(jω). Therefore steady state is independent of initial conditions. Frequency response is indeed obtained by evaluating G(s) on the jω axis (s = jω).

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