# Control Systems/Nonlinear Systems

## Nonlinear General Solution

A nonlinear system, in general, can be defined as follows:

${\displaystyle x'(t)=f(t,t_{0},x,x_{0})}$
${\displaystyle x(t_{0})=x_{0}}$

Where f is a nonlinear function of the time, the system state, and the initial conditions. If the initial conditions are known, we can simplify this as:

${\displaystyle x'(t)=f(t,x)}$

The general solution of this equation (or the most general form of a solution that we can state without knowing the form of f) is given by:

${\displaystyle x(t)=x_{0}+\int _{t_{0}}^{t}f(\tau ,x)d\tau }$

and we can prove that this is the general solution to the above equation because when we differentiate both sides we get the general solution.

### Iteration Method

The general solution to a nonlinear system can be found through a method of infinite iteration. We will define xn as being an iterative family of indexed variables. We can define them recursively as such:

${\displaystyle x_{n}(t)=x_{0}+\int _{t_{0}}^{t}f(\tau ,x_{n-1}(\tau ))d\tau }$
${\displaystyle x_{1}(t)=x_{0}}$

We can show that the following relationship is true:

${\displaystyle x(t)=\lim _{n\to \infty }x_{n}(t)}$

The xn series of equations will converge on the solution to the equation as n approaches infinity.

### Types of Nonlinearities

Nonlinearities can be of two types:

1. Intentional non-linearity: The non-linear elements that are added into a system. Eg: Relay
2. Incidental non-linearity: The non-linear behavior that is already present in the system. Eg: Saturation

## Linearization

Nonlinear systems are difficult to analyze, and for that reason one of the best methods for analyzing those systems is to find a linear approximation to the system. Frequently, such approximations are only good for certain operating ranges, and are not valid beyond certain bounds. The process of finding a suitable linear approximation to a nonlinear system is known as linearization.

This image shows a linear approximation (dashed line) to a non-linear system response (solid line). This linear approximation, like most, is accurate within a certain range, but becomes more inaccurate outside that range. Notice how the curve and the linear approximation diverge towards the right of the graph.

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