Linearization

In linearizing a non-linear function or equation is approximated by a linear function or differential equations. It is applied as linear functions and linear differential equations can be calculated very easily and the theory as opposed to non -linear systems is very well developed.

Applications

Common applications for the linearization, among others in the electrical and control engineering for the approximate description of nonlinear systems by linear systems.

The result of the network analyzer may be a non-linear equation system which can be converted into a linear equation system. Dynamic systems analysis based on linear differential equations is easier.

Tangent

The simplest method of linearization is drawing the line tangent to the graph. Then the parameters of the tangent can be read, and the resulting linear function ( point direction of the straight line equation )

Approximates the original function to the point. The increase is in point.

If the function is present in an analytical form, the equation of the tangent line can be specified directly.

Is the relative error of the approximation

For the function applies:

The determination of the tangent corresponds to the determination of the first Taylorpolynoms the function to be approximated.

Multiplication

Located in the signal flow diagram of a point multiplication can convert them by linearization in a summing point. The conversion of the underlying concept explains the calculation below.

Linearization of a multiplication in the operating point ( AP):

Multiplication approached by adding:

Example:

Division

It is

And

Or

For the geometric series

With

This is the linearized Division

Linearization of Ordinary Differential Equations

The best-known example of the linearization of a nonlinear differential equation is the pendulum. The equation is:

The non-linear part. This is for small fluctuations around an operating approximated by:

The operating point is valid:

Further details are described in state space representation.

Tangent plane

Is a given function in a point to be linearized, the Taylor formula is served. The result is the tangent plane at that point.

For the function is in the neighborhood of the point:

Example:

Gives the tangent plane