Superposition, derivative, and integral properties

From the principle of superposition, linear, time invariant (LTI) system responses to both initial conditions and nonzero forcing can be obtained by summing the free response \(y_{\textnormal{fr}}\) and forced response \(y_{\textnormal{fo}}\): \[\begin{aligned} y(t) = y_{\textnormal{fr}}(t) + y_{\textnormal{fo}}(t). \end{aligned}\] Moreover, superposition says that any linear combination of inputs yields a corresponding linear combination of outputs. That is, we can find the response of a system to each input, separately, then linearly combine (scale and sum) the results according to the original linear combination. That is, for inputs \(u_1\) and \(u_2\) and constants \(a_1,a_2\in\mathbb{R}\), a forcing function \[\begin{aligned} f(t) = a_1 u_1(t) + a_2 u_2(t)\end{aligned}\] would yield output \[\begin{aligned} y(t) = a_1 y_1(t) + a_2 y_2(t)\end{aligned}\] where \(y_1\) and \(y_2\) are the outputs for inputs \(u_1\) and \(u_2\), respectively.

This powerful principle allows us to construct solutions to complex forcing functions by decomposing the problem. It also allows us to make extensive use of existing solutions to common inputs.

There are two more LTI system properties worth noting here. Let a system have input \(u_1\) and corresponding output \(y_1\). If the system is then given input \(u_2(t) = \dot{u}_1(t)\), the corresponding output is \[\begin{aligned} y_2(t) = \dot{y}_1(t).\end{aligned}\] Similarly, if the same system is then given input \(u_3(t) = \int_0^t u_1(\tau) d\tau\), the corresponding output is \[\begin{aligned} y_3(t) = \int_0^t y_1(\tau) d\tau.\end{aligned}\] These are sometimes called the derivative and integral properties of LTI systems.