# Linear system

A system in which a change in an input parameter gives a proportional (linear) change in an output parameter (i.e., the slope of the performance curve for each parameter is constant). This linearity does not depend on the size of the input change or the initial state of the input. In a linear system, the performance at each operating point can be linearly extrapolated to any other operating point.

A linear system will be characterized by an equation —

\begin{align} \label{eq:14101a} Y = A_0 \, + A_1 \, X_1 \, + A_2 \, X_2 \, + ... \, + \, A_n \, X_n \end{align}

where —

 $$Y$$ = dependent (output performance) parameter $$X_1, \, X_2, \, ... X_n$$ = independent (input) parameters $$A_1, \, A_2, \, ... A_n$$ = proportionality constants that depend on the particular system

For example, the axial strain $$\epsilon$$ in a rod is related to a linear combination of input parameters ($$\sigma$$, $$T$$, and $$E'$$) by —

\begin{align} \label{eq:14102a} \epsilon = \frac{\sigma}{E } \, + \, \alpha(T \, - \, T_0) \, + \, d_{33} \, E' \end{align}

where —

 $$\epsilon$$ = axial strain $$\sigma$$ = axial stress $$E$$ = modululs of elasticity $$\alpha$$ = coefficient of thermal expansion $$T$$ = temperature $$T_0$$ = reference temperature $$d_{33}$$ = piezoelectric charge constant $$E'$$ = electric field strength

where $$E'$$, $$\alpha$$, and $$d_{33}$$ are the proportionality constants.

Many nonlinear systems can be approximated as linear when the input parameters are confined to a limited range.