mechatronics

1. 2.2.2 Linearization of Nonlinear Differential Equations
The above components and systems are supposed to be linear, so the mathematical models of them
are linear differential equations. Actually, all components or systems are nonlinear to some extent.
For example, rigidity of a spring is related to its formation. It is not always a constant.
Some parameters such as resistant R, inductance L and capacitance C are related to the
environment (temperature, humidity, pressure, etc) and the current going through. Thus, they are
not always constants. Friction, dead-zone or some other nonlinear factors will make the differential
equation complex and nonlinear. Strictly speaking, mathematical models for real systems are
always nonlinear.
Unfortunately, so far there is no universal solution for nonlinear differential equations. Thus,
nonlinear systems are usually linearized based on reasonable rules. Nonlinear systems usually can
be represented by linear differential equations in a small value range of variables so that they can
be analysis and design by linear system theories. Although it is an approximate solution, it is
convenient for analysis and calculation in practice.