Superposition Principle

Assume that a linear system has zero initial condition. If input $u_1(t)$ produces output $y_1(t)$ and input $u_2(t)$ produces output $y_2(t)$, then input ${\alpha }_1{u}_1(t)+{\alpha }_2{u}_2(t)$ produces output ${\alpha }_1{y}_1(t)+{\alpha }_2{y}_2(t)$ for all ${\alpha }_1,{\alpha }_2\in \mathbb{R}$.

Proof

With zero initial condition, input $u_1(t)$ produces output $y_1(t)$ and input $u_2(t)$ produces output $y_2(t)$, then there are ${\mathbf{x}}_1(t)$ and ${\mathbf{x}}_2(t)$ with ${\mathbf{x}}_1(0)=0$ and ${\mathbf{x}}_2(0)=0$ satisfying

$$\begin{array}{rl}{\dot{\mathbf{x}}}_1(t)& =\mathbf{A}(t){\mathbf{x}}_1(t)+\mathbf{b}(t)u_1(t)\\ y_1(t)& =\mathbf{c}(t)x_1(t)+d(t)u_1(t)\end{array}$$

and

$$\begin{array}{rl}{\dot{\mathbf{x}}}_2(t)& =\mathbf{A}(t){\mathbf{x}}_2(t)+\mathbf{b}(t)u_2(t)\\ y_2(t)& =\mathbf{c}(t)x_2(t)+d(t)u_2(t)\end{array}$$

If we add the two state equations and two output equations, and define $u(t)={\alpha }_1{u}_1(t)+{\alpha }_2{u}_2(t)$, $\mathbf{x}(t)={\alpha }_1{\mathbf{x}}_1(t)+{\alpha }_2{\mathbf{x}}_2(t)$, and $y(t)={\alpha }_1{y}_1(t)+{\alpha }_2{y}_2(t)$, then we obtain $\mathbf{x}(0)=0$ and

$$\begin{array}{r}\dot{\mathbf{x}}(t)=\mathbf{A}(t)\mathbf{x}(t)+\mathbf{b}(t)u(t)\\ y(t)=y\mathbf{c}(t)\mathbf{x}(t)+d(t)u(t)\end{array}$$

This implies that $y(t)$ is the output of the system with zero initial condition and input $u(t)$.