线性系统
指这样一种系统：输入参数的变化会引起输出参数成比例（线性）的变化（即每个参数的性能曲线斜率均为常数）。这种线性不取决于输入变化的大小，也不取决于输入的初始状态。在线性系统中，任一工作点处的性能都可以线性外推到任何其他工作点。
线性系统可用如下方程表征 —
\begin{align} \label{eq:14101a} Y = A_0 \, + A_1 \, X_1 \, + A_2 \, X_2 \, + ... \, + \, A_n \, X_n \end{align}
式中 —
| \( Y \) | = 因变量（输出性能）参数 |
| \( X_1, \, X_2, \, ... X_n \) | = 自变量（输入）参数 |
| \( A_1, \, A_2, \, ... A_n \) | = 取决于具体系统的比例常数 |
例如，杆中的轴向应变 \( \epsilon \) 与各输入参数（\( \sigma \)、\( T \) 和 \( E' \)）的线性组合相关 —
\begin{align} \label{eq:14102a} \epsilon = \frac{\sigma}{E } \, + \, \alpha(T \, - \, T_0) \, + \, d_{33} \, E' \end{align}
式中 —
| \( \epsilon \) | = 轴向应变 |
| \( \sigma \) | = 轴向应力 |
| \( E \) | = 弹性模量 |
| \( \alpha \) | = 热膨胀系数 |
| \( T \) | = 温度 |
| \( T_0 \) | = 参考温度 |
| \( d_{33} \) | = 压电电荷常数 |
| \( E' \) | = 电场强度 |
式中 \( E' \)、\( \alpha \) 和 \( d_{33} \) 为比例常数。
当输入参数被限制在一定范围内时，许多非线性系统可以近似为线性系统。
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.