Zero-input response

The zero-input response is the output due to a system's initial internal state when its externally applied input is set to zero. It describes memory left by earlier excitation. For a linear time-invariant system, linearity separates this from the zero-state response due to the subsequent input.

For constant matrices in $\dot{x}=Ax+Bu$, $y=Cx+Du$, setting $u=0$ and $x(0)=x_0$ givesThe matrix exponential solves the homogeneous state equation. With an arbitrary input, the complete solution adds the zero-state response:A zero initial output $C{x}_0=0$ alone does not establish that the whole state is zero.

For an RC circuit with capacitor difference $v(0)=V_0$ and no applied source, $RC\dot{v}+v=0$, so $v(t)=V_0{e}^{-t/(RC)}$. The capacitor energy falls as $C{V}_0^2{e}^{-2t/(RC)}/2$; the resistor receives the dissipated energy. The primary MIT lecture explicitly distinguishes the capacitor's initial state, zero external input and zero initial state.^[ref]

At a time constant $RC=2\mathrm{s}$ and $V_0=3\mathrm{V}$, the modeled difference after two seconds is $3/e\approx 1.104\mathrm{V}$. Zero input has not meant zero output or zero stored energy. A direct experiment charges the same capacitor to different known initial differences, removes the charging source and records the decay across a fixed resistor. The amplitudes should scale with $V_0$ while the normalized decay remains the same in the linear model.

A zero-input response need not decay in every system: an ideal undamped oscillator retains its initial motion, and an unstable homogeneous system can amplify a nonzero initial state. The transfer function describes the zero-state input-output map, so its product with a zero input cannot represent a separate nonzero initial-state contribution.