Editing Superposition principle (section)

==Additive state decomposition==
{{main|Additive state decomposition}}
Consider a simple linear system:
<math display="block">\dot{x} = Ax + B(u_1 + u_2), \qquad x(0) = x_0.</math>

By superposition principle, the system can be decomposed into
<math display="block">\begin{align}
 \dot{x}_1 &= Ax_1 + Bu_1, && x_1(0) = x_0,\\
 \dot{x}_2 &= Ax_2 + Bu_2, && x_2(0) = 0
\end{align}</math>
with
<math display="block">x = x_1 + x_2.</math>

Superposition principle is only available for linear systems. However, the [[additive state decomposition]] can be applied to both linear and nonlinear systems. Next, consider a nonlinear system
<math display="block">\dot{x} = Ax + B(u_1 + u_2) + \phi\left(c^\mathsf{T} x\right), \qquad x(0) = x_0,</math>
where <math>\phi</math> is a nonlinear function. By the additive state decomposition, the system can be additively decomposed into
<math display="block">\begin{align}
 \dot{x}_1 &= Ax_1 + Bu_1 + \phi(y_d), && x_1(0) = x_0, \\
 \dot{x}_2 &= Ax_2 + Bu_2 + \phi\left(c^\mathsf{T} x_1 + c^\mathsf{T} x_2\right) - \phi (y_d), && x_2(0) = 0
\end{align}</math>
with
<math display="block">x = x_1 + x_2.</math>

This decomposition can help to simplify controller design.