Zero-state response

The zero-state response of a system is the output produced purely by an input, with all internal state variables (capacitor voltages, inductor currents, integrator memories) initialized to zero. The system has “no stored energy at $t=0$” and the output comes entirely from the input.

Whenever we test a system’s properties — homogeneity, additivity, linearity, time-invariance — we work with the zero-state response. Otherwise, we’d be mixing two effects (input response and initial-condition response), and the property tests would be ambiguous.

Zero-state vs zero-input

The full response of a linear system splits cleanly:

$y(t)=y_{\text{zs}}(t)+y_{\text{zi}}(t).$

• $y_{\text{zs}}(t)$: zero-state response — the response to the input with zero initial conditions. Captured by h(t) and computed by convolution (or equivalently, transfer function multiplication in the s-domain).
• $y_{\text{zi}}(t)$: zero-input response — the response to nonzero initial conditions with no input. Captured by the homogeneous solution of the system’s differential equation; in the unilateral Laplace transform, it appears as the initial-condition terms.

Superposition says these add: the response with nonzero initial conditions and nonzero input is the sum of the two pieces. The transfer function only encodes the zero-state piece; the zero-input piece is the additional contribution from initial conditions.

In practice

For an LTI system $a{y}^{\prime \prime }+b{y}^{\prime }+cy=g(t)$:

• The zero-state response is $y_{\text{zs}}=h\ast g$, with $h(t)={\mathcal{L}}^{-1}\{1/(a{s}^2+bs+c)\}$.
• The zero-input response is the homogeneous solution with the given initial conditions, $y_{\text{zi}}(t)=c_1{\varphi }_1(t)+c_2{\varphi }_2(t)$ with $c_1,c_2$ pinned down by $y(0),y^{\prime }(0)$.

When the Unilateral Laplace transform is applied to the ODE, the initial-condition terms drop out of the differentiation property and contribute directly to $Y(s)$, which inverse-transforms to give both pieces at once.