Zero-state response ($Y=HX$)

The zero-state response is the output produced by an applied input when the system starts with zero internal state. For a linear time-invariant system it separates the effect of the input from the zero-input response caused by an initially stored state. A zero initial output alone need not mean zero internal state.

For a causal linear time-invariant system with impulse response $h$, the zero-state response is the convolutionAn instantaneous feedthrough term, if present, is represented by a Dirac delta term in $h$. Equivalently, for constant matrices in $\dot{x}=Ax+Bu$, $y=Cx+Du$ with $x(0)=0$,For nonzero $x(0)=x_0$, linearity adds the zero-input response $C{e}^{At}{x}_0$. Here the matrix exponential propagates the state left by the earlier input.

For example, $\tau \dot{y}+y=u$ with $\tau >0$, $y(0)=0$ and a constant input $u(t)=A$ for $t\ge 0$ has the step response $y_{\mathrm{zs}}=A(1-e^{-t/\tau })$. At one time constant it has reached about $63.2\%$ of its final value. If the initial value instead equals $y_0$, the total output is $A(1-e^{-t/\tau })+y_0{e}^{-t/\tau }$. A charged capacitor in an RC circuit provides a direct physical example: discharge it before repeating the same input step to isolate the zero-state response, and compare with a run from a known initial electric potential difference.^[ref]

With compatible transforms, the zero-state input-output relation is $Y=HU$, where $H$ is the transfer function. For a nonzero initial state, a unilateral Laplace transform includes additional initial-condition terms. Those terms cannot be absorbed into an input-independent transfer function multiplying $U$.