# Control Systems/Nonlinear Systems

## Nonlinear General Solution

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

$$x'(t) = f(t, t_0, x, x_0)$$

$$x(t_0) = x_0$$

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:

$$x'(t) = f(t, x)$$

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:

$$x(t) = x_0 + \int_{t_0}^t f(\tau, x)d\tau$$

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

### Iteration Method

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

$$x_n(t) = x_0 + \int_{t_0}^t f(\tau, x_{n-1}(\tau))d\tau$$

$$x_1(t) = x_0$$

We can show that the following relationship is true:

$$x(t) = \lim_{n \to \infty}x_n(t)$$

The *x*~n~ 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**.

![](Linear_Approximation_in_2D.svg "Linear_Approximation_in_2D.svg"){width="500"}

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.

